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Two sides of a triangle are 2 and $\sqrt{3}$ and the included angle is ${30}^{\prime }$ then the in-radius r of the triangle is equal

Find the equations of tangents and normal at the extremities of latus rectum of the parabola $y^2=4ax.$

Number of values of $x$, satisfying the equation $\sqrt{(x+8)+2\sqrt{(x+7)}}+\sqrt{(x+1)-\sqrt{x+7}}=4,$ is

Find the equation of the hyperbola satisfying the given conditions.
Foci $(0,\pm 13),$ the conjugate axis is of lenth 24.

The value of $co{s}^{-1}\left(cos10\right)$=

If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer.

[Hint: write aⁿ = (a – b + b)ⁿ and expand]

If the lines $\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}and\frac{7-7x}{3p}=\frac{y-x}{1}=\frac{6-z}{5}=$ and are at right angle, then the value of is

Let $f\left(x\right)={(\sin \left({\tan }^{-1}x\right)+\sin \left({\cot }^{-1}x\right))}^2-1$, where $|x|>1$. If $\frac{dy}{dx}=\frac{1}{2}\frac{d}{dx}\left({\sin }^{-1}\right(f\left(x\right)\left)\right)$ and $y\left(\sqrt{3}\right)=\frac{\pi }{6}$, then $y(-\sqrt{3})$ is equal to:

The range of $\theta$ for which the point $(\sqrt{3}\sin \theta ,\sqrt{4}\cos \theta )$ lies outside $\frac{x^2}{4}-\frac{y^2}{5}=1$ is

Determine order and degree(if defined)
of differential equation

The line drawn from (4, -1, 2) to the point (-3, 2, 3) meets a plane at right angles at the point (-10, 5, 4), then the equation of plane is
[DSSE 1985]

The number of straight lines equally inclined to both the axes are

A) 1

B) 0

C) 2

D) Infinite

If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a - b, d - c are in G.P.

Find the equation of the line passing through the intersection of the line $2x+y=5$ and $x+3y+8=0$ and parallel to the line $3x+4y=7.$

Sort the equations in ascending order from top to bottom based on the positive values of 'x'.

Area of the region bounded by rays $|x|+y=1$ and $X-$axis is

In a bolt factory, three machines $A,B$ and $C$ produce $25\%,35\%$ and $40\%$ of total output respectively. It was found that $5\%,4\%$ and $2\%$ are defective bolts in the production by machines $A,B,C$ respectively. If a bolt is chosen at random from the total output and is found to be defective, then the chance that the bolt comes from the machine:

Let A={a1,a2,a3,a4,a5} and B={b1,b2,b3,b4,} where ai's and bi's are school going students . Define a relation from A to B by xRy iff y is a true friend of x . If R={(a1,b1),(a2,b1),(a3,b3),(a4,b2},(a5,b2)} . Prove that R is neither one one nor onto

$5^{2n}-1$ is divisible by 24 for all n $ϵ$ N.

The minimum value of $\left|z-1+2i\right|+\left|4i-3-z\right|$ is

If $A$, $B$ and $C$ are three sets such that $A\cap B=A\cap CandA\cup B=A\cup C$, then

The integral $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{4}}{\int }}\frac{dx}{\sin 2x({\tan }^{5}x+co{t}^{5}x)}$ equals:

The maximum value of $y=6x-x^2-5$ is

A circle touches the line $y=x$ at a point $P$ such that $OP=4\sqrt{2}$, where $O$ is the origin. The circle makes an intercept of $6\sqrt{2}$ units on line $x+y=0.$ Then the equation of the circle(s) is/are

${lim}_{n\to \infty }\frac{1^3+2^3+3^3+....+n^3}{(n-1{)}^4}$