Property 1

Q.

If A ∝ B then :

1. B ∝ 1/A

2. B ∝ A

3. B = 1/A

4. B = A

Q. Prove that ;
${\int }_0^{\pi }\frac{xdx}{a^{2}co{s}^{2}x+b^{2}si{n}^{2}x}=\frac{{\pi }^2}{2ab}$

Q. If ${\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)=\theta$, then what is the value of $cose{c}^{-1}\left(\sqrt{5}\right)$?

1. $\left(\frac{\pi }{2}\right)$
2. $\left(\frac{\pi }{2}\right)-\theta$
3. $\left(\frac{\pi }{2}\right)+\theta$
4. $-\theta$

Q.

If $x\propto y$, then $x^3\propto y^2$

1. True

2. False

Q. If $\sec \theta =x+\frac{1}{4x}$, prove that $\sec \theta +\tan \theta =2x$ or $\frac{1}{2x}$

Q. Evaluate: $\sec h^{-1}\left(\sin \theta \right)$

1. $\log \left(\tan \frac{\theta }{2}\right)$
2. $\log \left(\sin \frac{\theta }{2}\right)$
3. $\log \left(\cos \frac{\theta }{2}\right)$
4. $\log \left(\cot \frac{\theta }{2}\right)$

Q. The value of $\sec \frac{2\pi }{3}+\text{cosec}\frac{5\pi }{6}$ is equal to :

1. -2
2. 4
3. 2
4. -4
5. 0

Q. Solve; $\log \left(x+1\right)-\log \left(x-1\right)=1$

Q.

If B ∝ C then :

1. C ∝ B

2. C ∝ 1/B

3. C = 1/B

4. C =B

Q. Find the area bounded by the parabola $y=x^2-4,$ the$X$ -axis and the lines$x=-1$ and $x=2$ ?

Q. Let $a_n,n\ge 1$, be an arithmetic progression with first term $2$ and common difference $4$. Let $M_n$ be the average of the first $n$ terms. Then the sum $\sum _{n=1}^{10}{M}_n$ is

1. $110$
2. $335$
3. $770$
4. $1100$

Q. Solve the following equations.
$x^2\log x=10x^2$

Q. Show that :
$\frac{\cos {30}^{\circ }+\sin {60}^{\circ }}{1+\sin {30}^{\circ }+\cos {60}^{\circ }}=\cos {30}^{\circ }$.

Q. Evaluate :$\frac{5{\sin }^2{30}^{\circ }+{\cos }^2{45}^{\circ }+4{\tan }^2{60}^{\circ }}{2\sin {30}^{\circ }\cos {60}^{\circ }+\tan {45}^{\circ }}$

1. 1
2. $9\frac{1}{6}$
3. $7\frac{3}{7}$
4. $\frac{47}{12}$

Q. The value of ${\sin }^2\left({\cos }^{-1}\frac{1}{2}\right)+{\cos }^2\left({\sin }^{-1}\frac{1}{3}\right)$ is

1. $\frac{17}{36}$
2. $\frac{59}{36}$
3. $\frac{36}{59}$
4. None of these

Q. $\text{Solve for}x\text{in the equation}$ $log\left[log\right(2+lo{g}_2(x+1)\left)\right]=0$

Q. Differentiate the following w.r.t.x:
${\cos }^{-1}\left(\sin x^2\right)$

Q. If $\sqrt{3}\tan \theta =3\sin \theta$, find the value of ${\sin }^2\theta -{\cos }^2\theta$.

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

Q. If ${\cos }^4\theta \cdot {\sec }^2\alpha ,\frac{1}{2}$and ${\sin }^4\theta \cdot co\sec {}^2\alpha$ are in AP, then $\cos {}^8\theta \cdot {\sec }^6\alpha$ , $\frac{1}{2}$ and $\sin {}^2\theta \cdot cosec{}^6\alpha$ are in

1. AP
2. GP
3. HP
4. none of these

Q. Solve: $\log 7^x=\log 9^{x-2}$

Q. Prove the following identity-
$\frac{cosec\theta }{cosec\theta -1}+\frac{cosec\theta }{cosec\theta +1}=2{\sec }^2\theta$

Q. ∫sin−1√xa+xdx

1. $\left(a+x\right)arc\tan \sqrt{\frac{x}{a}}-\sqrt{ax}+c$
2. $\left(a+x\right)arc\tan \sqrt{\frac{x}{a}}+\sqrt{ax}+c$
3. $\left(a-x\right)arc\tan \sqrt{\frac{x}{a}}-\sqrt{ax}+c$
4. $\left(a+x\right)arc\cot \sqrt{\frac{x}{a}}-\sqrt{ax}+c$

Q. Solve: $\int \frac{1}{(2x-3)(x-4)}dx$

Q. If $\tan \theta +\sec \theta =e^x$, then $\cos \theta$ equals-

1. $\frac{(e^x+e^{-x})}{2}$
2. $\frac{(e^x-e^{-x})}{2}$
3. $\frac{2}{(e^x+e^{-x})}$
4. $\frac{(e^x-e^{-x})}{(e^x+e^{-x})}$

Q. ${\cos }^2\frac{3\pi }{5}+{\cos }^2\frac{4\pi }{5}$ is equal to?

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