Addition Rule of Differentiation

Q. If $y=\sqrt{x}+\frac{1}{\sqrt{x}},$ then $\frac{dy}{dx}$ at x=1 is

1. 1
   
2. 0
3. $\frac{1}{2}$
   
4. $\frac{1}{\sqrt{2}}$
   

Q.

If $A,B$ and $C$are three sets such that $A\cap B=A\cap C$and $A\cup B=A\cup C$, then

1. $A=C$

2. $B=C$

3. $A\cap B=\varphi$

4. $A=B$

Q.

The value of $\left|\begin{array}{ccc}(a+1)(a+2)& a+2& 1\\ (a+2)(a+3)& a+3& 1\\ (a+3)(a+4)& a+4& 1\end{array}\right|$ is

1. $-2$

2. $(a+1)(a+2)(a+3)$

3. $0$

4. $(a+2)(a+3)(a+4)$

Q.

If $f\left(x\right)=\frac{x-4}{2\sqrt{x}},$ then $f^{\prime }\left(1\right)$ is

1. $\frac{5}{4}$
   
2. $\frac{4}{5}$
   
3. $1$
   
4. $-1$
   

Q.

If $y=\sin \left(\sqrt{\sin x+\cos x}\right)$, then $\frac{dy}{dx}=$

1. $\frac{\cos \left(\sqrt{\sin x+\cos x}\right)\left(\cos x-\sin x\right)}{2\sqrt{\sin x+\cos x}}$

2. $\frac{\cos \left(\sqrt{\sin x+\cos x}\right)}{2\sqrt{\sin x+\cos x}}$

3. $\frac{\cos \left(x\sqrt{\sin x+\cos x}\right)\left(\cos x-\sin x\right)}{2\sqrt{\sin x-\cos x}}$

4. none of these.

Q.

Find the simplified form of $co{s}^{-1}(\frac{3}{5}cosx+\frac{4}{5}sinx)wherex\in [\frac{-3\pi }{4},\frac{\pi }{4}].$

Q. If $f\left(x\right)=1-x+x^2-x^3+...-x^{99}+x^{100},$ then f'(1) equals

1. 150
   
2. -50
   
3. -150
   
4. 50

Q. If $f\left(x\right)=1+x+\frac{x^2}{2}+...+\frac{x^{100}}{100},$ then f'(1) is equal to

1. 50
   
2. 0
3. 100
   
4. $\frac{1}{100}$
   

Q.

If ${}^{2n+1}P_{n-1}:{}^{2n-1}{P}_n=3:5$, then the value of $n$ is equal to:

1. $4$

2. $3$

3. $2$

4. $1$

5. $5$

Q. If $f$ is a function satisfying $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,y\in N$ such that
$f\left(1\right)=3$ and ${\sum }_{x=1}^{n}f\left(x\right)=120$ find the value of $n$.

Q. If $\sin x+\cos x=\frac{1}{5},$ then sum of the possible value(s) of $|12\tan x|$ is equal to

1. $16$
2. $9$
3. $81$
4. $25$

Q.

Differentiate the function given below w.r.t. $x:$

$(2x-7{)}^2(3x+5{)}^3$

Q. The value of the expression $\tan \left(\frac{1}{2}{\cos }^{-1}\frac{2}{\sqrt{5}}\right)$ is

1. $2-\sqrt{5}$
2. $\sqrt{5}-2$
3. $\frac{\sqrt{5}-2}{2}$
4. $5+\sqrt{2}$

Q.

Let $F_1\left(A,B,C\right)=\left(A\wedge ~B\right)\vee \left[~C\wedge \left(A\vee B\right)\right]\vee ~A$ and $F_2\left(A,B\right)=\left(A\vee B\right)\vee \left(B\to ~A\right)$be two logical expressions. Then:

1. $F_1$ is not a tautology but $F_2$ is a tautology

2. $F_1$ is a tautology but $F_2$ is not a tautology

3. $F_1$ and $F_2$ both area tautologies

4. Both $F_1$ and $F_2$ are not tautologies

Q. If $5f\left(x\right)+3f\left(\frac{1}{x}\right)=x+2$ and $y=xf\left(x\right),then(\frac{dy}{dx}{)}_{x=1}$ is equal to

1. $14$
2. $1$
3. $\frac{7}{8}$
4. $Noneofthese$

Q. If $f$ and g are two real valued functions defined as

$f\left(x\right)=2x+1,g\left(x\right)=x^2+1$, then find:

(i) $f+g$

(ii) $f-g$

(iii) $fg$

(iv) $\frac{f}{g}$

Q.

Divide as directed:$52pqr(p+q)(q+r)(r+p)÷104pq(q+r)(r+p)$

Q.

What is the derivative of $y=(1-x)(2-x)....(n-x)$ at $x=1$?

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=\log \left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right)$ satisfies the equation

1. $\mathrm{f}\left(\mathrm{x}+2\right)-2\mathrm{f}\left(\mathrm{x}+1\right)+\mathrm{f}\left(\mathrm{x}\right)=0$

2. $\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}+1\right)=\mathrm{f}\left\{\mathrm{x}\left(\mathrm{x}+1\right)\right\}$

3. $\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right)=\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{y}}{1+\mathrm{xy}}\right)$

4. $\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)=\mathrm{f}\left(\mathrm{x}\right)·\mathrm{f}\left(\mathrm{y}\right)$

Q.

Prove that: $x^{2n}-y^{2n}$ is divisible by $x+y$.

Q.

Differentiate the following functions with respect to x :

$\frac{1}{sinx}+2^{x+3}+\frac{4}{lo{g}_{x}3}$

Q. If $f\left(x\right)={\log }_5{\log }_{3}x,$ then $f^{\prime }\left(e\right)$ is equal to

1. $e{\log }_{e}5$
2. $e{\log }_{e}3$
3. $\frac{1}{e{\log }_{e}5}$
4. $\frac{1}{e{\log }_{e}3}$

Q. If $y=\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...,$ then $\frac{dy}{dx}=$

1. $y^2$
2. y
   
3. y+1
   
4. y-1
   

Q. If $g\left(x\right)=\ln \left[f\right(x\left)\right]$ and $f(x+1)=xf\left(x\right),$ then $g^{\prime \prime }\left(\frac{5}{2}\right)-g^{\prime \prime }\left(\frac{1}{2}\right)$ is equal to

1. $-\frac{40}{9}$
2. $\frac{40}{9}$
3. $-\frac{20}{9}$
4. $\frac{20}{9}$

Q. If Sn be the sum of n terms of an AP and Spn/Sn is independent of n then the common difference is?

Q. Integral of $\sqrt{1+2cotx(cotx+cosecx)}$ with respect to x is:

1. In sin x - ln(cosecx+cot x) + c
2. 2 In $cos\frac{x}{2}+c$
3. 2 In $sin\frac{x}{2}+c$
4. $\frac{1}{2}$ In $cos\frac{x}{2}+c$

Q. (i) Find the following integral: \( \int \dfrac{x+2}{2x^2+6x+5}.dx\)

(ii) Find the following integral: \( \int \dfrac{x+3}{\sqrt{5-4x-x^2}}.dx\)

Q. Differentiate the following w.r.t. x :

Q. Integrate the function: $\frac{1}{x\sqrt{ax-x^2}}$

Q. $y=\frac{2x^9}{3}-\frac{5}{7}{x}^7+6x^3-x,$ find $\frac{dy}{dx}$ at x=1.