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Equation of a parabola whose vertex is (2,-3), axis is parallel to the x axis and lastusrectum 8 is

Given a G.P. with a = 729 and $7^{th}$ term 64. determine $S_7$

'The number of squares of side at least 6 with vertices in S = {(x, y)1 ≤ x, y ≤ 8} is

If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2) and (3, 2), then what type of triangle is this?

A line with positive direction cosines passes through the point P (2 , -1 , 2 ) and makes equal angles with the coordinate axes. The line meets the plane 2x+6y +z = 9 at point Q. The length of the line segment PQ equals

The range of $a$ for which $f\left(x\right)=\text{sgn}\left(\right(\sin x)+a)$ is continuous for all $x\in R$.

Let $f\left(x\right)$ be defined on $[-2,2]$ and is given by $f\left(x\right)=\{\begin{array}{cc}-1,& -2\le x\le 0\\ x-1,& 0<x\le 2\end{array},g\left(x\right)=|x|$. Then $fog\left(x\right)+gof\left(x\right)=$

A ray of light is incident along a line which meets another line, $7x–y+1=0$, at the point $(0,1)$. The ray is then reflected from this point along the line, $y+2x=1$. Then the equation of the line of incidence of the ray of light is :

$L1:x-2y+10=0$

$L2:x+2y-6=0$

What is the ratio in which the point of intersection of line L1 and L2 divides the line segment AB of L1?

A man starts repaying a loan as first instalment of Rs. 100. If he increases the instalments by Rs. 5 every month, what amount he will pay in the 30th instalment?

Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$, where $\alpha ϵR$. Suppose $Q=\left[q_{ij}\right]$ is a matrix such that PQ = kI, where $kϵR,k\ne 0$ and I is the identity matrix of order 3. If $q_{23}=-\frac{k}{8}anddet\left(Q\right)=\frac{k^2}{2}$ then

If $y=\cos ax,\text{then}\left|\begin{array}{ccc}y& y_1& y_2\\ y_3& y_4& y_5\\ y_6& y_7& y_8\end{array}\right|=$ (where $y_n=\frac{d^{n}y}{d{x}^n}$)

The value of the integral $\underset{0}{\overset{a}{\int }}\frac{dx}{x+\sqrt{a^2-x^2}}$ is

In column 1 different setup for Young’s double slit experiment is given. Distance between two slits $S_1$ and $S_2$ is ‘d’ and between slits and screen is ‘D’ in every setup. In column 2 location of central maxima is given. In column 3 path difference $\Delta P$ is given at point P which is $\theta$ angle above MO line (d << D)

Which of the following is correct when light is incident at an angle ${\theta }_0$

The latusrectum of a hyperbola subtends a right angle at its centre,then its e =

${lim}_{x\to \frac{\pi }{2}}\frac{co{s}^{2}x}{1-sinx}$

The value of $a$ for which the curves $y^2=4x$ and $x^2+y^2-2ax=0$ intersect exactly at one point is

If the equation in x, $x^4+p{x}^3+q{x}^2$ = 16(2x – 1), where p, q $ϵ$ R has all positive roots and A.M. of roots = H.M. of roots, then

If in $△ABC$, $\tan A+\tan B+\tan C=6$ and $\tan A\tan B=2$, then ${\sin }^{2}A:{\sin }^{2}B:{\sin }^{2}C$ can be

If $ln=\int (1+x+x^{-1})e^{x+x^{-1}}dx=$

If the third term in the binomial expansion of ${(1+x^{{\log }_{2}x})}^5$ equals $2560$, the a possible value of $x$ is :

if cosA+sinB=m and sinA+cosB=n, prove that 2sin(A+B)=m²+n²-2

If $\frac{x+\sqrt{x+2}}{x-\sqrt{x+2}}\ge 1,$ then

$\text{In a triangle ABC, if}\angle A={60}^0,a=5,b=4,\text{then c is a root of the equation}$

To obtain the graph of y = - sinx from the graph of y = sinx, phase shift required is

If the median of a triangle $ABC$ passing through $A$ is perpendicular to $AB,$ then $\tan A+2\tan B=$

The digits 1, 2, 3.... 9 are arranged in a random order, find the probability that 1, 2, 3 will appear as neighbors in the order mentioned.

The integral $\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}$ dx is equal to : where C is an arbitary constant.

A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish for sit on one particular side and two on the other side. In how many ways can they be seated ?