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Fourier transform of $x\left(t\right)=\delta \left(t\right)+3e^{-3|t|}$ is

Sketch the following graphs :
(i) $y=cos(x+\frac{\pi }{4})$
(ii) $y=cos(x-\frac{\pi }{4})$
(iii) $y=3cos(2x-1)$
(iv) $y=2cos(x-\frac{\pi }{2})$

The equation of the chord of the circle $x^2+y^2=a^2$ having $(x_1,y_1)$ as its mid-point is

The length and the midpoint of the chord 4x-3y+5=0 w.r.t circle $x^2+y^2-2x+4y-20=0$ is

Find the equation of the circle circumscribing the triangle formed by the lines $x+y=0,2x+y=4$ and $x+2y=5$

The graph of $f\left(x\right)=||x-1|-1|$ is

The possible values of $x$, which satisfy the trigonometric equation ${\tan }^{-1}\left(\frac{x-1}{x-2}\right)+{\tan }^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi }{4}$ are

The point of intersection of the tangents to the parabola $y^2=4x$ at the points where parameter 't' has value 3 and 5.

The value of $\int \frac{x^2}{(a+bx{)}^2}dx$ is:

(where $a\ne b$ and $C$ is constant of integration)

Check whether following function is strictly decreasing on $(0,\frac{\pi }{2})$ or not.

cos 2x

|x^2 -1| +x+1 =0

Urn $A$ contains $9$ red balls and $11$ white balls. Urn $B$ contains $12$ red balls and $3$ white balls. $A$ person is to roll a single fair die. If the result is a one or a two, then (s)he is to randomly select a ball from urn $A$. Otherwise (s)he is to randomly select a ball from urn $B$. The probability of obtaining a red ball is

The exhaustive set of values of $a^2$such that there exists a tangent to the ellipse$x^2+a^2{y}^2=a^2$such that the portion of the tangent intercepted by the hyperbola $a^2{x}^2-y^2=1$ subtends a right angle at the origin.

If ${}^n{C}_{12}{=}^n{C}_8$, then n is equal to

(a) 20 (b) 12 (c) 6 (d) 30

1. 6

The principal value of ${\sin }^{-1}(-\frac{\sqrt{3}}{2})$ is

If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{\lambda }$ and $\frac{x-1}{\lambda }=\frac{y-4}{2}=\frac{z-5}{1}$ intersect, then the value(s) of $\lambda$ can be

The probability of at least one double-six being thrown in n throws with two ordinary dice is greater than 99%. Calculate the least numerical value of n.___

$A=[a_{ij}{]}_{m\times n}$ is an square matix, if
(a)m < n
(b)m > n
(c)m = n
(d)None of these

Identify which one of the following is an eigen vector of the matrix $A=\left|\begin{array}{cc}1& 0\\ -1& -2\end{array}\right|$

If $\underset{x\to \infty }{lim}(\sqrt{x^2-x+1}-ax-b)=0,$ then for $k\ge 2,$ $\underset{n\to \infty }{lim}{\sec }^{2n}(k!\pi b)$ is equal to

If $\alpha ,\beta$ are the roots of the equation $2x^2+5x+6=0$, then find the equation whose roots are $\frac{1}{\alpha }and\frac{1}{\beta }$.

Let $a_1,a_2,a_3,...$ be an A.P. with $a_6=2.$Then the common difference of this A.P., which maximises the product $a_1{a}_4{a}_5,$ is :

Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\text{where}a>b>0,$be a hyperbola in the $xy-$plane whose conjugate axis $LM$ subtends an angle of ${60}^{\circ }$ at one of its vertices $N.$ Let the area of the triangle $LMN$ be $4\sqrt{3}.$ sq. unit

$\begin{array}{cccc}& \text{LIST-I}& & \text{LIST-II}\\ P.& \text{The length of the conjugate axis of}H\text{is}& 1.& 8\\ Q.& \text{The eccentricity of}H\text{is}& 2.& \frac{4}{\sqrt{3}}\\ R.& \text{The distance between the foci of}H\text{is}& 3.& \frac{2}{\sqrt{3}}\\ S.& \text{The length of the latus rectum of}H\text{is}& 4.& 4\end{array}$

The correct option is

Prove the following:

${\text{cos}}^2$ 2x - ${\text{cos}}^2$ 6x = sin 4x sin 8x

A farmer wants to buy some horses, and every horse he buys requires $2$ acres of land. If the farmer has $18$ acres of land, write an inequality representing the possible number of horses he can buy. (where $a$ is number of horses)

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be ${45}^{\circ }$ from a point $A$ on the plane. Let $B$ be the point $30$ m vertically above the point $A$. If the angle of elevation of the top of the tower from $B$ be ${30}^{\circ }$, then the distance (in m) of the foot of the tower from the point $A$ is :

The value of $\underset{n\to \infty }{\text{lim}}\sum _{x=1}^{20}{\cos }^{2n}(x-10)$ is equal to

The solution of differential equation $y^{2}xdx+ydx-xdy=0$ is

(where $C$ is constant of integration)

Choose personal pronoun for the blank.

Sam can laze away to glory because ___ does not have office in the morning.