Eccentric Angle : Hyperbola

Q. The parametric coordinates of the point $(8,3\sqrt{3})$ on the hyperbola $9x^2-16y^2=144$ is

1. $(2\sec {60}^{\circ },3\tan {60}^{\circ })$
2. $(3\sec {60}^{\circ },3\tan {60}^{\circ })$
3. $(4\sec {60}^{\circ },3\tan {60}^{\circ })$
4. $(4\sec {30}^{\circ },3\tan {30}^{\circ })$

Q. The parametric coordinates of the point $(8,3\sqrt{3})$ on the hyperbola $9x^2-16y^2=144$ is

1. $(2\sec {60}^{\circ },3\tan {60}^{\circ })$
2. $(3\sec {60}^{\circ },3\tan {60}^{\circ })$
3. $(4\sec {60}^{\circ },3\tan {60}^{\circ })$
4. $(4\sec {30}^{\circ },3\tan {30}^{\circ })$

Q. Compute the following:
(i) $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]+\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$
(ii) $\left[\begin{array}{cc}{a}^2+b^2& b^2+c^2\\ a^2+c^2& a^2+b^2\end{array}\right]+\left[\begin{array}{cc}2ab& 2bc\\ -2ac& -2ab\end{array}\right]$
(iii) $\left[\begin{array}{c}-1\\ 8\\ 2\end{array}\begin{array}{c}4\\ 5\\ 8\end{array}\begin{array}{c}-6\\ 16\\ 5\end{array}\right]+\left[\begin{array}{c}12\\ 8\\ 3\end{array}\begin{array}{c}7\\ 0\\ 2\end{array}\begin{array}{c}6\\ 5\\ 4\end{array}\right]$
(iv) $\left[\begin{array}{cc}{\cos }^{2}x& {\sin }^{2}x\\ {\sin }^{2}x& {\cos }^{2}x\end{array}\right]+\left[\begin{array}{cc}{\sin }^{2}x& {\cos }^{2}x\\ {\cos }^{2}x& {\sin }^{2}x\end{array}\right]$

Q. If $k+\mid k+z^2\mid =\mid z{\mid }^2\left(kϵR^{-}\right)$, then possible argument of $z$ is

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

Q. Write an anti derivative for each of the following functions using the method of inspection:
i) $\cos 2x$
ii) $3x^2+4x^3$
iii) $\frac{1}{x},x\ne 0$

Q. All possible values of eccentric angle of $(x_1,y_1)$ on $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance fromt the centre is 2.

1. $\pm$ 45.
2. 45
3. 90
4. -45

Q. The value of ${\int }_{-\pi }^{\pi }\frac{{\cos }^{2}x}{1+a^x}dx$ for $a>0$ is?

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

Q. $\underset{x\to 0}{lim}\frac{\sin x^n}{{\left(\sin x\right)}^m}(m<n)$ is equal to

1. $1$
2. None of he above
3. $0$
4. $\frac{n}{m}$

Q. Let $f$ be a continuous function on R. If $f\left(\frac{1}{4^n}\right)=\left(\sin e^n\right)e^{-n^2}+\frac{n^2}{n^2+1}$ then $f\left(0\right)$ is -

1. not unique
2. 1
3. data sufficient to find $f\left(0\right)$
4. data insufficient to find $f\left(0\right)$

Q. All possible values of eccentric angle of $(x_1,y_1)$ on $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance fromt the centre is 2.

1. $\pm$ 45.
2. 45
3. -45
4. 90

Q. ${\int }_0^{5/2}(\left[x\right]+[-x])dx$ equals

1. $\frac{5}{2}$
2. $-\frac{5}{2}$
3. $\frac{7}{2}$
4. none of these

Q. If $ABCD$ is a cyclic quadrilateral, then find which of the following statements is not correct.

1. $\sin (A+C)=0$
2. $\sin (A+B)=0$
3. $\cos (B+D)=-1$
4. $\sin (A+B+C+D)=0$

Q. Integrate the following with respect to $x$.
$\cos h3x$

Q. If $f\left(x\right)$ is continuous and $f\left(\frac{9}{2}\right)=\frac{2}{9}$, then $\underset{x\to 0}{lim}f\left(\frac{1-\cos 3x}{x^2}\right)$ is equal to :

1. $\frac{9}{2}$
2. $\frac{2}{9}$
3. 0
4. $\frac{8}{9}$

Q. If $A=\left[\begin{array}{cc}\cos A& \sin A\\ \sin A& -\cos A\end{array}\right]$ then find the negative of value of $|A{|}^5$.