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If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$, show that $A^2-5A+I=0.$

The value of the integral $\int \frac{x^3+x+1}{x^2-1}dx$ is

(where $C$ is an arbitrary constant)

The value of expression

$3(1!)-4(2!)+5(3!)-6(4!)+.....-2008(2006!)+(2007!)$ is

For the parabola $y^2=4px$ find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

${lim}_{x\to 0}\frac{x(2^x-1)}{1-cosx}$

What is 0 divided by 0?

In triangle ABC, right angled at B, if tan A =1/√3, find the value of : sinAcosC+cosAsinC.

If the sides $a,b,c$ of a triangle $ABC$ form successive terms of G.P. with common ratio $r(>1)$ then which of the following is correct?

If A and B are any two different square matrices of order n with A – B is non-singular $A^3=B^3$ and A(AB) = B(BA) , then

If from Lagrange's Mean Value Theorem, we have $f\text{'}\left(x_1\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$, then .

All $x$ satisfying the inequality $({\cot }^{-1}x{)}^2-7({\cot }^{-1}x)+10>0$, lie in the interval :

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (\alpha +2)x+\sin x}{x},& x<0\\ b,& x=0\\ \frac{(x+3x^2{)}^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}},& x>0\end{array}$ is continuous at $x=0$ then $a+2b$ is equal to :

If α
and β are different complex
numbers with
= 1, then find.

The general solution of the differential equation $\frac{dy}{dx}=\frac{r^2}{{(x+y)}^2},$ is (where $r$ is a fixed constant and $c$ is a constant of integration )

The foot of the perpendicular from P(1, 0, 2) to the line $\frac{x+1}{3}=\frac{y-2}{-2}=\frac{z+1}{-1}$ is

Let $PS$ be the median of the triangle with vertices $P(2,2),Q(6,-1)\text{and}R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to $PS$ is

f(x) and f’(x) are differentiable at x = c. Which of the following is the condition for f(x) to have a local minimum at x = c, if f’(c) = 0

The value of the integral $\int \frac{x}{(x-1{)}^2(x+2)}dx$

(where $m$ is an arbitrary constant)

Conisder the recurrance relation $a_k=-8a_{k-1}-15a_{k-2}$ with initial condition $a_0=0and{a}_1=2$. which of the following is an explicit solution to this recurrance relation ?

If $lo{g}_2(9^{x-1}+7)-lo{g}_2(3^{x-1}+1)=2$ then x values are

If $f\left(x\right)=max\{x^4,x^2,\frac{1}{4}\},\forall x\in [0,\infty ),$ then the sum of square of reciprocal of all the values of $x,$ where $f\left(x\right)$ is not differentiable is:

If $\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$ ; then for all

$\theta \in (\frac{3\pi }{4},\frac{5\pi }{4}),det\left(A\right)$ lies in the interval :

If the first of a G.P. $a_1,a_2,a_3,.......$ is unity such that $4a_2+5a_3$ is least, then the common ratio of G.P. is

Find the values of $co{s}^{-1}x$ in terms of given options.

Write the middle term in the expansion of ${(\frac{2x^2}{3}+\frac{3}{2x^2})}^{10}.$

In $\Delta ABC,$ the value of $(a-b{)}^2{\cos }^2\frac{C}{2}+(a+b{)}^2{\sin }^2\frac{C}{2}=$

By introducing a new variable t, putting x=cos t , the expression is $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y$transformed into :