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Using elementary transformations, find the inverse of matrix $\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$, if it exists.

Compute the following:
$\left(iv\right)\left[\begin{array}{cc}co{s}^{2}x& si{n}^{2}x\\ si{n}^{2}x& co{s}^{2}x\end{array}\right]+\left[\begin{array}{cc}si{n}^{2}x& co{s}^{2}x\\ co{s}^{2}x& si{n}^{2}x\end{array}\right]$

The value of ${\left(\frac{-1+i\sqrt{3}}{1-i}\right)}^{30}$ is:

Find the area of the region bounded by
the ellipse

If $\frac{dy}{dx}=y+3$ and $y\left(0\right)=2,$ then $y\left(\ln 2\right)$ is/are equal to:

Write the value of n for which nth terms of the A.P.s 3, 10, 7, ... and 63, 65, 67, .... are equal.

If, $(1+i\sqrt{3}{)}^{12}=a+ib,$ Here $a$ and $b$ are real, then the value of $b$ is

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

In $△ABC,A\left(z_1\right),B\left(z_2\right)$ and $C\left(z_3\right)$ are inscribed in the circle $|z|=5$. If $H\left(z_H\right)$ be the orthocentre of $△ABC$, then $z_H=$

Solve the given inequality graphically in two-dimensional plane: 3x + 4y ≤ 12

Let $P_k$ be a point on the curve $y=\ln x$ in $xy-$plane whose $x$ coordinate is $1+\frac{k}{n},$ $k=1,2,3,\dots ,n.$ If $A$ is $(1,0),$ then $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\left(A{P}_k\right)}^2$ equals

(Here, $A{P}_k$ is the distance between points $A$ and $P_k$)

Pair of tangents are drawn from a point $P$ on $x^2+y^2=4$ to $x^2+y^2-20x-20y+199=0$. If $A$ and $B$ are points of contact of these tangents, then the area bounded by the locus of circumcentre of $△PAB$ is (in sq. units)

The value of ${\sin }^{-1}\left(\sin 10\right)+{\cot }^{-1}\left(\cot \right(-2\left)\right)+{\cos }^{-1}\left(\cos \right(-5\left)\right)$ is

An ellipse with centre at (0,0) cuts x axis at (3,0) and (-3,0). If its eccentricity is $\frac{1}{2}$ then the length of its semiminor axis is

The benches of a Gallery in a cricket stadium are 1m wide and 1m high.A batsman hits the ball at a level one metre above the ground and hits a sixer .The ball starts at 35 ms^{​​​​​-1} at an angle of 53° with the horizontal . The benches are perpendicular to the plane of motion and the first bench is 110m from the batsman .On which bench will the ball hit?

Find the
principal and general solutions of the equation

The point $(1,2)$ is one extremity of focal chord of the parabola $y^2=4x$. The length of this focal chord is

${lim}_{x\to 0}\frac{\sqrt{a^2+x^2}-a}{x^2}$

The lines $(lx+my{)}^2-3(mx-ly{)}^2=0$ and lx + my + n = 0 form

1. 0.0803

${lim}_{x\to 3}\frac{\sqrt{x+3}-\sqrt{6}}{x^2-9}$

If the tangent at the point $P\left(\theta \right)$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15,$ then possible value(s) of $\theta$ is/are

The range of values of $x$ that satisfies the inequation ${\log }_2{\log }_{0.5}\left(2^x\frac{15}{16}\right)\le 2$ is

Two posts are $20$ metre apart and the height of one post is double that of the other. From the mid-point of the line segment joining their feet, an observer finds that the angular elevation of their tops are complementary. Then the height of the shorter post (in metre) is

The function f:[0, 3]→[1,29], defined by f(x)=2x³-15x²+36x+1, is

Find the general solution of the equation

If, $I=\int \frac{dx}{sin(x-\frac{\pi }{3})cosx}$ then I equals