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Locus of the point of intersection of any two perpendicular tangents to the parabola $x^2=4ay$ is

Match the following



$$\begin{array}{cc}\text{Column A}& \text{Column B}\\ 1.A=\{x:x\text{is an odd natural number}\}& a.\{\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{3}{4},\frac{3}{5},\frac{5}{7}\}\\ 2.B=\{x:x=(n^2)\},n\in N\text{and}x<100& b.\{1,4,9,16,25,........10000\}\\ 3.C=\{x:x=\frac{n}{n+2},\text{where}n\text{is a natural number and}1\le n\le 6\}& c.\{1,4,9,16,25,36,49,64,81\}\\ 4.D=\{x:x\text{is a letter of the word TRIGONOMETRY}\}& d.\{G,E,I,M,N,O,R,T,Y\}\\ & e.\{1,3,5,7,9,...............\}\end{array}$$

Let $f\left(\theta \right)=\sin \theta (\sin \theta +\sin 3\theta )$, then $f\left(\theta \right)$ is

Let $n$ be four digit positive integer in which all the digits are different. If $x$ is number of odd integers and $y$ is number of even integers, then

The mean of the numbers $a,b,8,5,10$ is $6$ and the variance is $6.80$. Then which one of the following gives possible values of $a$ and $b$?

If $P={\sec }^{6}A-{\tan }^{6}A$ and $Q=3{\sec }^{2}A{\tan }^{2}A,$ then $P-Q$ is equal to

If $|1+4i-2^{-x}|\le 5$ where $i=\sqrt{-1}$ then $xϵ$

The area (in sq. units) of the smaller region bounded by the curves $x^2+y^2=5$ and $y^2=4x$ is

How many four digit natural numbers not exceeding $4321$ can be formed with the digits $1,2,3$ and $4$, if the digits can repeat ?

If the matrix $A=\left[\begin{array}{cc}-1& 2\\ -3& 4\end{array}\right]$ satisfies the quadratic function $f\left(x\right)=(x-1)(x-\alpha ),$ then $\alpha$ is

A class has 30 students. The following prizes are to be awarded to the students of this class: first and second in Mathematics; first and second in Physics first in Chemistry and first in Biology. If N denote the number of ways in which this can be done, then

What is the total number of elementary events associated to the random experiment of throwing three dice together?

Equation of a common tangent to the parabola $y^2=4x$ and the hyperbola $xy=2$ is:

The solution set of $x^2-16\le 0$ and $x^2-9\ge 0$ is

Evaluate
the determinants in Exercises 1 and 2.

The value of $\underset{0}{\overset{\pi }{\int }}{e}^{{\cos }^{2}x}{\cos }^{5}3xdx$ is

The value of $\int \frac{(x+1)e^x}{1+x^2{e}^{2x}}\cdot {\tan }^{-1}\left(x{e}^x\right)dx$ is

(where $C$ is constant of integration)

The vectors $2\hat{i}+3\hat{j},5\hat{i}+6\hat{j}$ and $8\hat{i}+\alpha \hat{j}$ have their initial points at $(1,1)$. The value of $\alpha$ so that the vectors terminate on one straight line is

Differentiate the following functions with respect to x :

$\frac{cos(x-2)}{sinx}$

Let $f$ be a continuous function. If $\underset{0}{\overset{x}{\int }}f\left(t\right)dt=e^x-c{e}^{2x}\underset{0}{\overset{1}{\int }}f\left(t\right)e^{-t}dt,$ where $c$ is a non-zero constant, then

Find the area of the triangle formed by the lines

$\left(i\right)y=m_{1}x+c_1,y=m_{2}x+c_{2}andx=0$ $\left(ii\right)y=0,x=2andx+2y=3.$ $\left(iii\right)x+y-6=0,x-3y-2=0and5x-3y+2=0$

If $-3<\frac{2x-1}{3}\le 5$, then $x$ lies in the interval

The value of $\cos (n+1)\alpha \cos (n-1)\alpha +\sin (n+1)\alpha \sin (n-1)\alpha$ is

The Integrating Factor (IF) of the differential equation $x\frac{dy}{dx}-y=2x^2$ is
(a)$e^{-x}$
(b)$e^{-y}$
(c)$\frac{1}{x}$
(d)x

The straight line which is both tangent and normal to the curve $x=3t^2,y=2t^3$ is

If,
then prove
where
n is any
positive integer