Property 4

Q. Write the maximum value of 12 sin x − 9 sin² x.

Q. Let $f$ and $g$ be continuous function on $[0,a]$ such that $f\left(x\right)=f(a-x)$ and $g\left(x\right)+g(a-x)=4,$ then $\underset{0}{\overset{a}{\int }}f\left(x\right)g\left(x\right)dx$ is equal to :

1. $\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
2. $2\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
3. $4\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
4. $-3\underset{0}{\overset{a}{\int }}f\left(x\right)dx$

Q. Ques- Find the value of sin[2cos inverse root 5 by 3]

Q. For a function $f\left(x\right)\ge 0\forall xϵ(a,b);a<b\left|{\int }_a^{b}f\right(x\left)dx\right|=5thenfind{\int }_a^b|f\left(x\right)|dx$ ?

1. 3
2. 4
3. 5
4. 6

Q. If $I_1={\int }_0^{\pi }x\sin x{e}^{\cos 4x}dx\&I_2={\int }_0^{\frac{\pi }{2}}\cos x{e}^{\cos 4x}dx$, then the value of $[\frac{I_1}{I_2}]$ is (where [.] denotes the greatest integer function)

Q. If $f\left(x\right)=\int (\cot \frac{x}{2}-\tan \frac{x}{2})dx,$ where $f\left(\frac{\pi }{2}\right)=0,$ then which of the following statements is (are) CORRECT ?

$($Note : $\text{sgn}\left(y\right)$ denotes the signum function of $y.)$

1. $\underset{0}{\overset{\pi }{\int }}f\left(x\right)dx=-2\pi \ln 2$
2. $\underset{0}{\overset{\pi }{\int }}f\left(x\right)dx=-\pi \ln 2$
3. $\text{sgn}\left(f\left(\frac{2\pi }{3}\right)\right)=-1$
4. $\text{sgn}\left(f\left(\frac{2\pi }{3}\right)\right)=1$

Q. $\underset{x\to \sqrt{2}}{\lim }\frac{x^2-2}{x^2+\sqrt{2}x-4}$

Q. $\underset{x\to -1}{\lim }\frac{x^3+1}{x+1}$

Q. $\frac{\sec x-1}{\sec x+1}$

Q. Solve the following equations:
(i) $\tan \mathrm{\theta }+\tan 2\mathrm{\theta }+\tan 3\mathrm{\theta }=0$
(ii) $\tan \mathrm{\theta }+\tan 2\mathrm{\theta }=\tan 3\mathrm{\theta }$
(iii) $\tan 3\mathrm{\theta }+\tan \mathrm{\theta }=2\tan 2\mathrm{\theta }$

Q. Prove that $\underset{x\to a^{+}}{\lim }\left[x\right]=\left[a\right]$ for all a ∈ R. Also, prove that $\underset{x\to 1^{-}}{\lim }\left[x\right]=0.$

Q. Which of the following definite integrals reduces to $\frac{\pi }{2}$?

1. $\underset{0}{\overset{\pi }{\int }}\frac{dx}{1+(\sin x{)}^{\cos x}}$
2. $\underset{0}{\overset{\pi /2}{\int }}\frac{dx}{1+(\tan x{)}^5}$
3. $\underset{0}{\overset{\infty }{\int }}\frac{x^2+1}{x^4-x^2+1}dx$
4. $\underset{0}{\overset{\pi /2}{\int }}(\ln \left(\sec x\right))(e^{\ln \left(\ln 2\right)}{)}^{-1}dx$

Q. Show that $\underset{x\to 2^{-}}{\lim }\frac{x}{\left[x\right]}\ne \underset{x\to 2^{+}}{\lim }\frac{x}{\left[x\right]}.$

Q. $\underset{x\to 3}{\lim }\frac{x^2-x-6}{x^3-3x^2+x-3}$

Q.

Fill in the blank

If $2x+3=5$ , then $3x+2$ is________

Q.

If f(x) = x² -2 and g(x) = 2x. Find f/g.

1. 

2. 

3. 

4. 

Q.

If $x=5$, then find the value of the following algebraic expression.

$6x+2$

Q. The value of the integral
$\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x(1+\log \left(\frac{2+\sin x}{2-\sin x}\right))dx$ is :

1. $0$
2. $\frac{3}{4}$
3. $\frac{3}{8}\pi$
4. $\frac{3}{16}\pi$

Q. $\underset{x\to 1}{\lim }\frac{\sqrt{5x-4}-\sqrt{x}}{x^3-1}$

Q. If $f\left(x\right)=\frac{e^x}{1+e^x},I_1=\underset{f(-a)}{\overset{f\left(a\right)}{\int }}xg\left(x\right(1-x\left)\right)dx$ and $I_2=\underset{f(-a)}{\overset{f\left(a\right)}{\int }}g\left(x\right(1-x\left)\right)dx$, then the value of $\frac{I_2}{I_1}$ is

1. $-1$
2. $-2$
3. $2$
4. $1$

Q. Consider $f\left(x\right)=1-2p\cos x+p^2=g\left(p\right);$ $h\left(x\right)=g(-p).g\left(p\right);$ $k\left(p\right)=\underset{0}{\overset{\pi }{\int }}\ln g\left(p\right)dx$ for $p,x\in R.$ Then which of the following is/are correct ?

1. $k\left(x\right)=\frac{1}{4}k\left(x^2\right)$
2. $k\left(p\right)$ is an even function
3. $g\left(p^2\right)=h\left(\frac{x}{2}\right)$
4. $k\left(x\right)=\frac{1}{2}k\left(x^2\right)$

Q. Write the value of $\underset{x\to 0}{\lim }\frac{\sin x}{\sqrt{1+x}-1}.$

Q. $\frac{x\tan x}{\sec x+\tan x}$

Q.

Subtract : $-12\mathrm{xyz}$from $0$

Q. $\underset{x\to 2}{\lim }\left(\frac{x}{x-2}-\frac{4}{x^2-2x}\right)$

Q.

Find the value of the following expression when $x=2,y=-3,z=1$

$2x-3y$

Q.

Simplify the following expression and find the value when $x=3$

$8-6x+4x+3$

Q. $\underset{x\to \infty }{\lim }\left[\left\{\sqrt{x+1}-\sqrt{x}\right\}\sqrt{x+2}\right]$

Q. $\underset{x\to 1}{\lim }\frac{\sqrt{5x-4}-\sqrt{x}}{x-1}$

Q. f(x^3+1)=x^4+1 find f(x)