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There are n straight lines in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

Line L has intercepts a and b on the coordinate axes. when the axes are rotated through a given angle, keeping the origin fixed, the same line L has intercepts p and q. Then,

The sides of a right-angled triangle are integers. The length of one of the sides is $12.$ The largest possible radius of the incircle of such a triangle is

If sin x + cos x = a, find the value of $si{n}^{6}x+co{s}^{6}x$.

(sin
3x
+
sin x)
sin x
+
(cos 3x
–
cos x)
cos x
=
0

Let $R$ denote the set of real numbers. Let $f:R\times R\to R$ be a bijective function defined by $f(x,y)=(x+y,x-y)$. The inverse function of $f$ is given by

In a game a man wins 1 rupee for a six and loses one for any other number when a fair dice is thrown.man decided to throw dice thrice and quit as soon he wins.Find expected value he wins.

Won't we consider all loses in this ?

A wire of length $20$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is

If $3x+2y=1$ acts as a tangent to $y=f\left(x\right)$ at $x=\frac{1}{2}$ and if $p=\underset{x\to 0}{lim}\frac{x(x-1)}{f\left(\frac{e^{2x}}{2}\right)-f\left(\frac{e^{-2x}}{2}\right)},$ then $\sum _{r=1}^{\infty }{p}^r=$______

The range of $f\left(x\right)=-x^2+7x+60$ in $x\in [-3,2]$ is

If $\theta$ denotes the acute angle between the curves, $y=10-x^2$ and $y=2+x^2$ at a point of their intersection, then $|\tan \theta |$ is equal to :

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for $n\ge 3.$ And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal. Further consider a matrix ${\cup }_{3\times 3}$ with its column as ${\cup }_1,{\cup }_2,{\cup }_3$ such that $A^{50}{\cup }_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],A^{50}{\cup }_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],A^{50}{\cup }_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ Then,



The value of $|A^{50}|$ equals

If the top eight ranked teams make it to the quarter finals, then who, amongst the teams listed below, would definitely not play against Brazil in the final, in case Brazil reaches final?

For $x^2-(a+3)|x|+4=0$ to have real solutions, the range of $a$ is

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi }{4}$ (in the anti clockwise direction) is $a{X}^2+2hXY+b{Y}^2=20$ then which of the following is/are correct

The lengths of the axes of the hypberbola $9x^2-16y^2+72x-32y-16$ = 0 are

$Ifsi{n}^{3}xsin3x={\sum }_{m=0}^n{c}_{m}cosmxwhere{c}_0,c_1,c_2.\cdots \cdots ,c_{n}areconstantsand{c}_n\ne 0,thenthevalueofnis$

A capillary tube of radius $r$ and height $2h$ is immersed in water. The water rises up to height $h$. The mass of water in the tube is $10\text{g}$. If the whole system falls freely under gravity then the mass of water that will rise in the tube is

The minimum value of the expression $y=|x-2|+|x+4|+|x-6|$ is

If $({\alpha }^2,\alpha -2)$ be a point interior to the regions of the parabola $y^2=2x$ bounded by the chord joining the points $(2,2)$ and $(8,-4),$ then $\alpha$ belongs to the interval

The value of ${\cos }^2(\frac{\pi }{8}+x)-{\sin }^2(\frac{3\pi }{8}-x)$ is

Let $y=y\left(x\right)$ be solution of the differential equation,

$x{y}^{\prime }-y=x^2(x\cos x+\sin x),x>0.$ If $y\left(\pi \right)=\pi ,$ then $y^{\prime \prime }\left(\frac{\pi }{2}\right)+y\left(\frac{\pi }{2}\right)$ is equal to