ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5

More Exercises

Question 1.
Find the sum of:
(i) 20 terms of the series 2 + 6 + 18 + …
(ii) 10 terms of series 1 + √3 + 3 + …
(iii) 6 terms of the GP. 1, \(– \frac { 2 }{ 3 } \) , \(\\ \frac { 4 }{ 9 } \), …
(iv) 20 terms of the GP. 0.15, 0.015, 0.0015,…
(v) 100 terms of the series 0.7 + 0.07 + 0.007 +…
(vi) 5 terms and n terms of the series \(1+\frac { 2 }{ 3 } +\frac { 4 }{ 9 } +…\)
(vii) n terms of the G.P. √7, √21, 3√7, …
(viii)n terms of the G.P. 1, – a, a², – a³, … (a ≠ – 1)
(ix) n terms of the G.P. x3, x5 , x7, … (x ≠ ±1).
Solution:
(i) 2 + 6 + 18 + … 20 terms
Here, a = 2, r = 3, n = 20, r > 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q1.8

Question 2.
Find the sum of the first 10 terms of the geometric series
√2 + √6 + √18 + ….
Solution:
√2 + √6 + √18 + ….
Here, a = √2 , r = √3, r > 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q2.1

Question 3.
Find the sum of the series 81 – 27 + 9….\(– \frac { 1 }{ 27 } \)
Solution:
Given
81 – 27 + 9….\(– \frac { 1 }{ 27 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q3.1

Question 4.
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then find its first term.
Solution:
In a G.P.
Tn =128
Sn = 255
r = 2,
Let a be the first term, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q4.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q4.2

Question 5.
If the sum of first six terms of any G.P. is equal to 9 times the sum of the first three terms, then find the common ratio of the G.P.
Solution:
Sum of first 6 terms of a G.P. = 9 x The of first 3 terms
Let a be the first term and r be the common ratio
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q5.1

Question 6.
A G.P. consists of an even number of terms. If the sum of all the terms is 3 times the sum of the odd terms, then find its common ratio.
Solution:
Let the G.P. be a, ar, ar2, … ar2n – 1
These are 2n in number, which is an even number
A.T.Q.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q6.1

Question 7.
(i) How many terms of the G.P. 3, 32, 33, … are needed to give the sum 120?
(ii) How many terms of the G.P. 1, 4, 16, … must be taken to have their sum equal to 341?
Solution:
In G.P.
(i) 3, 32, 33, …
Sum = 120, Here, a = 3, r = \(\frac { { 3 }^{ 2 } }{ 3 } \) = 3, r > 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q7.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q7.3

Question 8.
How many terms of the GP. 1, √2 > 2, 2 √2 , … are required to give a sum of 1023( √2 + 1)?
Solution:
GP. 1, √2 > 2, 2 √2 , …
Sum = 1023 (√2 + 1)
Here, a = 1, r = √2 . r > 1
Let number of terms be n, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q8.2

Question 9.
How many terms of the \(\frac { 2 }{ 9 } -\frac { 1 }{ 3 } +\frac { 1 }{ 2 } +…\) will make the sum \(\\ \frac { 55 }{ 72 } \) ?
Solution:
G.P. is \(\frac { 2 }{ 9 } -\frac { 1 }{ 3 } +\frac { 1 }{ 2 } +…\)
sum \(\\ \frac { 55 }{ 72 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q9.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q9.2

Question 10.
The 2nd and 5th terms of a geometric series are \(– \frac { 1 }{ 2 } \) and sum \(\\ \frac { 1 }{ 16 } \) respectively. Find the sum of the series upto 8 terms.
Solution:
In a G.P.
a2 = \(– \frac { 1 }{ 2 } \) and a5 = \(\\ \frac { 1 }{ 16 } \)
Let a be the first term and r be the common ratio
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q10.2

Question 11.
The first term of a G.P. is 27 and 8th term is \(\\ \frac { 1 }{ 81 } \) . Find the sum of its first 10 terms.
Solution:
In a G.P.
First term (a) = 27
a8 = 81
Let r be the common ratio, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q11.1

Question 12.
Find the first term of the G.P. whose common ratio is 3, last term is 486 and the sum of whose terms is 728
Solution:
Common ratio of a G.P. = 3
and last term = 486
and sum of terms = 728
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q12.1

Question 13.
In a G.P. the first term is 7, the last term is 448, and the sum is 889. Find the common ratio.
Solution:
In a GP.
First term (a) = 7, last term (l) = 448
and sum = 889
Let r be the common ratio, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q13.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q13.2

Question 14.
Find the third term of a G.P. whose common ratio is 3 and the sum of whose first seven terms is 2186.
Solution:
In a G.P.
Common ratio = 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q14.1

Question 15.
If the first term of a G.P. is 5 and the sum of first three terms is \(\\ \frac { 31 }{ 5 } \), find the common ratio.
Solution:
In a G.P.
First term (a) = 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q15.2

Question 16.
The sum of first three terms of a GP. is to the sum of first six terms as 125 : 152. Find the common ratio of the GP.
Solution:
S3 ÷ S6 = 125 : 152
Let r be the common ratio and a be the first number, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q16.1

Question 17.
Find the sum of the products of the corresponding terms of the geometric progression 2, 4, 8, 16, 32 and 128, 32, 8, 2, \(\\ \frac { 1 }{ 2 } \)
Solution:
Sum of the product of corresponding terms of the G.M.s
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q17.1

Question 18.
Evaluate \(\sum _{ n=1 }^{ 50 }{ \left( { 2 }^{ n }-1 \right) } \)
Solution:
\(\sum _{ n=1 }^{ 50 }{ \left( { 2 }^{ n }-1 \right) } \)
Here n = 1, 2, 3,….,50
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q18.1

Question 19.
Find the sum of n terms of a series whose mth term is 2m + 2m.
Solution:
am = 2m + 2m
a1 = 21 + 2 x 1 = 2 + 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q19.1

Question 20.
Sum the series
x(x + y) + x2 (x2 + y2) + x3 (x3 + y3) … to n terms.
Solution:
Given
Sn = x(x + y) + x2 (x2 + y2) + x3 (x3 + y3) … n terms
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q20.1

Question 21.
Find the sum of the series
1 + (1 + x) + (1 + x + x2) + … to n terms, x ≠ 1.
Solution:
1 + (1 + x) + (1 + x + x2) +… n terms, x ≠ 1
Multiply and divide by (1 – x)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q21.2

Question 22.
Find the sum of the following series to n terms:
(i) 7 + 77 + 777 + …
(ii) 8 + 88 + 888 + …
(iii) 0.5 + 0.55 + 0.555 + …
Solution:
(i) 7 + 77 + 777 + … n terms
= 7[1 + 11 + 111 + … n terms]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q22.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5 Q22.3

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4

More Exercises

Question 1.
Can 0 be a term of a geometric progression?
Solution:
No, 0 is not a term of geometric progression.

Question 2.
(i) Find the next term of the list of numbers \(\frac { 1 }{ 6 } ,\frac { 1 }{ 3 } ,\frac { 2 }{ 3 } ,… \)
(ii) Find the next term of the list of numbers \(\frac { 3 }{ 16 } ,-\frac { 3 }{ 8 } ,\frac { 3 }{ 4 } ,-\frac { 3 }{ 2 } ,…\)
(iii) Find the 15th term of the series \(\sqrt { 3 } +\frac { 1 }{ \sqrt { 3 } } +\frac { 1 }{ 3\sqrt { 3 } } +…\)
(iv) Find the nth term of the list of numbers \(\frac { 1 }{ \sqrt { 2 } } ,-2,4\sqrt { 2 } ,-16,…\)
(v) Find the 10th and nth terms of the list of numbers 5, 25, 125, …
(vi) Find the 6th and the nth terms of the list of numbers \(\frac { 3 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 3 }{ 8 } ,…\)
(vii) Find the 6th term from the end of the list of numbers 3, – 6, 12, – 24, …, 12288.
Solution:
(i) Given
⇒ \(\frac { 1 }{ 6 } ,\frac { 1 }{ 3 } ,\frac { 2 }{ 3 } ,… \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q2.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q2.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q2.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q2.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q2.5

Question 3.
Which term of the G.P.
(i) 2, 2√2, 4, … is 128?
(ii) \(1,\frac { 1 }{ 3 } ,\frac { 1 }{ 9 } ,…is\quad \frac { 1 }{ 243 } ?\)
(iii) \(\frac { 1 }{ 3 } ,\frac { 1 }{ 9 } ,\frac { 1 }{ 27 } ,…is\quad \frac { 1 }{ 19683 } ? \)
Solution:
Given
(i) 2, 2√2, 4, … is 128?
Here a = 2, \(r=\frac { 2\sqrt { 2 } }{ 2 } =\sqrt { 2 } \), l = 128
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q3.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q3.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q3.3

Question 4.
Which term of the G.P. 3, – 3√3, 9, – 9√3, … is 729 ?
Solution:
G.P. 3, -3√3, 9, – 9√3, … is 729 ?
Here a = 3, \(r=\frac { -3\sqrt { 3 } }{ 3 } =\sqrt { -3 } \), l = 729
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q4.1

Question 5.
Determine the 12th term of a G.P. whose 8th term is 192 and common ratio is 2.
Solution:
In a G.P.
a8 = 192 and r = 2
Let a be the first term and r be the common ratio then.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q5.1
∴ a12 = 3072

Question 6.
In a GP., the third term is 24 and 6th term is 192. Find the 10th term
Solution:
In a GP.
a3 = 24 and a6 = 192, a10 = ?
Let a be the first term and r be the common ratio, therefore
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q6.1

Question 7.
Find the number of terms of a G.P. whose first term is \(\\ \frac { 3 }{ 4 } \), common ratio is 2 and the last term is 384.
Solution:
First term of a G.P. (a) = \(\\ \frac { 3 }{ 4 } \)
and common ratio (r) = 2
Last term = 384
Let number of terms is n
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q7.1

Question 8.
Find the value of x such that
(i) \(-\frac { 2 }{ 7 } ,x,-\frac { 7 }{ 2 } \) are three consecutive terms of a G.P.
(ii) x + 9, x – 6 and 4 are three consecutive terms of a G.P.
(iii) x, x + 3, x + 9 are first three terms of a G.P. Sol. Find the value of x
Solution:
Find the value of x
(i) \(-\frac { 2 }{ 7 } ,x,-\frac { 7 }{ 2 } \) are three consecutive terms of a G.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q8.2

Question 9.
If the fourth, seventh and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.
Solution:
In a G.P.
a4 = x, a7 = y, a10 = z
To prove : x, y, z are in G.P.
Let a be the first term and r be the common
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q9.1

Question 10.
The 5th, 8th and 11th terms of a G.P. are p, q and s respectively. Show that q² = ps.
Solution:
In a G.P.
a5 = p, a8 = q and a11 = s
To show that q² = px
Let a be the first term and r be the common
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q10.1

Question 11.
If a, b, c are in G.P., then show that a², b², c² are also in G.P.
Solution:
a, b, c are in G.P.
Show that a², b², c² are also in G.P
∵ a, b, c are in G.P., then
b² = ac …(i)
a², b², c² will be in G.P.
if (b²)² = a² x c²
⇒ (ac)² = a²c² [From (i)]
⇒ a²c² = a²c² which is true.
Hence proved.

Question 12.
If a, b, c are in A.P., then show that 3a, 3b, 3c are in G.P.
Solution:
a, b and c are in A.P.
Then, 2b = a + c
Now, 3a, 3b, 3c will be in G.P.
if (3b)2 = 3a.3c
if 32b = 3a+c
Comparing, we get
if 2b = a + c
Which are in A.P. is given

Question 13.
If a, b, c are in A.P., then show that 10ax + 10, 10bx + 10, 10cx + 10, x ≠ 0, are in G.P.
Solution:
a, b, c are in A.P.
To show that are in G.P.10ax + 10, 10bx + 10, 10cx + 10, x ≠ 0
∵ a, b, c are in A.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q13.1

Question 14.
If a, a2+ 2 and a3 + 10 are in G.P., then find the values(s) of a.
Solution:
a, a2 + 2 and a3 + 10 are in G.P.
∵ (a2 + 2)2 = a(a3 + 10)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q14.1

Question 15.
If k, 2k + 2, 3k + 3, … are in G.P., then find the common ratio of the G.P.
Solution:
k, 2k + 2, 3k + 3, … are in G.P.
then, (2k + 2)2 = k(3k + 3)
⇒ 4k2 + 8k + 4 = 3k2 + 3k
⇒ 4k2 + 8k + 4 – 3k2 – 3k = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q15.1

Question 16.
The first and the second terms of a GP. are x-4 and xm . If its 8th term is x52, then find the value of m.
Solution:
In a G.P.,
First term (a1) = x-4 …(i)
Second term (a2) = xm
Eighth term (a8) = x52
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q16.1

Question 17.
Find the geometric progression whose 4th term is 54 and the 7th Term is 1458.
Solution:
In a G.P.,
4th term (a4) = 54
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q17.1

Question 18.
The fourth term of a GP. is the square of its second term and the first term is – 3. Determine its seventh term.
Solution:
In a GP.
an is square of a2 i.e. an = (a2
a1 = – 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q18.1

Question 19.
The sum of first three terms of a G.P. is \(\\ \frac { 39 }{ 10 } \) and their product is 1. Find the common ratio and the terms.
Solution:
Sum of first three terms of G.P. = \(\\ \frac { 39 }{ 10 } \)
and their product = 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q19.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q19.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q19.3

Question 20.
Three numbers are in A.P. and their sum is 15. If 1, 4 and 19 are added to these numbers respectively, the resulting numbers are in G.P. Find the numbers.
Solution:
Given: Three numbers are in A.P. and their sum = 15
Let a – d, a, a + d be the three number in A.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q20.2

Question 21.
Three numbers form an increasing G.P. If the middle term is doubled, then the new numbers are in A.P. Find the common ratio of the G.P.
Solution:
Three numbers form an increasing G.P.
Let \(\\ \frac { a }{ r } \) ,a,ar be three numbers in G.P.
Double the middle term, we get
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q21.2

Question 22.
Three numbers whose sum is 70 are in GP. If each of the extremes is multiplied by 4 and the mean by 5, the numbers will be in A.P. Find the numbers.
Solution:
Three numbers are in G.P.
Let numbers be
\(\\ \frac { r }{ a } \), a, ar
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q22.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q22.3

Question 23.
There are four numbers such that first three of them form an A.P. and the last three form a GP. The sum of the first and third number is 2 and that of second and fourth is 26. What are these numbers?
Solution:
There are 4 numbers, such that
First 3 numbers are in A.P. and
last 3 numbers are in GP.
Sum of first and third numbers = 2
and sum of 2nd and 4th = 26
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q23.1

Question 24.
(i) If a, b, c are in A.P. as well in G.P., prove that a = b = c.
(ii) If a, b, c are in A.P as well as in G.P., then find the value of ab-c + bc-a + ca-b
Solution:
(i) a, b, c are in A.P. as well as in GP.
To prove: a = b = c
a, b, c are in A.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q24.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q24.2

Question 25.
The terms of a G.P. with first term a and common ratio r are squared. Prove that resulting numbers form a G.P. Find its first term, common ratio and the nth term.
Solution:
In a G.P.,
The first term = a
and common ratio = r
GP. is a, ar, ar²
Squaring we get
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q25.1

Question 26.
Show that the products of the corresponding terms of two G.P.’s a, ar, ar², …, arn-1 and A, AR, AR2, …, ARn-1 form a G.P. and find the common ratio.
Solution:
It has to be proved that the sequence
aA, arAR, ar²AR2, …, arn-1ARn-1 and forms a G.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q26.1

Question 27.
(i) If a, b, c are in G.P. show that \(\frac { 1 }{ a } ,\frac { 1 }{ b } ,\frac { 1 }{ c } \) are also in G.P.
(ii) If K is any positive real number and Ka, Kb Kc are three consecutive terms of a G.P., prove that a, b, c are three consecutive terms of an A.P.
(iii) If p, q, r are in A.P., show that pth, qth and rth terms of any G.P. are themselves in GP.
Solution:
(i) a, b, c are in G.P.
∴ b2 = ac
\(\frac { 1 }{ a } ,\frac { 1 }{ b } ,\frac { 1 }{ c } \) will be in G.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q27.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q27.2

Question 28.
If a, b, c are in GP., prove that the following are also in G.P.
(i) a3, b3, c3
(ii) a2 + b2, ab + bc, b2 + c2.
Solution:
(i) a, b, c are in G.P.
∴ b2 = ac
a3, b3, c3 are in G.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q28.1

Question 29.
If a, b, c, d are in G.P., show that
(i) a2 + b2, b2 + c2, c2 + d2 are in G.P.
(ii) (b – c)2 + (c – a)2 + (d – b)2 = (a – d)2.
Solution:
a, b, c, d are in G.P.
Let r be the common ratio, then a = a
b = ar, c = ar2, d = ar3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q29.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q29.2

Question 30.
The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nth hour?
Solution:
Bacteria in the beginning = 30 = a
After every hours, it doubles itself
After 1 hour it becomes = 30 x 2 = 60 = ar
After 2 hours it will becomes = 60 x 2 = 120 = a2
After 3 hours, it will becomes = 120 x 2 = 240 = a3
After 4 hours it will becomes = 240 x 2 = 480 = a4
∴ After n hour, it will become = arn

Question 31.
The length of the sides of a triangle form a G.P. If the perimeter of the triangle is 37 cm and the shortest side is of length 9 cm, find the lengths of the other two sides.
Solution:
Lengths of a triangle are in GP. and its sum is 37 cm
Let sides be a, ar, ar2
a + ar + ar2 = 37
a = 9
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.4 Q31.1

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

More Exercises

Question 1.
Find the sum of the following A.P.s :
(i) 2, 7, 12, … to 10 terms
(ii) \(\frac { 1 }{ 15 } ,\frac { 1 }{ 12 } ,\frac { 1 }{ 10 } ,… \) t0 11 terms
Solution:
(i) 2, 7, 12, … to 10 terms
Here a = 2, d = 7 – 2 = 5 and n = 10
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q1.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q1.2

Question 2.
How many terms of the A.P. 27, 24, 21, …, should be taken so that their sum is zero?
Solution:
A.P. = 27, 24, 21,…
a = 27
d = 24 – 27 = -3
Sn =0
Let n terms be there in A.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q2.1

Question 3.
Find the sums given below :
(i) 34 + 32 + 30 + … + 10
(ii) – 5 + ( – 8) + ( – 11) + … + ( – 230)
Solution:
(i) 34 + 32 + 30 + … + 10
Here, a = 34, d = 32 – 34 = -2, l = 10
Tn = a + (n – 1)d
10 = 34 + (n – 1)(-2)
-24 = -2 (n – 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q3.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q3.2

Question 4.
In an A.P. (with usual notations) :
(i) given a = 5, d = 3, an = 50, find n and Sn
(ii) given a = 7, a13 = 35, find d and S13
(iii) given d = 5, S9 = 75, find a and a9
(iv) given a = 8, an = 62, Sn = 210, find n and d
(v) given a = 3, n = 8, S = 192, find d.
Solution:
(i) a = 5, d = 3, an = 50
an = a + (n – 1 )d
50 = 5 + (n – 1) x 3
⇒ 50 – 5 = 3(n – 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.5

Question 5.
(i) The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
(ii) The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its 20th term.
Solution:
(i) First term of an A.P. (a) = 5
Last term (l) = 45
Sum = 400
l = a + (n – 1 )d
45 = 5 + (n – 1)d
⇒ (n – 1)d = 45 – 5 = 40 …(i)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q5.2

Question 6.
The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
Solution:
First term of an A.P. (a) = 17
and last term (l) = 350
d= 9
l = Tn = a + (n – 1 )d
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q6.1

Question 7.
Solve for x : 1 + 4 + 7 + 10 + … + x = 287.
Solution:
1 + 4 + 7 + 10 + .. . + x = 287
Here, a = 1, d = 4 – 1 = 3, n = x
l = x = a = (n – 1)d = 1 + (n – 1) x 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q7.2

Question 8.
(i) How many terms of the A.P. 25, 22, 19, … are needed to give the sum 116 ? Also find the last term.
(ii) How many terms of the A.P. 24, 21, 18, … must be taken so that the sum is 78 ? Explain the double answer.
Solution:
(i) A.P. is 25, 22, 19, …
Sum = 116
Here, a = 25, d = 22 – 25 = -3
Let number of terms be n, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.4

Question 9.
Find the sum of first 22 terms, of an A.P. in which d = 7 and a22 is 149.
Solution:
Sum of first 22 terms of an A.P. whose d = 7
a22 = 149 and n = 22
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q9.1

Question 10.
(i) Find the sum of first 51 terms of the A.P. whose second and third terms are 14 and 18 respectively.
(ii) If the third term of an A.P. is 1 and 6th term is – 11, find the sum of its first 32 terms.
Solution:
Sum of first 51 terms of an A.P. in which
T2 = 14, T3 = 18
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q10.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q10.3

Question 11.
If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.
Solution:
S6 = 36
S16 = 256
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q11.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q11.2

Question 12.
Show that a1, a2, a3, … form an A.P. where an is defined as an = 3 + 4n. Also find the sum of first 15 terms.
Solution:
an = 3 + 4n
a1 = 3 + 4 x 1 = 3 + 4 = 7
a2 = 3 + 4 x 2 = 3 + 8 = 11
a3 = 3 + 4 x 3 = 3 + 12 = 15
a4 = 3 + 4 x 4 = 3 + 16 = 19
and so on Here, a = 1 and d = 11 – 7 = 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q12.1

Question 13.
(i) If an = 3 – 4n, show that a1, a2, a3, … form an A.P. Also find S20.
(ii) Find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.
Solution:
(i) an = 3 – 4n
a1 = 3 – 4 x 1 = 3 – 4 = -1
a2 = 3 – 4 x 2 = 3 – 8 = -5
a3 = 3 – 4 x 3 = 3 – 12 = -9
a4 = 3 – 4 x 4 = 3 – 16 = -13 and so on
Here, a = -1, d = -5 – ( -1) = -5 + 1 = -4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q13.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q13.2

Question 14.
The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 equal to the sum of first 2n terms of another A.P. whose first term is – 30 and the common difference is 8. Find n.
Solution:
In an A.P.
Sn = S2n
For the first A.P. a = 8, d = 20
and for second A.P. a = -30, d = 8
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q14.2

Question 15.
The sum of first six terms of an arithmetic progression is 42. The ratio of the 10th term to the 30th term is \(\\ \frac { 1 }{ 3 } \). Calculate the first and the thirteenth term.
Solution:
T10 : T30 = 1 : 3, S6 = 42
Let a be the first term and d be a common difference, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q15.1

Question 16.
In an A.P., the sum of its first n terms is 6n – n². Find is 25th term.
Solution:
Sn = 6n – n²
T25 = ?
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q16.1

Question 17.
If the sum of first n terms of an A.P. is 4n – n², what is the first term (i. e. S1)? What is the sum of the first two terms? What is the second term? Also, find the 3rd term, the 10th term, and the nth terms?
Solution:
Sn = 4n – n²
Sn – 1 = 4(n – 1) – (n – 1)²
= 4n – 4 – (n² – 2n + 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q17.1

Question 18.
If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3(S20 – S10).
Solution:
Sn denotes the sum of first n terms of an A.P.
To prove: S30 = 3(S20 – S10)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q18.2

Question 19.
(i) Find the sum of first 1000 positive integers.
(ii) Find the sum of first 15 multiples of 8.
Solution:
(i) Sum of first 1000 positive integers
i. e., 1 + 2 + 3+ 4 + … + 1000
Here, a = 1, d = 1, n = 1000
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q19.1

Question 20.
(i) Find the sum of all two digit natural numbers which are divisible by 4.
(ii) Find the sum of all natural numbers between 100 and 200 which are divisible by 4.
(iii) Find the sum of all multiples of 9 lying between 300 and 700.
(iv) Find the sum of all natural numbers less than 100 which are divisible by 6.
Solution:
(i) Sum of two digit natural numbers which are divisible by 4
which are 12, 16, 20, 24, …, 96
Here, a = 12, d = 16 – 12 = 4, l = 96
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.5

Question 21.
(i) Find the sum of all two digit odd positive numbers.
(ii) Find the sum of all 3-digit natural numbers which are divisible by 7.
(iii) Find the sum of all two digit numbers which when divided by 7 yield 1 as the
Solution:
(i) Sum of all two-digit odd positive numbers which are 11, 13, 15, …, 99
Here, a = 11, d = 2, l = 99
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.4

Question 22.
A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows : Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day, etc., the penalty for each succeeding day being Rs 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work for 30 days ?
Solution:
Penalty for
First day = Rs 200
Second day = Rs 250
Third day = Rs 300
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q22.1

Question 23.
Kanika was given her pocket money on 1st Jan, 2016. She puts Rs 1 on Day 1, Rs 2 on Day 2, Rs 3 on Day 3, and continued on doing so till the end of the month, from this money into her piggy bank. She also spent Rs 204 of her pocket money, and was found that at the end of the month she still has Rs 100 with her. How much money was her pocket money for the month ?
Solution:
Pocket money for Jan. 2016
Out of her pocket money, Kanika puts
Rs 1 on the first day i.e., 1 Jan.
Rs 2 on second Jan
Rs 3 on third Jan
Rs 31 on 31st Jan
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q23.1

Question 24.
Yasmeen saves Rs 32 during the first month, Rs 36 in the second month and Rs 40 in the third month. If she continues to save in this manner, in how many months will she save Rs 2000?
Solution:
Savings for the first month = Rs 32
For the second month = Rs 36
For the third month = Rs 40
Total savings for the period = Rs 2000
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q24.1

Question 25.
The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books ? What is the maximum distance she travelled carrying a flag ?
Solution:
Total number of flags = 27
To fixed after every = 2 m
The flag is stored at the middlemost flag
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q25.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2

More Exercises

Question 1.
Find the A.P. whose nth term is 7 – 3K. Also find the 20th term.
Solution:
Tn = 7 – 3n
Giving values 1, 2, 3, 4, … to n, we get
T1 = 7 – 3 x 1 = 7 – 3 = 4
T2 = 7 – 3 x 2 = 7 – 6 = 1
T3 = 7 – 3 x 3 = 7 – 9 = -2
T4 = 7 – 3 x 4 = 7 – 12 = -5
T20 = 7 – 3 x 20 = 7 – 60 = -53
A.P. is 4, 1, -2, -5, …
20th term = -53

Question 2.
Find the indicated terms in each of following A.P.s:
(i) 1, 6, 11, 16, …; a20
(ii) – 4, – 7, – 10, – 13, …, a25, an
Solution:
(i) 1, 6, 11, 16, …
Here, a = 1, d = 6 – 1 – 5
a20 = a + (n – 1 )d
= 1 + (20 – 1) x 5
= 1 + 19 x 5
= 1 + 95
= 96
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q2.1

Question 3.
Find the nth term and the 12th term of the list of numbers: 5, 2, – 1, – 4, …
Solution:
5, 2, -1, -4, …
Here, a = 5 d = 2 – 5 = -3
(i) Tn = a + (n – 1)d
= 5 + (n – 1) (-3)
= 5 – 3n + 3
= 8 – 3n
(ii) T12 = a + 11d
= 5 + 11(-3)
= 5 – 33
= -28

Question 4.
Find the 8th term of the A.P. whose first term is 7 and common difference is 3.
Solution:
First term (a) = 7
and common difference (d) = 3
A.P. = 7, 10, 13, 16, 19, …
T8 = a + (n – 1)d
= 7 + (8 – 1) x 3
= 7 + 7 x 3
= 7 + 21
= 28

Question 5.
(i) If the common difference of an A.P. is – 3 and the 18th term is – 5, then find its first term.
(ii) If the first term of an A.P. is – 18 and its 10th term is zero, then find its common difference.
Solution:
(i) Common difference (d) = -3
T18 = -5
a + (n – 1 )d = Tn
a + (18 – 1) (-3) = -5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q5.1

Question 6.
Which term of the A.P.
(i) 3, 8, 13, 18, … is 78?
(ii) 7, 13, 19, … is 205 ?
(iii) 18, \(15 \frac { 1 }{ 2 } \), 13, … is – 47 ?
Solution:
(i) 3, 8, 13, 18, … is 78
Let 78 is nth term
Here, a = 3, d = 8 – 3 = 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q6.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q6.2

Question 7.
(i) Check whether – 150 is a term of the A.P. 11, 8, 5, 2, …
(ii) Find whether 55 is a term of the A.P. 7, 10, 13, … or not. If yes, find which term is it.
(iii) Is 0 a term of the A.P. 31,28, 25,…? Justify your answer.
Solution:
(i) A.P. is 11, 8, 5, 2, …
Here, a = 11, d = 8 – 11 = -3
Let -150 = n, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q7.2

Question 8.
(i) Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253.
(ii) Find the 12th from the end of the A.P. – 2, – 4, – 6, …; – 100.
Solution:
(i) A.P. is 3, 8, 13, …, 253
12th term from the end
Last term = 253
Here, a = 3, d = 8 – 3 = 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q8.2

Question 9.
Find the sum of the two middle most terms of the A.P.
\(-\frac { 4 }{ 3 } ,-1,-\frac { 2 }{ 3 } ,…,4\frac { 1 }{ 3 } \)
Solution:
Given
A.P. is \(-\frac { 4 }{ 3 } ,-1,-\frac { 2 }{ 3 } ,…,4\frac { 1 }{ 3 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q9.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q9.2

Question 10.
Which term of the A.P. 53, 48, 43,… is the first negative term ?
Solution:
Let nth term is the first negative term of the A.P. 53, 48, 43, …
Here, a = 53, d = 48 – 53 = -5
.’. Tn = a + (n – 1 )d
= 53 + (n – 1) x (-5)
= 53 – 5n + 5
= 58 – 5n
5n = 58
\(n= \frac { 58 }{ 5 } \)
= \(11 \frac { 3 }{ 5 } \)
∴ 12th term will be negative.

Question 11.
Determine the A.P. whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.
Solution:
In an A.P.,
T5 = 19
T13 – T8 = 20
Let a be the first term and d be the common difference
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q11.1

Question 12.
Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12
Solution:
T3 = 16
T7 – T5 = 12
Let a be the first term and d be the common difference
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q12.1

Question 13.
Find the 20th term of the A.P. whose 7th term is 24 less than the 11th term, first term being 12.
Solution:
T11 – T7 = 24
a= 12
Let a be the first term and d be the common difference, then
(a + 10d) – (a + 6d) = 24
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q13.1

Question 14.
Find the 31st term of an A.P. whose 11th term is 38 and 6th term is 73.
Solution:
T11 = 38, T6 = 73
Let a be the first term and d be the common difference, then
a + 10d = 38..(i)
a + 5d = 73…(ii)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q14.1

Question 15.
If the seventh term of an A.P. is \(\\ \frac { 1 }{ 9 } \) and its ninth term is \(\\ \frac { 1 }{ 7 } \), find its 63rd term.
Solution:
a7 = \(\\ \frac { 1 }{ 9 } \)
⇒ a + 6d = \(\\ \frac { 1 }{ 9 } \) ….(i)
a9 = \(\\ \frac { 1 }{ 7 } \)
⇒ a + 8d = \(\\ \frac { 1 }{ 7 } \)……(ii)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q15.2

Question 16.
(i) The 15th term of an A.P. is 3 more than twice its 7th term. If the 10th term of the A.P. is 41, find its nth term.
(ii) The sum of 5th and 7th terms of an A.P. is 52 and the 10th term is 46. Find the A.P.
(iii) The sum of 2nd and 7th terms of an A.P. is 30. If its 15th term is 1 less than twice its 8th term, find the A.P.
Solution:
(i) Let a be the first term and d be a common difference.
We have,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q16.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q16.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q16.3
∴ The A.P formed is 1, 6, 11, 16,….

Question 17.
If 8th term of an A.P. is zero, prove that its 38th term is triple of its 18th term.
Solution:
T8 = 0
To prove that T38 = 3 x T18
Let a be the first term and d be the common difference
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q17.1

Question 18.
Which term of the A.P. 3, 10, 17,… will be 84 more than its 13th term?
Solution:
A.P. is 3, 10, 17, …
Here, a = 3, d – 10 – 3 = 7
T13 = a + 12d
= 3 + 12 x 7
= 3 + 84
= 87
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q18.1

Question 19.
If the nth terms of the two A.g.s 9, 7, 5, … and 24, 21, 18, … are the same, find the value of n. Also, find that term
Solution:
nth term of two A.P.s 9, 7, 5,… and 24, 21, 18, … are same
In the first A.P. 9, 7, 5, …
a = 9 and d = 7 – 9 = -2
Tn = a + (n – 1)d
= 9 + (n – 1)(-2)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q19.1

Question 20.
(i) How many two digit numbers are divisible by 3 ?
(ii) Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.
(iii) How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?
Solution:
(i) Two digits numbers divisible by 3 are
12, 15, 18, 21, …, 99
Here, a = 13, d = 15 – 12 = 3 and l = 99
Let number divisible by 3 and n
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q20.2

Question 21.
If the numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n.
Solution:
n – 2, 4n – 1 and 5n + 2 are in A.P.
∴ 2(4n – 1) = n – 2 + 5n + 2
8n – 2 = 6n
⇒ 8n – 6n = 2
⇒ 2n = 1
⇒ \(n \frac { 2 }{ 2 } \) = 1
∴ n = 1

Question 22.
The sum of three numbers in A.P. is 3 and their product is – 35. Find the numbers.
Solution:
Sum of three numbers which are in A.P. = 3
Their product = -35
Let three numbers which are in A.P.
a – d, a, a + d
a – d + a + a + d = 3
⇒ 3a = 3 ,
⇒ a = \(\\ \frac { 3 }{ 3 } \) = 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q22.1

Question 23.
The sum of three numbers in A.P. is 30 and the ratio of first number to the third number is 3 : 7. Find the numbers.
Solution:
Sum of three numbers in A.P. = 30
Ratio between first and the third number = 3 : 7
Let numbers be
a – d, a, a + d, then
a – d + a + a + d = 30
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q23.1

Question 24.
The sum of the first three terms of an A.P.is 33. If the product of the first and the third terms exceeds the second term by 29, find the A.P.
Solution:
Let the three numbers in A.P. are
a – d, a, a + d
Now, a – d + a + a + d = 33
⇒ 3a = 33
⇒ a = \(\\ \frac { 33 }{ 3 } \) = 11
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q24.1

Question 25.
A man starts repaying a loan as first instalment of Rs 500. If he increases the instalment by Rs 25 every month, what,amount will he pay in the 30th instalment?
Solution:
First instalment of loan = Rs 500
Increases Rs 25 every month
Here, a = 500, d = 25
Total instalments (n) = 30
We have to find T30
T30 = a + (n – 1 )d = a + 29d
= 500 + 29 x 25
= 500 + 725
= Rs 1225

Question 26.
Ramkali saved Rs 5 in the first week of a year and then increased her savings by Rs 1.75. If in the rcth week, her weekly savings become Rs 20.75, find n.
Solution:
Savings in the first week = Rs 5
Increase every week = Rs 1.75
No. of weeks = n
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q26.1

Question 27.
Justify whether it is true to say that the following are the nth terms of an A.P.
(i) 2n – 3
(ii) n² + 1
Solution:
(i) 2n – 3
Giving the some difference values to n such as 1, 2, 3, 4, … then
2 x 1 – 3 = 2 – 3 = -1
2 x 2 – 3 = 4 – 3 = 1
2 x 3 – 3 = 6 – 3 = 3
2 x 4 – 3 = 8 – 3 = 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2 Q27.1

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1

More Exercises

Question 1.
For the following A.P.s, write the first term a and the common difference d:
(i) 3, 1, – 1, – 3, …
(ii) \(\frac { 1 }{ 3 } ,\frac { 5 }{ 3 } ,\frac { 9 }{ 3 } ,\frac { 13 }{ 3 } ,…. \)
(iii) – 3.2, – 3, – 2.8, – 2.6, …
Solution:
(i) 3, 1, -1, -3, …
Here first term (a) = 3
and the common difference (d)
= 1 – 3 = -2,
– 1 – 1 = -2,…
= -2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q1.1

Question 2.
Write first four terms of the A.P., when the first term a and the common difference d are given as follows :
(i) a = 10, d = 10
(ii) a = – 2, d = 0
(iii) a = 4, d = – 3
(iv) a = \(\\ \frac { 1 }{ 2 } \), d = \(– \frac { 1 }{ 2 } \)
Solution:
(i) a = 10, d = 10
∴ A.P. = 10, 20, 30, 40, …
(ii) a = -2, d = 0
∴ A.P. = -2, -2, -2, -2, ….
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q2.1

Question 3.
Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms :
(i) 4, 10, 16, 22,…
(ii) – 2, 2, – 2, 2,…..
(iii) 2, 4, 8, 16,….
(iv) 2, \(\\ \frac { 5 }{ 2 } \), 3, \(\\ \frac { 7 }{ 2 } \),……
(v) – 10, – 6, – 2, 2,….
(vi) 1², 3², 5², 7²,….
(vii) 1, 3, 9, 27,….
(viii) √2, √8, √18, √32,….
(ix) 3, 3 + √2, 3 + √2, 3 + 3√2,…..
(x) √3, √6, √9, √12,……
(xi) a, 2a, 3a, 4a,…….
(xii) a, 2a + 1, 3a + 2, 4a + 3,….
Solution:
(i) 4, 10, 16, 22,…
Here a = 4, d = 10 – 4 = 6, 16 – 10 = 6, 22 – 16 = 6
∵ common difference is same
∵ It is in A.P
and next three terms are 28, 34, 40
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1 Q3.7

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test

More Exercises

Question 1.
Find the values of a and below
\(\begin{bmatrix} a+3 & { b }^{ 2 }+2 \\ 0 & -6 \end{bmatrix}=\begin{bmatrix} 2a+1 & 3b \\ 0 & { b }^{ 2 }-5b \end{bmatrix}\)
Solution:
\(\begin{bmatrix} a+3 & { b }^{ 2 }+2 \\ 0 & -6 \end{bmatrix}=\begin{bmatrix} 2a+1 & 3b \\ 0 & { b }^{ 2 }-5b \end{bmatrix}\)
comparing the corresponding elements
a + 3 = 2a + 1
⇒ 2a – a =3 – 1
⇒ a = 2
b² + 2 = 3b
⇒ b² – 3b + 2 = 0
⇒ b² – b – 2b + 2 = 0
⇒ b (b – 1) – 2 (b – 1) = 0
⇒ (b – 1) (b – 2) = 0.
Either b – 1 = 0, then b = 1
or b – 2 = 0, then b = 2
Hence a = 2, b = 2 or 1

Question 2.
Find a, b, c and d if \(3\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 4 & a+b \\ c+d & 3 \end{bmatrix}+\begin{bmatrix} a & 6 \\ -1 & 2d \end{bmatrix}\)
Solution:
Given
\(3\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 4 & a+b \\ c+d & 3 \end{bmatrix}+\begin{bmatrix} a & 6 \\ -1 & 2d \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q2.1

Question 3.
Find X if Y = \(\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \) and 2X + Y = \(\begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \)
Solution:
Given
2X + Y = \(\begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \)
⇒ 2X = 2X + Y = \(\begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \) – Y
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q3.1

Question 4.
Determine the matrices A and B when
A + 2B = \(\begin{bmatrix} 1 & 2 \\ 6 & -3 \end{bmatrix} \) and 2A – B = \(\begin{bmatrix} 2 & -1 \\ 2 & -1 \end{bmatrix} \)
Solution:
A + 2B = \(\begin{bmatrix} 1 & 2 \\ 6 & -3 \end{bmatrix} \) …..(i)
2A – B = \(\begin{bmatrix} 2 & -1 \\ 2 & -1 \end{bmatrix} \) …….(ii)
Multiplying (i) by 1 and (ii) by 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q4.1

Question 5.
(i) Find the matrix B if A = \(\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \) and A² = A + 2B
(ii) If A = \(\begin{bmatrix} 1 & 2 \\ -3 & 4 \end{bmatrix} \), B = \(\begin{bmatrix} 0 & 1 \\ -2 & 5 \end{bmatrix} \)
and C = \(\begin{bmatrix} -2 & 0 \\ -1 & 1 \end{bmatrix} \) find A(4B – 3C)
Solution:
A = \(\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \)
let B = \(\begin{bmatrix} a & b \\ c & d \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q5.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q5.3

Question 6.
If A = \(\begin{bmatrix} 1 & 4 \\ 1 & 0 \end{bmatrix} \), B = \(\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix} \) and C = \(\begin{bmatrix} 2 & 3 \\ 0 & 5 \end{bmatrix} \) compute (AB)C = (CB)A ?
Solution:
Given
A = \(\begin{bmatrix} 1 & 4 \\ 1 & 0 \end{bmatrix} \),
B = \(\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix} \) and
C = \(\begin{bmatrix} 2 & 3 \\ 0 & 5 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q6.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q6.2

Question 7.
If A = \(\begin{bmatrix} 3 & 2 \\ 0 & 5 \end{bmatrix} \) and B = \(\begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} \) find the each of the following and state it they are equal:
(i) (A + B)(A – B)
(ii)A² – B²
Solution:
Given
A = \(\begin{bmatrix} 3 & 2 \\ 0 & 5 \end{bmatrix} \) and
B = \(\begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q7.2

Question 8.
If A = \(\begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \) find A² – 5A – 14I
Where I is unit matrix of order 2 x 2
Solution:
Given
A = \(\begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q8.2

Question 9.
If A = \(\begin{bmatrix} 3 & 3 \\ p & q \end{bmatrix} \) and A² = 0 find p and q
Solution:
Given
A = \(\begin{bmatrix} 3 & 3 \\ p & q \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q9.1

Question 10.
If A = \(\begin{bmatrix} \frac { 3 }{ 5 } & \frac { 2 }{ 5 } \\ x & y \end{bmatrix} \) and A² = I, find x,y
Solution:
Given
A = \(\begin{bmatrix} \frac { 3 }{ 5 } & \frac { 2 }{ 5 } \\ x & y \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q10.2

Question 11.
If \(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \) find a,b,c and d
Solution:
Given
\(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q11.1

Question 12.
Find a and b if
\(\begin{bmatrix} a-b & b-4 \\ b+4 & a-2 \end{bmatrix}\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}=\begin{bmatrix} -2 & -2 \\ 14 & 0 \end{bmatrix} \)
Solution:
Given
\(\begin{bmatrix} a-b & b-4 \\ b+4 & a-2 \end{bmatrix}\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}=\begin{bmatrix} -2 & -2 \\ 14 & 0 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q12.1

Question 13.
If A = \(\begin{bmatrix} { sec60 }^{ o } & { cos90 }^{ o } \\ { -3tan45 }^{ o } & { sin90 }^{ o } \end{bmatrix} \) and B = \(\begin{bmatrix} 0 & { cos45 }^{ o } \\ -2 & { 3sin90 }^{ o } \end{bmatrix} \)
Find (i) 2A – 3B (ii) A² (iii) BA
Solution:
Given
A = \(\begin{bmatrix} { sec60 }^{ o } & { cos90 }^{ o } \\ { -3tan45 }^{ o } & { sin90 }^{ o } \end{bmatrix} \) and
B = \(\begin{bmatrix} 0 & { cos45 }^{ o } \\ -2 & { 3sin90 }^{ o } \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q13.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test Q13.2

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Chapter Test are helpful to complete your math homework.

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS

More Exercises

Choose the correct answer from the given four options (1 to 14) :

Question 1.
If A = [aij]2×2 where aij = i + j, then A is equal to
(a) \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
(b) \(\begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \)
(c) \(\begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix} \)
(d) \(\begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix} \)
Solution:
A = [aij]2×2 where aij = i + j, then A is equal to
\(\begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \) (b)

Question 2.
If \(\begin{bmatrix} x+3 & 4 \\ y-4 & x+y \end{bmatrix}=\begin{bmatrix} 5 & 4 \\ 3 & 9 \end{bmatrix} \) then the values of x and y are
(a) x = 2, y = 7
(b) x = 7, y = 2
(c) x = 3, y = 6
(d) x = – 2, y = 7
Solution:
\(\begin{bmatrix} x+3 & 4 \\ y-4 & x+y \end{bmatrix}=\begin{bmatrix} 5 & 4 \\ 3 & 9 \end{bmatrix} \)
Comparing we get
x + 3 = 5
⇒ x = 5 – 3 = 2
and y – 4 = 3
⇒ y = 3 + 4 = 7
x = 2, y = 7 (a)

Question 3.
If \(\begin{bmatrix} x+2y & -y \\ 3x & 7 \end{bmatrix}=\begin{bmatrix} -4 & 3 \\ 6 & 4 \end{bmatrix} \) then the values of x and y are
(a) x = 2, y = 3
(b) x = 2, y = – 3
(c) x = – 2, y = 3
(d) x = 3, y = 2
Solution:
\(\begin{bmatrix} x+2y & -y \\ 3x & 7 \end{bmatrix}=\begin{bmatrix} -4 & 3 \\ 6 & 4 \end{bmatrix} \)
Comparing, we get
3x = 6
⇒ \(x= \frac { 6 }{ 3 } \) = 2
⇒ -y = 3
⇒ y = – 3
x = 2, y = -3 (b)

Question 4.
If \(\begin{bmatrix} x-2y & 5 \\ 3 & y \end{bmatrix}=\begin{bmatrix} 6 & 5 \\ 3 & -2 \end{bmatrix} \) then the value of x is
(a) – 2
(b) 0
(c) 1
(d) 2
Solution:
\(\begin{bmatrix} x-2y & 5 \\ 3 & y \end{bmatrix}=\begin{bmatrix} 6 & 5 \\ 3 & -2 \end{bmatrix} \)
Comparing, we get
y = -2
and x – 2y = 6
⇒ x – 2 x (-2) = 6
⇒ x + 4 = 6
⇒ x = 6 – 4 = 2 (d)

Question 5.
If \(\begin{bmatrix} x+2y & 3y \\ 4x & 2 \end{bmatrix}=\begin{bmatrix} 0 & -3 \\ 8 & 2 \end{bmatrix} \) then the value of x – y is
(a) – 3
(b) 1
(c) 3
(d) 5
Solution:
\(\begin{bmatrix} x+2y & 3y \\ 4x & 2 \end{bmatrix}=\begin{bmatrix} 0 & -3 \\ 8 & 2 \end{bmatrix} \)
Comparing, we get
3y = -3
⇒ \(y= \frac { -3 }{ 3 } \) = -1
4x = 8
⇒ \(x= \frac { 8 }{ 4 } \) = 2
x – y = 2 – (-1) = 2 + 1 = 3 (c)

Question 6.
If \(x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 6 \end{matrix} \right] \) then the values of x and y are
(a) x = 2, y = 6
(b) x = 2, y = – 6
(c) x = 3, y = – 4
(d) x = 3, y = – 6
Solution:
Given
\(x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 6 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q6.1

Question 7.
If B = \(\begin{bmatrix} -1 & 5 \\ 0 & 3 \end{bmatrix} \) and A – 2B = \(\begin{bmatrix} 0 & 4 \\ -7 & 5 \end{bmatrix} \)
then the matrix A is equal to
(a) \(\begin{bmatrix} 2 & 14 \\ -7 & 11 \end{bmatrix} \)
(b) \(\begin{bmatrix} -2 & 14 \\ 7 & 11 \end{bmatrix} \)
(c) \(\begin{bmatrix} 2 & -14 \\ 7 & 11 \end{bmatrix} \)
(d) \(\begin{bmatrix} -2 & 14 \\ -7 & 11 \end{bmatrix} \)
Solution:
Given
B = \(\begin{bmatrix} -1 & 5 \\ 0 & 3 \end{bmatrix} \) and
A – 2B = \(\begin{bmatrix} 0 & 4 \\ -7 & 5 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q7.1

Question 8.
If A + B = \(\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \) and A – 2B = \(\begin{bmatrix} -1 & 1 \\ 0 & -1 \end{bmatrix} \)
then A is equal to
(a) \(\frac { 1 }{ 3 } \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix} \)
(b) \(\frac { 1 }{ 3 } \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \)
(c) \(\begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix} \)
(d) \(\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \)
Solution:
A + B = \(\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \) and
A – 2B = \(\begin{bmatrix} -1 & 1 \\ 0 & -1 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q8.1

Question 9.
A = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) then A² =
(a) \(\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \)
(b) \(\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \)
(c) \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
(d) \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
Solution:
Given
A = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q9.1

Question 10.
If A = \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) , then A² =
(a) \(\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \)
(b) \(\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \)
(c) \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
(d) \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
Solution:
Given
A = \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q10.1

Question 11.
If A = \(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \) , then A² =
(a) A
(b) O
(c) I
(d) 2A
Solution:
Given
A = \(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q11.1

Question 12.
If A = \(\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \) , then A² =
(a) \(\begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix} \)
(b) \(\begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} \)
(c) \(\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \)
(d) none of these
Solution:
Given
A = \(\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q12.1

Question 13.
If A = \(\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \) , then A² =
(a) \(\begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix} \)
(b) \(\begin{bmatrix} 8 & -5 \\ 5 & 3 \end{bmatrix} \)
(c) \(\begin{bmatrix} 8 & -5 \\ -5 & -3 \end{bmatrix} \)
(d) \(\begin{bmatrix} 8 & -5 \\ -5 & 3 \end{bmatrix} \)
Solution:
A = \(\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \)
A² = A x A = \(\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \)\(\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q13.1

Question 14.
If A = \(\begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \) , then A² = pA, then the value of p is
(a) 2
(b) 4
(c) – 2
(d) – 4
Solution:
A = \(\begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \)
and A² = pA
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS Q14.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices MCQS are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

More Exercises

Question 1.
If A = \(\begin{bmatrix} 3 & \quad 5 \\ 4 & \quad -2 \end{bmatrix}\) and B = \(\left[ \begin{matrix} 2 \\ 4 \end{matrix} \right] \), is the product AB possible ? Give a reason. If yes, find AB.
Solution:
Yes, the product is possible because of
number of column in A = number of row in B
i.e., (2 x 2). (2 x 1) = (2 x 1) is the order of the matrix.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q1.1

Question 2.
If A = \(\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}\),B = \(\begin{bmatrix} 1 & -1 \\ -3 & 2 \end{bmatrix}\), find AB and BA, Is AB = BA ?
Solution:
A = \(\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}\),
B = \(\begin{bmatrix} 1 & -1 \\ -3 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q2.1

Question 3.
If P = \(\begin{bmatrix} 4 & 6 \\ 2 & -8 \end{bmatrix}\),Q = \(\begin{bmatrix} 2 & -3 \\ -1 & 1 \end{bmatrix}\)
Find 2PQ
Solution:
P = \(\begin{bmatrix} 4 & 6 \\ 2 & -8 \end{bmatrix}\),
Q = \(\begin{bmatrix} 2 & -3 \\ -1 & 1 \end{bmatrix}\)
\(2PQ=2\begin{bmatrix} 4 & \quad 6 \\ 2 & -8 \end{bmatrix}\times \begin{bmatrix} 2\quad & -3 \\ -1 & \quad 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q3.1

Question 4.
Given A = \(\begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}\) , evaluate A² – 4A
Solution:
A = \(\begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}\)
A² – 4A = \(\begin{bmatrix} 1 & \quad 1 \\ 8 & \quad 3 \end{bmatrix}\begin{bmatrix} 1\quad & 1 \\ 8\quad & 3 \end{bmatrix}-4\begin{bmatrix} 1\quad & 1 \\ 8\quad & 3 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q4.1

Question 5.
If A = \(\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}\), B = \(\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}\) and C = \(\begin{bmatrix} 1 & \quad -5 \\ -4 & \quad 6 \end{bmatrix}\)
Find AB – 5C
Solution:
A = \(\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}\), B = \(\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}\) and C = \(\begin{bmatrix} 1 & \quad -5 \\ -4 & \quad 6 \end{bmatrix}\)
AB = \(\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}\)\(\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q5.1

Question 6.
If A = \(\begin{bmatrix} 1 & \quad 2 \\ 2 & \quad 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & \quad 1 \\ 1 & \quad 2 \end{bmatrix}\), find A(BA)
Solution:
A = \(\begin{bmatrix} 1 & \quad 2 \\ 2 & \quad 1 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & \quad 1 \\ 1 & \quad 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q6.1

Question 7.
Given matrices:
A = \(\begin{bmatrix} 2 & \quad 1 \\ 4 & \quad 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & \quad 4 \\ -1 & \quad -2 \end{bmatrix}\), C = \(\begin{bmatrix} -3 & \quad 1 \\ 0 & \quad -2 \end{bmatrix}\)
Find the products of (i) ABC (ii) ACB and state whether they are equal.
Solution:
A = \(\begin{bmatrix} 2 & \quad 1 \\ 4 & \quad 2 \end{bmatrix}\)
B = \(\begin{bmatrix} 3 & \quad 4 \\ -1 & \quad -2 \end{bmatrix}\),
C = \(\begin{bmatrix} -3 & \quad 1 \\ 0 & \quad -2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q7.1

Question 8.
Evaluate : \(\begin{bmatrix} 4\sin { { 30 }^{ o } } & \quad 2cos{ 60 }^{ o } \\ sin{ 90 }^{ o } & \quad 2cos{ 0 }^{ o } \end{bmatrix}\begin{bmatrix} 4 & 5 \\ 5 & 4 \end{bmatrix}\)
Solution:
\(\begin{bmatrix} 4\sin { { 30 }^{ o } } & \quad 2cos{ 60 }^{ o } \\ sin{ 90 }^{ o } & \quad 2cos{ 0 }^{ o } \end{bmatrix}\begin{bmatrix} 4 & 5 \\ 5 & 4 \end{bmatrix}\)
\(sin{ 30 }^{ o }=\frac { 1 }{ 2 } ,cos{ 60 }^{ o }=\frac { 1 }{ 2 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q8.1

Question 9.
If A = \(\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\), B = \(\begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}\) find the matrix AB + BA
Solution:
A = \(\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\),
B = \(\begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}\)
\(AB=\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\times \begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q9.1

Question 10.
A = \(\begin{bmatrix} 1 & \quad 2 \\ 3 & \quad 4 \end{bmatrix}\) and B = \(\begin{bmatrix} 6 & \quad 1 \\ 1 & \quad 1 \end{bmatrix}\), C = \(\begin{bmatrix} -2 & \quad -3 \\ 0 & \quad 1 \end{bmatrix}\)
find each of the following and state if they are equal.
(i) CA + B
(ii) A + CB
Solution:
(i) CA + B
CA = \(\begin{bmatrix} -2 & \quad -3 \\ 0 & \quad 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & \quad 2 \\ 3 & \quad 4 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q10.1

Question 11.
If A = \(\begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}\)
Find 2B – A²
Solution:
A = \(\begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}\)
B = \(\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}\)
2B = \(2\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}\)
= \(\begin{bmatrix} 6 & 4 \\ -4 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q11.1

Question 12.
If A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\), C = \(\begin{bmatrix} 5 & 1 \\ 7 & 4 \end{bmatrix}\), compute
(i) A(B + C)
(ii) (B + C)A
Solution:
(i) A(B + C)
A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\),
C = \(\begin{bmatrix} 5 & 1 \\ 7 & 4 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q12.1

Question 13.
If A = \(\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}\), C = \(\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}\)
find the matrix C(B – A)
Solution:
A = \(\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}\),
C = \(\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q13.1

Question 14.
A = \(\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}\)
Find A² + AB + B²
Solution:
Given that
A = \(\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q14.2

Question 15.
If A = \(\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}\), C = \(\begin{bmatrix} -3 & 2 \\ -1 & 4 \end{bmatrix}\)
Find A² + AC – 5B
Solution:
A = \(\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix}\)
B = \(\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}\),
C = \(\begin{bmatrix} -3 & 2 \\ -1 & 4 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q15.1

Question 16.
If A = \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\), find A2 and A3.Also state that which of these is equal to A
Solution:
A = \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
A² = A x A = \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q16.1

Question 17.
If X = \(\begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}\), show that 6X – X² = 9I Where I is the unit matrix.
Solution:
Given that
X = \(\begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q17.1

Question 18.
Show that \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\) is a solution of the matrix equation X² – 2X – 3I = 0,Where I is the unit matrix of order 2
Solution:
Given
X² – 2X – 3I = 0
Solution = \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)
or
X = \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)
∴ X² = \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q18.1

Question 19.
Find the matrix X of order 2 × 2 which satisfies the equation
\(\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}+2X=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}\)
Solution:
Given
\(\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}+2X=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q19.1

Question 20.
If A = \(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\), find the value of x, so that A² – 0
Solution:
Given
A = \(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\)
A² = \(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\)\(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q20.1

Question 21.
If \(\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \) Find the value of x
Solution:
\(\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \)
⇒ \(\begin{bmatrix} 2 & -3 \\ 0 & 0 \end{bmatrix}=\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \)
⇒ \(\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \)
Comparing the corresponding elements
x = -1

Question 22.
(i) Find x and y if \(\begin{bmatrix} -3 & 2 \\ 0 & -5 \end{bmatrix}\left[ \begin{matrix} x \\ 2 \end{matrix} \right] =\left[ \begin{matrix} -5 \\ y \end{matrix} \right] \)
(ii) Find x and y if \(\begin{bmatrix} 2x & x \\ y & 3y \end{bmatrix}\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] =\left[ \begin{matrix} 16 \\ 9 \end{matrix} \right] \)
Solution:
(i) \(\begin{bmatrix} -3 & 2 \\ 0 & -5 \end{bmatrix}\left[ \begin{matrix} x \\ 2 \end{matrix} \right] =\left[ \begin{matrix} -5 \\ y \end{matrix} \right] \)
⇒ \(\begin{bmatrix} -3x & 4 \\ 0 & -10 \end{bmatrix}=\left[ \begin{matrix} -5 \\ y \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q22.2
Here x = 2, y = 1

Question 23.
Find x and y if
\(\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] \)
Solution:
Given
\(\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q23.1

Question 24.
If \(\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}\begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix} \) find the values of x and y
Solution:
Given
\(\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}\begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q24.1

Question 25.
If \(\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) write down the values of a,b,c and d
Solution:
Given
\(\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q25.1
Comparing the corresponding elements
a = 3, b = 4, c = 2, d = 5

Question 26.
Find the value of x given that A² = B
Where A = \(\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}\) and
B = \(\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}\)
Solution:
A = \(\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}\) and
B = \(\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}\)
A² = B
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q26.1

Question 27.
If A = \(\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 36 \\ 0 & 1 \end{bmatrix}\), find the value of x, given that A² – B
Solution:
Given
A² = \(\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}\)\(\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q27.1

Question 28.
If A = \(\begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 9 & 16 \\ 0 & -y \end{bmatrix}\) find x and y when A² = B
Solution:
Given
A = \(\begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 9 & 16 \\ 0 & -y \end{bmatrix}\) find x and y when A² = B
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q28.1

Question 29.
Find x, y if \(\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\left[ \begin{matrix} -1 \\ 2x \end{matrix} \right] +3\left[ \begin{matrix} -2 \\ 1 \end{matrix} \right] =2\left[ \begin{matrix} y \\ 3 \end{matrix} \right] \)
Solution:
Given
\(\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\left[ \begin{matrix} -1 \\ 2x \end{matrix} \right] +3\left[ \begin{matrix} -2 \\ 1 \end{matrix} \right] =2\left[ \begin{matrix} y \\ 3 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q29.1

Question 30.
If \(\begin{bmatrix} a & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix} \) find a,b and c
Solution:
\(\begin{bmatrix} a & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix} \)
⇒ \(\begin{bmatrix} 4a-3 & 3a+2 \\ 4+0 & 3+0 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q30.1

Question 31.
If A = \(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\) ,B = \(\begin{bmatrix} 2 & x \\ 0 & -\frac { 1 }{ 2 } \end{bmatrix} \) find the value of x if AB = BA
Solution:
Given
AB = \(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\)\(\begin{bmatrix} 2 & x \\ 0 & -\frac { 1 }{ 2 } \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q31.1

Question 32.
If A = \(\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\) find x and y so that A² – xA + yI
Solution:
Given
A² = \(\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\)\(\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q32.1

Question 33.
If P = \(\begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix}\), Q = \(\begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix}\)
find x and y such that PQ = 0
Solution:
Given
P = \(\begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix}\),
Q = \(\begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q33.1

Question 34.
Let \(M\times \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}=\left[ \begin{matrix} 1 & 2 \end{matrix} \right] \) where M is a matrix
(i) State the order of matrix M
(ii) Find the matrix M
Solution:
Given
(i) M is the order of 1 x 2
let M = [x y]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q34.1

Question 35.
Given \(\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}\) ,X = \(\left[ \begin{matrix} 7 \\ 6 \end{matrix} \right] \)
(i) the order of the matrix X
(ii) the matrix X
Solution:
We have
\(\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}\) , X = \(\left[ \begin{matrix} 7 \\ 6 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q35.1

Question 36.
Solve the matrix equation : \(\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right] \) ,X = \(\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}\)
Solution:
\(\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right] \) , X = \(\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}\)
Let matrix X = [x y]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q36.1

Question 37.
(i) If A = \(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\) and B = \(\left[ \begin{matrix} -3 \\ 2 \end{matrix} \right] \) find the matrix C such that AC = B
(ii) If A = \(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\) and B = [0 -3] find the matrix C such that CA = B
Solution:
(i) given
A = \(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\)
B = \(\left[ \begin{matrix} -3 \\ 2 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q37.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q37.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q37.3

Question 38.
If A = \(\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}\) , find matrix B such that BA = I,where I is unity matrix of order 2
Solution:
A = \(\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}\)
BA = I, where I is unity matrix of order 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q38.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q38.2

Question 39.
If B = \(\begin{bmatrix} -4 & 2 \\ 5 & -1 \end{bmatrix}\) and C = \(\begin{bmatrix} 17 & -1 \\ 47 & -13 \end{bmatrix}\)
find the matrix A such that AB = C
Solution:
B = \(\begin{bmatrix} -4 & 2 \\ 5 & -1 \end{bmatrix}\)
C = \(\begin{bmatrix} 17 & -1 \\ 47 & -13 \end{bmatrix}\)
and AB = C
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q39.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q39.2

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2

More Exercises

Question 1.
Given that M = \(\begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \) and N = \(\begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix}\),find M + 2N
Solution:
M = \(\begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \)
N = \(\begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q1.1

Question 2.
If A = \(\begin{bmatrix} 2 & 0 \\ -3 & 1 \end{bmatrix} \) and B = \(\begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \)
find 2A – 3B
Solution:
A = \(\begin{bmatrix} 2 & 0 \\ -3 & 1 \end{bmatrix} \)
B = \(\begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q2.1

Question 3.
If A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \)
Compute 3A + 4B
Solution:
A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \)
B = \(\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q3.1

Question 4.
Given A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix} \)
(i) find the matrix 2A + B
(ii) find a matrix C such that C + B = \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
Solution:
A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \)
B = \(\begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q4.1

Question 5.
A = \(\begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & -1 \\ 1 & 2 \end{bmatrix} \) , C = \(\begin{bmatrix} 0 & 3 \\ 2 & -1 \end{bmatrix} \)
Find A + 2B – 3C
Solution:
A = \(\begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & -1 \\ 1 & 2 \end{bmatrix} \) , C = \(\begin{bmatrix} 0 & 3 \\ 2 & -1 \end{bmatrix} \)
∴ A + 2B – 3C
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q5.1

Question 6.
If A = \(\begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix} \) and B = \(\begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix} \)
Find the matrix X if :
(i) 3A + X = B
(ii) X – 3B = 2A
Solution:
A = \(\begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix} \)
B = \(\begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix} \)
(i) 3A + X = B
⇒ X = B – 3A
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q6.1

Question 7.
Solve the matrix equation
\(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}-3X=\begin{bmatrix} -7 & 4 \\ 2 & 6 \end{bmatrix}\)
Solution:
\(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}-3X=\begin{bmatrix} -7 & 4 \\ 2 & 6 \end{bmatrix}\)
\(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}-\begin{bmatrix} -7 & 4 \\ 2 & 6 \end{bmatrix}=3X\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q7.1

Question 8.
If \(\begin{bmatrix} 1 & \quad 4 \\ -2 & \quad 3 \end{bmatrix}+2M=3\begin{bmatrix} 3 & \quad 2 \\ 0 & -3 \end{bmatrix}\), find the matrix M
Solution:
\(\begin{bmatrix} 1 & \quad 4 \\ -2 & \quad 3 \end{bmatrix}+2M=3\begin{bmatrix} 3 & \quad 2 \\ 0 & -3 \end{bmatrix}\)
2M =
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q8.1

Question 9.
A = \(\begin{bmatrix} 2 & -6 \\ 2 & 0 \end{bmatrix} \) and B = \(\begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix} \) , C = \(\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \)
Find the matrix X such that A + 2X = 2B + C
Solution:
A = \(\begin{bmatrix} 2 & -6 \\ 2 & 0 \end{bmatrix} \) and B = \(\begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix} \) , C = \(\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \)
let X = \(\begin{bmatrix} x & y \\ z & t \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q9.1

Question 10.
Find X and Y if X + Y = \(\begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}\) and X – Y = \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\)
Solution:
X + Y = \(\begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}\)…..(i)
X – Y = \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\)…….(ii)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q10.1

Question 11.
If \(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}\) Find the values of x and y
Solution:
\(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}\)
\(\begin{bmatrix} 6 & 8 \\ 10 & 2x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q11.1

Question 12.
If \(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} z & 0 \\ 10 & 5 \end{bmatrix}\) Find the values of x and y
Solution:
\(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} z & 0 \\ 10 & 5 \end{bmatrix}\)
\(\begin{bmatrix} 6 & 8 \\ 10 & 2x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} z & 0 \\ 10 & 5 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q12.1

Question 13.
If \(\begin{bmatrix} 5 & 2 \\ -1 & \quad y+1 \end{bmatrix}-2\begin{bmatrix} 1 & 2x-1 \\ 3 & -2 \end{bmatrix}=\begin{bmatrix} 3 & -8 \\ -7 & 2 \end{bmatrix}\) Find the values of x and y
Solution:
\(\begin{bmatrix} 5 & 2 \\ -1 & \quad y+1 \end{bmatrix}-2\begin{bmatrix} 1 & 2x-1 \\ 3 & -2 \end{bmatrix}=\begin{bmatrix} 3 & -8 \\ -7 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q13.1

Question 14.
If \(\begin{bmatrix} a & \quad 3 \\ 4 & \quad 2 \end{bmatrix}+\begin{bmatrix} 2 & \quad b \\ 1 & -2 \end{bmatrix}-\begin{bmatrix} 1\quad & 1 \\ -2\quad & c \end{bmatrix}=\begin{bmatrix} 5 & 0 \\ 7 & 3 \end{bmatrix}\)
Find the value of a,b and c
Solution:
\(\begin{bmatrix} a & \quad 3 \\ 4 & \quad 2 \end{bmatrix}+\begin{bmatrix} 2 & \quad b \\ 1 & -2 \end{bmatrix}-\begin{bmatrix} 1\quad & 1 \\ -2\quad & c \end{bmatrix}=\begin{bmatrix} 5 & 0 \\ 7 & 3 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q14.1

Question 15.
If A = \(\begin{bmatrix} 2 & a \\ -3 & 5 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & 3 \\ 7 & b \end{bmatrix} \) , C = \(\begin{bmatrix} c & 9 \\ -1 & -11 \end{bmatrix} \) and 5A + 2B = C, find the values of a,b,c
Solution:
A = \(\begin{bmatrix} 2 & a \\ -3 & 5 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & 3 \\ 7 & b \end{bmatrix} \) , C = \(\begin{bmatrix} c & 9 \\ -1 & -11 \end{bmatrix} \)
and 5A + 2B = C
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q15.1

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1

More Exercises

Question 1.
(i)\(\begin{bmatrix} 2 & -1 \\ 5 & 1 \end{bmatrix}\)
(ii)[2 3 – 7]
(iii)\(\left[ \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right] \)
(iv)\(\left[ \begin{matrix} \begin{matrix} 2 \\ 0 \\ 1 \end{matrix} & \begin{matrix} -4 \\ 0 \\ 7 \end{matrix} \end{matrix} \right] \)
(v)\(\left[ \begin{matrix} \begin{matrix} 2 & 7 & 8 \end{matrix} \\ \begin{matrix} -1 & \sqrt { 2 } & 0 \end{matrix} \end{matrix} \right] \)
(vi)\(\left[ \begin{matrix} \begin{matrix} 0 & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & 0 \end{matrix} \end{matrix} \right] \)
Solution:
(i) It is square matrix of order 2
(ii) It is row matrix of order 1 × 3
(iii) It is column matrix of order 3 × 1
(iv) It is matrix of order 3 × 2
(v) It is matrix of order 2 × 3
(vi) It is zero matrix of order 2 × 3

Question 2.
(i) If a matrix has 4 elements, what are the possible order it can have ?
(ii) If a matrix has 8 elements, what are the possible order it can have ?
Solution:
(i) It can have 1 × 4, 4 × 1 or 2 × 2 order
(ii) It can have 1 × 8, 8 × 1,2 × 4 or 4 × 2 order

Question 3.
Construct a 2 x 2 matrix whose elements aij are given by
(i) aij = 2i – j
(ii) aij = i.j
Solution:
(i) It can be \(\begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix}\)
(ii) It can be \(\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}\)

Question 4.
Find the values of x and y if : \(\left[ \begin{matrix} 2x+y \\ 3x-2y \end{matrix} \right] =\left[ \begin{matrix} 5 \\ 4 \end{matrix} \right] \)
Solution:
Comparing corresponding elements,
2x + y = 5 …(i)
3x – 2y = 4 …(ii)
Multiply (i) by 2 and (ii) by ‘1’ we get
4x + 2y = 10, 3x – 2y = 4
Adding we get, 7x = 14 ⇒ x = 2
Substituting the value of x in (i)
2 x 2 + y = 5 ⇒ 4 + y = 5
y = 5 – 4 = 1
Hence x = 2, y = 1

Question 5.
Find the value of x if \(\left[ \begin{matrix} \begin{matrix} 3x+y & \quad -y \end{matrix} \\ \begin{matrix} 2y-x & \quad \quad 3 \end{matrix} \end{matrix} \right] =\begin{bmatrix} 1 & 2 \\ -5 & 3 \end{bmatrix} \)
Solution:
\(\left[ \begin{matrix} \begin{matrix} 3x+y & \quad -y \end{matrix} \\ \begin{matrix} 2y-x & \quad \quad 3 \end{matrix} \end{matrix} \right] =\begin{bmatrix} 1 & 2 \\ -5 & 3 \end{bmatrix} \)
Comparing the corresponding terms, we get.
-y = 2
⇒ y = -2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1 Q5.1

Question 6.
If \(\left[ \begin{matrix} \begin{matrix} x+3 & \quad \quad 4 \end{matrix} \\ \begin{matrix} y-4 & \quad \quad x+y \end{matrix} \end{matrix} \right] =\begin{bmatrix} 5 & 4 \\ 3 & 9 \end{bmatrix} \) ,find values of x and y
Solution:
\(\left[ \begin{matrix} \begin{matrix} x+3 & \quad \quad 4 \end{matrix} \\ \begin{matrix} y-4 & \quad \quad x+y \end{matrix} \end{matrix} \right] =\begin{bmatrix} 5 & 4 \\ 3 & 9 \end{bmatrix} \)
Comparing the corresponding terms, we get.
x + 3 = 5
⇒ x = 5 – 3 = 2
⇒ y – 4 = 3
⇒ y = 3 + 4 = 7
x = 2, y = 7

Question 7.
Find the values of x, y and z if
\(\left[ \begin{matrix} \begin{matrix} x+2 & \quad \quad 6 \end{matrix} \\ \begin{matrix} 3 & \quad \quad \quad 5z \end{matrix} \end{matrix} \right] =\begin{bmatrix} -5 & \quad { y }^{ 2 }+y \\ 3 & -20 \end{bmatrix}\)
Solution:
Comparing the corresponding elements of equal determinents,
x + 2 = -5
⇒ x = -5 – 2 = -7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1 Q7.1

Question 8.
Find the values of x, y, a and b if
\(\begin{bmatrix} x-2 & y \\ a+2b & 3a-b \end{bmatrix}=\begin{bmatrix} 3 & 1 \\ 5 & 1 \end{bmatrix}\)
Solution:
Comparing corresponding elements
x – 2 = 3, y = 1
x = 3 + 2 = 5
a + 2b = 5 ……(i)
3a – b = 1 ……..(ii)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1 Q8.1

Question 9.
Find the values of a, b, c and d if
\(\begin{bmatrix} a+b & 3 \\ 5+c & ab \end{bmatrix}=\begin{bmatrix} 6 & d \\ -1 & 8 \end{bmatrix} \)
Solution:
\(\begin{bmatrix} a+b & 3 \\ 5+c & ab \end{bmatrix}=\begin{bmatrix} 6 & d \\ -1 & 8 \end{bmatrix} \)
Comparing the corresponding terms, we get.
3 = d ⇒ d = 3
⇒ 5 + c = – 1
⇒ c = -1 – 5
⇒ c = -6
a + b = 6 and ab = 8
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1 Q9.1

Question 10.
Find the values of x, y, a and b, if
\(\left[ \begin{matrix} \begin{matrix} 3x+4y & 2 & x-2y \end{matrix} \\ \begin{matrix} a+b & 2a-b & -1 \end{matrix} \end{matrix} \right] =\left[ \begin{matrix} \begin{matrix} 2 & \quad 2\quad & 4 \end{matrix} \\ \begin{matrix} 5 & -5 & -1 \end{matrix} \end{matrix} \right] \)
Solution:
Comparing the corresponding terms, we get.
3x + 4y = 2 ……(i)
x – 2y = 4 …….(ii)
Multiplying (i) by 1 and (ii) by 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.1 Q10.1

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test

More Exercises

Question 1.
Find the compound ratio of:
(a + b)2 : (a – b )2 ,
(a2 – b2) : (a2 + b2),
(a4 – b4) : (a + b)4
Solution:
(a + b)2 : (a – b )2 ,
(a2 – b2) : (a2 + b2),
(a4 – b4) : (a + b)4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q1.1

Question 2.
If (7 p + 3 q) : (3 p – 2 q) = 43 : 2 find p : q
Solution:
(7p + 3q) : (3p – 2q) = 43 : 2
⇒ \(\frac { 7p+3q }{ 3p-2q } =\frac { 43 }{ 2 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q2.1

Question 3.
If a : b = 3 : 5, find (3a + 5b): (7a – 2b).
Solution:
a : b = 3 : 5
⇒ \(\frac { a }{ b } =\frac { 3 }{ 5 } \)
⇒ 3a + 5n : 7a – 2b
Dividing each term by b
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q3.1

Question 4.
The ratio of the shorter sides of a right angled triangle is 5 : 12. If the perimeter of the triangle is 360 cm, find the length of the longest side.
Solution:
Let the two shorter sides of a right-angled triangle be 5x and 12x.
Third (longest side)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q4.1

Question 5.
The ratio of the pocket money saved by Lokesh and his sister is 5 : 6. If the sister saves Rs 30 more, how much more the brother should save in order to keep the ratio of their savings unchanged?
Solution:
Let the savings of Lokesh and his sister are 5x and 6x.
and the Lokesh should save Rs y more Now, according to the problem,
⇒ \(\frac { 5x+y }{ 6x+30 } =\frac { 5 }{ 6 } \)
⇒ 30x + 6y = 30x + 150
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q5.1

Question 6.
In an examination, the number of those who passed and the number of those who failed were in the ratio of 3 : 1. Had 8 more appeared, and 6 less passed, the ratio of passed to failures would have been 2 : 1. Find the number of candidates who appeared.
Solution:
Let number of passed = 3 x
and failed = x
Total candidates appeared = 3x + x = 4x.
In second case
No. of candidates appeared = 4 x + 8
and No. of passed = 3 x – 6
and failed = 4x + 8 – 3x + 6 = x + 14
then ratio will be = 2 : 1
Now according to the condition
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q6.1

Question 7.
What number must be added to each of the numbers 15, 17, 34 and 38 to make them proportional ?
Solution:
Let x be added to each number, then numbers will be
15 + x, 17 + x, 34 + x, and 38 + x.
Now according to the condition
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q7.1

Question 8.
If (a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion, prove that b is the mean proportional between a and c.
Solution:
(a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion
⇒ \(\frac { a+2b+c }{ a-c } =\frac { a-c }{ a-2b+c } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q8.1

Question 9.
If 2, 6, p, 54 and q are in continued proportion, find the values of p and q.
Solution:
2, 6, p, 54 and q are in continued proportional then
⇒ \(\frac { 2 }{ 6 } =\frac { 6 }{ p } =\frac { p }{ 54 } =\frac { 54 }{ q } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q9.1

Question 10.
If a, b, c, d, e are in continued proportion, prove that: a : e = a4 : b4.
Solution:
a, b, c, d, e are in continued proportion
⇒ \(\frac { a }{ b } =\frac { b }{ c } =\frac { c }{ d } =\frac { d }{ e } \) = k (say)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q10.1

Question 11.
Find two numbers whose mean proportional is 16 and the third proportional is 128.
Solution:
Let x and y be two numbers
Their mean proportion = 16
and third proportion = 128
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q11.1

Question 12.
If q is the mean proportional between p and r, prove that:
\({ p }^{ 2 }-{ 3q }^{ 2 }+{ r }^{ 2 }={ q }^{ 4 }\left( \frac { 1 }{ { p }^{ 2 } } -\frac { 3 }{ { q }^{ 2 } } +\frac { 1 }{ { r }^{ 2 } } \right) \)
Solution:
q is mean proportional between p and r
q² = pr
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q12.1

Question 13.
If \(\frac { a }{ b } = \frac { c }{ d } = \frac { e }{ f } \), prove that each ratio is
(i) \(\sqrt { \frac { { 3a }^{ 2 }-{ 5c }^{ 2 }+{ 7e }^{ 2 } }{ { 3b }^{ 2 }-{ 5d }^{ 2 }+{ 7f }^{ 2 } } } \)
(ii) \({ \left[ \frac { { 2a }^{ 3 }+{ 5c }^{ 3 }+{ 7e }^{ 3 } }{ { 2b }^{ 3 }+{ 5d }^{ 3 }+{ 7f }^{ 3 } } \right] }^{ \frac { 1 }{ 3 } } \)
Solution:
\(\frac { a }{ b } = \frac { c }{ d } = \frac { e }{ f } \) = k(say)
∴ a = k, c = dk, e = fk
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q13.1

Question 14.
If \(\frac { x }{ a } = \frac { y }{ b } = \frac { z }{ c } \), prove that
\(\frac { { 3x }^{ 3 }-{ 5y }^{ 3 }+{ 4z }^{ 3 } }{ { 3a }^{ 3 }-{ 5b }^{ 3 }+{ 4c }^{ 3 } } ={ \left( \frac { 3x-5y+4z }{ 3a-5b+4c } \right) }^{ 3 }\)
Solution:
\(\frac { x }{ a } = \frac { y }{ b } = \frac { z }{ c } \) = k (say)
x = ak, y = bk, z = ck
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q14.1

Question 15.
If x : a = y : b, prove that
\(\frac { { x }^{ 4 }+{ a }^{ 4 } }{ { x }^{ 3 }+{ a }^{ 3 } } +\frac { { y }^{ 4 }+{ b }^{ 4 } }{ { y }^{ 3 }+{ b }^{ 3 } } =\frac { { \left( x+y \right) }^{ 4 }+{ \left( a+b \right) }^{ 4 } }{ { \left( x+y \right) }^{ 3 }+{ \left( a+b \right) }^{ 3 } } \)
Solution:
\(\frac { x }{ a } = \frac { y }{ b } \) = k (say)
x = ak, y = bk
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q15.2

Question 16.
If \(\frac { x }{ b+c-a } =\frac { y }{ c+a-b } =\frac { z }{ a+b-c } \) prove that each ratio’s equal to :
\(\frac { x+y+z }{ a+b+c } \)
Solution:
\(\frac { x }{ b+c-a } =\frac { y }{ c+a-b } =\frac { z }{ a+b-c } \) = k(say)
x = k(b + c – a),
y = k(c + a – b),
z = k(a + b – c)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q16.1

Question 17.
If a : b = 9 : 10, find the value of
(i) \(\frac { 5a+3b }{ 5a-3b } \)
(ii) \(\frac { { 2a }^{ 2 }-{ 3b }^{ 2 } }{ { 2a }^{ 2 }+{ 3b }^{ 2 } } \)
Solution:
a : b = 9 : 10
⇒ \(\frac { a }{ b } = \frac { 9 }{ 10 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q17.2

Question 18.
If (3x² + 2y²) : (3x² – 2y²) = 11 : 9, find the value of \(\frac { { 3x }^{ 4 }+{ 25y }^{ 4 } }{ { 3x }^{ 4 }-{ 25y }^{ 4 } } \) ;
Solution:
\(\frac { { 3x }^{ 4 }+{ 25y }^{ 4 } }{ { 3x }^{ 4 }-{ 25y }^{ 4 } } =\frac { 11 }{ 9 } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q18.2

Question 19.
If \(x=\frac { 2mab }{ a+b } \) , find the value of
\(\frac { x+ma }{ x-ma } +\frac { x+mb }{ x-mb } \)
Solution:
\(x=\frac { 2mab }{ a+b } \)
⇒ \(\frac { x }{ ma } +\frac { 2b }{ a+b } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q19.1

Question 20.
If \(x=\frac { pab }{ a+b } \) ,prove that \(\frac { x+pa }{ x-pa } -\frac { x+pb }{ x-pb } =\frac { 2\left( { a }^{ 2 }-{ b }^{ 2 } \right) }{ ab } \)
Solution:
\(x=\frac { pab }{ a+b } \)
⇒ \(\frac { x }{ pa } +\frac { b }{ a+b } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q20.2

Question 21.
Find x from the equation \(\frac { a+x+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } }{ a+x-\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } =\frac { b }{ x } \)
Solution:
\(\frac { a+x+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } }{ a+x-\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } =\frac { b }{ x } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q21.2

Question 22.
If \(x=\frac { \sqrt [ 3 ]{ a+1 } +\sqrt [ 3 ]{ a-1 } }{ \sqrt [ 3 ]{ a+1 } -\sqrt [ 3 ]{ a-1 } } \), prove that :
x³ – 3ax² + 3x – a = 0
Solution:
\(x=\frac { \sqrt [ 3 ]{ a+1 } +\sqrt [ 3 ]{ a-1 } }{ \sqrt [ 3 ]{ a+1 } -\sqrt [ 3 ]{ a-1 } } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q22.2

Question 23.
If \(\frac { by+cz }{ b^{ 2 }+{ c }^{ 2 } } =\frac { cz+ax }{ { c }^{ 2 }+{ a }^{ 2 } } =\frac { ax+by }{ { a }^{ 2 }+{ b }^{ 2 } } \), prove that each of these ratio is equal to \(\frac { x }{ a } =\frac { y }{ b } =\frac { z }{ c } \)
Solution:
\(\frac { by+cz }{ b^{ 2 }+{ c }^{ 2 } } =\frac { cz+ax }{ { c }^{ 2 }+{ a }^{ 2 } } =\frac { ax+by }{ { a }^{ 2 }+{ b }^{ 2 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q23.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q23.2

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.