CBSE Class 10 Polynomials Sure Shot Questions Set J

Question. Every quadratic polynomial can have atmost
(a) one zero
(b) two zeroes
(c) no zero
(d) more than two zeroes
Answer: (b) two zeroes

Question. Suppose \( g(x) = ax^3 + bx^2 + cx + d \) is cubic polynomial, then which of the following is always true?
(a) \( b \neq 0 \)
(b) \( a \neq 0 \)
(c) \( c \neq 0 \)
(d) \( d \neq 0 \)
Answer: (b) \( a \neq 0 \)

Question. If 1 is a zero of polynomial \( p(x) = 3x^2 + 5x + a \), then the value of \( a \) is
(a) -8
(b) 8
(c) 7
(d) -7
Answer: (a) -8

Question. If \( p(x) = ax^2 + bx + c \) and \( a + b + c = 0 \), then one zero is
(a) \( \frac{c}{a} \)
(b) 1
(c) \( \frac{b}{a} \)
(d) Cannot determined
Answer: (b) 1

Question. A quadratic polynomial, whose sum and product of zeroes are 3 and 2, is
(a) \( x^2 + 3x + 2 \)
(b) \( x^2 - 3x - 2 \)
(c) \( x^2 - 3x + 2 \)
(d) None of the options
Answer: (c) \( x^2 - 3x + 2 \)

Question. The number of polynomials having zeroes as 3 and -7 is
(a) 11
(b) 2
(c) 3
(d) more than 2
Answer: (d) more than 2

Question. Find a quadratic polynomial, whose zeroes are 3 and \( \frac{1}{4} \).
(a) \( 2x^2 + 13x - 3 \)
(b) \( 4x^2 + 13x - 3 \)
(c) \( 4x^2 - 13x - 3 \)
(d) \( 4x^2 - 13x + 3 \)
Answer: (d) \( 4x^2 - 13x + 3 \)

Question. If 2 and 3 are zeroes of polynomial \( 3x^2 - 2kx + 2m \), then the values of \( k \) and \( m \) are respectively
(a) 15 and \( \frac{9}{2} \)
(b) \( \frac{15}{2} \) and -9
(c) \( \frac{15}{2} \) and 9
(d) None of the options
Answer: (c) \( \frac{15}{2} \) and 9

Question. If zeroes of the polynomial \( p(x) = -8x^2 + (k + 5)x + 36 \) are negative to each other, then the value of \( k \) is
(a) -5
(b) 5
(c) 4
(d) 3
Answer: (a) -5

Question. If one zero of the polynomial \( (a^2 + 4)x^2 + 9x + 4a \) is the reciprocal of the other, then the value of \( a \) is
(a) 2 and 3
(b) -2
(c) 2
(d) -2 and 3
Answer: (c) 2

Question. If \( \alpha \) and \( \beta \) are zeroes of the polynomial \( p(x) = x^2 - p(x + 1) + c \) such that \( (\alpha + 1)(\beta + 1) = 0 \), then the value of \( c \) is
(a) -1
(b) 1
(c) \( -\frac{1}{2} \)
(d) \( \frac{1}{2} \)
Answer: (a) -1

Question. If one zero of the polynomial \( p(x) = 2x^2 - 5x - (2k + 1) \) is twice the other zero, then the value of \( k \) is
(a) \( \frac{17}{9} \)
(b) \( -\frac{17}{9} \)
(c) \( \frac{9}{17} \)
(d) None of the options
Answer: (b) \( -\frac{17}{9} \)

Topics Covered

• Geometrical Meaning of the Zeroes of a Polynomial
• Relationship between Zeroes and Coefficients of a Polynomial

Introduction

Polynomial: An expression of the form \( p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( n \) is a non-negative integer, \( a_1, a_2, ..., a_n \) are constants (real numbers) and \( a_n \neq 0 \), is called a polynomial in \( x \) of degree \( n \).

Degree: If \( p(x) \) is a polynomial in \( x \), the highest power of \( x \) in \( p(x) \) is called the degree of the polynomial \( p(x) \).

Type of Polynomials: Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

1. Geometrical Meaning of the Zeroes of a Polynomial

Zeroes of a Polynomial: A real number ‘a’ is said to be a zeroes of a polynomial \( p(x) \) if \( p(a) = 0 \).

Note: A polynomial may have no zero, one or more than one zeroes. If \( p(a) \neq 0 \), then ‘a’ is not a zero of the polynomial \( p(x) \).

Question. The value of k for which (–4) is a zero of the polynomial \( x^2 – x – (2k + 2) \) is
(a) 2
(b) –6
(c) 9
(d) 8
Answer: (c) 9

Question. The solution of \( x^2 + 6x + 9 = 0 \) is
(a) –1
(b) 3
(c) –3
(d) 1
Answer: (c) –3

Question. The number of polynomials having zeroes as –2 and 5 is
(a) 1
(b) 2
(c) 10
(d) infinite
Answer: (d) infinite

Exercise 1

Question. The value of p, for which (–4) is a zero of the polynomial \( x^2 – 2x – (7p + 3) \) is
(a) 0
(b) 2
(c) 3
(d) None of the options
Answer: (c) 3

Question. If 2 is a zero of the polynomial \( ax^2 – 2x \), then the value of ‘a’ is
(a) 3
(b) 1
(c) 2
(d) 5
Answer: (b) 1

Question. What number should be added to the polynomial \( x^2 – 5x + 4 \), so that 3 is the zero of the polynomial?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. For what value of k, (– 4) is a zero of \( p(x) = x^2 – x – (2k – 2) \)?
(a) 8
(b) 11
(c) 13
(d) 15
Answer: (b) 11

Question. A teacher asked 10 of his students to write a polynomial in one variable on a paper and then to handover the paper. The following were the answers given by the students: \( 2x + 3 \), \( 3x^2 + 7x + 2 \), \( 4x^3 + 3x^2 + 2 \), \( x^3 + \sqrt{3}x + 7 \), \( 7x + \sqrt{7} \), \( 5x^3 – 7x + 2 \), \( 2x^2 + 3 – \frac{5}{x} \), \( 5x – \frac{1}{2} \), \( ax^3 + bx^2 + cx + d \), \( x + \frac{1}{x} \). How many of the above ten, are not polynomials?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. How many of the above ten (in the previous question), are quadratic polynomials?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. Assertion (A): \( x^2 + 4x + 5 \) has two zeroes.
Reason (R): A quadratic polynomial can have at the most two zeroes.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d) Assertion (A) is false but reason (R) is true.

Question. Assertion (A): A quadratic polynomial whose zeroes are \( 5 + \sqrt{2} \) and \( 5 – \sqrt{2} \) is \( x^2 – 10x + 23 \).
Reason (R): If \( \alpha \) and \( \beta \) are zeroes of the quadratic polynomial \( p(x) \), then \( p(x) = x^2 – (\alpha + \beta)x + \alpha\beta \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

2. Relationship between Zeroes and Coefficients of a Polynomial

• If \( \alpha, \beta \) are zeroes of a quadratic polynomial \( p(x) = ax^2 + bx + c \), where \( a \neq 0 \), then
  • (i) Sum of zeroes = \( \alpha + \beta = \frac{-b}{a} \Rightarrow \alpha + \beta = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2} \)
  • (ii) Product of zeroes = \( \alpha\beta = \frac{c}{a} \Rightarrow \alpha\beta = \frac{\text{Constant term}}{\text{Coefficient of } x^2} \)
• If \( \alpha, \beta \) are zeroes (or roots) of a quadratic polynomial \( p(x) \), then \( p(x) = x^2 – (\alpha + \beta)x + \alpha\beta \)
• \( p(x) = x^2 – (\text{Sum of zeroes})x + (\text{Product of zeroes}) \)

Question. A quadratic equation \( x^2 – 2x – 8 \) is given. The zeroes of it are
(a) –2 and 4
(b) 3 and 5
(c) 1 and 6
(d) None of the options
Answer: (a) –2 and 4

Question. The sum and product of the zeroes of the quadratic equation given in previous example are respectively
(a) 2, 4
(b) 5, –8
(c) 6, 8
(d) 2, –8
Answer: (d) 2, –8

Question. The zeroes of the quadratic equation \( 4s^2 – 4s + 1 \) are
(a) \( \frac{1}{2}, \frac{1}{4} \)
(b) \( \frac{1}{2}, \frac{1}{2} \)
(c) \( \frac{1}{4}, \frac{1}{14} \)
(d) \( \frac{1}{3}, \frac{1}{4} \)
Answer: (b) \( \frac{1}{2}, \frac{1}{2} \)

Question. The sum and product of zeroes of the quadratic equation \( 4s^2 – 4s + 1 \) are respectively
(a) \( 2, \frac{3}{4} \)
(b) \( 0, \frac{1}{8} \)
(c) \( 1, \frac{1}{4} \)
(d) None of the options
Answer: (c) \( 1, \frac{1}{4} \)

Question. The set of the zeroes of the polynomial \( x^2 – 25 \), their sum and product is
(a) 4, 3; 7; 12
(b) –3, 3; 0; –9
(c) 5, –5; 0; –25
(d) None of the options
Answer: (c) 5, –5; 0; –25

Question. If the product of the zeroes of the polynomial \( ax^2 – 6x – 6 \) is 4, then value of \( a \) is
(a) \( \frac{1}{8} \)
(b) \( -\frac{1}{4} \)
(c) \( -\frac{5}{3} \)
(d) \( -\frac{3}{2} \)
Answer: (d) \( -\frac{3}{2} \)

Question. The value of k, if the sum of the zeroes of the polynomial \( x^2 – (k + 6)x + 2(2k – 1) \) is half of their product is
(a) 7
(b) 11
(c) 12
(d) None of the options
Answer: (a) 7

Question. If the zeroes of the polynomial \( x^2 + px + q \) are double in value to the zeroes of \( 2x^2 – 5x – 3 \), the values of \( p \) and \( q \) are respectively
(a) 5, 6
(b) 4, 7
(c) –5, –6
(d) –4, –7
Answer: (c) –5, –6

Question. A quadratic polynomial whose zeroes are 1 and –3 is
(a) \( x^2 + 3x – 2 \)
(b) \( x^2 + 5x – 5 \)
(c) \( x^2 + 2x – 3 \)
(d) None of the options
Answer: (c) \( x^2 + 2x – 3 \)

Question. The quadratic polynomial, sum of whose zeroes is 8 and their product is 12, is given by
(a) \( x^2 – 8x + 12 \)
(b) \( x^2 + 8x – 12 \)
(c) \( x^2 – 5x + 7 \)
(d) \( x^2 + 5x – 7 \)
Answer: (a) \( x^2 – 8x + 12 \)

Question. If \( \alpha, \beta \) are the zeroes of the polynomial \( 2x^2 – 5x + 7 \), then a polynomial whose zeroes are \( 2\alpha + 3\beta, 3\alpha + 2\beta \) is
(a) \( k \left(x^2 - \frac{3}{5}x + 21\right) \)
(b) \( k \left(x^2 - \frac{25}{2}x + 41\right) \)
(c) \( k \left(x^2 + \frac{9}{2}x - 45\right) \)
(d) None of the options
Answer: (b) \( k \left(x^2 - \frac{25}{2}x + 41\right) \)

Question. A quadratic polynomial whose product and sum of zeroes are \( -\frac{13}{5} \) and \( \frac{3}{5} \), respectively.
(a) \( k(x^2 + 12x + 5) \)
(b) \( k[x^2 – (8x) + (–9)] \)
(c) \( k \left[x^2 - \left(\frac{1}{2}x\right) + \left(-\frac{7}{5}\right)\right] \)
(d) \( k \left[x^2 - \left(\frac{3}{5}x\right) + \left(-\frac{13}{5}\right)\right] \)
Answer: (d) \( k \left[x^2 - \left(\frac{3}{5}x\right) + \left(-\frac{13}{5}\right)\right] \)

Exercise 2

Question. Sum of the zeroes of the polynomial \( x^2 + 7x + 10 \) are
(a) 7
(b) – 7
(c) 10
(d) – 10
Answer: (b) – 7

Question. The quadratic polynomial, the sum of whose zeroes is –5 and their product is 6, is
(a) \( x^2 + 5x + 6 \)
(b) \( x^2 – 5x + 6 \)
(c) \( x^2 – 5x – 6 \)
(d) \( -x^2 + 5x + 6 \)
Answer: (a) \( x^2 + 5x + 6 \)

Question. Quadratic polynomial having zeroes \( \alpha \) and \( \beta \) is
(a) \( x^2 – (\alpha\beta)x + (\alpha + \beta) \)
(b) \( x^2 – (\alpha + \beta)x + \alpha\beta \)
(c) \( x^2 - \frac{\alpha}{\beta}x + \alpha\beta \)
(d) None of the options
Answer: (b) \( x^2 – (\alpha + \beta)x + \alpha\beta \)

Question. If one zero of the quadratic polynomial \( x^2 – 5x – 6 \) is 6, then other zero is
(a) 0
(b) 1
(c) –1
(d) 2
Answer: (c) –1

Question. A quadratic polynomial, the sum and product of whose zeroes and (–3) and 2 respectively is
(a) \( x^2 + 3x + 2 \)
(b) \( x^2 – 3x + 2 \)
(c) \( x^2 + 3x – 2 \)
(d) None of the options
Answer: (a) \( x^2 + 3x + 2 \)

Question. If the sum of the zeroes of the quadratic polynomial \( 3x^2 – kx + 6 \) is 3, then the value of \( k \) is
(a) 3
(b) 6
(c) 9
(d) 0
Answer: (c) 9

Question. A quadratic polynomial, whose sum of zeroes is 2 and product is –8 is
(a) \( x^2 – 3x – 3 \)
(b) \( x^2 + 2x + 8 \)
(c) \( x^2 + 3x + 3 \)
(d) \( x^2 – 2x – 8 \)
Answer: (d) \( x^2 – 2x – 8 \)

Question. A quadratic polynomial whose zeroes are \( 5 - 3\sqrt{2} \) and \( 5 + 3\sqrt{2} \) is
(a) \( x^2 – 10x + 7 \)
(b) \( x^2 + 10x + 7 \)
(c) \( x^2 – 5x + 9 \)
(d) \( x^2 + 5x – 9 \)
Answer: (a) \( x^2 – 10x + 7 \)

Question. The zeores of the quadratic polynomial \( 6x^2 – 3 – 7x \) are
(a) \( \frac{3}{2}, -\frac{1}{3} \)
(b) \( \frac{2}{3}, –3 \)
(c) \( \frac{3}{5}, -\frac{3}{7} \)
(d) None of the options
Answer: (a) \( \frac{3}{2}, -\frac{1}{3} \)

Question. The zeroes of the quadratic polynomial \( 4x^2 – 4x – 3 \) are
(a) \( \frac{3}{2}, -\frac{1}{3} \)
(b) \( \frac{3}{2}, -\frac{1}{2} \)
(c) \( \frac{2}{5}, -\frac{2}{5} \)
(d) None of the options
Answer: (b) \( \frac{3}{2}, -\frac{1}{2} \)

Question. The zeroes of the quadratic polynomial \( x^2 + 7x + 10 \) are
(a) –2, –5
(b) 2, 5
(c) –3, –8
(d) 3, 8
Answer: (a) –2, –5

Question. Assertion (A): If one zero of polynomial \( p(x) = (k^2 + 4)x^2 + 13x + 4k \) is reciprocal of each other, then \( k = 2 \).
Reason (R): If \( (x – \alpha) \) is a factor of \( p(x) \), then \( p(\alpha) = 0 \), i.e. \( \alpha \) is a zero of \( p(x) \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Question. Assertion (A): If both zeroes of the quadratic polynomial \( x^2 –2kx + 2 \) are equal in magnitude but opposite in sign, then value of \( k \) is \( \frac{1}{2} \).
Reason (R): Sum of zeroes of a quadratic polynomial \( ax^2 + bx + c \) is \( -\frac{b}{a} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d) Assertion (A) is false but reason (R) is true.