Mathematics · Textbook solutions

Polynomials

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 18 questions

2.2 Geometrical Meaning of the Zeroes of a Polynomial

2 q

Solved Examples

Worked · 1

2.1 Eg.1 $y = p(x)$

Exercise 2.1

Practice · 1

Ex 2.1 Q1 $y = p(x)$

2.3 Relationship between Zeroes and Coefficients of a Polynomial

16 q

Solved Examples

Worked · 4

2.2 Eg.2 $x^2 + 7x + 10$
2.2 Eg.3 $x^2 - 3$
2.2 Eg.4 $-3$
2.2 Eg.5 $3,\ -1,\ -\dfrac{1}{3}$

Exercise 2.2

Practice · 12

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
Ex 2.2 Q1 (i) $x^2 - 2x - 8$
Ex 2.2 Q1 (ii) $4s^2 - 4s + 1$
Ex 2.2 Q1 (iii) $6x^2 - 3 - 7x$
Ex 2.2 Q1 (iv) $4u^2 + 8u$
Ex 2.2 Q1 (v) $t^2 - 15$
Ex 2.2 Q1 (vi) $3x^2 - x - 4$
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Ex 2.2 Q2 (i) $\dfrac{1}{4},\ -1$
Ex 2.2 Q2 (ii) $\sqrt{2},\ \dfrac{1}{3}$
Ex 2.2 Q2 (iii) $0,\ \sqrt{5}$
Ex 2.2 Q2 (iv) $1,\ 1$
Ex 2.2 Q2 (v) $-\dfrac{1}{4},\ \dfrac{1}{4}$
Ex 2.2 Q2 (vi) $4,\ 1$

What the board has asked from this chapter

Questions per paper · 67 papers · 2022 to 2026

2026

26 q

3 $p(x)$
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Forming a Quadratic Polynomial from its Zeroes
4 $y = f(x)$
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Geometrical Meaning of the Zeroes of a Polynomial
4 $y = f(x)$
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Geometrical Meaning of the Zeroes of a Polynomial
5 $p(x)$
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Geometrical Meaning of the Zeroes of a Polynomial
5 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
6 $p(x)$
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Zeroes and Coefficients of a Quadratic Polynomial
7 $p(x)$
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Geometrical Meaning of the Zeroes of a Polynomial
8 $p(x) = (x - 2)(x + 3)$
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Geometrical Meaning of the Zeroes of a Polynomial
11 $(-3)$
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Forming a Quadratic Polynomial from its Zeroes
14 $p(x)$
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Forming a Quadratic Polynomial from its Zeroes
Question numbers 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below :
20 $p(y) = y^2 + 4y + 3$
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Geometrical Meaning of the Zeroes of a Polynomial
21 $\alpha$
Zeroes and Coefficients of a Quadratic Polynomial
21 $\alpha, \beta$
Zeroes and Coefficients of a Quadratic Polynomial
24 $\left(5 - 2\sqrt{3}\right)$
Forming a Quadratic Polynomial from its Zeroes
24 $\alpha, \beta$
Zeroes and Coefficients of a Quadratic Polynomial
24 $px^2 - 14x + 8$
Zeroes and Coefficients of a Quadratic Polynomial
25 $x^2 + 7x + 10$
Zeroes and Coefficients of a Quadratic Polynomial
25 $\alpha$
Zeroes and Coefficients of a Quadratic Polynomial
An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial $p(x) = -0.0025x^2 - 0.025x + 136$ . Observe the diagram and based on above information, answer the following questions :
36 (i) $Write the co-ordinates of point A.$
Geometrical Meaning of the Zeroes of a Polynomial
During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$ , where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions :
36 (i) $Determine the height of the arch.$
Geometrical Meaning of the Zeroes of a Polynomial
During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$ , where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions :
36 (ii) (a) $p(x)$
Geometrical Meaning of the Zeroes of a Polynomial
An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial $p(x) = -0.0025x^2 - 0.025x + 136$ . Observe the diagram and based on above information, answer the following questions :
36 (ii) $Find the span of the arch.$
Geometrical Meaning of the Zeroes of a Polynomial
An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial $p(x) = -0.0025x^2 - 0.025x + 136$ . Observe the diagram and based on above information, answer the following questions :
36 (iii) (a) $Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.$
Zeroes and Coefficients of a Quadratic Polynomial
During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$ , where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions :
36 (ii) (b) $Find the span of the arch on the stage floor.$
Geometrical Meaning of the Zeroes of a Polynomial
During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$ , where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions :
36 (iii) $Write the coordinates of the point of intersection of the above curve with the y-axis.$
Geometrical Meaning of the Zeroes of a Polynomial
An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial $p(x) = -0.0025x^2 - 0.025x + 136$ . Observe the diagram and based on above information, answer the following questions :
36 (iii) (b) $p(x)$
Geometrical Meaning of the Zeroes of a Polynomial

2025

27 q

1 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
4 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
4 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
4 $ax^2 + bx + \dfrac{2a}{b}$
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Zeroes and Coefficients of a Quadratic Polynomial
5 $q(x) = (p^2 + 4)x^2 + 65x + 4p$
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Zeroes and Coefficients of a Quadratic Polynomial
6 $p(y) = 7y^2 - \dfrac{11}{3}y - \dfrac{2}{3}$
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Zeroes and Coefficients of a Quadratic Polynomial
10 $Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is :$
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Geometrical Meaning of the Zeroes of a Polynomial
11 $p(x) = 5x - 7x^2 + 3$
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Zeroes and Coefficients of a Quadratic Polynomial
16 $-4$
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Zeroes and Coefficients of a Quadratic Polynomial
17 $Which of the following statements is true for a polynomial p(x) of degree 3?$
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Geometrical Meaning of the Zeroes of a Polynomial
17 $p(x) = x^2 - 3\sqrt{2}x + 4$
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Zeroes and Coefficients of a Quadratic Polynomial
21 (a) $p(x) = (p + 1)x^2 + (2p + 3)x + (3p + 4)$
Zeroes and Coefficients of a Quadratic Polynomial
21 $x^2 + ax + b$
Zeroes and Coefficients of a Quadratic Polynomial
21 $-\dfrac{7}{5}$
Forming a Quadratic Polynomial from its Zeroes
21 $p(y) = 21y^2 - y - 2$
Zeroes and Coefficients of a Quadratic Polynomial
21 (b) $\alpha$
Zeroes and Coefficients of a Quadratic Polynomial
22 $p(x) = x^2 + \dfrac{4}{3}x - \dfrac{4}{3}$
Zeroes and Coefficients of a Quadratic Polynomial
24 (b) $\alpha$
Zeroes and Coefficients of a Quadratic Polynomial
26 $\alpha$
Forming a Quadratic Polynomial from its Zeroes
26 $p(x) = 6x^2 - 5x - 1$
Forming a Quadratic Polynomial from its Zeroes
27 $7x^2 + 18x - 9$
Forming a Quadratic Polynomial from its Zeroes
27 $p(x) = 3x^2 - 4x - 4$
Forming a Quadratic Polynomial from its Zeroes
28 $\alpha$
Forming a Quadratic Polynomial from its Zeroes
29 $p(x) = 2x^2 - 5x - 3$
Forming a Quadratic Polynomial from its Zeroes
29 $q(x) = 8x^2 - 2x - 3$
Forming a Quadratic Polynomial from its Zeroes
30 $\alpha$
Forming a Quadratic Polynomial from its Zeroes
31 $r(x) = 4x^2 + 3x - 1$
Forming a Quadratic Polynomial from its Zeroes

2024

23 q

1 $p(x) = 2x^2 - k\sqrt{2}\,x + 1$
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Zeroes and Coefficients of a Quadratic Polynomial
3 $p(x) = x^2 - 5x + 6$
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Zeroes and Coefficients of a Quadratic Polynomial
5 $5x^2 - 6x + 21 = 0$
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Zeroes and Coefficients of a Quadratic Polynomial
6 $2x^2 - 3x - 9$
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Zeroes and Coefficients of a Quadratic Polynomial
6 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
9 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
10 $x^2 + px + q$
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Zeroes and Coefficients of a Quadratic Polynomial
12 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
13 $2\sqrt{3}$
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Forming a Quadratic Polynomial from its Zeroes
14 $x^2 - 5x + 4$
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Zeroes and Coefficients of a Quadratic Polynomial
15 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
16 $\alpha, \beta$
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Zeroes and Coefficients of a Quadratic Polynomial
17 $2 + \sqrt{5}$
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Forming a Quadratic Polynomial from its Zeroes
18 $x^2 - 16x + 30$
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Zeroes and Coefficients of a Quadratic Polynomial
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given below.
20 $Assertion (A) : Degree of a zero polynomial is not defined. Reason (R) : Degree of a non-zero constant polynomial is 0.$
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Geometrical Meaning of the Zeroes of a Polynomial
In Q. No. 19 and 20 a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.
20 $n(n > 1)$
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Geometrical Meaning of the Zeroes of a Polynomial
21 $\alpha$
Zeroes and Coefficients of a Quadratic Polynomial
23 $\alpha, \beta$
Zeroes and Coefficients of a Quadratic Polynomial
27 $x^2 - 15$
Zeroes and Coefficients of a Quadratic Polynomial
29 (a) $4x^2 + 4x - 3$
Zeroes and Coefficients of a Quadratic Polynomial
29 (b) $\alpha$
Zeroes and Coefficients of a Quadratic Polynomial
A ball is thrown in the air so that t seconds after it is thrown, its height h metre above its starting point is given by the polynomial $h = 25t - 5t^2$ . Observe the graph of the polynomial and answer the following questions :
36 (i) $Write zeroes of the given polynomial.$
Geometrical Meaning of the Zeroes of a Polynomial
A ball is thrown in the air so that t seconds after it is thrown, its height h metre above its starting point is given by the polynomial $h = 25t - 5t^2$ . Observe the graph of the polynomial and answer the following questions :
36 (ii) $Find the maximum height achieved by ball.$
Geometrical Meaning of the Zeroes of a Polynomial

2023

32 q

1 $Which of the following quadratic equations has sum of its roots as 4 ?$
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Zeroes and Coefficients of a Quadratic Polynomial
1 $x^2 - 3kx + 4k$
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Zeroes and Coefficients of a Quadratic Polynomial
1 $-3$
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Forming a Quadratic Polynomial from its Zeroes
1 $y = p(x)$
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Geometrical Meaning of the Zeroes of a Polynomial
3 $3x^2 + 11x - 4$
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Zeroes and Coefficients of a Quadratic Polynomial
3 $-5$
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Forming a Quadratic Polynomial from its Zeroes
5 $\alpha, \beta$
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Zeroes and Coefficients of a Quadratic Polynomial
5 $\sqrt{2}x^2 - 17$
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Zeroes and Coefficients of a Quadratic Polynomial
6 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
7 $\alpha, \beta$
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Zeroes and Coefficients of a Quadratic Polynomial
8 $\dfrac{5}{3}$
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Forming a Quadratic Polynomial from its Zeroes
8 $\alpha, \beta$
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Zeroes and Coefficients of a Quadratic Polynomial
8 $x^2 + 3x + k$
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Zeroes and Coefficients of a Quadratic Polynomial
8 $y = p(x)$
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Geometrical Meaning of the Zeroes of a Polynomial
9 $6x^2 + 37x - (k - 2)$
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Zeroes and Coefficients of a Quadratic Polynomial
12 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
14 $\dfrac{-2}{3}$
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Forming a Quadratic Polynomial from its Zeroes
14 $p(x) = x^2 + 4x + 3$
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Zeroes and Coefficients of a Quadratic Polynomial
16 $\alpha$
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Zeroes and Coefficients of a Quadratic Polynomial
16 $-1$
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Forming a Quadratic Polynomial from its Zeroes
17 $x^2 + (a + 1)\,x + b$
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Zeroes and Coefficients of a Quadratic Polynomial
17 $\alpha, \beta$
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Zeroes and Coefficients of a Quadratic Polynomial
Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below.
20 $p(x) = x^2 + 3x + 3$
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Geometrical Meaning of the Zeroes of a Polynomial
22 $p(x) = 6x^2 + 37x - (k - 2)$
Zeroes and Coefficients of a Quadratic Polynomial
23 (A) $2x^2 - 9x + 4 = 0$
Zeroes and Coefficients of a Quadratic Polynomial
26 $px^2 - 14x + 8 = 0$
Zeroes and Coefficients of a Quadratic Polynomial
27 (B) $x^2 + 12x - k = 0$
Zeroes and Coefficients of a Quadratic Polynomial
29 (B) $\alpha$
Forming a Quadratic Polynomial from its Zeroes
In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by $h = 20t - 16t^2$ . Based on the above, answer the following questions :
38 (i) $p(t) = 20t - 16t^2$
Zeroes and Coefficients of a Quadratic Polynomial
In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by $h = 20t - 16t^2$ . Based on the above, answer the following questions :
38 (ii) $p(t)$
Geometrical Meaning of the Zeroes of a Polynomial
In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by $h = 20t - 16t^2$ . Based on the above, answer the following questions :
38 (iii) (a) $t = \dfrac{3}{2}$
Geometrical Meaning of the Zeroes of a Polynomial
In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by $h = 20t - 16t^2$ . Based on the above, answer the following questions :
38 (iii) (b) $How much distance has the dolphin covered before hitting the water level again ?$
Zeroes and Coefficients of a Quadratic Polynomial

2022

1 q

3 (a) $ky^2 - 11y + (k - 23) = 0$
Zeroes and Coefficients of a Quadratic Polynomial

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 4 times: 2023*, 2024*, 2025*, 2026 (* = numbers or wording changed that year)
5 $\alpha$
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Question 2
Asked 3 times: 2023*, 2024*, 2026 (* = numbers or wording changed that year)
21 $\alpha, \beta$
Question 3
Asked 2 times: 2023, 2026
24 $px^2 - 14x + 8$
Question 4
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
25 $x^2 + 7x + 10$
Question 5
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
11 $(-3)$
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Question 6
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
25 $\alpha$
Question 7
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the options given below.
20 $p(y) = y^2 + 4y + 3$
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Question 8
Asked 2 times: 2023*, 2025 (* = numbers or wording changed that year)
5 $q(x) = (p^2 + 4)x^2 + 65x + 4p$
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Question 9
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
4 $\alpha$
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Question 10
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
6 $\alpha$
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2.2 Geometrical Meaning of the Zeroes of a Polynomial

Solved Examples

Exercise 2.1

2.3 Relationship between Zeroes and Coefficients of a Polynomial

Solved Examples

Exercise 2.2

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10