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|sin x| is not differentiable at the points

If f (x) is differentiable in the interval [2, 5], where $f\left(2\right)=\frac{1}{5}$ and $f\left(5\right)=\frac{1}{2}$, then there exists a number c, 2 < c < 5 for which f ' (c) is equal to

A sector $OABO$ of central angle $\theta$ is constructed in a circle with centre $O$ and radius $6.$ The radius of the circle that is circumscribed about the triangle $OAB,$ is

The value of $\underset{x\to 0}{lim}\left(\right[\frac{100x}{sinx}]+[\frac{99sinx}{x}\left]\right)$,where [.] denotes the greatest integer function, is

Evaluate $\int e^x{\left(\frac{1-x}{1+x^2}\right)}^{2}dx$

(where $C$ is constant of integration)

The sum of the first n terms of the series, $1^2+{2.2}^2+3^2+{2.4}^2+5^2+{2.6}^2+\dots .is\frac{n}{2}(n+1{)}^2$, when n is even. When n is odd, the sum is

If the points (0, 1, 2), (2, –1, 3) and (1, –3, 1) are the vertices of a triangle, then the triangle is

Let $\alpha ,\beta$ be distinct roots of $a{x}^2+bx+c=0$, then $\underset{x\to \alpha }{Lt}\frac{1-cos(a{x}^2+bx+c)}{(x-\alpha {)}^2}$=

If $(2\sin 3x\cos x-2\cos 3x\sin x{)}^2=2,$ then $x$ equals to

Consider the signal shown in below figure, corresponds to the second derivative of a given function f(t). Then the Fourier transform of f(t) is

The number of roots of the quadratic equation $8se{c}^2\theta -6sec\theta +1$ = 0 is

Coefficient of a32 in the binomial expansion of ${(a^4-\frac{1}{a^3})}^{15}$ is:

The coefficient of $x^n$ in the polynomial $(x+{}^n{C}_0)(x+3\cdot {}^n{C}_1)\cdots (x+(2n+1)\cdot {}^n{C}_n)$ is

Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.

If a tangent to a parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of the parabola. Then point of contact(s) is/are

Find the vector equation of the straight line passing through (1, 2, 3) and perpendicular to the plane $r.(\hat{i}+2\hat{j}-5\hat{k})+9=0$

If the major axis of a vertical ellipse is three times the minor axis, then its eccentricity is equal to

Choose the word that is most nearly opposite in meaning to the given word.

FRAGILE

Points $A,B,C,D$ are in the plane such that segments $AB,BC,CD,DA$ have lengths $2,7,5,12$ respectively. Let $m$ be the minimum possible value of the length of segment $AC$ and let $M$ be the maximum possible value of the length of segment $AC.$ The value of $m\cdot M$ is

If the function f(x)= x³-3ax²+b is strictly increasing derivative for x > 0, then which of the following is always true?

If the tangent drawn at point $P(t^2,2t)$ on the parabola $y^2=4x$ is same as the normal drawn at point $Q(\sqrt{5}\cos \theta ,2\sin \theta )$ on the ellipse $4x^2+5y^2=20,$ then

If $y={\log }_{10}e+{\log }_{10}x+{\log }_{e}x,$ then $\frac{dy}{dx}$ is equal to



[1 mark]

If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'.

The number of 5-digit numbers which are divisible by 3 that can be formed by using the digits 1,2,3,4,5,6,7,8 and 9, when repetition of digits is allowed, is

If $(x_r,y_r):r=1,2,3,4$ be the points of intersection of the parabola $y^2=4ax$ and the circle $x^2+y^2+2gx+2fy+c=0,$ then

If cos 3theta = $\alpha cos\theta +\beta co{s}^3\theta$, then $(\alpha ,\beta )$ =

Let $A,B,C$ are three angles such that
$\sin A+\sin B+\sin C=0,$ then the value of $\frac{\sin A.\sin B.\sin C}{\sin 3A+\sin 3B+\sin 3C}$ (wherever defined) is

If a function $f:[-2,\infty )\to R$ is such that $f\left(x\right)=x^2+4x-|x^2-4|$, then the value(s) $f\left(x\right)$ can have is (are)