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Equation of the tangent to $y^2=16x$ which is perpendicular to 2x-y+5=0 is

The equation of the plane through (1, 2, 3) and parallel to the plane 2x + 3y - 4z = 0 is

For natural number n, $(n!{)}^2>n^n$ if

An ellipse having foci at $(3,-3)$ and $(-4,-4)$ and passing through the origin. Then the length of latus rectum is

If $2\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}|.|\overrightarrow{b}|$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

Find the equations to the straight lines which pass through the point (h, k) and are inclined at angle $ta{n}^{-1}m$ to the straight line $y=mx+c$.

If ${(1+x+x^2)}^{25}={\sum }_{r=0}^{50}{a}_r{x}^r,$ then ${\sum }_{r=0}^{12}{a}_{4r}$ equals

In an equilateral $△ABC$ coordinates of the centroid of the triangle is ($1,2,3)$. Find the coordinates of the orthocenter of the triangle.

The general solution of the differential equation $\sqrt{1-x^2{y}^2}dx=ydx+xdy$ is

Sand is pouring from a pipe at the rate of $12c{m}^3/s$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one sixth of the radius of the base? How fast is the height of the sand cone increasing when the height is $4cm$?

the value of ${lim}_{x\to \infty }\frac{\sqrt{1+x^4}+(1+x^2)}{x^2}$ is

From a book containing 100 pages, one page is selected randomly. The probability that the sum of the digits of the page number of the selected page is 11, is

The system of equations
$\lambda x+y+3z=0$
$2x+\mu y-z=0$
$5x+7y+z=0$
has infinitely many solutions in $R$. Then,

$\int \frac{cos2x}{(sinx+cosx{)}^2}dx=$

The domain of the function $f\left(x\right)=\sqrt{2-2x-x^2}$ is

If f(x) is continuous for all real values of x, then ${\sum }_{r=1}^n{\int }_0^{1}f(r-1+x)dx=$

Let A = {1, 2, 3}, B = {1, 3, 5}. If relation R from A to B is given by

R = {(1, 3), (2, 5), (3, 3)}. Then, $R^{-1}$ is

If the latus rectum of an ellipse is equal to half of its minor axis, then its eccentricity is

In a triangle $\Delta ABC$
$3\sin A+4\cos B=6$ and $4\sin B+3\cos A=1$, then the angle $C$ is :

If a, b, c are in A.P. and a, b, d are in G.P. show that a, (a - b), (d - c) are in G.P.

For what value of k, is the function $f\left(x\right)=\{\begin{array}{c}\frac{k\cos x}{\pi -2x},\text{if}x\ne \frac{\pi }{2}\\ 3\text{if}x=\frac{\pi }{2}\end{array}$ continuous for all real $x?$

Choose the correct option of general solutions -
$$\begin{array}{cc}TrigonometricEquations& GeneralSolutions\\ 1.si{n}^2\theta =si{n}^{2}x& A.2n\pi \pm \alpha \\ 2.co{s}^2\theta =co{s}^{2}x& B.n\pi +(-1{)}^n\alpha \\ 3.ta{n}^2\theta =ta{n}^{2}x& C.n\pi \pm \alpha \\ & \end{array}$$

The area bounded by the normal at $(1,2)$ to the parabola $y^2=4x$, x-axis and the curve is given by

The smallest natural number $n,$ such that the cofficient of $x$ in the expansion of ${(x^2+\frac{1}{x^3})}^n$is ${}^n{C}_{23}$, is :

The coordinates of a point are (3, -2, -5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.

Let A = {1, 2, 3, 4, 5, 6}. Let R be a relation on A defined by

$R=\left\{\right(a,b):a,bϵA,b$ is exactly divisible by a}

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

Differentiate the following functions with respect to x :

$\frac{1}{sinx}+2^{x+3}+\frac{4}{lo{g}_{x}3}$

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is ${\left(\frac{3}{4}\right)}^{3}k$, then $k$ is equal to :

If two adjacent vertices of a regular hexagon are $(0,0)$ and $(1,2),$ then equation of the circumcircle of the hexagon is

Construct a $2\times 2$ matrix, where

$a_{ij}=\frac{(i-2j{)}^2}{2}$

Equation of the circle inscribed in $|x+1|+|y-6|=2$ is

Find the equation of a circle passing through the points (5, 7), (6, 6) and (2, -2). Find its centre and radius.

The area of the region for which $0<y<3-2x-x^2$ and $x>0$ is

4 persons are asked the same question by an interviewer. If each has independent probability $\frac{1}{6}$ of answering the question correctly. The probability that at least one answers correctly is:

In how many ways arrrangments of the word GOLDEN will the vowels never occur together?