Class 10 Maths Polynomials Question Paper | Chapter 2 Practice Paper

1The polynomials x² + ax + b and x² + bx + a (a ≠ b) have one zero in common. That common zero is:

2A quadratic polynomial with integer coefficients and leading coefficient 1 has 3 − 2√5 as one of its zeroes. Its constant term is:

3The zeroes of x² + bx + 12 are both integers. The number of possible values of b is:

4Both zeroes of the quadratic polynomial x² + px + q are negative. Then:

5The graph of a quadratic polynomial p(x) touches the x-axis at exactly one point, and p(0) > 0. Then the leading coefficient of p(x) is:

6If α and β are the zeroes of x² − x − 1, then α + β equals:

7For p(x) = x² − (k − 2)x − (k + 5), the value of k that makes the sum of the squares of the zeroes least is:

8Two distinct real numbers α and β satisfy α² = 5α + 3 and β² = 5β + 3. Then α/β + β/α equals:

9The number of integer values of k for which x² + kx + 9 has no zeroes is:

10The number of real values of a for which the zeroes of x² + ax + (a + 3) are equal in magnitude but opposite in sign is:

Directions: Choose the correct option for each question:

• (a) Both A and R are true and R is the correct explanation of A.
• (b) Both A and R are true but R is not the correct explanation of A.
• (c) A is true but R is false.
• (d) A is false but R is true.

11Assertion (A): If p(x) = x² + bx + c satisfies p(2) < 0, then p(x) has two distinct real zeroes, one less than 2 and the other greater than 2.

12Assertion (A): A quadratic polynomial with rational coefficients that has 2 + √3 as a zero must also have 2 − √3 as a zero.

13Assertion (A): Every quadratic polynomial ax² + bx + c with c = 0 has 0 as a repeated zero.

14Assertion (A): If a quadratic polynomial has two real zeroes of opposite signs, then its constant term and leading coefficient have opposite signs.

15Assertion (A): If a quadratic polynomial has equal zeroes, its graph touches the x-axis at exactly one point.

16Can a quadratic polynomial with all three coefficients positive have a positive zero? Justify your answer. 2 marks

17A parabola opens upward and its vertex lies in the second quadrant. How many real zeroes can the corresponding polynomial have? Justify. 2 marks

18Both zeroes of x² − 6x + k are positive real numbers. Find the range of values of k. 2 marks

19Without finding the zeroes, decide whether both zeroes of x² − 7x + 11 are greater than 1. Justify using the value of the polynomial at x = 1 and the position of the vertex. 2 marks

20For what values of k does x² + kx + (k + 3) have two distinct real zeroes? 2 marks

21If α and β are the zeroes of x² − px + (p − 1), find the value of p for which α² + β² is least. Also find the least value. 3 marks

22Prove that there is no real value of k for which x² + kx + (k² + 1) has real zeroes. 3 marks

23The graph of a quadratic polynomial passes through (0, −3), has its axis of symmetry at x = 2, and has one zero at x = 1. Without finding the polynomial fully, find the other zero and justify. 3 marks

24A quadratic polynomial p(x) with leading coefficient 1 satisfies p(2) = 0 and the line x − 1 = 0 acts as its axis of symmetry. Find the other zero and write the polynomial. 3 marks

25Find all values of k for which both zeroes of x² − 2kx + (k + 6) are positive. 3 marks

26Consider the family of polynomials p(x) = x² − 2tx + (t² − 3), where t is a real parameter.

1. Show that the difference of the zeroes is the same for every t.
2. Find the value of t for which the zeroes are equal.
3. For what t are both zeroes positive?

5 marks

27The graph of p(x) = ax² + bx + c cuts the x-axis at two points symmetric about x = 3, passes through (0, 5), and has minimum value −4.

1. Use the axis of symmetry to relate a, b and c.
2. Find a, b and c.
3. Find the zeroes.

5 marks