## RD Sharma Class 10 Solutions Chapter 2 Polynomials VSAQS

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 2 Polynomials VSAQS

**Other Exercises**

- RD Sharma Class 10 Solutions Chapter 2 Polynomials Ex 2.1
- RD Sharma Class 10 Solutions Chapter 2 Polynomials Ex 2.2
- RD Sharma Class 10 Solutions Chapter 2 Polynomials Ex 2.3
- RD Sharma Class 10 Solutions Chapter 2 Polynomials VSAQS
- RD Sharma Class 10 Solutions Chapter 2 Polynomials MCQS

**Answer each of the following questions in one word or one sentence or as per the exact requirement of the questions :**

**Question 1.**

Define a polynomial with real co-efficients.

**Solution:**

**Question 2.**

Define degree of a polynomial.

**Solution:**

The exponent of the highest degree term in a polynomial is known as its degree. A polynomial of degree O is called a constant polynomial.

**Question 3.**

Write the standard form of a linear polynomial with real co-efficients.

**Solution:**

ax + b is the standard form of a linear polynomial with real co-efficients and a ≠ 0

**Question 4.**

Write the standard form of a quadratic polynomial with real co-efficients.

**Solution:**

ax^{2} + bx + c is a standard form of quadratic polynomial with real co-efficients and a ≠ 0.

**Question 5.**

Write the standard form of a cubic polynomial with real co-efficients.

**Solution:**

ax^{3} + bx^{2} + cx + d is a standard form of cubic polynomial with real co-efficients and a ≠ 0.

**Question 6.**

Define value of a polynomial at a point.

**Solution:**

If f(x) is a polynomial and a is any real number then the real number obtained by replacing x by α in f(x) is called the value of f(x) at x = α and is denoted by f(α).

**Question 7.**

Define zero of a polynomial.

**Solution:**

A real number a is a zero of a polynomial f(x) if f(α) = 0.

**Question 8.**

The sum and product of the zeros of a quadratic polynomial are – \(\frac { 1 }{ 2 }\) and -3 respectively. What is the quadratic polynomial ?

**Solution:**

**Question 9.**

Write the family of quadratic polynomials having – \(\frac { 1 }{ 4 }\) and 1 as its zeros.

**Solution:**

**Question 10.**

If the product of zeros of the quadratic polynomial f(x) = x^{2} – 4x + k is 3, find the value of k.

**Solution:**

We know that a quadratic polynomial x^{2} – (sum of zeros) x + product of zeros

In the given polynomial f(x) = x^{2} – 4x + k is the product of zeros which is equal to 3

k = 3

**Question 11.**

If the sum of the zeros of a quadratic polynomial f(x) = kx^{2} – 3x + 5 is 1, write the value of k.

**Solution:**

f (x) = kx^{2} – 3x + 5

Here a = k, b = -3, c = 5

**Question 12.**

In the figure, the graph of a polynomial p (x) is given. Find the zeros of the polynomial.

**Solution:**

The graph of the given polynomial meets the x-axis at -1 and -3

Zero will be -1 and -3

Zero of polynomial is 3

**Question 13.**

The graph of a polynomial y = f(x) is given below. Find the number of real zeros of f (x).

**Solution:**

The curve touches x-axis at one point and also intersects at one point So number of zeros will be 3, two equal and one distinct

**Question 14.**

The graph of the polynomial f(x) = ax^{2} + bx + c is as shown below (in the figure) write the signs of ‘a’ and b^{2} – 4ac.

**Solution:**

The shape of parabola is up word a > 0

and b^{2} – 4ac >0 i.e., both are positive.

**Question 15.**

The graph of the polynomial f(x) = ax^{2} + bx + c is as shown in the figure write the value of b^{2} – 4ac and the number of real zeros of f(x).

**Solution:**

The curve parabola touches the x-axis at one point

It has two equal zeros

b^{2} – 4ac = 0

**Question 16.**

In **Q. No. 14**, write the sign of c

**Solution:**

The mouth of parabola is upward and intersect y-axis above x-axis

c > 0

**Question 17.**

In **Q. No. 15**, write the sign of c.

**Solution:**

The mouth of parabola is downward and intersects y-axis below x-axis

c < 0

**Question 18.**

The graph of a polynomial f (x) is as shown in the figure. Write the number of real zeros of f (x).

**Solution:**

The curves touches the x-axis at two distinct point

It has a pair of two equal zeros i.e., it has 4 real zeros

**Question 19.**

If x = 1, is a zero of the polynomial f(x) = x^{3} – 2x^{2} + 4x + k, write the value of k.

**Solution:**

**Question 20.**

State division algorithm for polynomials.

**Solution:**

If f(x) is a polynomial and g (x) is a non zero polynomial, there exist two polynomials q (x) and r (x) such that

f(x) = g (x) x q (x) + r (x)

where r (x) = 0 or degree r (x) < degree g (x)

This is called division algorithm

**Question 21.**

Give an example of polynomials f(x), g (x), q (x) and r (x) satisfying f(x) = g (x) . q (x) + r (x), where degree r (x) = 0.

**Solution:**

f (x) = x^{3} + x^{2} + x + 4

g (x) = x + 1

q (x) = x^{2} + 1

r (x) = 3

is an example of f (x) = g (x) x q (x) + r (x)

where degree of r (x) is zero.

**Question 22.**

Write a quadratic polynomial, sum of whose zeros is 2√3 and their product is 2.

**Solution:**

Sum of zeros = 2 √3

and product of zeros = 2

Quadratic polynomial will be f (x) = x2 – (sum of zeros) x + product of zeros

= x^{2} – 2 √3 x + 2

**Question 23.**

If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.

**Solution:**

Degree of the given polynomial = 4

and degree of divisor = 2

Degree of quotient will be 4 – 2 = 2

and degree of remainder will be less than 2 In other words equal to or less than one degree

**Question 24.**

If f(x) = x^{3} + x^{2} – ax + b is divisible by x^{2} – x, write the value of a and b.

**Solution:**

**Question 25.**

If a – b, a and a + b are zeros of the polynomial f(x) = 2x^{3} – 6x^{2} + 5x – 7, write the value of a.

**Solution:**

**Question 26.**

Write the coefficients of the polynomial p (z) = z^{5} – 2z^{2} + 4.

**Solution:**

p (z) = z^{5} + oz^{4} + oz^{3} – 2z^{2} + oz + 4

Coefficient of z^{5} = 1

Coefficient of z^{4} = 0

Coefficient of z^{3} = 0

Coefficient of z^{2} = – 2

Coefficient of z = 0

Constant = 4

**Question 27.**

Write the zeros of the polynomial x^{2} – x – 6. **(C.B.S.E. 2008)**

**Solution:**

**Question 28.**

If (x + a) is a factor of 2x^{2} + 2ax + 5x + 10, find a. **(C.B.S.E. 2008)**

**Solution:**

x + a is a factor of

**Question 29.**

For what value of k, -4 is a zero of the polynomial x^{2} – x – (2k + 2) ? **(CBSE 2009)**

**Solution:**

**Question 30.**

If 1 is a zero of the polynomial p (x) = ax^{2} – 3 (a – 1) x – 1, then find the value of a.

**Solution:**

**Question 31.**

If α, β are the zeros of a polynomial such that α + β = -6 and α β = -4, then write the polynomial. **[CBSE 2010]**

**Solution:**

**Question 32.**

If α, β are the zeros of the polynomial 2y^{2} + 7y + 5, write the value of α + β + αβ. **[CBSE 2010]**

**Solution:**

**Question 33.**

For what value of k, is 3 a zero of the polynomial 2x^{2} + x + k ? **[CBSE 2010]**

**Solution:**

3 is a zero of f(x) = 2x^{2} + x + k

It will satisfy the polynomial

f(x) = 0 ⇒ f(3) = 0

Now 2x^{2} + x + k = 0

=> 2 (3)^{2} + 3 + k = 0

=> 18 + 3 + k = 0

=> 21 + k = 0

=> k = -21

**Question 34.**

For what value of k, is -3 a zero of the polynomial x^{2} + 11x + k ? **[CBSE 2010]**

**Solution:**

-3 is a zero of polynomial f(x) = x^{2} + 11x + k

It will satisfy the polynomial

f (x) = 0 => f(-3) = 0

Now x^{2} + 11x + k = 0

=> (-3)^{2}+ 11 x (-3) + k = 0

⇒ 9 – 33 + k = 0

⇒ -24 + k = 0

⇒ k = 24

**Question 35.**

For what value of k, is -2 a zero of the polynomial 3x^{2} + 4x + 2k ?** [CBSE 2010]**

**Solution:**

-2 is a zero of the polynomial

f(x) = 3x^{2} + 4x + 2k

f(-2) = 0

=> 3 (-2)^{2} + 4 (-2) + 2k = 0

=> 12 – 8 + 2k = 0

=> 4 + 2k = 0

=> 2k = -4

=> k = -2

**Question 36.**

If a quadratic polynomial f(x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f (x) ?

**Solution:**

In a quadratic polynomial f(x) its degree is 2 and it can be factorised in to two distinct linear factors.

f(x) has two distinct zeros

**Question 37.**

If a quadratic polynomiaI f(x) is a square of a linear polynomial, then its two zeros are coincident. (True / False)

**Solution:**

In a quadratic polynomial f(x), it is the square of a linear polynomial It has two zeros which are equal i.e. coincident

It is true

**Question 38.**

If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True / False)

**Solution:**

A quadratic polynomial f(x) is not factorised into linear factors It has no real zeros It is true

**Question 39.**

If f(x) is a polynomial such that f(a) f(b) < 0, then what is the number of zeros lying between a and b ?

**Solution:**

f(x) is a polynomial such that f(a) f(b) < 0

At least one of its zeros will be between a and b

**Question 40.**

If graph of quadratic polynomial ax^{2} + bx + c cuts positive direction of y-axis, then what is the sign of c ?

**Solution:**

The graph of quadratic polynomial ax^{2} + bx + c cuts positive direction of y-axis Then sign of constant term c will be also positive.

**Question 41.**

If the graph of quadratic polynomial ax^{2} + bx + c cuts negative direction of y-axis, then what is the sigh of c ?

**Solution:**

The graph of quadratic polynomial ax^{2} + bx + c cuts negative side of y-axis

Then sign of constant term c will be negative

Hope given RD Sharma Class 10 Solutions Chapter 2 Polynomials VSAQS are helpful to complete your math homework.

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