2 Marks Questions

Question 12 Marks

If $x=a \sin p t, y=b \cos p t$, then find $\frac{d y}{d x}$ at $t=0$.

Answer

Given $x=a \sin p t$
$\therefore \quad \frac{d x}{d t}=a p \cos p t$
and $\quad y=b \cos p t$
$\therefore \quad \frac{d y}{d t}=-b p \sin p t$
$\therefore \quad\left(\frac{d y}{d x}\right)_{i=0}=-\frac{b}{a} \tan (p t)$
= 0

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Question 22 Marks

Differentiate $3^x+x^3+4 x-5$ with respect to $x$

Answer

Let, $f(x)=3^x+x^3+4 x-5$
$\therefore \quad \frac{d}{d x} f(x)=f^{\prime}(x)$
$\Rightarrow \quad f^{\prime}(x)=\frac{d}{d x}\left(3^x+x^3+4 x-5\right)$
$\begin{array}{l}=3^x \log _e 3+3 x^2+4 \times 1-0 \\ = 3^x \log _e 3+3 x^2+4\end{array}$

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Question 32 Marks

Differentiate $\frac{x}{\sin x}$ with respect to $x$.

Answer

Let, $f(x)=\frac{x}{\sin x}$
$\therefore \quad f^{\prime}(x)=\frac{d}{d x}\left(\frac{x}{\sin x}\right)$
$\begin{array}{l}=\frac{\sin x \frac{d}{d x} x-x \frac{d}{d x} \sin x}{(\sin x)^2} \\ =\frac{\sin x \cdot 1-x \cdot \cos x}{\sin ^2 x} \\ =\operatorname{cosec} x-x \cot x \operatorname{cosec} x \\ =(1-x \cot x) \cdot \operatorname{cosec} x\end{array}$

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Question 42 Marks

Find the value of $k$ for which the function.
$
f(x)=\left\{\begin{array}{cc}
\frac{x^2+3 x-10}{x-2}, & x \neq 2 \\
k, & x=2
\end{array}\right.
$
is continuous at $x=2$.

Answer

$\lim _{x \rightarrow 2} f(x)=f(2)=k$
$\lim _{x \rightarrow 2} \frac{(x+5)(x-2)}{x-2}=k \quad(x \neq 2)$
$\therefore \quad k=7$.

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Question 52 Marks

Show that $\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$ does not exist.

Answer

Let $x-4=n \Rightarrow$ As $x \rightarrow 4, n \rightarrow 0$
$
\begin{aligned}
\therefore \text { L.H.L } & =\lim _{n \rightarrow 0-} \frac{|n|}{n}=\lim _{n \rightarrow 0-} \frac{-n}{n} \\
& =\lim _{n \rightarrow 0-}(-1)=-1
\end{aligned}
$
$\begin{aligned} \text { R.H.L } & =\lim _{n \rightarrow 0+} \frac{|n|}{n}=\lim _{n \rightarrow 0+} \frac{n}{n} \\ & =\lim _{n \rightarrow 0+}(1)=1\end{aligned}$
Since L.H.L $\neq$ R.H.L
$\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$ does not exist.

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Question 62 Marks

Evaluate the left hand and right hand limits of the following function at $x=2$. Does $\lim _{x \rightarrow 2} f(x)$ exist ?
$
f(x)=\left\{\begin{array}{cll}
2 x+3, & \text { if } & x \leq 2 \\
x+5 & \text { if } & x>2
\end{array}\right.
$

Answer

$\begin{aligned} \text { L.H.L } & =\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{-}}(2 x+3) \\ & =4+3 \\ & =7\end{aligned}$
and
$\begin{aligned} \text { R.H.L } & =\lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2^{+}}(x+5) \\ & =2+5 \\ & =7\end{aligned}$
Since $\quad \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=7$
$\therefore \lim _{x \rightarrow 2} f(x)$ exists and is equal to 7.

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Question 72 Marks

Evaluate $\lim _{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}$.

Answer

$\begin{array}{l} \lim _{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1} \\ =\lim _{x \rightarrow 1} \frac{\left(x^5\right)^3-1^3}{\left(x^5\right)^2-1^2}\end{array}$
$\begin{array}{l}\quad \because a^3-b^3=(a-b)\left(a^2+b^2+a b\right) \\ =\lim _{x \rightarrow 1} \frac{\left(x^5-1\right)\left(x^{10}+x^5+1\right)}{\left(x^5-1\right)\left(x^5+1\right)} \quad[x \neq 1] \\ =\lim _{x \rightarrow 1} \frac{x^{10}+x^5+1}{x^5+1}\end{array}$
$\begin{array}{l}=\frac{1^{10}+1^5+1}{1^5+1} \\ =\frac{3}{2} .\end{array}$

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Question 82 Marks

Find the derivative of $\left(x^2+1\right) \cos x$.

Answer

Let $\quad y=\left(x^2+1\right) \cos x$
$\therefore \quad \frac{d y}{d x}=\frac{d}{d x}\left[\left(x^2+1\right) \cos x\right]$
$\begin{array}{l}=\cos x \frac{d}{d x}\left(x^2+1\right)+\left(x^2+1\right) \frac{d}{d x} \cos x \\ =\cos x \cdot 2 x+\left(x^2+1\right) \cdot(-\sin x) \\ =2 x \cos x-x^2 \sin x-\sin x\end{array}$

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Question 92 Marks

Find the derivative of $\operatorname{cosec} x \cot x$.

Answer

Let $\quad y=\operatorname{cosec} x \cot x$
$\therefore \quad \frac{d y}{d x}=\frac{d}{d x}[\operatorname{cosec} x \cot x]$
$\begin{array}{l}=\left(\frac{d}{d x} \operatorname{cosec} x\right) \cot x+\operatorname{cosec} x \frac{d}{d x} \cot x \\ =\cot x \cdot(-\cot x \operatorname{cosec} x)+\operatorname{cosec} x \\ =-\cot ^2 x \operatorname{cosec} x-\operatorname{cosec}^3 x \\ \left.=-\operatorname{cosec}^2 x\right) x\left(\cot ^2 x+\operatorname{cosec}^2 x\right)\end{array}$

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Question 102 Marks

Find the derivative of $f\left(e^{\tan x}\right)$ w.r.t. $x$ at $x=0$. It is given that $f^{\prime}(1)=5$.

Answer

Given $f\left(e^{\tan x}\right)$
$\therefore \quad y=f\left(e^{\tan x}\right)$
$\frac{d y}{d x}=f^{\prime}\left(e^{\tan x}\right) \frac{d}{d x}\left(e^{\tan x}\right)$
or $\quad \frac{d y}{d x}=f^{\prime}\left(e^{\tan x}\right) e^{\tan x} \sec ^2 x$
Now $\quad\left[\frac{d y}{d x}\right]_{x=0}=f^{\prime}\left(e^{\tan 0}\right) e^{\tan 0} \sec ^2 0$
$\begin{array}{l}=f^{\prime}\left(e^0\right) e^0 1 \\ =f^{\prime}(1)\end{array}$
$\left[\frac{d y}{d x}\right]_{x=0}=5 . \quad\left[\right.$ given $\left.f^{\prime}(1)=5\right]$

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Question 112 Marks

Differentiate $\sin ^2\left(x^2\right)$ w.r.t. $x^2$.

Answer

Let $x^2=t$
So,
$
\begin{aligned}
\frac{d\left[\sin ^2(x)^2\right]}{d x^2} & =\frac{d\left(\sin ^2 t\right)}{d t} \\
& =2 \sin t \times(\sin t)^{\prime} \\
& =2 \sin t \times \cos t \\
& =2 \sin \left(x^2\right) \cos \left(x^2\right)
\end{aligned}
$

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Question 122 Marks

If $f(x)=x+7$ and $g(x)=x-7, x \in R$, then find $\frac{d}{d x}(f o g)(x)$

Answer

Given, $f(x)=x+7 \text { and } g(x)=x-7$
Now, $f o g=f(g(x))$
$=f(x-7)=(x-7)+7=x$
$\therefore \quad \frac{d}{d x}[(f o g)(x)]=\frac{d}{d x}(x)$
=1

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Question 132 Marks

Determine the value of the constant ' $k$ ' so that the function $f(x)=\left\{\begin{array}{cll}\frac{k}{|x|}, & \text { if } & x<0 \\ 3, & \text { if } & x \geq 0\end{array}\right.$ is continuous at $x=0$.

Answer

Since $f$ is continuous at $x=0$,$
\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)
$
Here
$
\begin{aligned}
f(0) & =3, \\
LHL & =\lim _{x \rightarrow 0^{-}} f(x)
\end{aligned}
$
$=\lim _{x \rightarrow 0^{-}} \frac{k x}{|x|}=\lim _{x \rightarrow 0^{-}} \frac{k x}{-x}=-k$
$\therefore \quad-k=3$ or $k=-3$.

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Question 142 Marks

Evaluate $\lim _{x \rightarrow 3} \frac{\sqrt{2 x+3}}{x+3}$

Answer

$\lim _{x \rightarrow 3} \frac{\sqrt{2 x+3}}{x+3}$
$\begin{array}{l}=\frac{\sqrt{2 \cdot 3+3}}{3+3} \\ =\frac{\sqrt{9}}{6} \\ =\frac{3}{6}=\frac{1}{2} .\end{array}$

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Question 152 Marks

If the following function $f(x)$ is continuous at $x=0$, then write the value of $k$.
$f(x)=\left\{\begin{array}{cc}\frac{\sin \frac{3 x}{2}}{x}, & x \neq 0 \\ k, & x=0\end{array}\right.$

Answer

$\lim _{x \rightarrow 0} \frac{\sin \frac{3 x}{2}}{x}=\lim _{x \rightarrow 0} \frac{3}{2} \cdot \frac{\sin \frac{3 x}{2}}{\frac{3 x}{2}}=\frac{3}{2}$
or $k=\frac{3}{2} . \quad\left[\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\right]$

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Question 162 Marks

Find $n$, if $\lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80, n \in N$.

Answer

Given, $ \lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80$
$\Rightarrow n \cdot 2^{n-1}=80\left[\because \lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}\right]$
$
n \cdot 2^{n-1}=5 \cdot 2^{5-1}
$
$\therefore$ Value of $n$ is 5 .

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Question 172 Marks

Evaluate $\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots .+n}{n^2}$.

Answer

$\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots .+n}{n^2}$
$\begin{array}{l}=\lim _{n \rightarrow \infty} \frac{\frac{n(n+1)}{2}}{n^2} \\ =\frac{1}{2} \lim _{n \rightarrow \infty} \frac{n^2+n}{n^2} \\ =\frac{1}{2} \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right) \\ =\frac{1}{2}(1+0) \\ =\frac{1}{2} \times 1=\frac{1}{2} .\end{array}$

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Applied Maths 2 Marks Questions

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