Polynomials — Class 10 Maths PYQs

If one zero of the polynomial $x^2 + 3x + k$ is $2$, then the value of $k$.

What should be subtracted from the polynomial $x^2 - 16x + 30$, so that $15$ is the zero of the resulting polynomial ?

What should be added from the polynomial $x^2 - 5x + 4$, so that $3$ is the zero of the resulting polynomial ?

If a polynomial $p(x)$ is given by $p(x) = x^2 - 5x + 6$, then the value of $p(1) + p(4)$ is :

Assertion (A): Degree of a zero polynomial is not defined.
Reason (R): Degree of a non-zero constant polynomial is $0$.

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.
(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.
Assertion (A): Degree of a zero polynomial is not defined.
Reason (R): Degree of a non-zero constant polynomial is $0$.

If $-4$ is a zero of the polynomial $p(x) = x^2 - x - (2 + 2k)$, then the value of $k$ is:

Which of the following statements is true for a polynomial $p(x)$ of degree $3$?

Which of the following statements is true for a polynomial $p(x)$ of degree $3$?

If the sum of the zeroes of the polynomial $p(x) = (p + 1) x^2 + (2p + 3) x + (3p + 4)$ is $-1$, then find the value of 'p'.

The graph of $y = p(x)$ is given, for a polynomial $p(x)$. The number of zeroes of $p(x)$ from the graph is

The graph of $y = p(x)$ is given in the adjoining figure. Zeroes of the polynomial $p(x)$ are

Directions : In Question $19$ and $20$, Assertion (A) and Reason (R) are given. Select the correct option from the following :
(A) Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true. Reason (R) does not give correct explanation of (A).
(C) Assertion (A) is true but Reason (R) is not true.
(D) Assertion (A) is not true but Reason (R) is true.
Assertion (A): If the graph of a polynomial intersects the x-axis at exactly two points, then the number of zeroes of that polynomial is $2$.
Reason (R): The number of zeroes of a polynomial is equal to the number of points where the graph of the polynomial intersects x-axis.

Assertion (A): If the graph of a polynomial touches $x$-axis at only one point, then the polynomial cannot be a quadratic polynomial.
Reason (R) : A polynomial of degree $n(n >1)$ can have at most $n$ zeroes.

The graph of a polynomial intersects the y-axis at one point and the x-axis at two points. The number of zeroes of this polynomial are :

If the given figure shows the graph of polynomial $y = ax^2 + bx + c$, then :

If the given figure shows the graph of polynomial $y = ax^2 + bx + c$, then :

Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is:

The zeroes of the polynomial $3x^2 + 11x-4$ are:

The zeroes of the polynomial $3x^2 + 11x-4$ are:

The zeroes of the polynomial $p(x) = x^2 + 4x + 3$ are given by:

The zeroes of the polynomial $3x^2 + 11x - 4$ are :

The zeroes of the quadratic polynomial $2x^2 - 3x - 9$ are:

The zeroes of the quadratic polynomial $2x^2 - 3x - 9$ are:

The zeroes of the quadratic polynomial $x^2+ 99x + 127$ are :

Zeroes of the polynomial $p(x) = x^2 -3\sqrt{2}x+4$ are:

Zeroes of the polynomial $p(y) = 7y^2 - \frac{11}{3}y - \frac{2}{3}$ are:

Zeroes of the polynomial $p(x) = x^2 - 3\sqrt{2}x+4$ are:

The value of $x$, for which the polynomials $9 - x^2$ and $6x + x^2 + 9$ vanish simultaneously, is

Find the zeroes of the polynomial $p(x) = x^2 + \frac{4}{3}x - \frac{4}{3}$.

If $p$ and $q$ are zeroes of the polynomial $p(y) = 21y^2 - y - 2$, then find the value of $(1-p) . (1 - q)$.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = 2x^2- 7x + 3$, then the value of $\alpha^2 + \beta^2$ is :

If the sum of the zeroes of the quadratic polynomial $p(x) = kx^2+ 2x + 3k$ is equal to the product of its zeroes, then the value of $k$ is :

If $\alpha, \beta$ are the zeroes of a polynomial $p(x) = x^2+x-1$, then $\frac{1}{\alpha} + \frac{1}{\beta}$ equals to

If $\alpha, \beta$ are zeros of a polynomial $P(x) = 2x^2 -x-1$ then $\alpha^2 + \beta^2$ is equal to

If one zero of the polynomial $6x^2 + 37x - (k-2)$ is reciprocal of the other, then what is the value of k?

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - ax - b$, then the value of $\alpha^2 + \beta^2$ is :

If one zero of the polynomial $x^2-3kx + 4k$ be twice the other, then the value of $k$ is:

If '$\alpha$' and '$\beta$' are the zeroes of the polynomial $ax^2 - 5x + c$ and $\alpha + \beta = \alpha\beta = 10$, then :

The sum of zeroes of the polynomial $\sqrt{2}x^2 - 17$ are given as :

If $\alpha$, $\beta$ are zeroes of the polynomial $x^2-1$, then value of $(\alpha + \beta)$ is :

If $\alpha$, $\beta$ are the zeroes of the polynomial $p(x) = 4x^2 – 3x – 7$, then $(\frac{1}{\alpha} + \frac{1}{\beta})$ is equal to :

If $\alpha$, $\beta$ are the zeroes of the polynomial $p(x) = 4x^2 - 3x - 7$, then $\left(\frac{1}{\alpha} + \frac{1}{\beta}\right)$ is equal to :

If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - 1$, then the value of $(\alpha + \beta)$ is

If $\alpha$ and $\beta$ are the zeroes of the polynomial $2x^2 + 5x + 1$, then the value of $\alpha + \beta + 3\alpha\beta$ is :

If the sum of zeroes of the polynomial $p(x) = 2x^2 - k\sqrt{2}x+1$ is $\sqrt{2}$, then value of $k$ is :

The zeroes of a polynomial $x^2 + px + q$ are twice the zeroes of the polynomial $4x^2 - 5x - 6$. The value of $p$ is :

If $\alpha, \beta$ are the zeroes of the polynomial $6x^2 - 5x - 4$, then $\frac{1}{\alpha} + \frac{1}{\beta}$ is equal to :

If $\alpha$ and $\beta$ are zeroes of the polynomial $5x^2 + 3x - 7$, the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is

If $\alpha$ and $\beta$ are zeroes of the polynomial $2x^2 - 9x + 5$, then value of $\alpha^2 + \beta^2$ is

If $\alpha$ and $\beta$ ($\alpha > \beta$) are the zeroes of the polynomial $-x^2 + 8x + 9$, then $(\alpha - \beta)$ is equal to

The ratio of the sum and product of the roots of the quadratic equation $5x^2-6x+21 = 0$ is :

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = kx^2 - 30x + 45k$ and $\alpha + \beta = \alpha\beta$, then the value of $k$ is:

If one zero of the quadratic polynomial $4x^2+ 4x - m$ is $\frac{3}{2}$, then the other zero is :

If $\alpha$ and $\beta$ are the zeroes of polynomial $3x^2 + 6x + k$ such that $\alpha + \beta + \alpha\beta = -\frac{2}{3}$, then the value of $k$ is:

The sum of the zeroes of the polynomial $p(x) = 5x-7x^2 + 3$ is:

If one zero of the polynomial $q(x) = (p^2 + 4)x^2 + 65x + 4p$ is reciprocal of the other, then the value of 'p' is :

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = x^2-ax-b$, then the value of $(\alpha + \beta + \alpha\beta)$ is equal to :

If $\alpha$ and $\beta$ are zeroes of the polynomial $p(x) = kx^2 - 30x + 45k$ and $\alpha + \beta = \alpha\beta$, then the value of 'k' is :

If the zeroes of the polynomial $ax^2+bx+\frac{2a}{b}$ are reciprocal of each other, then the value of $b$ is

If the square of the difference of the zeroes of the quadratic polynomial $y^2 + py + 36$ is equal to $81$, then the values of $p$ are

If $p$ and $q$ are the zeroes of the quadratic polynomial $f(x) = 6x^2 + x - 2$, then find the value of $\frac{1}{p} + \frac{1}{q} - pq$.

If one zero of the polynomial $p(x) = 6x^2 + 37x – (k – 2)$ is reciprocal of the other, then find the value of $k$.

If $\alpha, \beta$ are zeroes of the polynomial $p(x) = 5x^2 - 6x + 1$, then find the value of $\alpha + \beta + \alpha\beta$.

If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $p(x) = x^2 - 5x + 4$, then find the value of $\frac{1}{\alpha} + \frac{1}{\beta} - 2\alpha\beta$.

If the zeroes of the polynomial $x^2 + ax + b$ are in the ratio $3: 4$, then prove that $12a^2 = 49b$.

If '$\alpha$' and '$\beta$' are the zeroes of the polynomial $p(y) = y^2 - 5y + 3$, then find the value of $\alpha^4\beta^3 + \alpha^3\beta^4$.

If $\alpha$ and $\beta$ are zeroes of the polynomial $p(x) = x^2 - 2x - 1$, then find the value of $\frac{1}{2\alpha} + \frac{1}{2\beta} + 3\alpha\beta$.

If $\alpha$ and $\beta$ are roots of the quadratic equation $x^2 - 7x + 10 = 0$, find the quadratic equation whose roots are $\alpha^2$ and $\beta^2$.

OR
If one root of the quadratic equation $x^2 + 12x - k = 0$ is thrice the other root, then find the value of $k$.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = x^2 – (k + 5)x + (5k + 1)$ such that, $\alpha + \beta = \frac{\alpha\beta}{3}$, then find the value of k.

If $\alpha, \beta$ are the zeroes of the polynomial $3x^2 - 13x - 10$, then find the value of $(3\alpha + 1) (3\beta + 1)$.

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = 6x^2 + 11x - 10$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

Find the zeroes of the polynomial $2t^2 - 9t - 45$ and verify the relationship between the zeroes and the coefficients of the polynomial.

Find the zeroes of the quadratic polynomial $x^2 - 15$ and verify the relationship between the zeroes and the coefficients of the polynomial.

Find the zeroes of the polynomial $4x^2 + 4x - 3$ and verify the relationship between zeroes and coefficients of the polynomial.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 + x - 2$, then find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = 6x^2 + 11x - 10$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

Find the zeroes of the polynomial $f(t) = t^2 + 4\sqrt{3}t - 15$ and verify the relationship between the zeroes and the coefficients of the polynomial.

$\alpha, \beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - 4x + k$, such that $\alpha - \beta = 8$. Find the value of $k$.

Zeroes of the quadratic polynomial $p(x) = (a^2 + 10)x^2 - 74x + 7a$ are reciprocal of each other and they are rational. Find the value of 'a'.

If $\alpha, \beta$ are the zeroes of the polynomial $p(x) = x^2 - 2x - 3$, then find a polynomial where zeroes are $(2\alpha + 3\beta)$ and $(3\alpha + 2\beta)$.

$\alpha$ and $\beta$ are zeroes of a quadratic polynomial $px^2+qx+1$. Form a quadratic polynomial whose zeroes are $\frac{2}{\alpha}$ and $\frac{2}{\beta}$.

$\alpha$ and $\beta$ are zeroes of a quadratic polynomial $x^2 - ax - b$. Obtain a quadratic polynomial whose zeroes are $3\alpha + 1$ and $3\beta + 1$.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $ax^2 - x + c$. Obtain a polynomial whose zeroes are $\alpha - 3$ and $\beta - 3$.

Obtain the zeroes of the polynomial $7x^2 + 18x - 9$. Hence, write a polynomial each of whose zeroes is twice the zeroes of given polynomial.

Obtain the zeroes of the polynomial $p(x) = 2x^2 - 5x - 3$. Hence, obtain a polynomial each of whose zeroes is one less than each of the zero of $p(x)$.

Find the zeroes of the polynomial $p(x) = 6x^2 - 5x - 1$. Hence, obtain a polynomial each of whose zeroes is three times the zeroes of $p(x)$.

Find the zeroes of the polynomial $p(x) = 3x^2 - 4x - 4$. Hence, write a polynomial whose each of the zeroes is 2 more than zeroes of $p(x)$.

Find the zeroes of the polynomial $q(x) = 8x^2 - 2x - 3$. Hence, find a polynomial whose zeroes are $2$ less than the zeroes of $q(x)$.

Find the zeroes of the polynomial $r(x) = 4x^2 + 3x - 1$. Hence, write a polynomial whose zeroes are reciprocal of the zeroes of polynomial $r(x)$.

Which of the following is a quadratic polynomial having zeroes $-\frac{2}{3}$ and $\frac{2}{3}$ ?

Which of the following is a quadratic polynomial with zeroes $\frac{5}{3}$ and 0?

The number of polynomials having zeroes $-3$ and $5$ is :

The number of polynomials having zeroes $-1$ and $2$ is :

The number of quadratic polynomials having zeroes $-5$ and $-3$ is

A quadratic polynomial, one of whose zeroes is $2 + \sqrt{5}$ and the sum of whose zeroes is $4$, is:

A quadratic polynomial whose one zero is $3$ and the product of zeroes is $0$, is :

Find a quadratic polynomial whose zeroes are $2$ and $-\frac{7}{5}$

Find a quadratic polynomial whose sum of the zeroes is $8$ and difference of the zeroes is $2$.

While playing in a garden, Samaira saw a honeycomb and asked her mother what is that. Her mother replied that it's a honeycomb made by honey bees to store honey. Also, she told her that the shape of the honeycomb formed is a mathematical structure. The mathematical representation of the honeycomb is shown in the graph.
Based on the above information, answer the following questions :
(i) How many zeroes are there for the polynomial represented by the graph given ?
(ii) Write the zeroes of the polynomial.
(iii) (a) If the zeroes of a polynomial $x^2 + (a + 1) x + b$ are $2$ and $-3$, then determine the values of $a$ and $b$.
OR
(iii) (b) If the square of difference of the zeroes of the polynomial $x^2 + px + 45$ is $144$, then find the value of $p$.

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 :
(i) Find zeroes of polynomial $p(t) = 20t - 16t^2$.
(ii) Which of the following types of graph represents $p(t)$?
(iii) (a) What would be the value of $h$ at $t = \frac{3}{2}$ ? Interpret the result.
OR
(iii) (b) How much distance has the dolphin covered before hitting the water level again?

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 :
(i) Write zeroes of the given polynomial.
(ii) Find the maximum height achieved by ball.
(iii) (a) After throwing upward, how much time did the ball take to reach to the height of $30$ m?
OR
(iii) (b) Find the two different values of $t$ when the height of the ball was $20$ m.

Assertion (A): The polynomial $p(x) = x^2 + 3x + 3$ has two real zeroes.
Reason (R): A quadratic polynomial can have at most two real zeroes.