MCQ

MCQ 11 Mark

If $x =\frac{e^t+e^{-t}}{2}, y =\frac{e^t-e^{-t}}{2}$ then $\frac{d y}{d x}=$ ?

A
$\frac{-y}{x}$
B
$\frac{y}{x}$
C
$\frac{-x}{y}$
✓
$\frac{x}{y}$

Answer

Correct option: D.

$\frac{x}{y}$

(d) $\frac{x}{y}$
Hint:
$
\begin{gathered}
\frac{d x}{d t}=\frac{1}{2}\left(e^t-e^{-t}\right), \frac{d y}{d t}=\frac{1}{2}\left(e^t+e^{-t}\right) \\
\therefore \frac{d y}{d x}=\frac{(d y / d t)}{(d x / d t)}=\left(\frac{e^t+e^{-t}}{2}\right) /\left(\frac{e^t-e^{-t}}{2}\right)=\frac{x}{y}
\end{gathered}
$

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MCQ 21 Mark

If $x ^4 \cdot y ^5=( x + y )^{( m +1)}$ and $\frac{d y}{d x}=\frac{y}{x}$ then $m =$ ?

✓
$8$
B
$4$
C
$5$
D
$20$

Answer

Correct option: A.

$8$

(a) 8
Hint:
If $x ^{ p } \cdot y ^{ q }=( x + y )^{ p + q }$, then $\frac{d y}{d x}=\frac{y}{x}$
$
\begin{aligned}
& \therefore m +1=4+5=9 \\
& \therefore m =8 .
\end{aligned}
$

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MCQ 31 Mark

If $a x^2+2 hxy + by ^2=0$, then $\frac{d y}{d x}=$ ?

A
$\frac{(a x+h y)}{(h x+b y)}$
✓
$\frac{-(a x+h y)}{(h x+b y)}$
C
$\frac{(a x-h y)}{(h x+b y)}$
D
$\frac{(2 a x+h y)}{(h x+3 b y)}$

Answer

Correct option: B.

$\frac{-(a x+h y)}{(h x+b y)}$

(b) $\frac{-(a x+h y)}{(h x+b y)}$

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MCQ 41 Mark

If $y =\log \left(\frac{e^x}{x^2}\right)$ then $\frac{d y}{d x}=$ ?

A
$\frac{2-x}{x}$
✓
$\frac{x-2}{x}$
C
$\frac{e-x}{e x}$
D
$\frac{x-e}{e x}$

Answer

Correct option: B.

$\frac{x-2}{x}$

(b) $\frac{x-2}{x}$
Hint:
$
\begin{aligned}
y & =\log \left(\frac{e^x}{x^2}\right)=\log e^x=\log x^2 \\
& =x-2 \log x \quad \ldots[\because \log e=1]
\end{aligned}
$
$\therefore \frac{d y}{d x}=1-\frac{2}{x}=\frac{x-2}{x}$

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MCQ 51 Mark

If $y =5^{ x } \cdot x ^5$, then $\frac{d y}{d x}=$ ?

A
$5^x \cdot x^4(5+\log 5)$
B
$5^x \cdot x^5(5+\log 5)$
✓
$5^x \cdot x^4(5+x \log 5)$
D
$5^x \cdot x^5(5+x \log 5)$

Answer

Correct option: C.

$5^x \cdot x^4(5+x \log 5)$

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MCQ 61 Mark

If $y=2 x^2+2^2+a^2$, then $\frac{d y}{d x}=$ ?

A
$x$
✓
$4 x$
C
$2 x$
D
$-2 x$

Answer

Correct option: B.

$4 x$

(b) $4 x$

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MCQ 71 Mark

If $y =e^{\log x}$ then $\frac{d y}{d x}=$ ?

✓
$\frac{e^{\log x}}{x}$
B
$\frac{1}{x}$
C
$0$
D
$\frac{1}{2}$

Answer

Correct option: A.

$\frac{e^{\log x}}{x}$

(a) $\frac{e^{\log x}}{x}$

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MCQ 81 Mark

If $y =\sqrt{x+\frac{1}{x}}$, then $\frac{d y}{d x}=$ ?

A
$\frac{x^2-1}{2 x^2 \sqrt{x^2+1}}$
B
$\frac{1-x^2}{2 x^2 \sqrt{x^2+1}}$
✓
$\frac{x^2-1}{2 x \sqrt{x} \sqrt{x^2+1}}$
D
$\frac{1-x^2}{2 x \sqrt{x} \sqrt{x^2+1}}$

Answer

Correct option: C.

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

(C) $\frac{x^2-1}{2 x \sqrt{x} \sqrt{x^2+1}}$
Hint:
$
\begin{gathered}
\frac{d y}{d x}=\frac{1}{2 \sqrt{x+\frac{1}{x}}} \cdot \frac{d}{d x}\left(x+\frac{1}{x}\right) \\
=\frac{\sqrt{x}}{2 \sqrt{x^2+1}}\left(1-\frac{1}{x^2}\right)=\frac{x^2-1}{2 x \sqrt{x} \sqrt{x^2+1}}
\end{gathered}
$

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MCQ 91 Mark

If $x =2 at ^2, y =4 at$, then $\frac{d y}{d x}=$ ?

A
$-\frac{1}{2 a t^2}$
B
$\frac{1}{2 a t^3}$
✓
$\frac{1}{t}$
D
$\frac{1}{4 a t^3}$

Answer

Correct option: C.

$\frac{1}{t}$

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MCQ 101 Mark

If $y =\left(5 x ^3-4 x ^2-8 x \right)^9$, then $\frac{d y}{d x}=$

✓
$9\left(5 x^3-4 x^2-8 x\right)^8\left(15 x^2-8 x-8\right)$
B
$9\left(5 x^3-4 x^2-8 x\right)^9\left(15 x^2-8 x-8\right)$
C
$9\left(5 x^3-4 x^2-8 x\right)^8\left(5 x^2-8 x-8\right)$
D
$9\left(5 x^3-4 x^2-8 x\right)^9\left(5 x^2-8 x-8\right)$

Answer

Correct option: A.

$9\left(5 x^3-4 x^2-8 x\right)^8\left(15 x^2-8 x-8\right)$

(a) $9\left(5 x^3-4 x^2-8 x\right)^8\left(15 x^2-8 x-8\right)$

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