Class 10th Maths Polynomials Hots Questions

Q 1 :
The zeroes of the polynomial $x^{2} - 3 x - m (m + 3)$ are

$m, m + 3$
$- m, m + 3$
$m, - (m + 3)$
$- m, - (m + 3)$

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(2)

Let $p (x) = x^{2} - 3 x - m (m + 3)$

$\Rightarrow p (x) = x^{2} - (m + 3) x + m x - m (m + 3)$

$= x {x - (m + 3)} + m {x - (m + 3)}$

For zeros of $p (x)$

$\Rightarrow p (x) = (x + m) {x - (m + 3)} = 0$ $\Rightarrow x = - m, m + 3$

$∴$ Its zeros are $- m, m + 3$

Q 2 :
If α and β are the zeros of a polynomial $f (x) = p x^{2} - 2 x + 3 p$ and α + β = αβ, then p is

$- 2 / 3$
$2 / 3$
$1 / 3$
$- 1 / 3$

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(2)

Given, $f (x) = p x^{2} - 2 x + 3 p$

Since $α$ and $β$ are the zeroes of the given polynomial.

$∴$ $α + β = - \frac{(- 2)}{p} = \frac{2}{p}, and α β = \frac{3 p}{p} = 3$

$∵$ $α + β = α β$ (given)

$∴$ $\frac{2}{p} = 3 \Rightarrow p = \frac{2}{3}$

Q 3 :
The zeroes of a polynomial $x^{2} + p x + q$ are twice the zeroes of the polynomial $4 x^{2} - 5 x - 6$ . The value of p is :

$- 5 / 2$
$5 / 2$
$- 5$
$10$

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(1)

Given polynomials : $x^{2} + p x + q$ ...(i)

and $4 x^{2} - 5 x - 6$ ...(ii)

Zero of polynomial $4 x^{2} - 5 x - 6$ are: $x = 2$ and $x = - 3 / 4$

Now, zero of polynomial $x^{2} + p x + q$ are 4 and $- 3 / 2$

$∴$ Sum of zeroes $= - \frac{p}{1}$

$\Rightarrow 4 - \frac{3}{2} = - \frac{p}{1} \Rightarrow - p = \frac{5}{2} \Rightarrow p = - \frac{5}{2}$

Q 4 :
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.

Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.

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(4)

The polynomials of the form ${(x + a)}^{2}$ and ${(x - a)}^{2}$ has only equal roots and graphs of these polynomials cut x-axis at only one point. These polynomials are quadratic Thus, Assertion is false Reason is true.

Q 5 :
If the sum of zeroes of the polynomial $p (x) = 2 x^{2} - k \sqrt{2} x + 1$ is $\sqrt{2}$ , then value of $k$ is :

$\sqrt{2}$
$2$
$2 \sqrt{2}$
$1 / 2$

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(2)

Sum of zeroes $= \sqrt{2} = \frac{- (- k \sqrt{2})}{2} \Rightarrow k = 2$

Q 6 :
If $α$ and $β$ are zeroes of the polynomial $5 x^{2} + 3 x - 7$ , the value of $\frac{1}{α} + \frac{1}{β}$ is

−3/7
3/5
3/7
−5/7

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(3)

$α + β = \frac{- b}{a} = \frac{- 3}{5}, α β = \frac{c}{a} = \frac{- 7}{5}$

Now, $\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β} = \frac{\frac{- 3}{5}}{\frac{- 7}{5}} = \frac{3}{7}$

Q 7 :
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.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.

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(1) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

Q 8 :
Consider the polynomial $p (x) = x ³ - 3 x ² + 2 x .$ Which of the following statements are true about its graph and zeros?

(i) The polynomial has a zero at x = 0.
(ii) The polynomial has a zero at x = 1.
(iii) The graph touches the x-axis at x = 1.
(iv) The graph intersects the x-axis at x = 2.

Choose the correct option from the following:

both (i) and (ii)
both (ii) and (iii)
(i), (ii) and (iv)
both (i) and (iv)

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(3)

$p (x) = x ³ - 3 x ² + 2 x \Rightarrow p (x) = x (x ² - 3 x + 2) \Rightarrow p (x) = x (x - 2) (x - 1)$

Factoring p(x) gives x(x – 1)(x – 2), indicating zeros at x = 0, 1, and 2. The graph intersects the x-axis at these points.

Q 9 :
For the polynomial r(x) = x² – 4x + 4, which of the following statements are correct about its graph and zeros?

(i) The graph touches the x-axis at x = 2.
(ii) The graph intersects the x-axis at x = 2.
(iii) The polynomial is equivalent to (x – 2)².
(iv) The vertex of the graph is at the point (2, 0).

Choose the correct option from the following:

(i), (iii) and (iv)
(ii) and (iii)
(i), (ii) and (iv)
(i) and (iv)

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(1)

Given polynomial r(x) can be rewritten as (x – 2)², indicating a repeated zero at x = 2. The graph touches the x-axis at this point and does not cross it. The vertex of the graph is at point (2, 0).

Q 10 :
For the polynomial t(x) = x³ – 6x² + 11x – 6, which statements are true about its graph and zeros?

(i) The polynomial has a zero at x = 1.
(ii) The polynomial has a zero at x = 2.
(iii) The graph of the polynomial intersects the x-axis at x = 3.
(iv) The polynomial can be factorised as (x – 1)(x – 2)(x – 3).

Choose the correct option from the following:

(i), (iii) and (iv)
(i), (ii), (iii) and (iv)
(i), (ii) and (iv)
(i) and (iv)

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(2)

At x = 1, t(1) = 1 – 6 + 11 – 6 = 0

So (x – 1) is a factor of t(x).

∴ t(x) = (x – 1)(x² – 5x + 6)

⇒ t(x) = (x – 1)(x – 2)(x – 3)

Factoring t(x) as (x – 1)(x – 2)(x – 3) reveals zeros at x = 1, 2, 3. The graph intersects the x-axis at these points.

Topic Question Set

Q 1 :
The zeroes of the polynomial $x^{2} - 3 x - m (m + 3)$ are

Q 2 :
If α and β are the zeros of a polynomial $f (x) = p x^{2} - 2 x + 3 p$ and α + β = αβ, then p is

Q 3 :
The zeroes of a polynomial $x^{2} + p x + q$ are twice the zeroes of the polynomial $4 x^{2} - 5 x - 6$ . The value of p is :

Q 4 :
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.

Q 5 :
If the sum of zeroes of the polynomial $p (x) = 2 x^{2} - k \sqrt{2} x + 1$ is $\sqrt{2}$ , then value of $k$ is :

Q 6 :
If $α$ and $β$ are zeroes of the polynomial $5 x^{2} + 3 x - 7$ , the value of $\frac{1}{α} + \frac{1}{β}$ is

Q 7 :
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.

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Topic Question Set

Q 1 : The zeroes of the polynomial x2–3x–m(m+3) are

Q 2 : If α and β are the zeros of a polynomial f(x)=px2–2x+3p and α + β = αβ, then p is

Q 3 : The zeroes of a polynomial x2+px+q are twice the zeroes of the polynomial 4x2–5x–6. The value of p is :

Q 4 : 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.

Q 5 : If the sum of zeroes of the polynomial p(x)=2x2-k2 x+1 is 2, then value of k is :

Q 6 : If α and β are zeroes of the polynomial 5x2+3x–7, the value of 1α+1β is

Q 7 : 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.

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Q 1 :
The zeroes of the polynomial $x^{2} - 3 x - m (m + 3)$ are

Q 2 :
If α and β are the zeros of a polynomial $f (x) = p x^{2} - 2 x + 3 p$ and α + β = αβ, then p is

Q 3 :
The zeroes of a polynomial $x^{2} + p x + q$ are twice the zeroes of the polynomial $4 x^{2} - 5 x - 6$ . The value of p is :

Q 4 :
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.

Q 5 :
If the sum of zeroes of the polynomial $p (x) = 2 x^{2} - k \sqrt{2} x + 1$ is $\sqrt{2}$ , then value of $k$ is :

Q 6 :
If $α$ and $β$ are zeroes of the polynomial $5 x^{2} + 3 x - 7$ , the value of $\frac{1}{α} + \frac{1}{β}$ is

Q 7 :
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.