M.C.Q

Question 11 Mark

Write the correct answer in the following:
Which one of the following is a polynomial?

$\frac{\text{x}^2}{2}-\frac{2}{\text{x}^2}$
$\sqrt{2\text{x}-1}$
$\text{x}^2+\frac{3\text{x}^{\frac{3}{2}}}{\sqrt{\text{x}}}$
$\frac{\text{x}-1}{\text{x}+1}$

Answer

$\text{x}^2+\frac{3\text{x}^{\frac{3}{2}}}{\sqrt{\text{x}}}$

Solution:

Now, $\frac{\text{x}^2}{2}-\frac{2}{\text{x}^2}=\frac{\text{x}^2}{2}-2\text{x}^{-2},$ it is not a polynomial, because exponent of x is -2 which is not a whole number.
Now, $\sqrt{2\text{x}-1}=\sqrt{\text{2}\text{x}}^{\frac{1}{2}}-1, $ it is not a polynomial, because exponent of x is $-\frac{1}{2}$ which is not a whole number.
Now, $\text{x}^2+\frac{3\text{x}^{\frac{3}{2}}}{\sqrt{\text{x}}}=\text{x}^2+3\text{x}^{\frac{3}{2}-\frac{1}{2}}=\text{x}^2+3\text{x}^{\frac{2}{2}}=\text{x}^2+3\text{x},$ it is not a polynomial, because exponent of x is which is a whole number.
$\frac{\text{x}-1}{\text{x}+1},$ it is not a polynomial because it is a rational function.

View full question & answer→

Question 21 Mark

Write the correct answer in the following:
$\sqrt{2}$ is a polynomial of degree.

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

Answer

Solution:
$\sqrt{2}$ is a constant polynomial. The only term here is $\sqrt{2}$ which can be written as $\sqrt{2}\text{x}^\circ.$ So, the exponent of x is zero. Therefore, the degree of the polynomial is 0.

View full question & answer→

MCQ 31 Mark

Write the correct answer in the following: One of the zeroes of the polynomial $2x^2 + 7x - 4$ is.

A
$2$
✓
$\frac{1}{2}$
C
$-\frac{1}{2}$
D
$-2$

Answer

Correct option: B.

$\frac{1}{2}$

Let $p(x) = 2x^2 + 7x - 4$
$= 2x^2 + 8x - x- 4 \ [$by splitting middle term$]$
$= 2x(x + 4) -1(x + 4)$
$=(2x - 1)(x + 4)$
For zeroes of $p(x),$ put $p(x) = 0$
$\Rightarrow (2x - 1)(x + 4) = 0$
$\Rightarrow 2x - 1 = 0 $ and $x + 4 = 0$
$\Rightarrow \text{x}=\frac{1}{2}$ and $\text{x}=-4$
Hence, one of the zeroes of the polynomial $p(x)$ is $\frac{1}{2}.$

View full question & answer→

MCQ 41 Mark

Write the correct answer in the following: The coefficient of $x$ in the expansion of $(x + 3)^3$ is.

A
$1$
B
$9$
C
$18$
✓
$27$

Answer

Correct option: D.

$27$

Now,$ (x + y)^3 = x^{3 }+ 3^{3 }= 3.x.3(x + 3)$
$[$Using identity$, (a + b)^3 = a^3 + b^3 + 3ab\ (a + b)]$
$= x^3 + 27 + 9x(x + 3)$
$= x^3 + 27 + 9x^2 + 27x$
Hence, the coefficient of $x$ in $(x + 3)^3$ is $27.$

View full question & answer→

MCQ 51 Mark

Write the correct answer in the following: 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 $
$= (249 + 248)(249 - 248) [(a)^2 - (b)^2 $
$= (a + b)(a - b)]$
$= (497)(1) $
$= 497$

View full question & answer→

MCQ 61 Mark

Write the correct answer in the following: If $a + b + c = 0,$ then $a^3 + b^3 + c^3$ is equal to.

A
$0$
B
$abc$
✓
$3abc$
D
$2abc$

Answer

Correct option: C.

$3abc$

Now, $a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - be - ca) + 3abc$
$[$Using identity,$ a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - be - ca)] = 0 + 3abc$
$\therefore a + b + c = 0,$ given
$a^3 + b^3 + c^3 = 3abc$

View full question & answer→

MCQ 71 Mark

Write the correct answer in the following: If $x^{51} + 51$ is divided by $x + 1,$ the remainder is.

A
$0$
B
$1$
C
$49$
✓
$50$

Answer

Correct option: D.

$50$

If $p(x)$ is divided by $x + a,$ then the remainder is $p(-a).$
Here $p(x) = x^{51} + 51$ is divided by $ x + 1,$ then
$x = -1$
Remainder $= p(-1) = (-1)^{51} + 51 $
$= 50 = -1 + 51 = 50$

View full question & answer→

MCQ 81 Mark

Write the correct answer in the following: The value of the polynomial $5x - 4x^2 + 3,$ when $x = -1$ is.

✓
$-6$
B
$6$
C
$2$
D
$-2$

Answer

Correct option: A.

$-6$

Let $p(x) = 5x - 4x^2 + 3 ... (i)$
On putting $x= -1$ in $eq. (i),$ we get
$p(-1) = 5(-1) - 4(-1)^2 + 3$
$ = -5 - 4 + 3 $
$= -6$

View full question & answer→

MCQ 91 Mark

Write the correct answer in the following: 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.

$3xy$

$(x + y)^3 - (x^{3 }+ y^3) = x^{3 }+ y^3 + 3xy(x + y) - x^3 - y^3$
$[(a + b)^3 = a^3 + b^3 + 3ab(a + b)]$
$= 3xy(x + y)$
So$, 3xy$ is a factor of $(x + y)^3 - (x^{3 }+ y^3).$

View full question & answer→

MCQ 101 Mark

Write the correct answer in the following: If $x + 1$ is a factor of the polynomial $2x^2 + kx$, then the value of $k$ is.

A
$-3$
B
$4$
✓
$2$
D
$-2$

Answer

Correct option: C.

$2$

Let $p(x) = 2x^2 + kx$
Since$, (x + 1)$ is a factor of $p(x),$ then
$p(-1) = 0$
$2(-1)^2 + k(-1) = 0$
$\Rightarrow 2 - k = 0$
$\Rightarrow k = 2$

View full question & answer→

Question 111 Mark

Write the correct answer in the following:
Zero of the polynomial p(x) = 2x + 5 is.

$-\frac{2}{5}$
$-\frac{5}{2}$
$\frac{2}{5}$
$\frac{5}{2}$

Answer

$-\frac{5}{2}$

Solution:
Finding a zero of p(x) is the same as solving an equation p(x) = 0.
Now, p(x) = 0 ⇒ 2x + 5 = 0,
2x = -5
Which give us $\text{x}=-\frac{5}{2}.$
Therefore, $-\frac{5}{2}$ is the zero of the polynomial.

View full question & answer→

Question 121 Mark

Write the correct answer in the following:
If p(x) = x + 3, then p(x) + p(-x) is equal to.

Answer

Solution:
We have p(x) = x + 3, then
p(-x) = -x + 3
Therefore, p(x) + p(-x) = x + 3 + (-x + 3) = x + 3 - x + 3 = 6

View full question & answer→

Question 131 Mark

Write the correct answer in the following: If $\text{p}\text{(x)}=\text{x}^2-2\sqrt{2\text{x}}+1,$ then is $\text{p}(2\sqrt{2})$ equal to.

$0$
$1$
$4\sqrt{2}$
$8\sqrt{2}+1$

Answer

Solution:
We have,
$\text{p}\text{(x)}=\text{x}^2-2\sqrt{2}\text{x}+1$
$\text{p}(2\sqrt{2})=(2\sqrt{2})^2-2\sqrt{2}(2\sqrt{2})+1$
$= 8 - 8 + 1$
$= 1$

View full question & answer→

MCQ 141 Mark

Write the correct answer in the following: The factorisation of $4x^2 + 8x + 3$ is.

A
$(x + 1)(x + 3)$
✓
$(2x + 1)(2x + 3)$
C
$(2x + 2)(2x + 5)$
D
$(2x –1)(2x –3)$

Answer

Correct option: B.

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

Now$, 4x^2 + 8x + 3= 4x^2 + 6x + 2x + 3 \ [$by splitting middle term$]$
$= 2x(2x + 3) + 1 (2x + 3)$
$= (2x + 3)(2x + 1)$

View full question & answer→

Question 151 Mark

Write the correct answer in the following:
If $\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{x}}=-1 \ (\text{x},\text{y}\neq0),$ the value of $\text{x}^3-\text{y}^3$ is.

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

Answer

Solution:
Given, $\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{x}}=-1$
$\Rightarrow\frac{\text{x}^2+\text{y}^2}{\text{xy}}=-1$
$\Rightarrow\text{x}^2+\text{y}^2=-\text{xy}$
$\Rightarrow\text{x}^2+\text{y}^2+\text{xy}=0$
Now, $\text{x}^3-\text{y}^3=(\text{x}-\text{y})(\text{x}^2+\text{xy}+\text{y}^2) \ ...(\text{i})$
$[\text{a}^3-\text{b}^3=(\text{a}-\text{b})(\text{a}^2+\text{ab}+\text{b}^2)]$
$=(\text{x}-\text{y})\times0=0$ [From Eq. (i)]

View full question & answer→

MCQ 161 Mark

Write the correct answer in the following: Degree of the zero polynomial is.

✓
$0$
B
$1$
C
Any natural number.
D
Not defined.

Answer

Correct option: A.

$0$

The degree of zero polynomial is not defined, because in zero polynomial, the coefficient of any variable is zero$ i.e., 0x^2$ or $0x^5,$ etc.
Hence, we cannot exactly determine the degree of variable.

View full question & answer→

MCQ 171 Mark

Write the correct answer in the following$: x + 1$ is a factor of the polynomial.

✓
$x^3 + x^2 - x + 1$
B
$x^3 + x^2 + x + 1$
C
$x^4 + x^3 + x^2 + 1$
D
$-x^4 + 3x^3 + 3x^2+ x + 1$

Answer

Correct option: A.

$x^3 + x^2 - x + 1$

Let assume $(x + 1)$ is a factor of $x^3 + x^2 + x + 1$
So$, x = -1$ is zero of $x^3 + x^2 + x + 1$
$(-1)^3 + (-1)^2 + (-1) + 1 = 0$
$\Rightarrow -1 + 1 - 1 + 1 = 0$
$\Rightarrow 0 = 0$
Hence, our assumption is true.

View full question & answer→

MCQ 181 Mark

Write the correct answer in the following: 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 $
$= (5x)^2 - 1^2 + (5x + 1)^2$
$= (5x - 1)(5x - 1) + (5x + 1)^2$
$ = (5x + 1)(5x - 1 + 5x + 1)$
$= (5x + 1)(10x) = 10x(5x + 1)$
Hence, one of the factors of $(25x^2 - 1) + (1 + 5x)^2$ is $10x.$

View full question & answer→

MCQ 191 Mark

Write the correct answer in the following: Degree of the polynomial $4x^4 + 0x^3 + 0x^5 + 5x + 7$ is.

✓
$4$
B
$5$
C
$3$
D
$7$

Answer

Correct option: A.

$4$

The height power of the variable in a polynomial is called the degree of the polynomial.
In this polynomial, the term with highest power of $x$ is $4x^4.$
Highest power of $x$ is $4$, so the degree of the given polynomial is $4.$

View full question & answer→

Question 201 Mark

Write the correct answer in the following:
Zero of the zero polynomial is.

0
1
Any real number.
Not defined.

Answer

Any real number.

Solution:
Zero of the zero polynomial is any real number.
e.g., Let us consider zero polynomial be 0(x - k), where k is a real number. For determining the zero, put x - k = 0 ⇒ x = k Hence, zero of the zero polynomial be any real number.

View full question & answer→

Question 211 Mark

Write the correct answer in the following:
If $49\text{x}^2 -\text{b}=\Big(7\text{x}+\frac{1}{2}\Big)\Big(7\text{x}-\frac{1}{2}\Big),$ the value of b is.

$0$
$\frac{1}{\sqrt2}$
$\frac{1}{4}$
$\frac{1}{2}$

Answer

$\frac{1}{4}$

Solution:
$49\text{x}^2 -\text{b}=\Big(7\text{x}+\frac{1}{2}\Big)\Big(7\text{x}-\frac{1}{2}\Big)$
$\Rightarrow49\text{x}^2 -\text{b}=\Big(7\text{x}\Big)^2-\Big(\frac{1}{2}\Big)^2$
$49^2-\frac{1}{4} [\therefore(\text{a}+\text{b})(\text{a}-\text{b})=\text{a}^2-\text{b}^2]$
So, we get $\text{b}=\frac{1}{4}.$

View full question & answer→

MATHS M.C.Q

Test yourself on this topic