Mastersheet: Polynomials | Class 10 CBSE | Vardaan Learning Institute

✅ Rationalised Syllabus Note: This mastersheet strictly follows the CBSE Rationalised Syllabus (2024–25 onwards). The Division Algorithm for Polynomials and problems on cubic/higher-degree polynomial zeroes have been deleted from the syllabus and are NOT included here. Focus areas: Geometrical Meaning of Zeroes • Finding Zeroes of a Quadratic Polynomial • Relationship between Zeroes & Coefficients • Forming Quadratic Polynomial from Given Zeroes.

Topic 1: MCQ / 1-Mark Questions

The number of zeroes of a polynomial whose graph is a straight line intersecting the $x$-axis at exactly one point is: [1M]

If the graph of $y = p(x)$ is a parabola that touches the $x$-axis at one point, then the number of zeroes of $p(x)$ is: [1M]

The zeroes of the polynomial $p(x) = x^2 - 5x + 6$ are: [1M]

$2$ and $3$
$-2$ and $-3$
$2$ and $-3$
$-2$ and $3$

If $\alpha$ and $\beta$ are zeroes of $p(x) = x^2 - 4x + 3$, then $\alpha + \beta$ equals: [1M]

$-4$
$4$
$3$
$-3$

If $\alpha$ and $\beta$ are zeroes of $p(x) = 2x^2 - 7x + 3$, then $\alpha \cdot \beta$ equals: [1M]

$\dfrac{7}{2}$
$\dfrac{3}{2}$
$-\dfrac{3}{2}$
$-\dfrac{7}{2}$

The quadratic polynomial whose sum of zeroes is $3$ and product of zeroes is $2$ is: [1M]

$x^2 + 3x + 2$
$x^2 - 3x - 2$
$x^2 - 3x + 2$
$x^2 + 3x - 2$

If one zero of the polynomial $p(x) = 5x^2 + 13x + k$ is the reciprocal of the other, the value of $k$ is: [1M]

$0$
$5$
$\dfrac{1}{5}$
$6$

The graph of $y = p(x)$ does not intersect or touch the $x$-axis at any point. Then $p(x)$ has: [1M]

One zero
Two zeroes
No real zeroes
Infinitely many zeroes

The zeroes of the polynomial $p(x) = x^2 - 1$ are: [1M]

$1$ and $1$
$-1$ and $-1$
$1$ and $-1$
$0$ and $1$

10.

If the sum of zeroes of $p(x) = kx^2 + 2x + 3k$ is equal to the product of its zeroes, then $k$ equals: [1M]

$\dfrac{1}{3}$
$-\dfrac{1}{3}$
$\dfrac{2}{3}$
$-\dfrac{2}{3}$

11.

The degree of the polynomial $p(x) = \sqrt{3}\,x^2 - 8$ is: [1M]

$1$
$2$
$3$
$\dfrac{1}{2}$

12.

If $x = 2$ is a zero of $p(x) = x^2 + kx - 6$, then $k$ equals: [1M]

$1$
$-1$
$2$
$-2$

Topic 2: Geometrical Meaning of Zeroes (Graph-Based)

13.

The graph of $y = p(x)$ is given. In each of the following cases, find the number of zeroes of $p(x)$: [2M]

The graph touches the $x$-axis at exactly one point and does not cross it.
The graph cuts the $x$-axis at two distinct points.
The graph is entirely below the $x$-axis and never touches it.
The graph cuts the $x$-axis at three points.

14.

Look at the graph of $y = p(x)$. The curve crosses the $x$-axis at $x = -2$ and $x = 3$. [2M]

How many zeroes does $p(x)$ have?
Write the zeroes of $p(x)$.
Write a possible quadratic polynomial for $p(x)$.

15.

Explain the geometrical meaning of zeroes of a polynomial. How many zeroes can a quadratic polynomial have at most? Justify using a graph. [2M]

16.

Can a quadratic polynomial have three zeroes? Justify your answer with the help of a graph. [2M]

17.

The graph of the quadratic polynomial $y = ax^2 + bx + c$ opens upward and intersects the $x$-axis at two points. What can you say about the value of $a$ and the nature of zeroes? [2M]

Topic 3: Finding Zeroes of a Quadratic Polynomial

18.

Find the zeroes of $p(x) = x^2 - 4$. [1M]

19.

Find the zeroes of $p(x) = x^2 - 2x - 8$. [2M]

20.

Find the zeroes of $p(x) = 6x^2 - 3 - 7x$ and verify the relationship between the zeroes and the coefficients. [3M]

21.

Find the zeroes of $p(x) = 4s^2 - 4s + 1$ and verify the relationship between the zeroes and coefficients. [3M]

22.

Find the zeroes of $p(x) = \sqrt{3}\,x^2 + 10x + 7\sqrt{3}$ and verify the relationship between the zeroes and the coefficients. [3M]

23.

Find the zeroes of $p(x) = x^2 - (\sqrt{3}+1)x + \sqrt{3}$ and verify the relationship between the zeroes and the coefficients. [3M]

24.

Find the zeroes of $p(x) = 4x^2 + 5\sqrt{2}\,x - 3$. [2M]

25.

Find the zeroes of $p(x) = x^2 + \dfrac{1}{6}x - 2$ and verify the relationship between the zeroes and coefficients. [3M]

26.

Find the zeroes of $p(x) = 2x^2 + 5x - 12$ and verify the relationship between the zeroes and coefficients. [3M]

27.

Find the zeroes of $p(x) = abx^2 + (b^2 - ac)x - bc$. [3M]

Topic 4: Relationship between Zeroes & Coefficients (Without finding zeroes)

28.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - 5x + 6$, find the value of: (i) $\alpha + \beta$, (ii) $\alpha\beta$, (iii) $\alpha^2 + \beta^2$. [3M]

29.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = 2x^2 + 5x - 3$, find the value of $\dfrac{1}{\alpha} + \dfrac{1}{\beta}$. [2M]

30.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - 6x + 8$, find the value of $\alpha^2\beta + \alpha\beta^2$. [2M]

31.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - px + q$, find the value of $\alpha^2 + \beta^2$ in terms of $p$ and $q$. [2M]

32.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - 5x + 6$, find the value of $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}$. [3M]

33.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - 5x + 6$, find the value of $\dfrac{\alpha^2}{\beta} + \dfrac{\beta^2}{\alpha}$. [3M]

34.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - 1$, find the value of $\alpha^3 + \beta^3$. [2M]

35.

If $\alpha$ and $\beta$ are the zeroes of $f(x) = x^2 - 3x - m(m+3)$, show that $\alpha^2 + \beta^2 = (m+3)^2 + m^2$. [3M]

36.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 + px + q$, find the value of $\left(\dfrac{\alpha}{\beta} - \dfrac{\beta}{\alpha}\right)^2$. [3M]

37.

If the product of zeroes of $p(x) = ax^2 - 6x - 6$ is 4, find the value of $a$. [1M]

38.

If the sum of zeroes of $p(x) = 3x^2 - kx + 6$ is 3, find the value of $k$. [1M]

39.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = 5x^2 - 7x + 2$, find the value of $\dfrac{\alpha}{\beta^2} + \dfrac{\beta}{\alpha^2}$. [3M]

40.

If one zero of $p(x) = x^2 - 5x + k$ is 3, find the other zero and the value of $k$. [2M]

Topic 5: Finding Unknown Constants Using Zeroes

41.

If one zero of the polynomial $p(x) = 5x^2 + 13x + k$ is the reciprocal of the other, find the value of $k$. [2M]

42.

If one zero of $p(x) = 4x^2 - 8kx - 9$ is the negative of the other, find the value of $k$. [2M]

43.

If $\alpha$ and $\beta$ are zeroes of $p(x) = x^2 - 6x + k$ and $3\alpha + 2\beta = 20$, find the value of $k$. [3M]

44.

If $\alpha$ and $\beta$ are zeroes of $p(x) = x^2 + px + 45$ and $(\alpha - \beta)^2 = 144$, find the value of $p$. [3M]

45.

If $\alpha$ and $\beta$ are zeroes of $p(x) = kx^2 + 4x + 4$ such that $\alpha^2 + \beta^2 = 24$, find the value(s) of $k$. [3M]

46.

If one zero of $p(x) = 3x^2 - kx - 4$ is the double of the other, find the value of $k$. [3M]

47.

Find the sum and product of zeroes of $p(x) = x^2 - 3$, without actually finding the zeroes. [1M]

Topic 6: Forming a Quadratic Polynomial from Given Zeroes

48.

Find a quadratic polynomial whose sum of zeroes is $5$ and product of zeroes is $6$. [2M]

49.

Find a quadratic polynomial whose sum of zeroes is $-\dfrac{3}{2}$ and product of zeroes is $-1$. [2M]

50.

Find a quadratic polynomial whose zeroes are $\sqrt{3}$ and $-\sqrt{3}$. [2M]

51.

Find a quadratic polynomial whose zeroes are $3 + \sqrt{2}$ and $3 - \sqrt{2}$. [2M]

52.

Find a quadratic polynomial whose zeroes are $\dfrac{1}{4}$ and $-1$. [2M]

53.

If $\alpha$ and $\beta$ are the zeroes of $p(x) = x^2 - 2x + 3$, find a quadratic polynomial whose zeroes are $\alpha + 2$ and $\beta + 2$. [3M]

54.

If $\alpha$ and $\beta$ are zeroes of $p(x) = x^2 - 5x + 4$, find a quadratic polynomial whose zeroes are $\dfrac{\alpha}{\beta}$ and $\dfrac{\beta}{\alpha}$. [3M]

55.

If $\alpha$ and $\beta$ are zeroes of $p(x) = 3x^2 + 4x - 4$, find a quadratic polynomial whose zeroes are $\dfrac{1}{\alpha}$ and $\dfrac{1}{\beta}$. [3M]

Topic 7: Assertion-Reason (A-R) Questions

56.

Assertion (A): The zeroes of the polynomial $p(x) = x^2 - 3x + 2$ are $1$ and $2$.
Reason (R): The zeroes of a polynomial $ax^2 + bx + c$ are the values of $x$ where the graph of $y = p(x)$ intersects the $x$-axis. [1M]

Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is NOT the correct explanation of A.
A is true but R is false.
A is false but R is true.

57.

Assertion (A): A quadratic polynomial can have at most 2 zeroes.
Reason (R): The number of zeroes of a polynomial is equal to the degree of the polynomial. [1M]

Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is NOT the correct explanation of A.
A is true but R is false.
A is false but R is true.

58.

Assertion (A): If the sum and product of zeroes of a quadratic polynomial are $-3$ and $2$ respectively, then the polynomial is $x^2 + 3x + 2$.
Reason (R): A quadratic polynomial with zeroes $\alpha$ and $\beta$ is given by $k\bigl[x^2 - (\alpha+\beta)x + \alpha\beta\bigr]$, $k \neq 0$. [1M]

Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is NOT the correct explanation of A.
A is true but R is false.
A is false but R is true.

59.

Assertion (A): $p(x) = x^2 + 4$ has no real zeroes.
Reason (R): A polynomial has real zeroes only when its graph intersects the $x$-axis. [1M]

Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is NOT the correct explanation of A.
A is true but R is false.
A is false but R is true.

Topic 8: Case Study Questions (4-Mark / Competency-Based)

60.

Case Study: The Parabolic Path
During a school science project, students observed the path of a ball thrown in the air. The height $h$ (in metres) of the ball at horizontal distance $x$ (in metres) is modelled by the polynomial $p(x) = -x^2 + 5x - 6$. [4M]

Find the zeroes of $p(x)$ and interpret them in the context of the ball's journey.
Verify the relationship between the zeroes and the coefficients of $p(x)$.
At what horizontal distance does the ball reach maximum height?
Find the maximum height of the ball.

61.

Case Study: The Garden Design
A rectangular garden is to be designed such that its area (in sq. m) is represented by the polynomial $A(x) = x^2 - 7x + 10$, where $x$ is the length of one side in metres. [4M]

Find the zeroes of $A(x)$.
The zeroes represent the possible dimensions of the garden. Which zero is more meaningful in the context of the problem? Why?
Using the relationship between zeroes and coefficients, find the sum and product of dimensions.
Form a new quadratic polynomial whose zeroes are the squares of the dimensions of this garden.

62.

Case Study: The Architecture Arch
An architect designs a parabolic arch whose shape is described by the polynomial $p(x) = -2x^2 + 8x$, where $x$ is the horizontal distance (in metres) from one end of the arch. [4M]

Find the zeroes of $p(x)$. What do these zeroes represent?
Find the sum and product of zeroes using the relationship formula.
At what point does the arch reach its maximum height? Find the maximum height.
If the arch is to be widened by moving each zero outward by 1 metre, write the new quadratic polynomial.