RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2I.

Other Exercises

Question 1.
Solution:
(i)(3x+2)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I Q1.1

Question 2.
Solution:
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I Q2.1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I Q2.2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I Q2.3

Question 3.
Solution:
(i)(95)3 = (100 – 5)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2I Q3.1

Hope given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2I are helpful to complete your math homework.

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2F

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2F

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2F.

Other Exercises

Factorize :

Question 1.
Solution:
25x2 – 64y2
= (5x)2 – (8y)2
= (5x + 8y) (5x – 8y) Ans.

Question 2.
Solution:
100 – 9x2
(10)2 – (3x)2
= (10 + 3x) (10 – 3x) Ans.

Question 3.
Solution:
5x2 – 7y2
=(√5x)2 +(√7y)2
=(√5x – √7y)(√5x – √7y) Ans

Question 4.
Solution:
(3x + 5y)2 – 4z2
= (3x + 5y)2 – (2z)2
= (3x + 5y + 2z) (3x + 5y – 2z) Ans.

Question 5.
Solution:
150 – 6x2
= 6(25 – x2)
= 6{(5)2 – (x2)}
= 6 (5 + x) (5 – x) Ans.

Question 6.
Solution:
20x2 – 45
= 5 (4x2 – 9)
= 5{(2x)2 – (3)2}
= 5 (2x + 3) (2x – 3) Ans.

Question 7.
Solution:
3x3 – 48x
= 3x (x2 – 16)
= 3x {(x)2 – (4)2}
= 3x (x + 4) (x – 4) Ans.

Question 8.
Solution:
2 – 50x2
= 2(1 – 25x2) = 2 {(1)2 – (5x)2}
= 2 (1 + 5x) (1 – 5x) Ans.

Question 9.
Solution:
27a2 – 48b2
= 3(9a2 – 16b2)
= 3 {(3a)2 – (4b)2}
= 3(3a + 4b) (3a – 4b) Ans.

Question 10.
Solution:
x – 64x3
= x (1 – 64x2)
= x{(1)2 – (8x)2}
= x (1 + 8x) (1 – 8x) Ans.

Question 11.
Solution:
8ab2 – 18a3
= 2a (4b2 – 9a2)
= 2a {(2b)2 – (3a)2}
= 2a (2b + 3a) (2b – 3a) Ans

Question 12.
Solution:
3a3b – 243ab3
= 2ab (a2 -81 b2)
= 3ab {(a)2 – (9b)2}
= 3ab (a + 9b) (a – 9b) Ans.

Question 13.
Solution:
(a + b)3 – a – b
= (a + b)3 – 1 (a + b)
= (a + b) {(a + b)2 – 1}
= (a + b) {(a + b)2 – (1)2}
= (a + b) (a + b + 1) (a + b – 1) Ans.

Question 14.
Solution:
108 a2 – 3(b – c)2
= 3 {36a2 – (b – c)2}
= 3 {(6a)2 – (b – c)2}
= 3 {(6a + (b – c)} {6a – (b – c)}
= 3 (6a + b – c) (6a – b + c) Ans.

Question 15.
Solution:
x3 – 5x2 – x + 5
= x2(x – 5) -1(x – 5)
= (x – 5) (x2 – 1)
= (x – 5) {(x)2 – (1)2}
= (x – 5) (x + 1) (x – 1) Ans.

Question 16.
Solution:
a2 + 2ab + b2 – 9c2
= (a + b)2 – (3c)2
= {a2 + b2 + 2ab – (a + b)2}
= (a + b + 3c) (a + b – 3c) Ans.

Question 17.
Solution:
9 – a2 + 2ab – b2
= 9 – (a2 – 2ab + b2)
= (3)2 -(a- b)2
{ ∴a2 + b2 – 2ab = (a – b)2}
= (3 + a – b) (3 – a + b) Ans.

Question 18.
Solution:
a– b2 – 4ac + 4c2
= a2 – 4ac + 4c2 – b2
= (a)2 – 2 x a x 2c + (2c)2 – (b)2
= (a – 2c)2 – (b)2
= (a – 2c + b) (a – 2c – b)
= (a + b – 2c) (a – b – 2c) Ans.

Question 19.
Solution:
9a2 + 3a – 8B – 64b2
= 9a2 – 64b2 + 3a – 8b
= (3a)2 – (8b)2 + 1(3a – 8b)
= (3a + 8b) (3a – 8b) + 1 (3a – 8b)
= (3a – 8b) (3a + 8b + 1) Ans.

Question 20.
Solution:
x2 – y2 + 6y _ 9
= (x)2 – (y2 – 6y + 9)
= (x)2 – {(y)2 – 2 x y x 3 + (3)2}
= (x)2 – (y – 3)2
= (x + y – 3) (x – y + 3) Ans.

Question 21.
Solution:
4x2 – 9y2 – 2x – 3y
= (2x)2 – (3y)2 – 1(2x + 3y)
= (2x + 3y) (2x – 3y) – 1( 2x + 3y)
= (2x + 3y) (2x – 3y – 1) Ans.

Question 22.
Solution:
x4 – 1
= (x2)2 – (1)2
= (x2 + 1) (x2 – 1)
= (x2 + 1) {(x)2 – (1)2}
= (x2 + 1) (x + 1) (x – 1) Ans.

Question 23.
Solution:
a – b – a2+ b2
= 1 (a – b) – (a2 – b2)
= 1 (a – b) – (a + b) (a – b)
= (a – b) (1 – a – b) Ans.

Question 24.
Solution:
x4 – 625
= (x2)2 – (25)2
= (x2 + 25) (x2 – 25)
= (x2 + 25) {(x)2 – (5)2}
= (x2 + 25) (x + 5) (x – 5) Ans.

Hope given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2F 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.

RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A

RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A.

Question 1.
Solution:
In ∆ ABC, ∠A = 70° and AB = AC (given)
∴ ∠C = ∠B
(Angles opposite to equal sides)
But ∠A + ∠B + ∠C = 180°
(Sum of angles of a triangle)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 1
=> 70° + ∠B + ∠B = 180°
(∴ ∠B = ∠C)
=> 2∠B = 180°- 70° = 110°
∠B = \(\frac { { 110 }^{ o } }{ 2 } \) = 55° and
Hence ∠B = 55°,∠C = 55° Ans.

Question 2.
Solution:
In ∆ ABC, ∠A= 100°
It is an isosceles triangle
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 2
∴AB = AC
∠B = ∠C
(Angles opposite to equal sides)
But ∠A + ∠B + ∠C = 180°
(Sum of angles of a triangle)
=> 100° + ∠B + ∠B = 180°
(∴ ∠B = ∠C)
=> 2∠B + 100° = 180°
=> 2∠B = 180°- 100° = 80°
∠B = \(\frac { { 80 }^{ o } }{ 2 } \) = 40°
and ∠C = 40°
∴ Base angles are 40°, 40° Ans.

Question 3.
Solution:
In ∆ ABC,
AB = AC
∴∠C = ∠B
(Angles opposite to equal sides)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 3
But ∠B = 65°
∴ ∠ C = 65°
But ∠A + ∠B + ∠C = 180°
(Sum of angles of a triangle)
=> ∠A + 65° + 65° = 180°
=> ∠ A + 130° = 180°
=> ∠ A = 180° – 130°
=> ∠ A = 50°
Hence ∠ A = 50° and ∠ C = 65° Ans.

Question 4.
Solution:
In ∆ ABC
AB = AC
∴ ∠C = ∠B
(Angles opposite to equal sides)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 4
But ∠A = 2(∠B + ∠C)
=> ∠B + ∠C = \(\frac { 1 }{ 2 } \) ∠A 2
But ∠A + ∠B + ∠C = 180°
(sum of angles of a triangle)
=> ∠A+ \(\frac { 1 }{ 2 } \) ∠A = 180°
=> \(\frac { 3 }{ 2 } \) ∠A = 180°
=> ∠A = 180° x \(\frac { 2 }{ 3 } \) = 120°
and ∠B + ∠C = \(\frac { 1 }{ 2 } \) ∠A = \(\frac { 1 }{ 2 } \) x 120°
= 60°
∴ ∠ B = ∠ C
∴ ∠ B = ∠ C = \(\frac { { 60 }^{ o } }{ 2 } \) = 30°
Hence ∠ A = 120°, ∠B = 30°, ∠C = 30° Ans.

Question 5.
Solution:
In ∆ ABC,
AB = BC and ∠B = 90°
∴ AB = BC
∴ ∠ C = ∠ A
(Angles opposite to equal sides)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 5
Now ∠A + ∠B + ∠C = 180°
(Sum of angles of a triangle)
=> ∠ A + 90° + ∠ A = 180°
(∴ ∠C = ∠A)
=> 2∠A + 90° – 180°
=> 2∠ A = 180° – 90° = 90°
∠ A = \(\frac { { 90 }^{ o } }{ 2 } \) = 45°
∴ ∠ C = 45° (∴ ∠ C = ∠ A)
Hence, each of the equal angles is 45° Ans.

Question 6.
Solution:
Given : ∆ ABC is an isosceles triangle in which AB = AC
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 6
Base BC is produced to both sides upto D and E respectively forming exterior angles
∠ ABD and ∠ ACE
To Prove : ∠ABD = ∠ACE
Proof : In ∆ ABC
∴ AB = AC (given)
∴ ∠C = ∠B
(Angles opposite to equal sides)
=> ∠ ABC = ∠ACB
But ∠ ABC + ∠ABD = 180° (Linear pair)
Similarly ∠ACB + ∠ACE = 180°
∠ ABC + ∠ABD = ∠ACB + ∠ACE
But ∠ ABC = ∠ ACB (proved)
∠ABD = ∠ACE
Hence proved.

Question 7.
Solution:
∆ ABC is an equilateral triangle
∴ AB = BC = CA
and ∠A = ∠B = ∠C = 60°
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 7
The sides of the ∆ ABC are produced in order to D,E and F forming exterior angles
∠ACP, ∠BAE and ∠CBF
∆ACD + ∠ACB = 180°
=> ∠ACD + 60° = 180°
=> ∠ACD = 180° – 60°
=> ∠ACD = 120°
Similarly, ∠BAE + ∠BAC = 180°
=> ∠BAE + 60° = 180°
=> ∠BAE = 180° – 60° = 120°
and ∠BAF + ∠ABC = 180°
=> ∠BAF + 60° = 180°
=> ∠BAF = 180° – 60° = 120°
Hence each exterior angle of an equilateral triangle is 120°.

Question 8.
Solution:
Given : In the figure,
O is mid-point of AB and CD.
i.e. OA = OB and OC = OD
To Prove : AC = BD and AC || BD.
Proof : In ∆ OAC and ∆ OBD,
OA = OB {given}
OC = OD {given}
∠AOC = ∠BOD
(Vertically opposite angles)
∴∆ OAC ≅ ∆ OBD (S.A.S. axiom)
∴AC = BD (c.p.c.t.)
and ∠ C = ∠ D
But these are alt. angles
∴AC || BD
Hence proved.

Question 9.
Solution:
Given : In the figure,
PA ⊥ AB, QB ⊥ AB and
PA = QB, PQ intersects AB at O.
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 9
To Prove : O is mid-point of AB and PQ.
Proof : In ∆ AOP and ∆ BOQ,
∠ A = ∠ B
AP = BQ (given)
∠ AOP = ∠BOQ
(Vertically opposite angles)
∴ ∆AOP ≅ ∆ BOQ (A.A.S. axiom)
∴OA = OB (c.p.c.t)
and OP = OQ (c.p.c.t)
Hence, O is the mid-point of AB as well as PQ

Question 10.
Solution:
Given : Two line segments AB and CD intersect each other at O and OA = OB, OC = OD
AC and BD are joined.
To Prove : AC = BD
Proof : In ∆ AOC and ∆ BOD,
OA = OB {given}
OC = OD
∠AOC = ∠BOD
(vertically opposite angles)
∴ ∆ AOC ≅ ∆ BOD (S.A.S. axiom)
∴ AC = BD (c.p.c.t)
and ∠A = ∠D (c.p.c.t.)
Hence AC ≠ BD
Hence proved.

Question 11.
Solution:
Given : In the given figure,
l || m, m is mid-point of AB
CD is another line segment, which intersects AB at M.
To Prove : M is mid-point of CD
Proof : l || m
∴ ∠ CAM = ∠ MBD (Alternate angles)
Now, in ∆ AMC and ∆ BMD,
AM = MB (Given)
∠ CAM = ∠ MBD (proved)
∠ AMC = ∠BMD
(vertically opposite angles)
∆ AMC ≅ ∆ BMD (ASA axiom)
∴ CM = MD (c.p.c.t.)
Hence, M is mid-point of CD.
Similarly we can prove that M is mid point of any other line whose end points are on l and m.
Hence proved.

Question 12.
Solution:
Given : In ∆ ABC, AB = AC and in ∆ OBC,
OB = OC
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 12
To Prove : ∠ABO = ∠ACO
Construction. Join AO.
Proof : In ∆ ABO and ∆ ACO,
AB = AC (Given)
OB = OC (Given)
AO = AO (Common)
∴ ∆ ABO ≅ ∆ ACO (S.S.S. Axiom)
∴ ∠ABO = ∠ACO (c.p.c.t.)
Hence proved.

Question 13.
Solution:
Given : In ∆ ABC, AB = AC
D is a point on AB and a line DE || AB is drawn.
Which meets AC at E
To Prove : AD = AE
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 13
Proof : In ∆ ABC
AB = AC (given)
∴ ∠ C = ∠ B (Angles opposite to equal sides)
But DE || BC (given)
∴ ∠ D = ∠ B {corresponding angles}
and ∠ E = ∠ C
But ∠C = ∠B
∴ ∠E = ∠D
∴ AD = AE (Sides opposite to equal angles)
Hence proved.

Question 14.
Solution:
Given : In ∆ ABC,
AB = AC
X and Y are two points on AB and AC respectively such that AX = AY
To Prove : CX = BY
Proof : In ∆ AXC and ∆ AYB
AC = AB (given)
AX = AY (given)
∠ A = ∠ A (common)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 14
∴ ∆ AXC ≅ ∆ AYB (S.A.S. axiom)
∴ CX = BY (c.p.c.t.)
Hence proved.

Question 15.
Solution:
Given : In the figure,
C is mid point of AB
∠DCA = ∠ECB and
∠DBC = ∠EAC
To Prove : DC = EC
Proof : ∠ DCA = ∠ FCB (given)
Adding ∠ DCE both sides,
∠DCA +∠DCE = ∠DCE + ∠ECB
=> ∠ACE = ∠ BCD
Now, in ∆ ACE and ∆ BCD,
AC = BC (C is mid-point of AB)
∠EAC = ∠DBC (given)
∠ACE =∠ BCD (proved)
∴ ∆ ACE ≅ ∆ BCD (ASA axiom)
∴ CE = CD (c.p.c.t.)
=> EC = DC
or DC = EC
Hence proved.

Question 16.
Solution:
Given : In figure,
BA ⊥ AC, DE ⊥ EF .
BA = DE and BF = DC
To Prove : AC = EF
Proof : BF = DC (given)
Adding FC both sides,
BF + FC = FC + CD
=> BC = FD.
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 16
Now, in right-angled ∆ ABC and ∆ DEF,
Hyp. BC = Hyp. FD (proved)
Side AB = side DE (given)
∴ ∆ ABC ≅ ∆DEF (RHS axiom)
∴ AC = EF (c.p.c.t.)
Hence proved.

Question 17.
Solution:
To prove : AE = CD
Proof: x° + ∠ BDC = 180° (Linear pair)
Similarly y°+ ∠AEB = 180°
∴ x° + ∠BDC = y° + AEB
But x° = y° (given)
∠ BDC = ∠ AEB
Now, in ∆ AEB and ∆ BCD,
AB = CB (given)
∠B = ∠B (common)
∠ AEB = ∠ BDC (proved)
∴ ∆ AEB ≅ ∆ BCD (AAS axiom)
∴ AE = CD. (c.p.c.t.)
Hence proved.

Question 18.
Solution:
Given : In ∆ ABC,
AB = AC.
Bisectors of ∠ B and ∠ C meet AC and AB in D and E respectively
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 18
To Prove : BD = CE
Proof : In ∆ ABC,
AB = AC
∠ B = ∠ C (Angles opposite to equal sides)
Now, in ∆ ABD and ∆ ACE,
AB = AC (given)
∠ A = ∠ A (common)
∠ ABD = ∠ACE
(Half of equal angles)
∴ ∆ A ABD ≅ ∆ ACE (ASA axiom)
∴ BD = CE (c.p.c.t)
Hence proved.

Question 19.
Solution:
Given : In ∆ ABC,
AD is median. BL and CM are perpendiculars on AD and AD is produced to E
To prove : BL = CM.
Proof : In ∆ BLD and ∆ CMD,
BD = DC (D is mid-point of BC)
∠LDB = ∠CDM
(Vertically opposite angles)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 19
∠L = ∠M (each 90°)
∴ ∆ BLD ≅ ∆ CMD (A.A.S. axiom)
Hence, BL = CM (c.p.c.t)
Hence proved.

Question 20.
Solution:
Given : In ∆ ABC, D is mid-point of BC. DL ⊥ AB and DM ⊥ AC and DL = DM
To prove : AB = AC
Proof: In right angled ∆ BLD and ∆ CMD
Hyp. DL = DM (given)
Side BD = DC (D is mid-point of BC)
∴ ∆ BLD ≅ ∆ CMD (R.H.S. axiom)
∴ ∠B = ∠C (c.p.c.t.)
∴ AC = AB
(sides opposite to equal angles)
Hence AB = AC
Hence proved.

Question 21.
Solution:
Given : In ∆ AB = AC and bisectors of ∠B and ∠C meet at a point O. OA is joined.
To Prove : BO = CO and Ray AO is the bisector of ∠ A
Proof : AB = AC (given)
∴ ∠C = ∠B
(Angles opposite to equal sides)
=> \(\frac { 1 }{ 2 } \) ∠C = \(\frac { 1 }{ 2 } \) ∠B
=>∠OBC = ∠OCB
∴ in ∆OBC,
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 21
OB = OC (Sides opposite to equal angles)
Now in ∆ OAB and ∆ OCA
OB = OC (proved)
OA = OA (common)
AB = AC (given)
∴ ∆ OAB ≅ ∆ OCA (S.S.S. axiom)
∴ ∠OAB = ∠OAC (c.p.c.t.)
Hence OA is the bisector of ∠ A.
Hence proved.

Question 22.
Solution:
Given : ∆ PQR is an equilateral triangle and QRST is a square. PT and PS are joined.
To Prove : (i) PT = PS
(ii) ∠PSR = 15°
Proof : In ∆ PQT and ∆ PRS,
PQ = PR (Sides of equilateral ∆ PQR)
QT = RS (sides of in square PRST) .
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 22
∠PQT = ∠PRS
(each angle = 90° + 60° = 150°)
∴ ∆ PQT ≅ ∆ PRS (S.A.S. axiom)
∴ PT = PS (c.p.c.t)
In ∆ PRS, ∠PRS = 60° + 90° = 150°
∠ RPS + ∠ PSR = 180° – 150° = 30°
But ∠RPS = ∠PSR ( ∴PR = RS)
∴∠PSR + ∠PSR = 30°
=> 2∠PSR = 30°
∴ ∠PSR = \(\frac { { 30 }^{ o } }{ 2 } \) = 15°
Hence proved.

Question 23.
Solution:
Given : In right angle ∆ ABC, ∠B is right angle. BCDE is square on side BC and ACFG is also a square on AC.
AD and BF are joined.
To Prove : AD = BF
Proof : ∠ACF = ∠BCD (Each 90°)
Adding ∠ ACB both sides,
∠ ACF + ∠ ACB = ∠ BCD + ∠ ACB
=> ∠ BCF = ∠ ACD
Now in ∆ ACD and ∆ BCF,
AC = CF (sides of a squares)
CD = BC (sides of a square)
∴∠ ACD = ∠ BCF (proved)
∴ ∆ ACD ≅ ∆ BCF (S.A.S. axiom)
AD = BF (c.p.c.t)
Hence proved.

Question 24.
Solution:
Given : ∆ ABC is an isosceles in which AB = AC. and AD is the median which meets BC at D.
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 24
To Prove : AD is the bisector of ∠ A
Proof : In ∆ ABD and ∆ ACD,
AD = AD (Common)
AB = AC (Given)
BD = CD (D is mid-point of BC)
∴ ∆ ABD ≅ ∆ ACD (S.S.S. axiom)
∠ BAD = ∠ CAD (c.p.c.t.)
Hence, AD in the bisector of ∠ A.

Question 25.
Solution:
Given : ABCD is a quadrilateral in which AB || DC. P is mid-point of BC. AP and DC are produced to meet at Q.
To Prove : (i) AB = CQ.
(ii) DQ = DC + AB
Proof:
(i) In ∆ ABP and ∆ CPQ,
BP = PC (P is mid-point of BC)
∠ BAP = ∠ PQC (Alternate angles)
∠ APB = ∠ CPQ
(Vertically opposite angles)
∴ ∆ ABP ≅ ∆ CPQ (A.A.S. axiom)
∴ AB = CQ. (c.p.c.t.)
(ii) Now DQ = DC + CQ
=> DQ = DC + AB ( CQ = AB proved)
Hence proved.

Question 26.
Solution:
Given : In figure,
OA = OB and OP = OQ.
To Prove : (i) PX = QX (ii) AX = BX
Proof : In ∆OAQ and ∆OBP,
OA = OB. (Given)
OQ = OP (Given)
∠O = ∠O (Common)
∴ ∆ OAQ ≅ ∆ OBP (SAS axiom)
∴ ∠ A = ∠ B (c.p.c.t)
Now OA = OB and OP = OQ
Subtracting
OA – OP = OB – OQ
=>PA = QB
Now, in ∆ XPA and ∆ XQB,
PA = QB (Proved)
∠AXP = ∠BXQ
(Vertically opposite angles)
∠ A = ∠ B (Proved)
∴ ∆ XPA ≅ ∆ XQB (A.A.S. axiom)
∴ AX = BX (c.p.c.t.) and PX = QX (c.p.c.t.)
Hence proved.

Question 27.
Solution:
Given : ABCD is a square in which a point P is inside it. Such that PB = PD.
To prove : CPA is a straight line.
Proof : In ∆ APB and ∆ ADP,
AB = AD (Sides of a square)
AP = AP (Common)
PB = PD (Given)
∴ ∆ APB ≅ ∆ ADP (S.S.S. axiom)
∠APD = ∠ APB (c.p.c.t) …(i)
Similarly, in ∆ CBP and ∆ CPD,
CB = CD (Sides of a square)
CP – CP (Common)
PB = PD (Given)
∴ ∆ CBP ≅ ∆ CPD (S.S.S. axiom)
∴ ∠BPC = ∠CPD (c.p.c.t.) …(i)
Adding (i) and (ii),
∴ ∠ APD + ∠ CPD = ∠ APB + ∠ BPC
∠APC = ∠APC
∠APC = 180°
(v sum of angles at a point is 360°)
APC or CPA is a straight line.
Hence proved.

Question 28.
Solution:
Given :∆ ABC is an equilateral triangle PQ || AC and AC is produced to R such that CR = BP
To prove : QR bisects PC
Proof : ∴ PQ || AC
∴ ∠BPQ = ∠BCA
(Corresponding angles)
But ∠ BCA = 60°
(Each angle of the equilateral triangle)
and ∠ ABC or ∠QBP = 60°
∴ ∆ BPQ is also an equilateral triangle.
∴ BP = PQ
But BP = CR (Given)
∴ PQ = CR
Now in ∆ PQM and ∆ RMC,
PQ = CR (proved)
∠QMP = ∠RMC
(vertically opposite angles)
∠ PQM = ∠ MRC (alternate angles)
∴ ∆ PQM ≅ ∆ RMC (AAS axiom)
∴ PM = MC (c.p.c.t.)
Hence, QR bisects PC.
Hence proved

Question 29.
Solution:
Given : In quadrilateral ABCD,
AB = AD and BC = DC
AC and BD are joined.
To prove : (i) AC bisects ∠ A and ∠ C
(ii) AC is perpendicular bisector of BD.
Proof : In ∆ ABC and ∆ ADC,
AB = AD (given)
BC = DC (given)
AC = AC (common)
∴ ∆ ABC ≅ ∆ ADC (S.S.S. axiom)
∴ ∠BCA = ∠DCA (c.p.c.t.)
and ∠BCA = ∠DAC (c.p.c.t.)
Hence AC bisects ∠ A and ∠ C
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 29
(ii) In ∆ ABO and ∆ ADO,
AB = AD (given)
AO = AO (common)
∠BAO = ∠DAO
(proved that AC bisects ∠ A)
∴ ∆ ABO ≅ ∆ ADO (SAS axiom)
∴ BO = OD and ∠AOB = ∠AOD (c.p.c.t.)
But ∠ AOB + ∠ AOD = 180° (linear pair)
∠AOB = ∠AOD = 90°
Hence AC is perpendicular bisector of BD. Hence proved.

Question 30.
Solution:
Given : In ∆ ABC,
Bisectors of ∠ B and ∠ C meet at I
From I, IP ⊥ BC. IQ ⊥ AC and IR ⊥ AB.
IA is joined.
To prove : (i) IP = IQ = IR
(ii) IA bisects ∠ A.
Proof : (i) In ∆ BIP and ∆ BIR
BI = BI (Common)
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 30
and ∠P = ∠R (Each 90°)
∴ ∠ IBP ≅ ∠ IBR
( ∴ IB is bisector of ∠ B)
∴ ∆ BIP = ∆ BIR (A.A.S. axiom)
∴ IP = IR (c.p.c.t) …(i)
Similarly, in ∆ CIP and ∆ CIQ,
CI = CI (Common)
∠P = ∠Q (each = 90°) and ∠ICP = ∠ ICR
( IC is bisector of ∠ C)
∴ ∆CIP ≅ ∆CIQ (A.A.S. axiom)
∴ IP = IQ (c.p.c.t.) …(ii)
From (i) and (ii),
IP = IQ = IR.
(i) In right angled ∆ IRA and ∆ IQA,
Hyp. IA = IA (Common)
side IR = IQ (Proved)
∴ ∆ IRA ≅ ∆ IQA (R.H.S. axiom)
∴ ∠IAR = ∠ IRQ (c.p.c.t.)
Hence IA is the bisector of ∠ A
Hence proved.

Question 31.
Solution:
Given. P is a point in the interior of ∠ AOB
PL ⊥ OA and PM ⊥ OC are drawn and PL = PM.
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 31
To prove : OP is the bisector of ∠ AOB
Proof: In right angled ∆ OPL and ∆ OPM,
Hyp. OP OP (Common)
Side PL = PM (Given)
∴ ∆ OPL ≅ ∆ OPM (R.H.S. axiom)
∴ ∠POL = ∠POM (c.p.c.t.)
Hence, OP is the bisector of ∠AOB.
[ Hence proved.

Question 32.
Solution:
Given : ABCD is a square, M is midpoint of AB.
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 32
PQ is perpendicular to MC which meets CB produced at Q. CP is joined.
To prove : (i) PA = BQ
(ii) CP = AB + PA
Proof : (i) In ∆ PAM and ∆ QBM,
AM = MB (M is midpoint of AB)
∠AMP = ∠BMQ
(vertically opposite angles)
∠ PAM = ∠ QBM
(Each = 90° angles of a square)
∴ ∆ PAM ≅ ∆ QBM (A.S.A. axiom)
∴ AP = BQ
=> PA = BQ (c.p.c.t.)
and PM = QM (c.p.c.t.)
Now, in ∆ CPM and ∆ CQM,
CM = CM (Common)
PM = QM (Proved)
∠CMP =∠CMQ (Each 90° as PQ ⊥ MC)
∴ ∆ CPM ≅ ∆ CQM (SAS axiom)
∴ CP = CQ (c.p.c.t.)
= CB + BQ
= AB + PA
( CB = AB sides of squares and BQ = PA proved)
Hence, CP = AB + PA.

Question 33.
Solution:
Construction. Let AB be the breadth of the river. Mark any point M on the bank on which B is situated. Let O be the midpoint of BM. From M move along the path MN perpendicular to BM to a point N such that A, O, N are on the same straight line. Then MN is the required breadth of the river.
Proof : In ∆ ABO and ∆ MNO,
BO = OM (const.)
∠ AOB = ∠ MON (vertically opposite angle)
∠B = ∠M (each 90°)
∴ ∆ ABO ≅ ∆ MNO (A.S.A. axiom)
∴ AB = MN. (c.p.c.t.)
Hence, MN is the required breadth of the river.

Question 34.
Solution:
In ∆ ABC,
∠A = 36°, ∠B = 64°
But ∠A + ∠B + ∠C = 180°
(Sum of angles of a triangle)
=> 36° + 64° + ∠C = 180°
=> 100° + ∠C = 180°
=> ∠C = 180° – 100° = 80°
∴ ∠ C = 80° which is the greatest angle.
∴ The side AB, opposite to it is the longest side.
∴∠ A is the shortest angle
∴ BC is the shortest side.

Question 35.
Solution:
In ∆ ABC,
∴ ∠A = 90°
∴ ∠B + ∠C 180° – 90° = 90°
Hence, ∠ A is the greatest angle of the triangle.
Side BC, opposite to this angle be the longest side.

Question 36.
Solution:
In ∆ ABC,
∠ A = ∠ B = 45°
∴ ∠A + ∠B = 45° + 45° = 90°
and ∠C= 180°-90° = 90°
∴ ∠ C is the greatest angle.
∴ Side AB, opposite to ∠ C will be the longest side of the triangle.

Question 37.
Solution:
Given : In ∆ ABC, side AB is produced to D such that BD = BC.
∠B = 60°, ∠A = 70°
To Prove : (i) AD > CD (ii) AD > AC
Proof : In ∆ BCD,
Ext. ∠ B = 60°
∴ ∠ CBD = 180° – 60° = 120°
and ∠ BCD + ∠ BDC = Ext. ∠ CBA = 60°
But ∠BCD = ∠ BDC ( BC = BD)
∴ ∠BCD = ∠BDC = \(\frac { { 60 }^{ o } }{ 2 } \) = 30°.
∴ ∠ ACD = 50° + 30° = 80°
(i) Now in ∆ ACD,
∴ ∠ ACD > ∠ CAD ( 80° > 70°)
AD > CD .
(ii) and ACD > ∠D (80° > 30°)
∴ AD > AC
Hence proved.

Question 38.
Solution:
Given : In ∆ ABC, ∠B = 35°and ∠C = 65°
AX is the bisector of ∠ BA C meeting BC in X
∠A + ∠B + ∠C = 180°
(Sum of angles of a triangle)
=> ∠ A + 35° + 65° = 180°
=> ∠A + 100° = 180°
=> ∠A = 180° – 100° = 80°
∴ AX is the bisector of ∠ BAC
∴ ∠ BAX = ∠ CAX = 40°
In ∆ ABX,
∠BAX = 40° and ∠B = 35°
∴ ∠BAX > ∠B
∴ BX > AX …(i)
and in ∆ AXC,
∠C = 65° and ∠CAX = 40°
∴ ∠C > ∠CAX
∴ AX > XC …(ii)
From (i) and (ii),
BX > AX > XC
or BX > AX > CX

Question 39.
Solution:
Given : In ∆ ABC,
AD is the bisector of ∠ A
To prove : (i) AB > BD and
(ii) AC > DC
Proof : (i) In ∆ ADC,
Ext. ∠ ADB > ∠ CAD
=>∠ ADB > ∠BAD ,
( AD is bisector of ∠ A)
In ∆ ABD,
AB > BD.
(ii) Again, in ∆ ADB,
Ext. ∠ADC > ∠BAD
=> ∠ADC > ∠CAD
( ∠ CAD = ∠BAD)
Now in ∆ ACD.
AC > DC.
Hence proved

Question 40.
Solution:
Given : In ∆ ABC, AB = AC
BC is produced to D and AD is joined.
To Prove : AD > AC
Proof : In ∆ ABC,
AB = AC (given)
∴ ∠B = ∠C (Angles opposite to equal sides)
Ext. ∠ ACD > ∠ ABC
∠ACB = ∠ABC
∴ ∠ABC > ∠ADC
Now, in ∆ ABD,
∴ ∠ABC > ∠ADC or ∠ADB
∴ AD > AC
Hence proved.

Question 41.
Solution:
Given : In ∆ ABC,
AC > AB and AD is the bisector of ∠ A which meets BC in D.
To Prove : ∠ADC > ∠ADB
Proof : In ∆ ABC,
AC > AB
∴∠B > ∠C
∴∠ 1 = ∠ 2 (AD is the bisector of ∠ A)
∴ ∠B + ∠2 = ∠C + ∠1
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 41
But in ∆ ADB
Ext. ∠ADC = ∠B + ∠2
and in ∆ ADC,
Ext. ∠ADB > ∠C + ∠1
∴ ∠B + ∠2 > ∠C + ∠1 (Proved)
∴ ∠ ADC > ∠ ADB Hence proved.

Question 42.
Solution:
Given : In ∆ PQR,
S is any point on QR and PS is joined
To Prove : PQ + QR + RP > 2PS
Proof : In ∆ PQS,
PQ + QS > PS
(Sum of two sides of a triangle is greater than the third side) …(i)
Similarly, in ∆ PRS.
PQ + SR > PS …(ii)
Adding (i), and (ii),
PQ + QS + PR + SR > PS + PS
=> PQ + QS + SR + PR > 2PS
=> PQ + QR + RP > 2PS
Hence proved.

Question 43.
Solution:
Given : O is the centre of the circle and XOY is its diameter.
XZ is the chord of this circle
To Prove : XY > XZ
Const. Join OZ
Proof : OX, OZ and OY are the radii of the circle
∴ OX = OZ = OY
In ∆ XOZ,
OX + OZ > XZ (Sum of two sides of a triangle is greater than its third side)
=> OX + OY > XZ (∴ OZ = OY)
=> OXY > XZ
Hence proved.

Question 44.
Solution:
Given : In ∆ ABC, O is any point within it OA, OB and OC are joined.
To Prove : (i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + OB + OC
(iii) OA + OB + OC > \(\frac { 1 }{ 2 } \)
(AB + BC + CA)
Const. Produce BO to meet AC in D.
RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A 44
Proof : (i) In ∆ ABD,
AB + AD > BD …(i)
(Sum of two sides of a triangle is greater than its third side)
=> AB + AD > BO + OD …(i)
Similarly in ∆ ODC
OD + DC > OC. …(ii)
Adding (i) and (ii)
AB + AD + OD + DC > OB + OD + OC
=> AB + AD + DC > OB + OC
AB + AC > OB + OC.
(ii) In (i) we have proved that
AB + AC > OB + OC
Similarly, we can prove that
AC + BC > OC + OA
and BC + AB > OA + OB
Adding, we get:
AB + AC + AC + BC + B + AB > OB + OC + OC + OA + OA + OB
=> 2(AB + BC + CA) > 2(OA + OB + OC)
=> AB + BC + CA > OA + OB + OC.
(iii) In ∆ AOB,
OA + OB > AB
Similarly, in ∆ BOC
OB + OC > BC
and in ∆ COA
OC + OA > CA
adding we get:
OA + OB + OB + OC + OC + OA > AB + BC + CA
=> 2(OA + OB + OC) > (AB + BC + CA)
=> OA + OB + OC > \(\frac { 1 }{ 2 } \) (AB + BC + CA)
Hence proved.

Question 45.
Solution:
Sides of ∆ ABC are AB = 3cm, BC = 3.5cm and CA = 6.5cm
We know that sum of any two sides of a triangle is greater than its third side.
Here, AB = 3 cm and BC = 3.5 cm
∴ AB + BC = 3cm + 3.5 cm = 6.5 cm and CA = 6.5 cm
∴ AB + BC = CA
Which is not possible to draw the triangle.
Hence, we cannot draw the triangle with the given data.

Hope given RS Aggarwal Class 9 Solutions Chapter 5 Congruence of Triangles and Inequalities in a Triangle Ex 5A are helpful to complete your math homework.

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2J.

Other Exercises

Factorize :

Question 1.
Solution:
x3 + 27
= (x)3 + (3)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 1

Question 2.
Solution:
8x3 + 27y3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 2

Question 3.
Solution:
343 + 125b3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 3

Question 4.
Solution:
(1)3+(4x)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 4

Question 5.
Solution:
125a3+ \(\frac { 1 }{ 8 } \)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 5

Question 6.
Solution:
216x3+\(\frac { 1 }{ 125 } \)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 6

Question 7.
Solution:
16x4 + 54x
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 7

Question 8.
Solution:
7a3 + 56b3
=7(a3+8b3)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 8

Question 9.
Solution:
x5 + x2
=x2(x3+1)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 9

Question 10.
Solution:
a3 + 0.008
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 10

Question 11.
Solution:
x6 + y6
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 11

Question 12.
Solution:
2a3 + 16b3 – 5a – 10b
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 12

Question 13.
Solution:
x3 – 512
=(x)– (8)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 13

Question 14.
Solution:
64x3 – 343
=(4x)– (7)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 14

Question 15.
Solution:
1 – 27x3
=(1)3– (3x)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 15

Question 16.
Solution:
x3 – 125y3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 16

Question 17.
Solution:
8x3 – \(\frac { 1 }{ { 27y }^{ 3 } } \)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 17

Question 18.
Solution:
a3 – 0.064
=(a)– (0.4)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 18

Question 19.
Solution:
(a + 6)3 – 8
=(a+b)– (2)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 19

Question 20.
Solution:
x6 – 729
=(x2)– (9)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 20

Question 21.
Solution:
(a + b)3 – (a – b)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 21

Question 22.
Solution:
x – 8xy3
=x(1 – 8y3)
=x{(1)– (2y)3}
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 22

Question 23.
Solution:
32x4 – 500x
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 23

Question 24.
Solution:
3a7b – 81a4 b4
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 24

Question 25.
Solution:
\({ a }^{ 3 }-\frac { 1 }{ { a }^{ 3 } } -2a+\frac { 2 }{ a } \)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 25

Question 26.
Solution:
8a3 – b3 – 4ax + 2bx
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 26

Question 27.
Solution:
a3 + 3a2b + 3ab2 + b3 – 8
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2J 27

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2K.

Other Exercises

Question 1.
Solution:
125a3 + b3 + 64c3 – 60abc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q1.1

Question 2.
Solution:
a3 + 8b3 + 64c3 – 24abc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q2.1

Question 3.
Solution:
1 + b3 + 8c3 – 6bc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q3.1

Question 4.
Solution:
216 + 27b3 + 8c3 – 108bc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q4.1

Question 5.
Solution:
27a3 – b3 + 8c3 + 18abc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q5.1

Question 6.
Solution:
8a3 + 125b3 – 64c3 + 120abc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q6.1

Question 7.
Solution:
8 – 27b3 – 343c3 – 126bc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q7.1

Question 8.
Solution:
125 – 8x3 – 27y3 – 90xy
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q8.1

Question 9.
Solution:
2√2a3 + 16√2b3 + c3 – 12abc
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q9.1

Question 10.
Solution:
x3 + y3 – 12xy + 64
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q10.1

Question 11.
Solution:
(a – b)3 + (b – c)3 + (c – a)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q11.1

Question 12.
Solution:
(3a – 2b)3 + (26 – 5c)3 + (5c – 3a)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q12.1

Question 13.
Solution:
a3 (b – c)3 + b3 (c – a)3 + c3 (a – b)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q13.1

Question 14.
Solution:
(5a – 7b)3 + (9c – 5a)3 + (7b – 9c)3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q14.1

Question 15.
Solution:
(x + y – z) (x2 + y2 + z2 – xy + yz + zx)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q15.1

Question 16.
Solution:
(x – 2y + 3) (x2 + 4y2 + 2xy -3x + 6y + 9)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q16.1

Question 17.
Solution:
(x – 2y – z) (x2 + 4y2 + z2 + 2xy + zx- 2yz)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q17.1

Question 18.
Solution:
x + y + 4 = 0,
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q18.1

Question 19.
Solution:
x = 2y + 6
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2K Q19.1

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2E

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2E

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2E.

Other Exercises

Factorize:

Question 1.
Solution:
9x2 + 12xy
= 3x (3x + 4y) Ans.

Question 2.
Solution:
18x2y – 24xyz
= 6xy (3x – 4z) Ans.

Question 3.
Solution:
27a3b3 – 45a4b2
= 9a3b2 (3b – 5a) Ans.

Question 4.
Solution:
2a (x + y) – 3b(x + y)
= (x + y) (2a – 3b) Ans.

Question 5.
Solution:
2x (p2 + q2) + 4y (p2 + q2)
= 2(p2 + q2) (x + 2y) Ans

Question 6.
Solution:
x (a – 5) + y (5 – a)
= x (a – 5) -y (a – 5)
= (a – 5) (x – y) Ans.

Question 7.
Solution:
4(a + b) – 6 (a + b)2
= 2(a + b) {2 – 3 (a + b)}
= 2(a + b) (2 – 3a – 3b) Ans.

Question 8.
Solution:
8(3a – 2b)2 – 10 (3a – 2b)
= 2(3a – 2b) {4 (3a – 2b) – 5}
= 2(3a – 2b) (12a – 8b – 5) Ans.

Question 9.
Solution:
x (x + x)3 – 3x2 y (x + y)
= x (x + y) {(x + y)2 – 3xy}
= x (x + y) [x2 + y2 + 2xy – 3xy)
= x (x + y) (x2 + y2 – xy) Ans.

Question 10.
Solution:
x3 + 2x2 + 5x + 10
= x2 (x + 2) + 5 (x + 2)
= (x + 2) (x2 + 5) Ans.

Question 11.
Solution:
x2 + xy – 2xz – 2yz
= x (x + y) -2z (x + y)
= (x + y) (x – 2z) Ans.

Question 12.
Solution:
a3 b – a2b + 5ab – 5b.
= b (a3 – a2 + 5a – 5)
= b {(a2 (a – 1) + 5 (a – 1)}
= b (a – 1) (a2 + 5) Ans.

Question 13.
Solution:
8 – 4a – 2a3 + a4
= 4 (2 – a) – a3 (2 – a)
= (2 – a) (4 – a3) Ans.

Question 14.
Solution:
x3 – 2x2y + 3xy2 – 6y3
= x2 (x – 2y) + 3y2 (x – 2y)
= (x – 2y) (x2 + 3y2) Ans

Question 15.
Solution:
px – 5q + pq – 5x
= px – 5x + pq – 5q
= x(p – 5) + q(p – 5)
= {p – 5) (x + q) Ans.

Question 16.
Solution:
x2 + y – xy – x
= x2 – x – xy + y
= x (x – 1) – y (x – 1)
= (x – 1) (x – y) Ans.

Question 17.
Solution:
(3a – 1)2 – 6a + 2
= (3a – 1)2 – 2 (3a – 1)
= (3a – 1) (3a – 1 – 2)
= (3a – 1) (3a – 3)
= 3(3a – 1) (a – 1) Ans.

Question 18.
Solution:
(2x – 3)2 – 8x + 12
= (2x – 3)2 – 4(2x – 3)
= (2x – 3) (2x – 3 – 4)
= (2x – 3) (2x – 7) Ans.

Question 19.
Solution:
a3 + a – 3a2 – 3
= a3 – 3a2 + a – 3
= a2 (a – 3) + 1 (a – 3)
= (a – 3) (a2 + 1) Ans.

Question 20.
Solution:
3ax – 6ay – 8by + 4bx
= 3ax – 6ay + 4bx – 8by
= 3a (x – 2y) + 4b (x – 2y)
= (x – 2y) (3a + 4b) Ans

Question 21.
Solution:
abx2 + a2x + b2x + ab
= ax (bx + a) + b (bx + a)
= (bx + a) (ax + b) Ans.

Question 22.
Solution:
x3 – x2 + ax + x – a – 1
= x3 – x2 + ax – a + x – 1
= x2 (x – 1) + a (x – 1) + 1 (x – 1)
= (x – 1) (x2 + a + 1) Ans.

Question 23.
Solution:
2x + 4y – 8xy – 1
= 2x – 8xy -1+4y
= 2x (1 – 4y) -1 (1 – 4y)
= (1 – 4y) (2x – 1) Ans.

Question 24.
Solution:
ab (x2 +y2) – xy (a2 + b2)
= abx2 + aby2 – a2xy – b2xy
= abx2 – a2xy – b2xy + aby2
= ax (bx – ay) – by (bx – ay)
= (bx – ay) (ax – by) Ans.

Question 25.
Solution:
a2 + ab (b + 1) + b3
= a2 + ab2 + ab + b3
= a (a + b2) + b (a + b2)
= (a + b2) (a + b) Ans

Question 26.
Solution:
a3 + ab (1 – 2a) – 2b2
= a3 + ab – 2a2b – 2b2
= a3 – 2a2b + ab – 2b2
= a2 (a – 2b) + b (a – 2b)
= (a – 2b) (a2 + b) Ans.

Question 27.
Solution:
2a2 + bc – 2ab – ac
= 2a2 – 2ab – ac + bc
= 2a (a – b) – c (a – b)
= (a – b) (2a – c) Ans.

Question 28.
Solution:
(ax + by)2 + (bx – ay)2
= a2x2 + b2y2 + 2abxy + b2x2 + a2y2 – 2bxy
= a2x2 + b2y2 + b2x2 + a2y2
= a2x2 + b2x2 + a2y2 + b2y2
= x2(a2 + b2) + y2(a2 + b2)
= (a2 + b2) (x2 + y2) Ans.

Question 29.
Solution:
a (a + b – c) – bc
= a2 + ab – ac – bc
= a (a + b) – c (a + b)
(a + b) (a – c) Ans.

Question 30.
Solution:
a(a – 2b – c) + 2bc
= a2 – 2ab – ac +2bc
= a2 – ac – 2ab + 2bc
= a (a – c) – 2b (a – c)
= (a – c) (a – 2b) Ans.

Question 31.
Solution:
a2x2 + (ax2 + 1) x + a
= a2x2 + ax3 + x + a
= a2x2 + ax3 + a + x
= ax2 (a + x) + 1 (a + x)
– (a + x) (ax2 + 1) Ans

Question 32.
Solution:
ab (x2 + 1) + x (a2 + b2)
= abx2 + ab + a2x + b2x
= abx2 + a2x + b2x + ab
= ax (bx + a) + b (bx + a)
= (bx + a) (ax + b) Ans.

Question 33.
Solution:
x2 – (a + b) x + ab
= x2 – ax – bx + ab
= x (x – a) – b (x – a)
= (x – a) (x – b) Ans.

Question 34.
Solution:
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2E 34

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RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4B

RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4B

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4B.

Other Exercises

RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4A
RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4B
RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4C
RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4D

Question 1.
Solution:
AOB is a straight line
∠AOC + ∠BOC = 180° (Linear pair)
=> 62° + x= 180°
=> x = 180° – 62°
=> x = 118°
Hence, x = 118° Ans.

Question 2.
Solution:
AOB is straight line
∠AOC + ∠COD + ∠DOB – 180°
=> (3x – 5)° + 55° + (x + 20)° = 180°
=> 3x – 5° + 55° + x + 20° = 180°
=> 4x – 5° + 75° = 180°
=> 4x + 70° = 180°
=> 4x = 180° – 70°
=> 4x = 110°
=> \(x=\frac { { 110 }^{ o } }{ 4 } =27.{ 5 }^{ o }\)
Hence x = 27.5°
and ∠AOC = 3x – 5° = 3 x 27.5° – 5°
= 82.5° – 5° = 77.5°
∠BOD = x + 20° = 27.5° + 20°
= 47.5° Ans.

Question 3.
Solution:
AOB is a straight line
∠AOC + ∠COD + ∠DOB = 180°
{angles on the same side of line AB}
=> (3x + 7)° + (2x – 19)° + x = 180°
=> 3x + 7° + 2x – 19° + x = 180°
=> 6x – 12° – 180°
=> 6x = 180° + 12° = 192°
=> \(x=\frac { { 192 }^{ o } }{ 6 } =32^{ o }\)
Here x = 32°
∠AOC = 3x + 7° = 3 x 32° + 7°
= 96° + 7°= 103°
∠COD = 2x – 19° = 2 x 32° – 19°
= 64 – 19° = 45°
and ∠BOD = x = 32° Ans.

Question 4.
Solution:
In the given figure,
x + y + z = 180°
But x : y : z = 5:4:6
Let ∠XOP = x° – 5a
∠POQ =y° = 4a
and ∠QOY = z = 6a
then 5a + 4a + 6a = x + y + z = 180°
=> 15a = 180°
=> a = \(\frac { { 180 }^{ o } }{ 15 } ={ 12 }^{ o }\)
=> x = 5a = 5 x 12° = 60°
y = 4a = 4 x 12° = 48°
and z = 6x = 6 x 12° = 72° Ans.

Question 5.
Solution:
AOB will be a straight line
If ∠AOC + ∠COB = 180°
If (3x + 20)° + (4x – 36)° = 180°
If 3x + 20 + 4x – 36 = 180°
If 7x – 16 = 180°
If 7x = 180° + 16 = 196°
If \(x=\frac { { 196 }^{ o } }{ 7 } ={ 28 }^{ o }\)
Hence, if x = 28°, then AOB will be a straight line.

Question 6.
Solution:
AB and CD intersect each other at O
AOC = ∠BOD and ∠BOC = ∠AOD (vertically opposite angles)
But ∠AOC = 50°
∠BOD = ∠AOC = 50°
But ∠AOC + ∠BOC = 180° (Linear pair)
=> 50° + ∠BOC = 180°
=> ∠BOC = 180° – 50° = 130°
∠AOD = ∠BOC = 130°
Hence,∠AOD = 30°,∠BOD = 50° and ∠BOC = 130° Ans.

Question 7.
Solution:
In the figure,
AB, CD and EF are coplanar lines intersecting at O.
∠AOF = ∠BOE
∠DOF = ∠COE and ∠BOD = ∠AOC (Vertically opposite angles)
x = y,
z = 50°
t = 90°
But AOF + ∠DOF + ∠BOD = 180° (Angles on the same side of a st. line)
=> x + 50° + 90° = 180°
=> x° + 140° + 180°
=> x = 180° – 140° = 40°
Hence, x = 40°, y = x = 40°, z = 50° and t = 90° Ans.

Question 8.
Solution:
Three coplanar lines AB, CD and EF intersect at a point O
∠AOD = ∠BOC
∠DOF = ∠COE
and ∠AOE = ∠BOF
(Vertically opposite angles)
But ∠AOD = 2x
∠BOC = 2x
and ∠BOF = 3x
∠AOE = 3x
and ∠COE = 5x
∠DOF = 5x
But ∠AOD + ∠DOF + ∠BOF + ∠BOC + ∠COE + ∠AOE = 360° (Angles at a point)
=> 2x + 5x + 3x + 2x + 5x + 3x = 360°
=> 20x = 360° => x = \(\frac { { 360 }^{ o } }{ 20 } \) = 18°
Hence x = 18°
∠AOD = 2x = 2 x 18° = 36°
∠COE = 5x = 5 x 18° = 90°
and ∠AOE = 3x = 3 x 18° = 54° Ans.

Question 9.
Solution:
AOB is a line and CO stands on it forming ∠AOC and ∠BOC
But ∠AOC : ∠BOC = 5:4
Let ∠AOC = 5x and ∠BOC = 4x
But ∠AOC + ∠BOC = 180° (Linear pair)
=> 5x + 4x = 180° => 9x = 180°
=> x = \(\frac { { 180 }^{ o } }{ 9 } \) = 20°
∠AOC = 5x = 5 x 20° = 100°
and ∠BOC = 4x = 4 x 20° = 80° Ans.

Question 10.
Solution:
Two lines AB and CD intersect each other at O and
∠AOC = 90°
∠AOC = ∠BOD
(Vertically opposite angles)
∠BOD = 90°
But ∠AOC + ∠BOC = 180° (Linear pair)
=> 90° + ∠BOC – 180°
=> ∠BOC = 180° – 90° = 90°
But ∠AOD = ∠BOC
(Vertically opposite angles)
∠AOD = 90°
Hence each of the remaining angle is 90°.

Question 11.
Solution:
Two lines AB and CD intersect each other at O and
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4B 1
∠BOC + ∠AOD = 280°
∠AOD = ∠BOC
(vertically opposite angles)
∠BOC + ∠BOC = 280°
(∠AOD = ∠BOC)
=> 2 ∠BOC = 280°
=> ∠BOC = \(\frac { { 280 }^{ o } }{ 2 } \) = 140°
But ∠BOC + ∠AOC = 180° (Linear pair)
=> 140° + ∠AOC = 180°
=> ∠AOC = 180° – 140° = 40°
But ∠BOD = ∠AOC
(vertically opposite angles)
∠BOD = 40°
Hence ∠AOC = 40°, ∠BOC = 140°,
∠BOD = 40°
and ∠AOD = 140° Ans.

Question 12.
Solution:
OC is the bisector of ∠AOB. and OD is the ray opposite to OC.
Now ∠AOC = ∠BOC (OC is bisector of ∠AOB)
But ∠BOC + ∠BOD = 180° (Linear pair)
Similarly, ∠AOD + ∠AOC = 180°
=> ∠BOC + ∠BOD = ∠AOD + ∠AOC
But ∠AOC = ∠BOC (Given)
∠BOD = ∠AOD
=> ∠AOD = ∠BOD
Hence proved.

Question 13.
Solution:
AB is the mirror.
PQ is the incident ray, QR is its reflected ray.
=> ∠BQR = ∠PQA
But ∠BQR + ∠PQR + ∠PQA = 180° (Angles on one side of a straight line)
=> ∠PQA + ∠PQA + 112° = 180°
=> 2∠PQA + 112° = 180°
=> 2∠PQA = 180° – 112° = 68°
∠PQA = \(\frac { { 68 }^{ o } }{ 2 } \) = 34° Ans.

Question 14.
Solution:
Given. Two lines AB and CD intersect each other at O.
OE is the bisector of ∠BOD and EO is produced to F.
To Prove : OF bisects ∠AOC.
Proof : AB and CD intersect each other at O
∠AOC = ∠BOD
(Vertically opposite angles)
OE is the bisector of ∠BOD
∠1 = ∠2
But ∠1 = ∠3
and ∠2 = ∠4 (Vertically opposite angles)
and ∠1 = ∠2 (proved)
∠3 = ∠4
Hence, OF is the bisector of ∠AOC.
Hence proved.

Question 15.
Solution:
Given ∠AOC and ∠BOC are supplementary angles
OE is the bisector of ∠BOC and OF is the bisector of ∠AOC
To Prove : ∠EOF = 90°
Proof : ∠1 = ∠2
∠3 = ∠4
{OE and OF are the bisectors of ∠BOC and ∠AOC respectively}
But ∠AOC + ∠BOC = 180°
(Linear pair)
=> ∠1 + ∠2 + ∠3 + ∠4 = 180°
=> ∠1 + ∠1 + ∠3 + ∠3 = 180°
=> 2∠1 + 2∠3 = 180°
=> 2(∠1 + ∠3) = 180°
=> ∠1 + ∠3 = \(\frac { { 180 }^{ o } }{ 2 } \) 90°
=> ∠EOF = 90°
Hence proved.

Hope given RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4B are helpful to complete your math homework.

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2C.

Other Exercises

Using remainder theorem, find the remainder when :

Question 1.
Solution:
f(x) = (x3 – 6x2 + 9x + 3)
Let x-1 = 0, then x = 1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q1

Question 2.
Solution:
f(x) = 2x3 – 5x2 + 9x – 8)
Let x-3 = 0, then x = 3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q2.1

Question 3.
Solution:
f(x) = (3x4 – 6x2 – 8x + 2)
Let x-2 = 0, the x = 2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q2.3.1

Question 4.
Solution:
f(x) = (x3 – 7x2 + 6x + 4)
Let x-6 = 0,then x=6
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q2.4.1

Question 5.
Solution:
f(x)=(x3 – 6x2 + 13x + 60)
Let x+2=0,then x=-2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q5.1

Question 6.
Solution:
f(x)=(2x4 + 6x3 + 2x2 + x – 8)
Let x+3=0,then x=-3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q6.1

Question 7.
Solution:
f(x)=(4x3 – 12x2 + 11x – 5)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q7.1

Question 8.
Solution:
f(x)=(81x4 + 54x3 – 9x2 – 3x + 2)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q8.1

Question 9.
Solution:
f(x)=(x3 – ax2 + 2x – a)
let x-a=0, then x=a
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q9.1

Question 10.
Solution:
f(x)=(ax3+ 3x2 – 3)
g(x)=(2x3 -5x + a)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q10.1

Question 11.
Solution:
f(x)=x4 – 2x3 + 3x2 – ax + b
let x-1=0, then x=1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q11.1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2C Q11.2

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2H.

Other Exercises

Question 1.
Solution:
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q1.1

Question 2.
Solution:
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q2.1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q2.2

Question 3.
Solution:
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q3.1

Question 4.
Solution:
9x2 + 16y2 + 4z2 – 24xy + 16yz – 12xz
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q4.1

Question 5.
Solution:
25x2 + 4y2 + 9z2 – 20xy – 12yz + 30xz
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q5.1

Question 6.
Solution:
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2H Q6.1

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RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C

RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4C.

Other Exercises

RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4A
RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4B
RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4C
RS Aggarwal Solutions Class 9 Chapter 4 Lines and Triangles Ex 4D

Question 1.
Solution:
AB || CD and a line t intersects them at E and F forming angles ∠1, ∠ 2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8.
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q1.1

RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q1.3

Question 2.
Solution:
AB || CD and a transversal t intersects them at E and F respectively forming angles ∠l, ∠2, ∠3, ∠4, ∠5, ∠ 6, ∠ 7 and ∠ 8
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q1.2
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q2.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q2.2

Question 3.
Solution:
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q3.1
Given. In quadrilateral ABCD, AB || DC and AD || BC
To Prove : ∠ADC = ∠ABC
Proof : AB || DC and AD intersects their
∠DAB + ∠ADC = 180°
(sum of co-interior angles)
Similarly ∴ AD || BC
∠DAB + ∠ABC = 180° …(ii)
from (i) and (ii),
∠ DAB + ∠ ADC = ∠DAB + ∠ABC
∴∠ADC = ∠ABC. Hence proved.

Question 4.
Solution:
(i) In the figure, AB || CD
∠ABE = 35° and ∠EDC = 65°
Draw FEG || AB or CD
∴ AS || FG (const.)
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q4.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q4.2
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q4.3
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q4.5

Question 5.
Solution:
In the figure, AB || CD || EF,
∠ ABC = 70° and ∠ CEF = 130°
∴EF || CD (given)
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q5.1
∴∠ CEF +∠1 – 180°
(sum of Co-interior angles)
=> 130° + ∠ 1 = 180°
=> ∠ 1 = 180° – 130° = 50°
Again, AB || CD (given)
∴ ∠ ABC = ∠ BCD (Alternate angles)
=> 70° = ∠ BCD = x + ∠ 1
=> x + 50° – 70°
=> x = 70° – 50° = 20°
Hence x = 20° Ans.

Question 6.
Solution:
In the figure, AB || CD.
∠DCE = 130°
and ∠ CEA – 20°
From E, draw EF || AB or CD.
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q6.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q6.2

Question 7.
Solution:
Given. In the given figure, AB || CD.
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q7.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q7.2

Question 8.
Solution:
In the figure, AB || CD
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q8.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q8.2

Question 9.
Solution:
In the figure, AB || CD, ∠AEF = p
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q9.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q9.2

Question 10.
Solution:
In the figure, AB || PQ.
A transversal LM cuts them at E and F
∠ LEB = 75°
∠BEG = 20°
∠ EFG = 25°
∠ EGF = x° and ∠ GFD = y°
∴∠ LEB + ∠ BEF = 180° (Linear pair)
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q10.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q10.2

Question 11.
Solution:
In the figure
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q11.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q11.2

Question 12.
Solution:
In the figure, AB || CD
∠PEF = 85°, ∠QHC = 115°
∴ ∠ GHF = ∠ QHC
(Vertically opposite angles)
∠ GHF = 115°
∴AB || CD
∴∠ PEF = ∠ EGH
(Corresponding angles)
∴∠EGH = 85°
But ∠ QGH + ∠ EGH = 180°, (Linear pair)
=> ∠QGH + 85° = 180°
=> QGH = 180° – 85° = 95°
In ∆ GHQ,
Ext. ∠GHF = ∠QGH + ∠GQH
=> 115° = 95° + x
=> x = 115° – 95°
Hence, x = 20° Ans.

Question 13.
Solution:
In the figure, AB || CD
∠BAD = 75°, ∠ BCD = 35°
∴AB || CD
∴∠ABC = ∠BCD (Alternate angles)
=> x = 35°
and ∠ BAD = ∠ ADC (Alternate angles)
=> 75° = z
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q13.1

Question 14.
Solution:
In the figure, AB || CD
∠APQ = 75°, ∠ PRD = 125°
∴AB || CD.
∠ APQ = ∠ PQR (Alternate angles)
∴75° = y°
=> y° = 75°
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q14.1

Question 15.
Solution:
In the figure, AB || CD and EF || GH
∠APR = 110°, ∠LRF = 60°
∴∠PRQ = ∠LRF
(vertically opposite angles)
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q15.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q15.2

Question 16.
Solution:
(i) l is parallel to m
if 3x – 20° = 2x +10°
(Alternate angles are equal)
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q16.1
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q16.2
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q16.3

Question 17.
Solution:
Given. Two lines AB and CD are perpendiculars on EF
To Prove : AB ⊥ CD.
RS Aggarwal Class 9 Solutions Chapter 4 Lines and Triangles Ex 4C Q17.1

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RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D

RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 2 Polynomials Ex 2D.

Other Exercises

Using factor theorem, show that :

Question 1.
Solution:
By factor theorem, x – 2 will be a factor of f(x) = x3 – 8 if f(2) = 0
(∴ x-2 = 0=>x = 2)
Now f(2) = (2)3 – 8 = 8- 8 = 0
Hence (x – 2) is a factor of f(x) Ans.

Question 2.
Solution:
By factor theorem,
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q2.1

Question 3.
Solution:
By factor theorem,
(x – 1) is a factor of f(x)=(2x4 + 9x3 + 6x2– 11x – 6)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q3.1

Question 4.
Solution:
By factor theorem, (x + 2) will
a factor of f (x) = x4 – x4 + 2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q4.1

Question 5.
Solution:
By factor theorem, (x + 5) will be a factor of f(x) = 2x3 + 9x2 – 11x – 30 if f(-5) = 0
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q5.1

Question 6.
Solution:
By factor theorem, (2x – 3) is a factor of f(x) = 2x4 + x3 – 8x2 – x + 6
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q6.1

Question 7.
Solution:
By factor theorem, (x – √2 ) will be a factor of f(x) = 7x2 – 4√2x – 6
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q7.1

Question 8.
Solution:
By factor theorem, (x + √2) will be a factor of f(x) = 2√2 x3 + 5x + √2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q8.1

Question 9.
Solution:
Let f(x) = 2x3 + 9x2 + x + k and x – 1 is a factor of f(x)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q9.1

Question 10.
Solution:
Let f(x) = 2x3 – 3x2 – 18x + a and x – 4 is its factor
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q10.1

Question 11.
Solution:
Let f(x) – x4 – x3 – 11x2 – x + a
f(x) is divisible by (x + 3)
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q11.1

Question 12.
Solution:
Let f(x) = 2x3 + ax2 + 11x + a + 3
and (2x – 1) is its factor
Let 2x – 1 = 0 then 2x = 1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q12.1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q12.2

Question 13.
Solution:
Let f(x) = x3 – 10x2 + ax + b and (x – 1) and (x – 2) are its factors
∴ x – 1 = 0 =>x=1
and x – 2 = 2 =>x=2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q13.1

Question 14.
Solution:
Let f(x) = x4 + ax3 – 7x2 – 8x + b ,
and (x + 2) and (x + 3) are its factors
∴x + 2 = 0 => x = -2
and x + 3= 0 => x = -3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q14.1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q14.2

Question 15.
Solution:
Let f(x) = x3 – 3x2 – 13x + 15
Now x2 + 2x – 3 = x2 + 3x – x – 3
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q15.1

Question 16.
Solution:
Let f(x) = x3 + ax2 + bx + 6 and (x – 2) is its factor
Let x – 2 = 0 then x = 2
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q16.1
RS Aggarwal Class 9 Solutions Chapter 2 Polynomials Ex 2D Q16.2

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