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If $y=3^{x\log x},$ then $\frac{dy}{dx}=$

The angle between the straight lines

$\frac{x+1}{2}=\frac{y-2}{5}=\frac{x+3}{4}and\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{-3}is$

If $\sin 2\beta$ is the G.M. between $\sin \alpha$ and $\cos \beta$, then $\cos 4\beta$ is equal to

The set of all those points, where the function

$f\left(x\right)=\frac{x}{1+|x|}$

is differentiable, is

If the adjoint of a $3\times 3$ matrix P is $\left[\begin{array}{ccc}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right]$, then the possible value(s) of the determinant of P is (are)

$(A-B)-(B-C)=$___

The value of integral $\underset{-2}{\overset{2}{\int }}\frac{e^{2x}+1}{e^{2x}-1}\cdot x^{6}dx$ is

A standard hyperbola is given as below. Which among the points given would lie on the auxiliary circle of the hyperbola?

By usingproperties of determinants, show that:
(i)$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$$=(a+b+c{)}^3$

(ii)$\left|\begin{array}{ccc}x+y+2z& x& y\\ z& y+z+2x& y\\ z& x& z+x+2y\end{array}\right|$$=2(x+y+c{)}^3$

A variable straight line is drawn through the point of intersection of the straight lines $\frac{x}{2}+\frac{y}{3}=1$ and $\frac{x}{3}+\frac{y}{2}=1$ and meets the coordinate axes at $A$ and $B$. Then the locus of the mid-point of $AB$ is

Answer the following questions briefly.

(a) In 1953, Hooper was a favoured young man. Explain.

(b) They said that they would create a desk job for Hooper at
headquarters.

i) Who are ‘they’?

ii) Why did they decide to do this?

(c) Duke was an extraordinary dog. What special qualities did he
exhibit to justify this? Discuss.

(d) What problems did Chuck present when he returned to the
company headquarters?

(e) Why do you think Charles Hooper’s appointment as
Assistant National Sales Manager is considered to be a tribute to
Duke?

If a,b and c are three positive real numbers, which one of the following are true?

Area of the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 3 is:

If the chord through the points whose eccentric angles are $\theta$ and $\varphi$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ passes through a focus, then possible value(s) of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$ is/are

In $\Delta ABC$, if $\angle A={30}^{\circ },a=7,b=8$ then B has

If the length of subnormal is four times the length of subtangent at a point $(3,4)$ on the curve $y=f\left(x\right)$. The tangent at $(3,4)$ to $y=f\left(x\right)$ meets the coordinate axes at $P$ and $Q$ and the maximum area of triangle $OPQ$ (where $O$ is origin) is $4b^2$, then $b=$

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is

Which of the following is/are the roots of the quadratic equation

$2x^2+5x-3=0$ ?

If $\frac{(a^2+1{)}^2}{2a-1}=x+iy$ find the value of $x^2+y^2$.

If S is the set of distinct values of b for which the folowing system of linear equations

$x+y+z=1$,

$x+ay+z=1$

and $ax+by+z=0$ has no solution, then S is

If $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ then $A^{16}$ =

If $x=2cost-cot2t,y=2sint-sin2t$, then $\frac{d^{2}y}{d{x}^2}$ at t = $\frac{\pi }{2}$ is

Point M moved along the circle $(x-4{)}^2+(y-8{)}^2=20$. Then it broke away from it and moving along a tangent to the circle, cuts the x-axis at the point (–2, 0). The coordinates of the point on the circle at which the moving point broke away can be

If $(x-2{)}^{\circ }$ and $(2x+5{)}^{\circ }$ are supplementary angles, then the value of $x$ is

Write quadratic equation the arithmetic and geometric means of whose roots are A and G respectively.

If $y=\left(\frac{ax+b}{cx+d}\right)$ , then $2\frac{dy}{dx}.\frac{d^{3}y}{d{x}^3}$ is equal to

The triangle formed by the tangent to the parabola y²=4x at the point whose abscissa lies in the interval [a², 4a²], the ordinate and the X-axis, has greatest area equal to

The Taylor's series expansion of sin x is

Find the
derivative of the following functions (it is to be understood that a,
b, c, d, p, q, r and s are
fixed non-zero constants and m and n are integers):