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How to prove the point A B C D are colliear

The equation of tangents to the ellipse $9x^2+16y^2=144$ at the ends of the latus rectum are

Three numbers whose product is 512 are in GP. If 8 is added to the first and 6 to the second, the numbers will be in AP. The number is,

If $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$, then $A^2=$

A coin whose faces marked 2 and 3 is thrown 5 times, then chance of obtaining a total of 12 is

If $x\in (\frac{\pi }{2},\pi ),$ then $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to

If $\theta \in (\frac{\pi }{2},\frac{3\pi }{2})$, then the value of $\sqrt{4{\cos }^4\theta +{\sin }^{2}2\theta }+4\cot \theta {\cos }^2(\frac{\pi }{4}-\frac{\theta }{2})$ is

If the mean deviation about the median of the numbers $a,2a,......,50a$ is $50$, then $|a|$ equals

If a+b+c=0 and |a|=3, |b|=4 and $|c|=\sqrt{37}$, the angle between a and b is

Statement (1): Maximum value of $si{n}^2$x + $co{s}^2$y is 2

Statement (2): Maximum value of 2 $se{c}^2$z is 2

Show that $(a-b)\times (a+b)=2(a\times b)$

If the roots of the quadratic equation $x^2+px+q=0$ are $tan{30}^o$ and $tan{15}^o$ then the value of 2 + q - p is

Let $ABC$ be a triangle in which $AB=BC.$ Let $X$ be a point on $AB$ such that $AX:XB=AB:AX,$ if $AC=AX$ then the measure of $\angle ABC$ equals

If ${\log }_3(x^3-x^2-x+1)-{\log }_3(x-1)-{\log }_3(x+1)=2$, then $x=$

Find $\lambda$ and $\mu$, if $(2\hat{i}+6\hat{j}+27\hat{k})\times (\hat{i}+\lambda \hat{j}+\mu \hat{k})=0$

Find the equation of the parabola that satisfies the given conditons:
Focus (6,0); directrix x = -6

In the parabola,$y^2-2y+8x-23=0$, the length of double ordinate at a distance of 4 units from its vertex is

If ${\cos }^{-1}x+{\cos }^{-1}y=\frac{\pi }{2}$ and ${\tan }^{-1}x-{\tan }^{-1}y=0$ then $x^2+xy+y^2$ is equal to

$\int e^x\left(\sin \left(x\right)+\cos \left(x\right)\right)dx$ is equal to -

15 coupons are numbered 1....15 respectively.7 coupons are selected at random 1 at a time with replacement.The probability that the largest number appearing is 9?

If A = {1, 2, 3} and B = {2, 4}, what are $A\times B,B\times A,A\times A,B\times B,$ and $(A\times B)\cap (B\times A)$ ?

If AB is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\Delta OAB$ is an equilateral triangle O being the origin, then the eccentricity of the hyperbola satisfies

If $y=y\left(x\right)$ is the solution of the differential equation, $x\frac{dy}{dx}+2y=x^2$ satisfying $y\left(1\right)=1$, then $y\left(\frac{1}{2}\right)$ is equal to:

A variable circle passes through the fixed point A(p, q) and touches the x-axis. The locus of the other end of the diameter through A is

The equation of the ellipse, whose axes are of lengths $6$ and $2\sqrt{6}$ and their equations are $x-3y+3=0$ and $3x+y-1=0$ respectively, is

$\sqrt{lo{g}_{2}x}-0.5=lo{g}_2\sqrt{x}$, then x equals

Find the derivative of the following function
$f\left(x\right)=\sin \left(2x^2+4^x+3\right)$

Differentiate the following functions with respect to x :

$\frac{4x+5sinx}{3x+7cosx}$

Distance between two parallel planes 2x+y+2z=8 and 4x+2y+4z+5=0 is

​​​​​​A circle $C_1$ with centre at the origin meets $x$-axis at $A$ and $B$ (where $A\&B$ lies on negative and positive $x-$axis respectively). Two points $P\left(a\right)$ and $Q\left(b\right)$ are on the circle such that $b-a$ is a constant, where $a$ and $b$ are the parametric angles of the points. $BP$ and $AQ$ meets at $R$. Locus of $R$ is a circle $C_2$.
Let $c$ be the radius of $C_1$ and $d$ be the radius of $C_2$.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(1)}& \text{For}b-a=\frac{\pi }{2}\text{, the value of}\frac{d^2}{c^2}\text{is}& \text{(P)}& 0\\ \text{(2)}& \text{For}b-a=\frac{\pi }{2}\text{and}c=\sqrt{2}\text{, circle}{C}_2\text{intersects}& \text{(Q)}& 1\\ & \text{the coordinate axes at four points}L,M,N,O.& & \\ & \text{Let the area of the quadrilateral LMNO is}2\sqrt{2}p.& & \\ & \text{Then the value of}p\text{is}& & \\ \text{(3)}& \text{Let}{m}_1,m_2\text{be the slopes of the line BQ,AP}& \left(R\right)& 2\\ & \text{respectively.}\text{If}{m}_1{m}_2=-1\text{, then}\frac{a}{b}\text{is}& & \\ \text{(4)}& \text{Let}{m}_1,m_2\text{be the slopes of the line BQ,AP}& \left(S\right)& 3\\ & \text{respectively.}\text{If}{m}_1=m_2\text{, then}\frac{3|b-a|}{\pi }\text{is}& & \\ & & \left(T\right)& 4\end{array}$

Then the CORRECT option is :

If the length of the perpendicular from the point (1, 1) to the line $ax-by+c=0$ be unity, show that $\frac{1}{c}+\frac{1}{a}-\frac{1}{b}=\frac{c}{2ab}$

Find the integral of the function $\frac{tan\left(3ln\right(x\left)\right)}{x}$

“Two vector A and B whose tails touch each other make an angle of 120^• find component of vector A along direction of B A||B and in perpendicular direction of B” The solution to this answer involves the use of sin and cos 120^• and I did’nt understand how?

Find the equation of plane which is at a distance of 8 units from the origin and which is normal to the line having direction ratios 2,1,2 ?

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

The area enclosed by the curves y = |sin x|, x axis and |x| = $\pi$ is (in sq units)

If ${}^m{C}_1={}^n{C}_2,$ then

${lim}_{x\to \infty }\sqrt{x^2+7x}-x$

$Thedomainofthefunctionf\left(x\right)=si{n}^{-1}\left(lo{g}_2\left(\frac{x^2}{2}\right)\right)is$

If α, β are the roots of the equation $a{x}^2+bx+c=0$, then $\frac{\alpha }{a\beta +b}+\frac{\beta }{a\alpha +b}=$

Solve the system of equations 3x – 2y + 3z = 8, 2x + y – z = 1 and 4x – 3y + 2z = 4 by matrix method.