Rectangular Cartesian Coordinate System

Q. A pair of tangents are drawn to the parabola $y^2=4ax$ which are equally inclined to a straight line $y=mx+c$, whose inclination to the axis is $\alpha$ then locus of their point of intersection is

1. $y=(x+a)\tan 2\alpha$
2. $y=(x-a)\tan 2\alpha$
3. $y=(x+a)\tan \alpha$
4. $y=(x-a)\tan 2\alpha$

Q. How do you simplify $\cos \left(si{n}^{-1}x\right)$ ?

Q. The set of all possible values of $\theta$ in the interval $(0,\pi )$ for which the points $(1,2)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y=1$ is:

1. $(0,\frac{\pi }{4})$
2. $(0,\frac{\pi }{2})$
3. $(0,\frac{3\pi }{4})$
4. $(\frac{\pi }{4},\frac{3\pi }{4})$

Q. For hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant with change in '$\alpha$'

1. Eccentricity
2. Abscissae of foci
3. Abscissae of vertices
4. Directrix

Q. If the distance from $(1,0)$ to two distinct points $A(\sin \theta ,\cos \theta )$ and $B({\sin }^2\theta ,-\cos \theta )$ is equal$(\ne 0)$ then the distance $AB$ (in units) is

Q. Let $A=(\sin \theta +1,\cos \theta )$ and $B=(1-\cos \theta ,-\sin \theta )$. Then the maximum value of $AB$ is

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

Q.

Find the length of the perpendicular from the point $(x^1,y^1)$ to the straight line Ax + By + C = 0, the axes being inclined at an angle $\omega$, and the equation being written such that C is a negative quantity.

1. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.cos\omega$

2. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.sin\omega$

3. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABcos\omega }}.sin\omega$

4. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABsin\omega }}.cos\omega$

Q. If $\sin \alpha =\frac{5}{13}$ and $\frac{\pi }{2}<\alpha <\pi$, then the value of $\frac{\sec \alpha +\tan \alpha }{cosec\alpha -\cot \alpha }$ is

1. $0.3$
2. $0.5$
3. $-0.3$
4. $-0.4$

Q. Let $\alpha ,\beta$ be the roots of the quadratic equation $x^2+px+p^3=0(p\ne 0).If(\alpha ,\beta )$ is a point on the parabola $y^2=x$, then the roots of the quadratic equation are

1. –4, –2
2. 4, ­2
3. 4, 2
4. –4, 2

Q. If $1+\sin \theta +{\sin }^2\theta +....upto\infty =2\sqrt{3}+4$, then $\theta =$

1. $\frac{\pi }{3}$
2. $\frac{3\pi }{4}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{4}$

Q. The length of projection of the line segment joining the point $(1,0,-1)$ and $(-1,2,2)$ to the plane $x+3y-5z=6$, is equal to -

1. 2
2. $\sqrt{\frac{271}{53}}$
3. $\sqrt{\frac{472}{31}}$
4. $\sqrt{\frac{474}{35}}$

Q. A curve is governed by the equation $y=\cos x$.Then what is the area enclosed by the curve and $x$ axis between $x=0$ and $x=\frac{\pi }{2}$ is shaded region?

Q. Let $A(\alpha ,\beta ),B({\alpha }^2,\beta ),C(\alpha ,{\beta }^2)$ are three ditinct points which are at same distance from origin. Then the sum of all possible value of ${}^{\prime }{\theta }^{\prime }$ such that $(\sin \theta ,\cos \theta )$ is equidistant to any of these points taken pairwise is
where $(0\le \theta \le \frac{\pi }{2})$

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

Q. Let $A(1,2),B(\text{cosec}\alpha ,-2)$ and $C(2,\sec \beta )$ are $3$ points such that $(OA{)}^2=OB\cdot OC$, ($O$ is the origin) then the value of $2{\sin }^2\alpha -{\tan }^2\beta$ is

1. $2$
2. $1$
3. $3$
4. $0$

Q. If $\sin {\theta }_1-\sin {\theta }_2=a$ and $\cos {\theta }_1-\cos {\theta }_2=b$, then

1. $a^2+b^2\ge 4$
2. $a^2+b^2\le 4$
3. $a^2+b^2\ge 3$
4. $a^2+b^2\le 2$

Q. If $\Delta =\left|\begin{array}{ccc}1& sin\theta & 1\\ -sin\theta & 1& sin\theta \\ -1& -sin\theta & 1\end{array}\right|;0\le \theta <2\pi$ then $\Delta \in [a,b]$ Find $\frac{b}{a}?$

Q. If the distance between the points $(2,1)$ and $(\alpha ,3)$ is equal to minimum value of the quadratic equation $y=x^2-4x+6$ i.e. $\beta$ and which is possible at $x=\gamma$, then $\alpha +\beta +\gamma$ is:

1. $4$
2. $2$
3. $6$
4. $8$

Q. Let the distance of a point on the line $x=3$ to the point $(1,-2)$ is twice that of from the point $(4,0)$ . Then the integral value for the ordinate is

Q.

$y=0$ is the equation of _____ axis.

Q. If the maximum and the minimum values of $1+\sin (\frac{\pi }{4}+\theta )+2\cos (\frac{\pi }{4}-\theta )$ for all real values of $\theta$ are $\lambda$ and $\mu$ respectively, then $\lambda -\mu$ is

1. $2$
2. $4$
3. $6$
4. $8$

Q. If $\Delta =\left|\begin{array}{ccc}2& -\sin \theta & 1\\ -\sin \theta & 2& \sin \theta \\ -1& -\sin \theta & 2\end{array}\right|$ then

1. Minimum of $\Delta =8$
2. Maximum of $\Delta =12$
3. Minimum of $\Delta =10$
4. Minimum of $\Delta =$ Minimum of $\sin \theta$

Q. Find the distance between the following pairs of points:
$(a\cos \theta ,a\sin \theta )$ and $(a\cos \varphi ,a\sin \varphi )$

Q. If $\varphi$ is an acute angle such that $\tan \varphi =\frac{2}{3}$, then evaluate
$\left(\frac{1+\tan \varphi }{\sin \varphi +\cos \varphi }\right)\left(\frac{1-\cot \varphi }{\sec \varphi +\text{cosec}\varphi }\right)$

1. $\frac{-1}{5}$
2. $\frac{-4}{\sqrt{13}}$
3. $\frac{1}{5}$
4. $\frac{4}{\sqrt{13}}$

Q. If abscissae and ordinates of the points $A(x_1,y_1)$ and $B(x_2,y_2)$ are the roots of the quadratic equation $x^2-x-1=0$ and $y^2-2y=0$ respectively, then the distance $AB$(in units) is

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

Q. $\frac{cosecA}{cosecA-1}+\frac{cosecA}{cosecA+1}=$

1. ${\sec }^{2}A$
2. $2{\sec }^{2}A$
3. ${\cos }^{2}A$
4. $2{\cos }^{2}A$

Q. Let $f\left(\theta \right)=sin\theta .(sin\theta +sin3\theta )$, then $f\left(\theta \right)$ is

1. $\ge 0$ only when $\theta \ge 0$
2. $\le 0$ for all real $\theta$
3. $\le 0$ only when $\theta \le 0$
4. $\ge 0$ for all real $\theta$

Q. Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2}sin\theta -x(sin\theta cos\theta +1)+cos\theta =0$
$(0<\theta <{45}^o)$, and $\alpha <\beta$. Then $\stackrel{\infty }{\underset{n=0}{\Sigma }}$ $(a^n+\frac{(-1{)}^n}{{\beta }^n})$ is equal to:

1. $\frac{1}{1+cos\theta }+\frac{1}{1-sin\theta }$
2. $\frac{1}{1-cos\theta }-\frac{1}{1+sin\theta }$
3. $\frac{1}{1+cos\theta }-\frac{1}{1-sin\theta }$
4. $\frac{1}{1-cos\theta }+\frac{1}{1+sin\theta }$

Q. If the distance from $(1,0)$ to two distinct points $A(\sin \theta ,\cos \theta )$ and $B({\sin }^2\theta ,-\cos \theta )$ is equal$(\ne 0)$ then the distance $AB$ (in units) is