### Mathematics Assignments | PDF | 8 to 12

PDF Files of Mathematics Assignments From VIII Standard to XII Standard PDF of mathematics Assignments for the students from VIII standard to XII standard.These assignments are strictly according to the CBSE and DAV Board Final question Papers

### Math Assignment Class XII Ch-5 | Derivatives

Math Assignment Class XII Chapter 5 Derivatives

Question 1 Use chain rule to find the derivative of

$y=\left(\frac{2x-1}{2x+1}\right)^{2}\;\;\;\:Ans:\frac{8(2x-1)}{(2x+1)^{3}}$

Question 2. Differentiate the following w.r.t. x

$y=log_{10}x+log_{x}10+log_{x}x+log_{10}10$

$\frac{dy}{dx}=\frac{1}{xlog10}-\frac{log10}{x(logx)^{2}}$

Solution Hint

$\frac{logx}{log10}+\frac{log10}{logx}+\frac{logx}{logx}+\frac{log10}{log10}$

$=\frac{logx}{log10}+\frac{log10}{logx}+1+1$

Now differentiating w. r. t. x and taking log10 as constant.

Question 3. Differentiate the following w. r. t. x at x = 1

$y=e^{x(1+logx)}$
Question 4

$If\:\:\frac{x}{x-y}=log\frac{a}{x-y}$ ,  Then prove that :   $\frac{dy}{dx}=\frac{2y-x}{y}$
Question 5: Differentiate log(x ex)  w. r. t.   xlogx.

Answer 5:   $\frac{1+x}{x(1+logx)}$

Question 6: Differentiate x2 w. r. t. x3.        ...........    Ans:  2/3x.

Question 7$If\:e^{x}+e^{y}=e^{x+y}$, Prove that   $\:\:\frac{dy}{dx}+e^{y-x}=0$

Solution Hint:

$e^{x}+e^{y}=e^{x+y}$
Dividing on both side by   $e^{x+y}$

$\frac{e^{x}}{e^{x+y}}+\frac{e^{y}}{e^{x+y}}=1$

$\Rightarrow e^{-y}+e^{-x}=1$
Now differentiating w. r. t. x we get the required answer.
Question 8: Differentiate  (xx) w. r. t. x
$\frac{dy}{dx}=x^{x^{2}}[x+2xlogx]$
Solution Hint:
$(x^{x})^{x}=x^{x^{2}}$  Now differentiating w. r. t. x
Question9.   Differentiate x16y9 = (x2 + y)17 w. r. t. x ,
Answer :    $\frac{dy}{dx}=\frac{2y}{x}$
Question10) Differentiate  (logx)x + xlogx, w. r. t. x.
$(logx)^{x}\left[\frac{1}{logx}+log(logx)\right]+2x^{logx-1}.logx$

Question11
$If\:\:f(x)=|cosx|,find\:f'\left(\frac{3\pi}{4}\right)$
Answer: $\frac{1}{\sqrt{2}}$
Question 12
$If\:\:y=\sqrt{cosx+\sqrt{cosx+\sqrt{cos+....\infty}}}$
Then prove that :   $(2y-1)\frac{dy}{dx}=-sinx$

Solution 12:

$y=\sqrt{cosx+\sqrt{cosx+\sqrt{cos+....\infty}}}$

$y=\sqrt{cosx+y}\:\:\Rightarrow\:\:y^{2}=cosx+y$

Differentiating both side w. r. t. x we get
$2y\frac{dy}{dx}=-sinx+\frac{dy}{dx}$

$(2y-1)\frac{dy}{dx}=-sinx$

Question 13. Find dy/dx at x = 1, y = Ï€/4, If Sin2y + cosxy = k
Answer 13.  $\frac{\pi(\sqrt{2}+1)}{4}$

Question 14. If x = a(2Î¸ - sin2Î¸) and y = a(1-cos2Î¸), find dy/dx, when Î¸ = Ï€/4.
Question 15. Differentiate Log(cosex) w. r. t x
Ans[-ex tanx].
Question 16.
$If\:y=\sqrt{tanx+\sqrt{tanx+\sqrt{tanx+......\infty}}}$  then prove that

$(2y-1)\frac{dy}{dx}=Sec^{2}x$
Question 17.
$If\:y=\frac{sin^{-1}x}{\sqrt{1-x^{2}}},$  then prove that   $(1-x^{2})\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}-y=0$

Question 18.
$If\:x=cost+log\:tan\frac{t}{2},\:y=sint$ , then find the value of   $\frac{d^{2}y}{dx^{2}}\:at\:\:t=\frac{\pi}{4}$
$Ans:\: 2\sqrt{2}$
Question 19: Differentiate the following w. r. t. x

$y=sin^{-1}\left(\frac{2^{x+1}}{1+4^{x}}\right)$
Ans:  $\frac{2^{x+1}}{1+4^{x}}log\:2$

Question 20. Differentiate the following w. r. t. x

$y=sin^{-1}\left\{\frac{2^{x+1}.3^{x}}{1+(36)^{x}}\right\}$     Ans:   $\frac{2(log6)6^{x}}{1+(36)^{x}}$
Question 21. Differentiate:  $y=e^{e^{x}}\:w.\:r.\:t.\:x.$
Ans:  $e^{e^{x}}\times e^{x}$
Question 22.
Find the value of k if the following function is continuous.
$\inline \LARGE \bg{black}\mathbf{f(x)=\left\{\begin{matrix}\frac{kcosx}{\pi-2x}\;\;,\;\;x<\frac{\pi}{2}\\\\\frac{3tan2x}{2x-\pi}\;\;,\;\;x>\frac{\pi}{2}\end{matrix}\right.}$
Question 23.
Find the value of k for which the following function is continuous at x = 0
$\inline \LARGE \bg{black}\mathbf{f(x)=\left\{\begin{matrix}\frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}\;\;,\;\;-1\leq x<0\\\\\frac{2x+1}{x-2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:,\:\:\:\:\:\:0\leq x<1\end{matrix}\right.}$
Question 24:
Show that f(x) = |x - 2| is continuous but not differentiable at x = 2.

Question 25:
Discuss the continuity and differentiability of the following function.
$\inline \LARGE \bg{black}\mathbf{f(x)=\left\{\begin{matrix}1-x\:\:\:\:\:\:\:\:,\:\:x<1\\(1-x)(2-x),\:1\leq x\leq 2\\3-x\:\:\:\:\:,\:\:\:\:\:x>2\end{matrix}\right.}$
f(x) is continuous at x = 1 but not continuous at x = 2
f(x) is differentiable at x = 1.
As f(x) is not continuous at x = 2 so f(x) is not differentiable at x = 2.

Question 26
Check the differentiability of the function

$f(x)=\left\{\begin{matrix}x,&x<1\\2-x&1\leq x\leq 2&\;\;at\;\;\;x=2\\-2+3x-x^{2},&x>2\\\end{matrix}\right.$
Solution Hint

$LHD=\displaystyle\lim_{x\to a^{-}}\frac{f(x)-f(a)}{x-a}=-1$

$RHD=\displaystyle\lim_{x\to a^{+}}\frac{f(x)-f(a)}{x-a}=-1$

LHD = RHD  ⇒ f(x) is differentiable at x = 2

Question 27
Question Check the differentiability of
$\inline \large \bg{black}\mathbf{f(x)=\left\{\begin{matrix}x^{2}+1,&0\leq x<1\\3-x,&1\leq x\leq 2\\\end{matrix}\right.\;\;at\;x=1}$
Solution Hint:
At x = 2,  LHD = 2 and RHD = -1
As LHD ≠ RHD, this function is not differentiable
Question 28

If f(x) = (x + 1)cotx  is continuous at x = 0 then find f(0).
Solution:  We know that

$\displaystyle\lim_{x\to 0}f(x)=f(0)$
Now, f(x) = (x+1)cotx

Log[f(x)] = Log[(x+1)cotx ]

Log[f(x)] = cotx Log(x+1)

$\displaystyle\lim_{x\to 0}Log[f(x)]=\displaystyle\lim_{x\to 0}cotx\;log(x+1)$

$Log\displaystyle\lim_{x\to 0}f(x)=\displaystyle\lim_{x\to 0}\left(\frac{x}{tanx}\right)\;\left(\frac{log(x+1)}{x}\right)$

$Log\displaystyle\lim_{x\to 0}f(x)=1\times 1=1$
$\displaystyle\lim_{x\to 0}f(x)=e^{1}$
$\displaystyle\lim_{x\to 0}f(x)=e=f(0)$

Question 29 :  Find   $\frac{d}{dx}|x|$
Solution:  $\frac{d}{dx}|x|=\frac{d}{dx}\sqrt{x^{2}}$
$\frac{d}{dx}|x|=\frac{1}{2\sqrt{x^{2}}}\times 2x=\frac{x}{|x|}$

Question 30 :

Check whether the function f(x) = x2 |x| is differentiable at x = 0 or not.

Solution Hint

This function can be written as  $\inline \large \bg{black} f(x)=\left\{\begin{matrix}x^{3},&x\geq 0\\-x^{3},&x\leq 0\\\end{matrix}\right.$

Find LHD and RHD we get

LHD = RHD = 0

Yest this function is differentiable.

Question 31 :

If  $\inline \large \bg{black}\mathbf{y=\sqrt{tan\sqrt{x}}}$ , then find dy/dx

Answer:  $\inline \LARGE \bg{black}\mathbf{\frac{dy}{dx}}=\frac{1+y^{4}}{4y}$

Question 33 :

If  $\inline \large \bg{black}\mathbf{\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)}$  then prove that   $\inline \large \bg{black}\mathbf{\frac{dy}{dx}=\sqrt{\frac{1-b^{2}}{1-x^{2}}}}$

Question 34 :

If  y = cosec(cot-1x), then prove that

$\inline \large \bg{black}\mathbf{\sqrt{1+x^{2}}\frac{dy}{dx}-x=0}$

Question 35 :

If x = ecos3t  and y = esin3t  , prove that   $\inline \large \bg{black}\mathbf{\frac{dy}{dx}=\frac{-ylogx}{xlogy}}$

Question 36 :

If x = ex/y, then prove that  $\inline \large \bg{black}\mathbf{\frac{dy}{dx}=\frac{logc-1}{(logx)^{2}}}$

Question 37:

If y = cos3(sec22t), find dy/dt

Question 38

If xy = ex - y, prove that  $\mathbf{\frac{dy}{dx}=\frac{logx}{(1+logx)^{2}}}$

Question 39

Given that y = (sinx)x . xsinx + ax, find dy/dx

Question 40

If x = a sin3Î¸, y = bcos3Î¸, then find $\bg{black}\mathbf{\frac{d^{2}y}{dx^{2}}}$   at Î¸ = Ï€/4

## PDF form of Assignment

1. Good and better also
Mujhe bohot help hua hai isse
Thz for help