M.C.Q (1 Marks)

MCQ 2011 Mark

If roots of the quadratic equation
$3 a x^2+2 b x+c=0$ are in the ratio $2: 3$, then
Ram and Shyam said the following :
Ram $: 8 a c=25 b$
Shyam : $8 b^2=9 a c$
Which of them is/are correct?

A
Only Ram
B
Only Shyam
C
Both Ram and Shyam
✓
Neither of them

Answer

Correct option: D.

Neither of them

Let $\alpha$ and $\beta$ be the roots of $3 a x^2+2 b x+c=0$.
Then, $\alpha+\beta=\frac{-2 b}{3 a}, \alpha \beta=\frac{c}{3 a}$.
Given $, \frac{\alpha}{\beta}=\frac{2}{3} $
$\Rightarrow \alpha=\frac{2}{3} \beta$
$\therefore \frac{2}{3} \beta+\beta=-\frac{2 b}{3 a} $
$\Rightarrow \frac{5 \beta}{3}=\frac{-2 b}{3 a} $
$\Rightarrow \beta=\frac{-2 b}{5 a}$
$\therefore \alpha=\frac{2}{3} \times \frac{-2 b}{5 a}=\frac{-4 b}{15 a}$
Now, $\alpha \beta=\frac{c}{3 a}$
$ \Rightarrow\left(\frac{-4 b}{15 a}\right) \times\left(\frac{-2 b}{5 a}\right)=\frac{c}{3 a} $
$\Rightarrow 8 b^2=25 a c$

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MCQ 2021 Mark

The roots of the equation $2 x-\frac{3}{x}=1 \ (x \neq 0)$ are

A
$\frac{1}{2},-1$
B
$\frac{3}{2}, 1$
✓
$\frac{3}{2},-1$
D
none of these

Answer

Correct option: C.

$\frac{3}{2},-1$

$2 x-\frac{3}{x}=1 $
$\Rightarrow 2 x^2-3=x$
$\Rightarrow 2 x^2-x-3=0 $
$\Rightarrow 2 x^2-3 x+2 x-3=0$
$\Rightarrow x(2 x-3)+1(2 x-3)=0$
$\Rightarrow(2 x-3)(x+1)=0 $
$\Rightarrow x=-1,3 / 2$

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MCQ 2031 Mark

The quadratic equation $a x^2-4 a x+2 a+1=0$ has repeated roots, if $a=$

A
$0$
✓
$1 / 2$
C
$2$
D
$4$

Answer

Correct option: B.

$1 / 2$

The roots are said to be repeated, if
$B^2-4 A C=0 $
$\Rightarrow(-4 a)^2-4 \times a \times(2 a+1)=0$
$\Rightarrow 16 a^2-4 a(2 a+1)-0$
$ \Rightarrow 16 a^2-8 a^2-4 a-0$
$\Rightarrow 8 a^2-4 a=0 $
$\Rightarrow a(8 a-4)=0 $
$\Rightarrow a=0 \text { or } 8 a-4=0$
$\therefore a=1 / 2 \ (\because a \neq 0)$

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Maths M.C.Q (1 Marks)