M.C.Q

MCQ 11 Mark

$4a^2 + b^2 + 4ab + 8a + 4b + 4 =?$

✓
$(2a + b + 2)^2$
B
$(2a - b + 2)^2$
C
$(a + 2b + 2)^2$
D
None of these

Answer

Correct option: A.

$(2a + b + 2)^2$

$4a^2 + b^2 + 4ab + 8a + 4b + 4$
$= 4a^2 + b^2 + 4 + 4ab + 4b + 8a$
$= (2a)^2 + b^2 + 2^2 + 2 \times 2a \times b + 2 \times b \times 2 + 2 \times 2a \times 2$
$= (2a + b +2)^2$

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MCQ 21 Mark

If $\frac{\text{a}}{\text{b}}+\frac{\text{b}}{\text{a}}=-1$ then $(a^3 - b^3) =$ ?

A
$-3$
B
$-2$
C
$-1$
✓
$0$

Answer

Correct option: D.

$0$

$\frac{\text{a}}{\text{b}}+\frac{\text{b}}{\text{a}}=-1$
$\Rightarrow\frac{\text{a}^2+\text{b}^2}{\text{ab}}=-1$
$\Rightarrow a^2 + b^2 = -ab$
$\Rightarrow a^2 + b^2 + ab = 0$
Thus, we have:
$(a^3 - b^3) = (a - b)(a^2 + b^2 + ab)$
$= (a - b) \times 0$
$= 0$

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MCQ 31 Mark

If $x + y + z = 9$ and $xy + yz + zx = 23,$ the value of $(x^3 + y^3 + z^3 - 3xyz) =$ ?

✓
$108$
B
$207$
C
$669$
D
$729$

Answer

Correct option: A.

$108$

$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$
$= (x + y + z)[(x + y + z)^2 - 3(xy + yz + zx)]$
$= 9 \times (81 - 3 \times 23)$
$= 9 \times 12$
$= 108$

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MCQ 41 Mark

$3x^3 + 2x^2 + 3x + 2 =$ ?

A
$(3x - 2)(x^2 - 1)$
B
$(3x - 2)(x^2 + 1)$
C
$(3x + 2)(x^2 - 1)$
✓
$(3x + 2)(x^2 + 1)$

Answer

Correct option: D.

$(3x + 2)(x^2 + 1)$

$3x^3 + 2x^2 + 3x + 2$
$= x^2(3x + 2) + 1(3x + 2)$
$= (3x + 2)(x^2 + 1)$

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MCQ 51 Mark

$6x^2 + 17x + 5 =$ ?

A
$(2x + 1)(3x + 5)$
✓
$(2x + 5)(3x + 1)$
C
$(6x + 5)(x + 1)$
D
None of these

Answer

Correct option: B.

$(2x + 5)(3x + 1)$

$6x^2 + 17x + 5$
$= 6x^2 + 15x + 2x + 5$
$= 3x(2x + 5) + 1(2x + 5)$
$= (2x + 5)(3x + 1)$

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MCQ 61 Mark

$(4x^2 + 4x - 3) =$ ?

A
$(2x - 1)(2x - 3)$
B
$(2x + 1)(2x - 3)$
✓
$(2x + 3)(2x - 1)$
D
None of these

Answer

Correct option: C.

$(2x + 3)(2x - 1)$

$4x^2 + 4x - 3$
$= 4x^2 + 6x - 2x - 3$
$= 2x(2x + 3) - 1(2x + 3)$
$= (2x + 3)(2x - 1)$

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MCQ 71 Mark

If $(x + 1)$ is a factor of the polynomial $(2x^2 + kx)$ then the value of $k$ is:

A
$-2$
B
$-3$
✓
$2$
D
$3$

Answer

Correct option: C.

$2$

$(x + 1)$ is a factor of $2x^2 + kx$
So, $-1$ is a zero of $2x^2 + kx$
Thus, we have:
$2 \times (-1)^2 + k \times (-1) = 0$
$\Rightarrow 2 - k = 0$
$\Rightarrow k = 2$

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MCQ 81 Mark

The value of $(249)^2 - (248)^2$ is:

A
$1^2$
B
$477$
C
$487$
✓
$497$

Answer

Correct option: D.

$497$

$(249)^2 - (248)^2$
We know
$a^2 - b^2 = (a + b)(a - b)$
So,
$(249)^2 - (248)^2$
$(249 - 248)(249 + 248)$
$= 497$

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MCQ 91 Mark

The coefficient of $x$ in the expansion of $(x + 3)^3$ is:

A
$1$
B
$9$
C
$18$
✓
$27$

Answer

Correct option: D.

$27$

$(x + 3)^3$
$= x^3 + 3^3 + 9x(x + 3)$
$= x^3 + 27 + 9x^2 + 27x$
So, the coefficient of $x$ in $(x + 3)^3$ is $27.$

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MCQ 101 Mark

$(207 \times 193) =$ ?

A
$39851$
✓
$39951$
C
$39961$
D
$38951$

Answer

Correct option: B.

$39951$

$207 \times 193$
$= (200 + 7)(200 - 7)$
$= (200)^2 - (7)^2$
$= 40000 - 49$
$= 39951$

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MCQ 111 Mark

Which of the following is a factor of $(x + y)^3 - (x^3 + y^3)$?

A
$x^2 + y^2 + 2xy$
B
$x^2 + y^2 - xy$
C
$xy^2$
✓
$3xy$

Answer

Correct option: D.

$3xy$

$(x + y)^3 - (x^3 + y^3)$
$= x^3 + y^3 + 3xy(x + y) - (x^3 + y^3)$
$= 3xy(x + y)$
Thus, the factors of $(x + y)^3 - (x^3 + y^3)$ are $3xy$ and $(x + y)$

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MCQ 121 Mark

$(x^2 - 4x - 21) =$ ?

A
$(x - 7)(x - 3)$
B
$(x + 7)(x - 3)$
✓
$(x - 7)(x + 3)$
D
None of these

Answer

Correct option: C.

$(x - 7)(x + 3)$

$x^2 - 4x - 21$
$x^2 - 7x + 3x - 21$
$= x(x - 7) + 3(x - 7)$
$= (x - 7)(x + 3)$

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MCQ 131 Mark

$(x + 1)$ is a factor of the polynomial:

A
$x^3 - 2x^2 + x + 2$
B
$x^3 + 2x^2 + x - 2$
✓
$x^3 + 2x^2 - x - 2$
D
$x^3 + 2x^2 - x + 2$

Answer

Correct option: C.

$x^3 + 2x^2 - x - 2$

Let:
$f(x) = x^3 - 2x^2 + x + 2$
By the factor theorem, $(x + 1)$ will be a factor of $f(x)$ if $f(-1) = 0.$
We have:
$f(-1) = (-1)^3 - 2 \times (-1)^2 + (-1) + 2$
$= -1 - 2 - 1 + 2$
$=-2\neq0$
Hence, $(x + 1)$ is not a factor of $f(x) = x^3 - 2x^2 + x + 2.$
Now,
Let:
$f(x) = x^3 + 2x^2 + x - 2$
By the factor theorem, $(x + 1)$ will be a factor of $f(x)$ if $f(-1) = 0.$
We have:
$f(-1) = (-1)^3 + 2 \times (-1)^2 + (-1) - 2$
$= -1 + 2 - 1 - 2$
$=-2\neq0$
Hence, $(x + 1)$ is not factor of $f(x) = x^3 + 2x^2 + x - 2.$
Now,
Let:
$f(x) = x^3 + 2x^2 - x - 2$
By the factor theorem, $(x + 1)$ will be a factor of $f(x)$ if $f(-1) = 0.$
We have:
$f(-1) = (-1)^3 + 2 \times (-1)^2 - (-1) - 2$
$= -1 + 2 + 1 - 2$
$= 0$
Hence, $(x + 1)$ is a factor of $f(x) = x^3 + 2x^2 - x - 2.$

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MCQ 141 Mark

If $a + b + c = 0$ then $(a^3 + b^3 + c^3)$ is:

A
$0$
B
$abc$
C
$2abc$
✓
$3abc$

Answer

Correct option: D.

$3abc$

$a + b + c = 0$
$\Rightarrow a + b = -c$
$\Rightarrow (a + b)^3 = (-c)^3$
$\Rightarrow a^3 + b^3 + 3ab(a + b) = -c^3$
$\Rightarrow a^3 + b^3 + 3ab(-c) = -c^3$
$\Rightarrow a^3 + b^3 + c^3 = 3abc$

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MCQ 151 Mark

If $\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{x}}=+1,$ where $\text{x},\ \text{y}\neq0$ then the value of $(x^3 - y^3)$ is:

A
$1$
B
$-1$
✓
$0$
D
$\frac{1}{2}$

Answer

Correct option: C.

$0$

$\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{x}}=-1$
$\Rightarrow\frac{\text{x}^2+\text{y}^2}{\text{xy}}=-1$
$\Rightarrow x^2 + y^2 = -xy$
$\Rightarrow x^2 + y^2 + xy = 0$
Thus, we have:
$(x^3 - y^3) = (x - y)(x^2 + y^2 + xy)$
$= (x - y) \times 0$
$= 0$

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MCQ 161 Mark

One of the factors of $(25x^2 - 1) + (1 + 5x)^2$ is

A
$5 + x$
B
$5 - x$
C
$5x - 1$
✓
$10x$

Answer

Correct option: D.

$10x$

$(25x^2 - 1) + (1 + 5x)^2$
$= (5x - 1)(5x + 1) + (1 + 5x)^2$
$= (5x + 1)[(5x - 1) + (1 + 5x)]$
$= (5x + 1)(10x)$
So, the factors of $(25x^2 - 1) + (1 + 5x)^2$ are $(5x + 1)$ and $10x$

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MCQ 171 Mark

If $a + b + c = 0$ then $\Big(\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}\Big)=$?

A
$1$
B
$0$
C
$-1$
✓
$3$

Answer

Correct option: D.

$3$

$a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc$
Thus, we have:
$\Big(\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}\Big)=\frac{\text{a}^3+\text{b}^3+\text{c}^3}{\text{abc}}$
$=\frac{3\text{abc}}{\text{abc}}$
$=3$

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MCQ 181 Mark

$(305 \times 308) = ?$

A
$94940$
B
$93840$
✓
$93940$
D
$94840$

Answer

Correct option: C.

$93940$

$305 \times 308 = (300 + 5)(300 + 8)$
$= (300)^2 + 300 \times (5 + 8) + 5 \times 8$
$= 90000 + 3900 + 40$
$= 93940$

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MCQ 191 Mark

$(104 \times 96) = ?$

A
$9894$
✓
$9984$
C
$9684$
D
$9884$

Answer

Correct option: B.

$9984$

$104 \times 96 = (100 + 4)(100 - 4)$
$= 100^2 - 4^2$
$= (10000 - 16)$
$= 9984$

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MCQ 201 Mark

If $(x + 2)$ and $(x - 1)$ are factor of $(x^3 + 10x^2 + mx + n)$ then:

A
$m = 5, n = -3$
✓
$m = 7, n = -18$
C
$m = 17, n = -8$
D
$m = 23, n = -19$

Answer

Correct option: B.

$m = 7, n = -18$

Let:
$p(x) = x^3 + 10x^2 + mx + n$
Now,
$x + 2 = 0 \Rightarrow x = -2$
$(x + 2)$ is a factor of $p(x).$
So, we have $p(-2)^2 + m \times (-2) + n = 0$
$\Rightarrow (-2)^3 + 10 \times (-2)^2 + m \times (-2) + n = 0$
$\Rightarrow -8 + 40 - 2m + n = 0$
$\Rightarrow 32 - 2m + n = 0$
$\Rightarrow 2m - n = 32 ...(i)$
Now,
$x - 1 = 0 \Rightarrow x = 1$
Also,
$(x - 1)$ is a factor of $p(x)$
We have:
$p(1) = 0$
$\Rightarrow 1^3 + 10 \times 1^2 + m \times 1 + n = 0$
$\Rightarrow 1 + 10 + m + n = 0$
$\Rightarrow 11 + m + n = 0$
$\Rightarrow m + n = -11 ...(ii)$
From $(i)$ and $(ii),$
We get:
$3m = 21 \Rightarrow m = 7$
By substituting the value of $m$ in $(i),$ we get $n = -18$
$\therefore\ m = 7$ and $n = -18$

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MCQ 211 Mark

If $(x + 5)$ is a factor of $p(x) = x^3 - 20x + 5k$ then $k = ?$

A
$-5$
✓
$5$
C
$3$
D
$-3$

Answer

Correct option: B.

$5$

$(x + 5)$ is a factor of $p(x) = x^3 - 20x + 5k$
$\therefore\ p(-5) = 0$
$\Rightarrow (-5)^3 - 20 \times (-5) + 5k = 0$
$\Rightarrow -125 + 100 + 5k = 0$
$\Rightarrow 5k = 25$
$\Rightarrow k = 5$

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Question 221 Mark

If $\Big(3\text{x}+\frac{1}{2}\Big)\Big(3\text{x}-\frac{1}{2}\Big)=9\text{x}^2-\text{p}$ then the value of p is:

$0$
$-\frac{1}{4}$
$\frac{1}{4}$
$\frac{1}{2}$

Answer

$\frac{1}{4}$

Solution:
$\Big(3\text{x}+\frac{1}{2}\Big)\Big(3\text{x}-\frac{1}{2}\Big)=9\text{x}^2-\text{p}$
$9\text{x}^2-\frac{1}{4}$ $\Big(\therefore\ \big(\text{a}^2-\text{b}^2\big)=(\text{a}+\text{b})(\text{a}-\text{b})\Big)$
$=9\text{x}^2-\text{p}$
$\Rightarrow\text{p}=\frac{1}{4}$

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