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The third derivative of a function f(x) vanishes for all x. If f(0)=1, f'1(1)=2 and f”(1) = – 1, then f(x) is equal to

Find the expansion of
using
binomial theorem.

Write the length of the intercept made oy the circle $x^2+Y^2+2x-4y-5=0$
on y-axis.

The value of the limit ${lim}_{x\to 0}{\left(\frac{x}{sinx}\right)}^{6/x^2}$ is

Find the mean and
variance for the data

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is:

Express $\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]$ as the sum of a symmetric and a skew symmetric matrix .

Which of the following statements are correct?

1. The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is called an equilateral hyperbola.

2. Eccentricity of equilateral hyperbola = $\sqrt{2}$

3. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=0$

4. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=-a^2$

If -1,$\alpha$,${\alpha }^3$,${\alpha }^5$,$\overline{\alpha }$,$\overline{{\alpha }^3}$,$\overline{{\alpha }^5}$ are the roots of the equation $z^7$ + 1 = 0. Find the value of $cos\frac{\pi }{7}$ $\times$ $cos\frac{3\pi }{7}$ $\times$ $cos\frac{5\pi }{7}$.

$\frac{sin2A}{1+cos2A}$ .

$\frac{cosA}{1+cosA}$ =

Find
the inverse of each of the matrices, if it exists.

If $f\left(x\right)=\left|\begin{array}{cc}2x& 3\\ x^2& 1\end{array}\right|,$ then the value of $f^{\prime }\left(1\right)$ is

The value of $I=\underset{-10}{\overset{3}{\int }}\mathrm{sgn}({\cot }^{-1}x+e^{-x}+1-\sin x)dx$ is

The letters of the word 'FORTUNATES' are arranged at random in row.
What is the chance that the two 'T' come together?

Find out the value of K_c
for each of the following equilibria from the value of K_p:

The solution of differential equation $(3y-7x+7)dx+(7y-3x+3)dy=0$ is:

(where $C$ is constant of integration)

To check whether the matrix B is an matrix A, we need to check whether

The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point (1, 1) is

If the length of the tangents from $(a,b)$ to the circles $x^2+y^2-4x-5=0$ and $x^2+y^2+6x-2y+6=0$ are equal then $10a-2b$ is equal to

The points $z_1=3+\sqrt{3}i\text{and}{z}_2=2\sqrt{3}+6i$ are given on a complex plane. The complex number lying on the bisector of the angle formed by the vector $z_1$ and $z_2$ is

Consider a plane $x+y-z=1$ and point $A(1,2,-3).$ A line $L$ has the equation $x=1+3r,y=2-r$ and $z=3+4r.$



The coordinate of a point $B$ of Line $L$ such that $AB$ is parallel to the plane is

The domain of the function $\frac{1}{\sqrt{[x{]}^2-5[x]+6}}$ is

(where [.] denotes the greatest integer function)

If the points of intersections of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the circle $x^2+y^2=4b,$ $b>4$ lie on the curve $y^2=3x^2,$ then $b$ is equal to :

The critical point(s) of the function $f\left(x\right)=(x-2{)}^{2/3}(2x+1)$ is(are)

The value of (3 + 4i) (6 - 7i) is

If two tangents drawn from the point $(\alpha ,\beta )$ to the parabola $y^2=4x$ such that the slope of one tangent is double of the other, then

If the function $f\left(x\right)=a{x}^3+b{x}^2+11x-6$ satisfies conditions of Rolle's theorem in $[1,3]$ for $x=2+\frac{1}{\sqrt{3}}$, then value of $a$ and $b$ respectively, is

In $\Delta ABC,(a-b{)}^2{\cos }^2\frac{C}{2}+(a+b{)}^2{\sin }^2\frac{C}{2}$ is equal to

Which of the following functions is inverse to itself?

Match the following:

$$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1. {x: x}\in \text{R, -5 < x}\le \text{15 }}& \text{a. 1}\\ \text{2. A =}\varphi \text{, then the power set P(A) has}\underset{–}{}\text{elements}& b.8\\ \text{3. Universal set of Rational Numbers}\&\text{Irrational Numbers}& c.\text{Real Numbers}\\ \text{4. No. of elements in Power Set of A}=\{a,b,c\}& d.\text{Integers}\\ & e.(-5,15]\\ & f.[-5,15)\end{array}$$