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 $0$.

One of the zeroes of the polynomial $p(x) = kx^2 - 9x + 3$ is $(-\frac{3}{2})$. The value of $k$ is :

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'.

While playing badminton Ravi has set the barrier chain hung between two posts at the edge of the walkway of a street. It is hung in the shape of a parabola.
Based on the above information answer the following questions :
(a) Which type of the polynomial (linear, quadratic, cubic etc.) is graphically represented by a parabola ?
(b) If the polynomial represented by a parabola, intersects the $x$-axis at $-2$ and $3$ and $y$-axis at $-3$, then write the zeroes of the parabola.
(c) Find the expression for the above polynomial.
OR
(c) If the zeroes of the polynomial are $-5$ and $3$, find its expression.

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

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 :

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

The graph of $y = f(x)$ is given.
The number of zeroes of $f(x)$ is:

The graph of $y = f(x)$ is given.
The number of zeroes of $f(x)$ is :

The graph of $y = f(x)$ is given. The number of distinct zeroes of $y = f(x)$ is :

Observe the graph of polynomial $p(x)$. Number of zeroes of $p(x)$ is

Observe the graph of polynomial $p(x)$. The zeroes of the polynomial are

The graph of the polynomial $ax^2 + bx + c$ is a downward parabola if

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 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:

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

How many zeroes does $p(x) = (x-2)(x+3)$ have?

At which of the following points, the quadratic polynomial $p(x) = 3x + 18x^2 - 1$ intersects the positive $x$-axis ?

The graph of a quadratic polynomial $f(x)$ passes through $(5,0)$, $(0, -1)$ and $(-2, 0)$. The two factors of the polynomial are

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 $(1-p) . (1 - q)$.

Find the zeroes of the polynomial $p(x) = 15x^2 – 19x + 6$.

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 $\alpha$ 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 $\alpha$ 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 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 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 $\alpha$ and $\beta$ are two zeroes of a polynomial $f(x) = px^2 - 2x + 3p$ and $\alpha + \beta = \alpha\beta$, then value of $p$ is:

A polynomial $p(x)$, which has sum of its zeroes equal to their product, is :

$\alpha, \beta$ are zeroes of the polynomial $2x^2 + 5x + 1$. The value of $(\frac{1}{\alpha} + \frac{1}{\beta})$ is :

If $\alpha, \beta$ are zeroes of the polynomial $3x^2 + 14x - 5$, then the value of $3\left(\frac{\alpha + \beta}{\alpha\beta}\right)$ is :

If $\alpha$ and $\beta$ are zeroes of the polynomial $x^2 - x - 4$, then the value of $(\frac{1}{\alpha} + \frac{1}{\beta})$ is

The value of $k$ for which sum of the zeroes of the polynomial $p(x) = 3x^2 - kx + 6$ is $2$, is

If the zeroes of the polynomial $p(x) = 2x^2- 7x + 6$ are $\alpha$ and $\beta$, then the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is

If $\alpha$ and $\beta$ are two zeroes of the quadratic polynomial $p(x) = x^2 - 11x + 30$, then $\frac{1}{\alpha} + \frac{1}{\beta}$ is equal to:

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 $\frac{1}{2\alpha} + \frac{1}{2\beta} + 3\alpha\beta$.

$\alpha$ and $\beta$ are the zeroes of the polynomial $5x^2 – 16x - 10$. Find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

Verify the relationship between the zeroes and the coefficients of the quadratic polynomial $25x^2 - 16$.

Verify the relationship between the zeroes and the coefficients of the quadratic polynomial $9x^2 - 25$.

Verify the relationship between the zeroes and the coefficients of the polynomial $p(x) = 3x^2 – 5x$.

Verify the relationship between the zeroes and the coefficients of the polynomial $2x^2 – 3x$.

Verify the relationship between the zeroes and the coefficients of the polynomial $5x^2 + 2x$.

If $\alpha$, $\beta$ are the zeroes of polynomial $p(x) = 6x^2-5x - 3$, then find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

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

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 $p$.