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 :

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 5 :

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 6 :

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 7 :

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 8 :

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.

Q 9 :

Shown below is a part of the graph of a polynomial h(x).

On dividing h(x) by which of the following will the remainder be zero?

(i) (x – 2)
(ii) (x + 2)
(iii) (x – 4)
(iv) (x + 4)

Choose the correct option from the following:

only (ii)
only (i) and (iii)
only (ii) and (iv)
cannot be determined without knowing the polynomial h(x).

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(1)

Since the given graph intersects the x-axis at x = –2, therefore (x + 2) is a factor of h(x).

So, on dividing h(x) by (x + 2), we get remainder zero

Q 10 :

If the sum of zeros of the polynomial $p (x) = 2 x ² - k \sqrt 2 x + 1 i s \sqrt 2$ , then value of k is:

√2
2
2√2
1/2

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(2)

Given polynomial is p(x) = 2x² – k√2x + 1.

Let α, β be its zeros then

α + β = –b/a = –(–k√2)/2 = k√2/2 = √2

[Here a = 2, b = –k√2, c = 1]

⇒ k√2 = 2√2 ⇒ k = 2

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 :

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 5 :

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

(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.

Q 9 :

Shown below is a part of the graph of a polynomial h(x).

On dividing h(x) by which of the following will the remainder be zero?

(i) (x – 2)
(ii) (x + 2)
(iii) (x – 4)
(iv) (x + 4)

Choose the correct option from the following:

Q 10 :

If the sum of zeros of the polynomial $p (x) = 2 x ² - k \sqrt 2 x + 1 i s \sqrt 2$ , then value of k is:

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Topic Question Set

Q 1 : The zeroes of the polynomial x2–3x–m(m+3) are .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 5 : If α and β are zeroes of the polynomial 5x2+3x–7, the value of 1α+1β is .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

(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.

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Q 1 :

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

Q 5 :

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

(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.