Property 6

Q. Find the value of k, if f(x) is continuous at x = 0: f(x) = (Sin3x/2)/x , x not equal to 0 k , x = 0

Q. The value of $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\sin }^{3}x}{\sin x+\cos x}dx$ is:

1. $\frac{\pi -1}{2}$
2. $\frac{\pi -1}{4}$
3. $\frac{\pi -2}{8}$
4. $\frac{\pi -2}{4}$

Q. The product of all positive real values of $x$ satisfying the equation $x^{\left(16\right({\log }_{5}x{)}^3-68{\log }_{5}x)}=5^{-16}$ is

Q.

Evaluate :${\int }_{-2}^2\left|\left[x\right]\right|dx$

1. $1$

2. $2$

3. $3$

4. $4$

5. $5$

Q. If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, then for any natural number, find the value of Det(Aⁿ).

Q. $\underset{0}{\overset{2a}{\int }}{x}^3\sqrt{2ax-x^2}dx$ is equal to

1. $\frac{7}{4}{a}^5$
2. $\frac{7\pi }{8}{a}^5$
3. $\frac{7\pi }{8}{a}^4$
4. $\frac{7\pi }{8}{a}^3$

Q. If $\sum _{i=1}^{10}{\sin }^{-1}{x}_i=5\pi$, then the value of $\sum _{i=1}^{10}{x}_i^2$ is

Q. The value of the integral $\underset{0}{\overset{1}{\int }}x{\cot }^{-1}(1-x^2+x^4)dx$ is :

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

Q. If $a_1,a_2$ and $a_3$ are the three value of $‘a^{\prime }$ which satisfy the equation $\underset{0}{\overset{\pi /2}{\int }}(\sin x+a\cos x{)}^{3}dx-\frac{4a}{\pi -2}\underset{0}{\overset{\pi /2}{\int }}x\cos xdx=2$, then the value of $a_1^2+a_2^2+a_3^2$ is

1. $\frac{21}{2}$
2. $21$
3. $7$
4. $\frac{21}{4}$

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

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

Q. Evaluate the definite integral as limit of sums:
${\int }_a^{b}xdx$

Q. If the $9^{\text{th}}$ term in the expansion of ${(3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{-\frac{1}{8}{\log }_3(5^{x-1}+1)})}^{10}$ is $180$, then the value(s) of $x$ is/are

1. $2$
2. $1$
3. ${\log }_{5}3$
4. ${\log }_{5}15$

Q.

the question given below is of matrix :-

If A=[cos2theta sin2theta]

[-sin2theta cos2theta] find A^2

Q. $f\left(x\right)=\{\begin{array}{c}x|x|x\le -1\\ [1+x]+[1-x]-1<x<1\\ -x|x|x\ge 1\end{array}$, then $f\left(x\right)$ is

1. both even as well as odd function
2. an even function
3. an odd function
4. neither even nor odd function

Q. The value of $\sum _{x=0}^4{\sin }^{-1}\left(\sin x\right)$ is equal to

1. $3\pi -8$
2. $3\pi -7$
3. $3\pi -9$
4. $3\pi -6$

Q. $\int \sqrt{1-4x-x^2}dx$ is equal to
$($where $C$ is integration constant$)$

1. $\frac{x+2}{2}\sqrt{1-4x-x^2}+\frac{5}{2}{\sin }^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+C$
2. $\frac{x}{2}\sqrt{1-4x-x^2}+\frac{5}{2}{\sin }^{-1}\left(\frac{x}{\sqrt{5}}\right)+C$
3. $\frac{5}{2}{\sin }^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+C$
4. $\frac{x}{2}\sqrt{1-4x-x^2}+\frac{5}{2}\ln |x+2+\sqrt{1-4x-x^2}|+C$

Q. The area bounded by the x-axis and the curve $y=4x–x^2–3$ is

1. $\frac{2}{3}$
2. $\frac{1}{3}$
3. $\frac{8}{3}$
4. $\frac{4}{3}$

Q. The value of integral $\underset{-1/\sqrt{3}}{\overset{1/\sqrt{3}}{\int }}\frac{x^4}{1-x^4}\cdot {\cos }^{-1}\left(\frac{2x}{1-x^2}\right)dx$ is

1. $\frac{\pi }{12}[\pi +3\ln (2+\sqrt{3})-4\sqrt{3}]$
2. $\frac{\pi }{6}[\pi +\ln (2+\sqrt{3})-2\sqrt{3}]$
3. $\frac{\pi }{4}[\pi +\ln (2+\sqrt{3})+\sqrt{3}]$
4. $\frac{\pi }{3}[\pi +2\ln (2+\sqrt{3})+3\sqrt{3}]$

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. The value of integral $\underset{0}{\overset{\pi /2}{\int }}\frac{{\sin }^{5/2}x}{{\sin }^{5/2}x+{\cos }^{5/2}x}dx$ is

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

Q. $\underset{0}{\overset{1}{\int }}{x}^4(1-x^2{)}^{\frac{3}{2}}dx$ is equal to

1. $\frac{3\pi }{256}$
2. $\frac{\pi }{256}$
3. $\frac{3\pi }{128}$
4. $\frac{\pi }{64}$

Q. Give an example of a statement P(n) which is true for all $n\ge 4$ but P(1), P(2) and P(3) are not true. Justify your answer.

Q. The value of the definite integral $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\ln (1+\tan x)dx$ is:

1. $\frac{\pi }{4}\ln 2$
2. $\frac{\pi }{8}\ln 2$
3. $\pi \ln 2$
4. $\ln 2$

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. If $I\left(a\right)=\underset{0}{\overset{\pi /2}{\int }}\ln \left(\frac{1+a\sin x}{1-a\sin x}\right)\frac{dx}{\sin x},$ then value of $\frac{dI\left(a\right)}{da},$ is (where $|a|<1$)

1. $-\frac{\pi }{\sqrt{1-a^2}}$
2. $-\pi \sqrt{1-a^2}$
3. $\sqrt{1-a^2}$
4. $\frac{\pi }{\sqrt{1-a^2}}$

Q. The value of $\underset{0}{\overset{\pi /2}{\int }}\left[\frac{\left({\sin }^{1/3}x\right)}{({\sin }^{1/3}x+{\cos }^{1/3}x)}\right]dx$ is

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

Q.

If $n$ is a positive integer, prove that
$|\mathrm{Im}\left(z^n\right)|\le n|\mathrm{Im}\left(z\right)|{\left|z\right|}^{n-1}$

Q. Prove that lim(h->0) {sin(x+h)-sin x}/h = lim(h->0) {2sin(h/2) cos (x+h/2)}/2(h/2)

Q. Let the curve $y=y\left(x\right)$ be the solution of the differential equation. $\frac{dy}{dx}=2(x+1).$. If the numerical value of area bounded by the curve $y=y\left(x\right)$ and $x-$axis is $\frac{4\sqrt{8}}{3}$, then the value of $y\left(1\right)$ is equal to