Grade 10 : Mathematics - 2. Polynomials (Set 3)

Question 1

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - 7x + 12$, find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

Accepted Answer

A) $\frac{7}{12}$

Answer

B) $\frac{12}{7}$

Answer

C) $-\frac{7}{12}$

Answer

D) $\frac{7}{6}$

Question 2

Find a quadratic polynomial whose zeroes are $3 + \sqrt{5}$ and $3 - \sqrt{5}$.

Accepted Answer

B) $x^2 - 6x + 4$

Answer

A) $x^2 - 6x - 4$

Answer

C) $x^2 + 6x + 4$

Answer

D) $x^2 - 4x + 6$

Question 3

If one zero of the polynomial $p(x) = (k-1)x^2 + kx + 1$ is -3, then the value of $k$ is:

Accepted Answer

A) $\frac{4}{3}$

Answer

B) $-\frac{4}{3}$

Answer

C) $\frac{2}{3}$

Answer

D) $-\frac{2}{3}$

Question 4

A fountain jet shoots water in a parabolic path modeled by $h(t) = -t^2 + 6t - 5$, where $h(t)$ is height in meters and $t$ is time in seconds.

At what time values is the water at ground level ($h(t) = 0$)?

Accepted Answer

A) $t = 1	ext{ s}$ and $t = 5	ext{ s}$

Answer

B) $t = 2	ext{ s}$ and $t = 4	ext{ s}$

Answer

C) $t = 0	ext{ s}$ and $t = 6	ext{ s}$

Answer

D) $t = 1	ext{ s}$ and $t = 6	ext{ s}$

Question 5

If $\alpha$ and $\beta$ are the zeroes of $f(x) = 2x^2 + 5x + k$ such that $\alpha^2 + \beta^2 + \alpha\beta = \frac{21}{4}$, find the value of $k$.

Accepted Answer

B) 2

Answer

A) 1

Answer

C) 3

Answer

D) 4

Question 6

The graph of $y = p(x)$ crosses the x-axis at 3 distinct points and touches the x-axis at 1 point without crossing. What is the total number of real zeroes of $p(x)$?

Accepted Answer

C) 4

Answer

A) 2

Answer

B) 3

Answer

D) 5

Question 7

If $x = 2$ and $x = -3$ are zeroes of the cubic polynomial $p(x) = x^3 + ax^2 + bx - 6$, find the third zero.

Accepted Answer

B) -1

Answer

A) 1

Answer

C) 3

Answer

D) -2

Question 8

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

Accepted Answer

B) -10

Answer

A) 10

Answer

C) -7

Answer

D) 5

Question 9

If the sum of the zeroes of the quadratic polynomial $kx^2 + 2x + 3k$ is equal to their product, then $k$ equals:

Accepted Answer

B) $-\frac{2}{3}$

Answer

A) $\frac{2}{3}$

Answer

C) $\frac{3}{2}$

Answer

D) $-\frac{3}{2}$

Question 10

Polynomial	Zeroes ($\alpha, \beta$)	Polynomial Form
$P_1(x)$	2, 5	$x^2 - 7x + 10$
$P_2(x)$	-1, 4	$x^2 - 3x - 4$
$P_3(x)$	$\alpha, \beta$	$x^2 - px + q$

If $\alpha$ and $\beta$ are the zeroes of $P_3(x) = x^2 - px + q$, what is the value of $\alpha^2 + \beta^2$?

Accepted Answer

B) $p^2 - 2q$

Answer

A) $p^2 + 2q$

Answer

C) $p^2 - q$

Answer

D) $p^2 + q$

Question 11

If $\alpha$ and $\beta$ are the zeroes of $p(x) = 3x^2 - 6x + 4$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} + 2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) + 3\alpha\beta$.

Accepted Answer

C) 8

Answer

A) 6

Answer

B) 7

Answer

D) 9

Question 12

On dividing $x^3 - 3x^2 + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$, respectively. Find $g(x)$.

Accepted Answer

A) $x^2 - x + 1$

Answer

B) $x^2 + x - 1$

Answer

C) $x^2 - x - 1$

Answer

D) $x^2 + x + 1$

Question 13

If $6x^4 + 8x^3 + 17x^2 + 21x + 7$ is divided by $3x^2 + 4x + 1$, the remainder is $ax + b$. Find $a$ and $b$.

Accepted Answer

A) $a = 1, b = 2$

Answer

B) $a = 2, b = 1$

Answer

C) $a = 1, b = -2$

Answer

D) $a = -1, b = 2$

Question 14

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

Accepted Answer

A) $1 - c$

Answer

B) $1 + c$

Answer

C) $c - 1$

Answer

D) $1 + p + c$

Question 15

A rectangular park has length $(x + 5)$ meters and width $(x - 2)$ meters. Its area is represented by $A(x) = x^2 + kx - 10$.

What is the value of $k$?

Accepted Answer

A) 3

Answer

B) -3

Answer

C) 7

Answer

D) -10

Question 16

Statement I: A quadratic polynomial whose zeroes are 2 and -3 is $x^2 + x - 6$.

Statement II: If $\alpha$ and $\beta$ are zeroes of $ax^2 + bx + c$, then $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.

Choose the correct option regarding Statement I and Statement II.

Accepted Answer

A) Both Statement I and Statement II are true.

Answer

B) Statement I is false, but Statement II is true.

Answer

C) Statement I is true, but Statement II is false.

Answer

D) Both Statement I and Statement II are false.

Question 17

If $\alpha$ and $\beta$ are zeroes of $p(x) = 2x^2 - 4x + 5$, form a quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.

Accepted Answer

A) $x^2 - 4x + 10$

Answer

B) $x^2 + 4x + 10$

Answer

C) $x^2 - 8x + 20$

Answer

D) $x^2 - 4x + 5$

Question 18

A graph of $y = p(x)$ opens downwards (an inverted parabola) with its vertex at $(2, 9)$ and intersects the x-axis at $x = -1$ and $x = 5$.

What is the quadratic polynomial $p(x)$ represented by this graph?

Accepted Answer

A) $-x^2 + 4x + 5$

Answer

B) $x^2 - 4x - 5$

Answer

C) $-x^2 - 4x + 5$

Answer

D) $-x^2 + 4x - 5$

Question 19

If $(x + a)$ is a factor of $2x^2 + 2ax + 5x + 10$, find the value of $a$.

Accepted Answer

A) 2

Answer

B) -2

Answer

C) 5

Answer

D) -5

Question 20

If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - 5x + k$ such that $\alpha - \beta = 1$, find the value of $k$.

Accepted Answer

C) 6

Answer

A) 4

Answer

B) 5

Answer

D) 7

Set 4Grade 10 : Mathematics - Polynomials

Grade 10 : Mathematics - Polynomials

Examination Instructions:

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Set 4Grade 10 : Mathematics - Polynomials

Grade 10 : Mathematics - Polynomials

Examination Instructions:

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Set 4Grade 10 : Mathematics - Polynomials

Grade 10 : Mathematics - Polynomials

Examination Instructions:

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Set 4Grade 10 : Mathematics - Polynomials

Grade 10 : Mathematics - Polynomials

Examination Instructions:

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Discussion 0

Rating & Discussion Locked

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