Class 10 Maths Extra Questions Chapter 2 Polynomials

(iii) Polynomial and its graph : In general a polynomial p(x) of degree n crosses the
x-axis at most n points.
y

x' x
0

y'

VERY SHORT ANSWER TYPE QUESTIONS
1. If one zero of the polynomial P(x) = 5x2 + 13x + K is reciprocal of the other, then
value of k is
1
(a) 0 (b) 5 (c) (d) 6
6
2. If  and  are the zeroes of the polynomial p(x) = x2 – p(x + 1) – c such that
( + 1) ( + 1) = 0, the c = _______ .
3. If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is
(a) 10 (b) – 10 (c) 5 (d) – 5
2
4. If the zeroes of the quadratic polynomial x + (a + 1)x + b are 2 and – 3, then
(a) a = – 7, b = – 1 (b) a = 5, b = – 1
(c) a = 2, b = – 6 (d) a = 0, b = – 6
5. What should be added to the polynomial x2 – 5x + 4, so that 3 is the zero of the
resulting polynomial:
(a) 1 (b) 2 (c) 4 (d) 5
6. If  and  are the zeros of the polynomial
1 1
f (x) = x2 + x + 1, then  
 
7. If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real
zero. (True/False)
8. If a quadratic polynomial f(x) is a square of a linear polynomial, then its two zeros are
coincident. (True/False).
9. If p(x) = x3 – 2x2 – x + 2 = (x + 1) (x – 2) (x – d) then what is the value of d?

18 Mathematics-X

10. The quadratic polynomial ax2 + bx + c, a  0 is represented by this graph then a is

(a) Natural no. (b) Whole no. (c) Negative Integer (d) Irrational no.
11. What will be the number of zeros of a linear polynomial p(x) if its graph (i) passes
through the origin. (ii) doesn’t intersect or touch x-axis at any point?
12. Find the quadratic polynomial whose zeros are
5 + 2 3 and 5 – 2 3
13. If one zero of p(x) = 4x2 – (8k2 – 40k) x – 9 is negative of the other, find values of
k.
14. What number should be subtracted to the polynomial x2 – 5x + 4, so that 3 is a zero
of polynomial so obtained.
15. How many (i) maximum (ii) minimum number of zeroes can a quadratic polynomial
have?
16. What will be the number of real zeros of the polynomial x2 + 1?
17. If  and  are zeros of polynomial 6x2 – 7x – 3, then form a quadratic polynomial
where zeros are 2 and 2 (CBSE)
1
18. If  and are zeros of 4x2 – 17x + k – 4, find the value of k.

19. What will be the number of zeros of the polynomials whose graphs are parallel to (i)
y-axis (ii) x-axis?
20. What will be the number of zeros of the polynomials whose graphs are either touching
or intersecting the axis only at the points:
(i) (–3, 0), (0, 2) & (3, 0) (ii) (0, 4), (0, 0) and (0, –4)

SHORT ANSWER TYPE (I) QUESTIONS
21. For what value of k, x2 – 4x + k touches x-axis.
22. If the product of zeros of ax2 – 6x – 6 is 4, find the value of a. Hence find the sum of
its zeros.
23. If zeros of x2 – kx + 6 are in the ratio 3 : 2, find k.
24. If one zero of the quadratic polynomial (k2 + k)x2 + 68x + 6k is reciprocal of the
other, find k.
Mathematics-X 19

25. If  and  are the zeros of the polynomial x2 – 5x + m such that –  = 1, find m.
(CBSE)
26. If the sum of squares of zeros of the polynomial x2 – 8x + k is 40, find the value of k.
27. If  and  are zeros of the polynomial t2 – t – 4, form a quadratic polynomial whose

1 1
zeros are and .
 

28. What should be added to the polynomial x3 – 3x2 + 6x – 15, so that it is completely
divisible by x – 3 ? (CBSE 2016)
m n
29. If m and n are the zeros of the polynomial 3x2 + 11x – 4, find the value of  .
n m
(CBSE, 2012)

3 5 3 5
30. Find a quadratic polynomial whose zeros are and .
5 5
(CBSE, 2013)

SHORT ANSWER TYPE (II) QUESTIONS
31. If (k+ y) is a factor of each of the polynomials y2 + 2y – 15 and y3 + a , find the
values of k and a.

32. Obtain zeros of 4 3 x 2  5 x – 2 3 and verify relation between its zeroes and
coefficients.
33. If x4 + 2x3 + 8x2 + 12x + 18 is divided by (x2 + 5) , remainder comes out to be (px
+ q) , find values of p and q.
34. –5 is one of the zeros of 2x2 + px – 15, zeroes of p(x2 + x) + k are equal to each
other. Find the value of k.
35. Find the value of k such that 3x2 + 2kx + x – k– 5 has the sum of zeros as half of
their product.
36. If zeros of the polynomial ax2 + bx – c, a  0 are additive inverse of each other then
what is the value of b?
37. If  and  are zeros of x2 – x – 2, find a polynomial whose zeros are (2+ 1) and
(2 + 1)

20 Mathematics-X

38. Find values of a and b so that x4 + x3 + 8x2 + ax + b is divisible by x2 + 1.
39. What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that the resulting
polynomial is exactly divisible by 4x2 + 3x – 2 ?
40. What must be added to 4x4 + 2x3 – 2x2 + x – 1 so that the resulting polynomial is
divisible by x2 – 2x – 3 ?

LONG ANSWER TYPE QUESTIONS
41. Find all zeros of the polynomial 2x3 + x2 – 6x – 3 if two of its zeroes are
3 and – 3 .

42. If 2 is a zero of (6 x  2 x – 10 x – 4 2) , find its other zeroes.
3 2

43. If two zeros of x4 – 6x3 – 26x2 + 138 x – 35 are (2  3) , find other zeroes.
44. On dividing the polynomial x3 – 5x2 + 6x – 4 by a polynomial g(x), quotient and
remainder are (x –3) and (– 3x + 5) respectively. Find g(x)
45. Obtain all zeros of the polynomial 2x4 – 2x3 – 7x2 + 3x + 6 if two factors of this

 3
polynomial are  x  .
 2 

46. If the polynomial x4 – 3x3 – 6x2 + kx – 16 is exactly divisible by x2 – 3x + 2, then
find the value of k. (CBSE, 2014)
47. If the polynomial x – 6x + 16x – 25x + 10 is divided by x – 2x + k, the
4 3 2 2

reaminder is (x + a) then find the value of k and a. (CBSE)
48. If  and  are zeros of the polynomial x2 + 4x + 3, find the polynomial whose zeros
 
are 1  and 1  . (CBSE)
 
49. Find K, so that x2 + 2x + K is a factor of 2x4 + x3 – 14x2 + 5x + 6. Also find all the
zeros of the two polynomials: (Exempler, HOTS)
50. If x  5 is a factor of the cubic polynomial x3  3 5x2  13x  3 5 , then find
all the zeros of the polynomial.
51. If zeros of x2 – 5kx + 24 are in the ratio 3 : 2, find k.
52. Form a quadratic polynomial one of whose zero is 2  5 and sum of the zeros is 4.

Mathematics-X 21

53. Form a polynomial whose zeros are the reciprocal of the zeros of p(x) = ax2 + bx +
c, a  0.
54. If (x + 2) is a factor of x2 + px + 2q and p + q = 4 then what are the values of
p and q?
55. What should be subtracted from x3 – 3x2 + 6x – 15, so that it is completely divisible
by (x – 3)?
56. If x2 + 1 is a factor of x4 + x3 + 8x2 + ax + b then what are the values of a and b.
57. If sum of the zeros of 5x2 + (p + q + r)x + pqr is zero, then find p3 + q3 + r3.
58. If the zeros of x2 + px + q are double in value to the zeros of 2x2 – 5x – 3 find p and
q.

ANSWERS AND HINTS

1. (b) 5 2. 1
3. (b) –10 4. (d) a = 0, b = –6
5. (b) 2 6. – 1
7. True 8. True
9. 1 10. (c) Negative Integer
11. (i) 1 (ii) 0 12. x2 – 10x + 13
13. k = 0, 5 14. (– 2)
15. (i) 2 (ii) 0 16. 0
17. [3x2 – 7x – 6] k 18. k = 8
19. (i) 1 (ii) 0 20. (i) 2 (ii) 1

3
21. 4 22. a   , sum of zeroes   4
2
23. – 5, 5 24. 5
25. 6 26. 12
27. 4t2 + t – 1
28. On dividing x3 – 3x2 + 6x – 15 by x – 3, remainder is + 3, hence – 3 must be
added to x3 – 3x2 + 6x – 15.

22 Mathematics-X

2
 11  4
    2   
m n m n 2  2
( m  n)  2mn 3 3   145
29.    = 4 12
n m mn mn 
3
6 4
30.     ,   , 31. k = –3, 5 and a = –27, 125
5 25
25x2 – 30x + 4
2 3
32.  , 33. p = 2, q = 3
3 4
7
34. 35. 1
4
36. b = 0 37. x2 – 4x – 5
38. a = 1, b = 7 39. 14x – 10
1
40. 61x – 65 41. 3,  3, 
2

2 2 2
42.  , 43. – 5, 7
2 3

3
44. x2 – 2x + 3 45. 2,  1, 
2
46. x2 – 3x + 2 = (x – 2) (x – 1)
P(1) = 0, K = 24.
47. On dividing x4 – 6x3 + 16x2 – 25x + 10 by x2 – 2x + k we get remainder
(2k – 9)x + (10 – 8k + k2)
Given remainder = x + a
2k – 9 = 1  k5
10 – 8k + k2 = a  a = 10 – 40 + 25 = – 5
a = – 5, k = 5
2 16 16 1
48. x  x or (3 x 2  16 x  16)
3 3 3

Mathematics-X 23

49. On dividing 2x4 + x3 – 14x2 + 5x + 6 by x2 + 2x + k
We get (7k + 21)x + 2k2 + 8k + 6 as remainder is zero.
 7k + 21 = 0 and 2k2 + 8k + 6 = 0
 k = – 3 and k = –1 or – 3
 k = –3
quotient = 2x2 – 3x – (2k + 8)
= 2x2 – 3x – 2
1
Zeros of x2 + 2x – 3 are 1, – 3 and 2x4 + x3 – 14x2 + 5x + 6 are 1,  3, 2, 
2
50. 5, 5  2 , 5 2
51. k = 2
52. 2 – 5

 2 b a
53. k  x  x  
 c c
54. p = 4, q = 0
55. 3
56. a = 1, b = 7
57. Product of the zeros = 3 pqr
5 3
58. p = – and q = 
4 8

24 Mathematics-X

PRACTICE-TEST
Polynomials
Time : 1 Hr. M.M. : 20
SECTION- A
1. If  and  are zeros of a quadratic polynomial p(x), then factorize p(x). 1
 1
2. If  and  are zeros of x2 – x – 1, find the value of  . 1
 

3. If one of the zeros of quadratic polynomial (K –1) x2 + kx + 1 is – 3 then the value

of K is, 1

4 4 2 2
(a) (b)  (c) (d) 
3 3 3 3
4. A quadratic polynomial, whose zeros are – 3 and 4, is 1
(a) x2 – x + 12 (b) x2 + x + 12

x2 x
(c)  6 (d) 2x2 + 2x – 24
2 2

SECTION-B

1
5. If  and  are zeros of x2 – (k + 6)x + 2(2k –1). find the value of k if  +  =  .
2
2
6. Find a quadratic polynomial one of whose zeros is (3  2) and the sum
of its zeroes is 6. 2
2
7. If zeros of the polynomial x2 + 4x + 2a are  and then find the value of a. 2


Mathematics-X 25

SECTION-C
8. Find values of a and b if (x2 + 1) is a factor of the polynomial x4 + x3 + 8x2 + ax
+ b. 3
9. If truth and lie are zeros of the polynomial px2 + qx + r, (p  0) and zeros are
reciprocal to each other, Find the relation between p and r. 3

SECTION-D
10. On dividing the polynomial x3 + 2x2 + kx + 7 by (x – 3), remainder comes out to be
25. Find quotient and the value of k. Also find the sum and product of zeros of the
quotient so obtained. 4

26 Mathematics-X