M.C.Q (1 Marks) - Maths STD 10 Questions - Vidyadip

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M.C.Q (1 Marks)

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18 questions · timed · auto-graded

MCQ 11 Mark

The roots of the quadratic equation $2 x^2-x-6=0$ are

A
$-2, \frac{3}{2}$
✓
$2,-\frac{3}{2}$
C
$-2,-\frac{3}{2}$
D
$2, \frac{3}{2}$

Answer

Correct option: B.

$2,-\frac{3}{2}$

Using factorization method of splitting the middle term, we can solve the quadratic equation as follows:
$2 x^2-x-6=0$
$\Rightarrow 2 x^2-4 x+3 x-6=0$
$\Rightarrow\left(2 x^2-4 x\right)+(3 x-6)=0$
$\Rightarrow 2 x(x-2)+3(x-2)=0$
$\Rightarrow(x-2)(2 x+3)=0$
$\Rightarrow x-2=0 \text { or } 2 x+3=0$
$\Rightarrow x=2 \text { or } x=-\frac{3}{2}$
Option $(b)$ is correct.

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MCQ 21 Mark

The roots of the equation $x^2+x-p(p+1)=0$ where $p$ is a constant, are

A
$p, p+1$
B
$-p, p+1$
✓
$p,-(p+1)$
D
$-p,-(p+1)$

Answer

Correct option: C.

$p,-(p+1)$

We have, $x^2+x-p(p+1)=0$
$\Rightarrow x^2+(p+1-p) x-p(p+1)=0$
$\Rightarrow x^2+(p+1) x-p x-p(p+1)=0$
$\Rightarrow x(x+p+1)-p(x+p+1)=0$
$\Rightarrow(x+p+1)(x-p)=0$
$\Rightarrow(x+p+1)=0 \text { or }(x-p)=0$
$\Rightarrow x=-(p+1) \text { or } x=p$
The roots of the given quadratic equation are $p$ and $-(p+1)$.
Hence, the correct option is $(c).$

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MCQ 31 Mark

If the quadratic equation $a x^2+b x+c=0$ has two real and equal roots, then $'c\ '$ is equal to

A
$\frac{- b }{2 a }$
B
$\frac{ b }{2 a }$
C
$\frac{-b^2}{4 a}$
✓
$\frac{ b ^2}{4 a }$

Answer

Correct option: D.

$\frac{ b ^2}{4 a }$

For the equation to have equal roots, the discriminent must be equal to zero.
$ D = b ^2-4 ac =0$
$b^2=4 ac $
$c=\frac{b^2}{4 a}$

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MCQ 41 Mark

The roots of the quadratic equation $x^2-0.04=0$ are

✓
$\pm 0.2$
B
$\pm 0.02$
C
0.4
D
2

Answer

Correct option: A.

$\pm 0.2$

(a)
$x^2-0.04=0$
$
\begin{array}{ll}
\Rightarrow & x^2=0.04 \\
\Rightarrow & x=\sqrt{0.04} \\
\Rightarrow & x= \pm 0.2
\end{array}
$

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MCQ 51 Mark

A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option :
Assertion (A) : If one root of the quadratic equation $4 x^2-10 x+(k-4)=0$ is reciprocal of the other, then value of k is 8 .
Reason $( R )$ : Roots of the quadratic equation $x ^2- x +1=0$ are real.

A
(a) Both Assertion (A) and Reason (R) are true and Reason (R) gives the correct explanation of Assertion (A).
B
(b) Both Assertion (A) and Reason (R) are true but Reason (R) does not give the correct explanation of Assertion (A).
✓
(c) Assertion (A) is true but Reason (R) is false.
D
(d) Assertion (A) is false but Reason (R) is true.

Answer

Correct option: C.

(c) Assertion (A) is true but Reason (R) is false.

(c)
Assertion is correct but the reason is false.

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MCQ 61 Mark

Which of the following is not a quadratic equation?

A
$2(x-1)^2=4 x^2-2 x+1$
B
$2 x-x^2=x^2+5$
✓
$(\sqrt{2} x+\sqrt{3})^2+x^2=3 x^2-5 x$
D
$\left(x^2+2 x\right)^2=x^4+3+4 x^3$

Answer

Correct option: C.

$(\sqrt{2} x+\sqrt{3})^2+x^2=3 x^2-5 x$

(c)
The standard form of a quadratic equation is $a x^2+b x+c=0$ in variable $x$.
Where $a, b$, and $c$ are real numbers and $a \neq 0$.
We need to check if the degree of the given equations is 2 .

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MCQ 71 Mark

Aquadratic equation whose one root is 2 and the sum of whose roots is zero, is :

A
$x ^2+4=0$
B
$x^2-2=0$
C
$4 x ^2-1=0$
✓
$x^2-4=0$

Answer

Correct option: D.

$x^2-4=0$

(d)
The general equation is of the form
$x ^2+$ (sum of the roots) $x +$ product of the roots $=0$
$x ^2-4=0$ is the required equation

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MCQ 81 Mark

The discriminant of the quadratic equation $4 x^2-6 x+3=0$ is

A
12
B
84
C
$2 \sqrt{3}$
✓
-12

Answer

Correct option: D.

-12

(d)
$
\begin{aligned}
4 x^2-6 x+3 & =0 \\
D & =b^2-4 ac \\
& =(-b)^2-4 \times 4 \times 3 \\
& =36-48=-12
\end{aligned}
$

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MCQ 91 Mark

If the sum of the roots of equation $kx ^2+2 x +3 k$ $=0$ is equal to their product then thevalue of k is

A
$\frac{1}{3}$
B
$-\frac{1}{3}$
C
$\frac{2}{3}$
✓
$-\frac{2}{3}$

Answer

Correct option: D.

$-\frac{2}{3}$

(d)
Sum of roots $=\frac{-2}{ k }$
and product of roots $=\frac{ c }{ a }=\frac{3 k }{ k }=3$
$\because$ Sum of roots $=$ product of roots
$
\Rightarrow \frac{-2}{k}=3 \Rightarrow k=\frac{-2}{3}
$

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MCQ 101 Mark

The roots of a quadratic equation are $5$ an $-2 ,$ Then, the equation is.

A
$x ^2-3 x +10=0$
✓
$x ^2-3 x -10=0$
C
$x ^2+3 x -10=0$
D
$x^2+3 x+10=0$

Answer

Correct option: B.

$x ^2-3 x -10=0$

Sum of roots $=5-2=3$
Product of roots $=5 \times(-2)=-10$
$\therefore$ Quadratic equation
$x^2-(\text { sum of roots }) x+\text { product of roots }=0$
$\Rightarrow x^2-3 x-10 x=0$

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MCQ 111 Mark

The roots of the equation $a x^2+b x+c=0$ will be reciprocal of each other if

A
$a = b$
B
$b = c$
✓
$c = a$
D
None of these

Answer

Correct option: C.

$c = a$

Let the roots $\alpha \ \frac{1}{\alpha}$.
Then, product of roots $=\frac{ c }{ a }$
$\alpha=\frac{1}{\alpha}=\frac{c}{a}$
$\Rightarrow 1=\frac{c}{a}$
$\Rightarrow c=a$

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MCQ 121 Mark

If the sum of the roots of a quadratic equation is 6 and their product is 6 , the equation is.

✓
$x ^2-6 x +6=0$
B
$x^2+6 x-6=0$
C
$x ^2-6 x -6=0$
D
$x ^2+6 x +6=0$

Answer

Correct option: A.

$x ^2-6 x +6=0$

(a)
Required quadratic equation
$
\begin{aligned}
& \Rightarrow x^2-(\text { sum of roots }) x+\text { product of roots }=0 \\
\Rightarrow & x^2-6 x+6=0
\end{aligned}
$

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MCQ 131 Mark

If one root of $5 x^2+13 x+k=0$ be the reciprocal of the other root then the value of $k$ is

A
$0$
B
$1$
C
$2$
✓
$5$

Answer

Correct option: D.

$5$

Let the roots be a and $\frac{1}{ a }$.
Then, product of roots $=\frac{ c }{ a }$
$\alpha \times \frac{1}{\alpha}=\frac{k}{5}$
$\Rightarrow k=5 \times 1=5$

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MCQ 141 Mark

If one root of the equation $3 x^2-10 x+3=0$ is $\frac{1}{3}$ then the other root is

A
$-\frac{1}{3}$
B
$\frac{1}{3}$
C
$-3$
✓
$3$

Answer

Correct option: D.

$3$

Let the other root be $a.$
Then, product of root $=\frac{ c }{ a }$
$\frac{1}{3} \times a=\frac{3}{3}$
$\Rightarrow a=\frac{3 \times 3}{3}=3$

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MCQ 151 Mark

The ratio of the sum and product of the roots of the equation $7 x ^2-12 x +18=0$ is

A
$7: 12$
B
$7: 18$
C
$3: 2$
✓
$2: 3$

Answer

Correct option: D.

$2: 3$

(d)
Given equation, $7 x^2-15 x+18=0$On comparing $ax ^2+ bx + c =0$
$a =7, b=-12$ and $c =18$
Sum of roots $=\frac{-b}{a}=-\left(\frac{-12}{7}\right)=\frac{12}{7}$
Product of roots $=\frac{ c }{ a }=\frac{18}{7}$
$\therefore$ Required ratio $=\frac{12}{7}: \frac{18}{7}=12: 18$

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MCQ 161 Mark

If the product of the roots of the equation $x ^2-3 x + k =10$ is $-2$ then the value of $k$ is

A
$-2$
B
$-8$
✓
$8$
D
$12$

Answer

Correct option: C.

$8$

$x^2-3 x+k=10$
$\Rightarrow x^2-3 x+(k-10)=0$
$\text { On comparing } a x^2+b x+c=0$
$a=1, b=-3 \text { and } c=k-10$
$\therefore \text { Product of the roots }=\frac{c}{a}$
$-2=\frac{k-10}{1}$
$\Rightarrow-2=k-10$
$\Rightarrow k=-2+10=8$

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MCQ 171 Mark

The sum of the roots of the equation $x ^2-6 x +2=0$ is

A
2
B
-2
✓
6
D
-6

Answer

Correct option: C.

6

(c)
Given equation $x^2-6 x+2=0$
On comparing $ax ^2+ bx + c =0$
$a =1, b=-6$ and $c =2$
$\therefore$ Sum of the roots $=\frac{-b}{a}=\frac{-(-6)}{1}=6$

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MCQ 181 Mark

If one root of the equation $2 x^2+a x+6=0$ is $2$ then $a =$ ?

A
$7$
✓
$-7$
C
$\frac{7}{2}$
D
$-\frac{7}{2}$

Answer

Correct option: B.

$-7$

$2 x^2+a x+6=0$
$\because 2$ is a root of the given equation
$\therefore$ Putting $x =2$
$\ 2(2)^2+a(2)+6=0$
$\Rightarrow 8+2 a+6=0$
$\Rightarrow 2 a=-14 $
$\Rightarrow a=\frac{-14}{2}=-7$

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M.C.Q (1 Marks) - Maths STD 10 Questions - Vidyadip