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The coordinate of foot of the perpendicular drawn from the point $A(1,0,3)$ to the line joining the points $B(4,7,1)$ and $C(3,5,3)$ is

The inverse of matrix $\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$ is

Let $P(h,k)$ be a point on the curve $y=x^2+7x+2,$ nearest to the line, $y=3x-3$. Then the equation of the normal to the curve at $P$ is:

Let $p\left(x\right)$ be a polynomial such that $\frac{p(x+1)}{p\left(x\right)}=\frac{x^2+x+1}{x^2-x+1}$ and $p\left(2\right)=3$, then ${\int }_0^1{\tan }^{-1}\left(p\right(x\left)\right)\cdot {\tan }^{-1}\sqrt{\frac{x}{1-x}}dx$ is

The vector z=3-4i is turned anticlockwise through an angle of ${180}^{\circ }$ and stretched 2.5 times. The complex number corresponding to the newly obtained vector is

The area of triangle whose vertices are (0, 0), (a², 0) and (0, b²) is:

If $A=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right],$ Then the value of $\alpha$ for which $A^2=B$ is

Let $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$ be two real functions. Find (f + g), (f - g) (x), (fg) (x) and $\left(\frac{f}{g}\right)\left(x\right)$

Two dice are thrown simultaneously. The probability of obtaining a score less than 11 is:

If θ and ø are the roots of the equation $8x^2+22x+5=0$, then

The value of $sin\left(2si{n}^{-1}0.8\right)$ is

The possible values of the expression $\frac{1}{x^2-2x+5}$

Find the angle between planes 2x +7y +11z - 3 = 0 & 5x +3y +9z +1 = 0

The vertices of a triangle are (6, 0) (0, 6) and (6, 6). The distance between its circumcentre and centroid is

There are four machines and it is known that exactly two of them are faulty machines and are to be identified. Then the probability that only two tests are needed is:

In a triangle tan A + tan B + tan C =

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

The order and degree of the differential equation ${[4+{\left(\frac{dy}{dx}\right)}^2]}^{2/3}=\frac{d^{2}y}{d{x}^2}$ are

The tangent at a point whose eccentric angle is ${60}^{\circ }$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b),$ meets the auxiliary circle at $L$ and $M$. If $LM$ subtends a right angle at the centre, then eccentricity of the ellipse is

If $y=\sqrt{\sin x+y}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

Express the complex numbers in the form of a + ib:

$(\frac{1}{5}+\frac{2}{5}i)-(4+\frac{5}{2}i)$

Let $f\left(x\right)=\underset{n\to \infty }{lim}\frac{1}{{\left(\frac{3}{\pi }{\tan }^{-1}2x\right)}^{2n}+5}.$ Then the complete set of values of $x$ for which $f\left(x\right)=0$ is

Integrate the following functions.
$\int \frac{(x+1)(x+logx{)}^2}{x}dx.$

State whether the given table is not the probability distributions of a random variable. Give reasons for your answer.
$$\begin{array}{cccccc}Z& 3& 2& 1& 0& -1\\ P\left(Z\right)& 0.3& 0.2& 0.4& 0.1& 0.05\end{array}$$

The value of $\left|\begin{array}{ccc}x& x^2& yz\\ y& y^2& xz\\ z& z^2& xy\end{array}\right|$ is

For a positive integer n, let $f_n\left(\theta \right)=\left(tan\frac{\theta }{2}\right)(1+sec\theta )(1+sec2\theta )(1+sec4\theta )......(1+sec{2}^n\theta ).Then$

8^th term of the series $2\sqrt{2}+\sqrt{2}+0+....$ will be

Let X and Y be two arbitary, 3 $\times$ 3 non – zero, skew – symmetric matrices and Z be an arbitary, 3 $\times$ 3 non zero, symmetric matrix. Then, which of the following matrices is/are skew-symmetric?