Property 6

Q. Area enclosed by curve $y^3-9y+x=0$ and Y - axis is -

1. $81$
2. $\frac{9}{2}$
3. $9$
4. $\frac{81}{2}$

Q. Find the area enclosed by the curve $x=y^2+2$, ordinates y = 0 & y = 3 and the Y - axis.

1. 5
2. 10
3. 15
4. 20

Q. For any integer n the integral ${\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3(2n+1)xdx$ has the value

1. None of these
2. $1$
3. $0$
4. $\pi$

Q. The value of ${\int }_0^{\pi }\frac{1}{1+e^{\cos x}}\mathrm{dx}$ is

1. $\pi$
2. \N
3. $\frac{\pi }{2}$
4. $-\frac{\pi }{2}$

Q. If $I={\int }_0^{\pi /2}{\cos }^{n}x{\sin }^{n}x\mathrm{dx}=k{\int }_0^{\pi /2}{\sin }^{n}x\mathrm{dx}$, then k equals

1. $2^{-n+1}$
2. $2^{-n-1}$
3. $2^{-n}$
4. $2^{-1}$

Q. The value of ${\int }_0^{2\pi }{\sin }^{100}x{\cos }^{99}x\mathrm{dx}$is equal to

Q. If $I={\int }_0^1{\cot }^{-1}(1-x+x^2)\mathrm{dx}=k{\int }_0^1{\tan }^{-1}x\mathrm{dx}$, then the value of k equals

1. 1
2. 2
3. $\pi$
4. $2\pi$

Q. If $A_n=\underset{0}{\overset{\pi /2}{\int }}\frac{\sin (2n-1)x}{\sin x}dx,B_n=\underset{0}{\overset{\pi /2}{\int }}{\left(\frac{\sin nx}{\sin x}\right)}^{2}dx,$ for $n\in N,$ then

1. $A_{n+1}=A_n$
2. $B_{n+1}=B_n$
3. $A_{n+1}-A_n=B_{n+1}$
4. $B_{n+1}-B_n=A_{n+1}$

Q. $\begin{array}{cc}\text{List I}& \text{List II}\\ \text{(I)}{\int }_{-1}^1\frac{dx}{(1+x^2)(1+3^x)}\text{is}& \text{(P)}\frac{\pi }{2}\\ \text{(II)}7\pi {\int }_0^1(1-x^2{)}^{\frac{3}{2}}{x}^{3}dx\text{is}& \text{(Q)}\frac{\pi }{4}\\ \text{(III)}\frac{\pi }{4}.\frac{{\int }_0^{10}\left[x\right]dx}{{\int }_0^{10}\left\{x\right\}dx},& \\ \text{(where [.] {.} denotes}& \text{(R)}\frac{9\pi }{4}\\ \text{greatest integer and}& \\ \text{fractional part function)}& \\ \text{(IV)}{\int }_0^{\frac{\pi }{4}}[\tan x+[\sin x+[\cos x+[\sec x\left]\right]\left]\right]dx,& \text{(S)}\frac{2\pi }{5}\\ \text{(where [.] denotes greatest integer function)}& \end{array}$

Which of the following is only CORRECT combination

1. (II)-(R)
2. (IV)-(P)
3. (IV)-(Q)
4. (II)-(Q)

Q. Let $f\left(x\right)$ be a function satisfying $f\left(x\right)+f(x+2)=20\forall x\in R$, then

1. $f\left(x\right)$ is a periodic function
2. ${\int }_{-3}^{13}f\left(x\right)dx=80$
3. ${\int }_{-1}^{7}f\left(x\right)dx=80$
4. $f\left(x\right)$ is a many one function