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The set of values of $a$ for which $a^2-a-2<0$ and $\underset{x\to \infty }{lim}{a}^x=0$ is

Let $a_1,a_2,a_3,\dots$ be an A.P. If $\frac{a_1+a_2+\cdots +a_{10}}{a_1+a_2+\cdots +a_p}=\frac{100}{p^2},$ $p\ne 10,$ then $\frac{a_{11}}{a_{10}}$ is equal to

Choose the correct answer.

$\int \frac{cos2x}{(sinx+cosx{)}^2}dxisequalto\left(a\right)\frac{-1}{sinx+cosx}+C\left(b\right)log|sinx+cosx|+C\left(c\right)log|sinx-cosx|+C\left(d\right)\frac{1}{(sinx+cosx{)}^2}+C$

Let $z=cos\theta +isin\theta$. The value of ${\sum }_{m=1}^{15}Im\left(z^{2m-1}\right)$ at $\theta =2^{\circ }$

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type

$\int \left[f\right(x\left)h\right(x)+g(x\left)\right]dx$

Let $\int f\left(x\right)h\left(x\right)dx=I_1$ and $\int g\left(x\right)dx=I_2$

Integrating $I_1$ by parts, we get

$I_1=f\left(x\right)\int h\left(x\right)dx-\int \left\{f^{\prime }\right(x)\int h(x\left)dx\right\}dx$

Now find $\int (\frac{1}{logx}-\frac{1}{(logx{)}^2})dx$ (x > 0)

Three numbers are chosen at random without replacement from {1, 2, ......, 15}. Let $E_1$ be the event that minimum of the chosen numbers is 5 and $E_2$ be that their maximum is 10 then:

In YDSE experiment with identical slit, the distance from central maxima where we get intensity half of maximum is [`D` , `d` and $\lambda$ have usual meaning]

If $x=3t$ and $y=t^3,$ then $\frac{dy}{dx}$ at $t=1$ is equal to

If and;

0° < A + B ≤ 90°, A > B find A and B.

The value of $\underset{1}{\overset{e}{\int }}({\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x)\ln xdx$

If 3X + 2Y = I and 2X - Y = O, where I and O are unit and null matrices of order 3 respectively, then

If ${\log }_2(5\times 2^x+1),{\log }_4(2^{1-x}+1)$ and $1$ are in A.P., then $x$ equals

Find the inverse of the given matrix
$\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]$

1. $X_1{Y}_1+X_0{Y}_0$

Differential equation of the family of circles touching the line $y=2$ at $(0,2)$ is

Let $f\left(x\right)$ be a differentiable function at $x=a$ with $f^{\prime }\left(a\right)=2$ and $f\left(a\right)=4$. Then $\underset{x\to a}{lim}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ equals:

If$\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}+\overrightarrow{D}$, then select the correct alternative:

If $cosecx-sinx=a^{3}andsecx-cosx=b^3;thenwhatisthevalueof{a}^2{b}^2\left(a^2+b^2\right)-$

If $f\left(x\right)=\cos \left({\pi }^2\right]x+\cos \left(-{\pi }^2\right]x,$ where [x] stands for the greatest integer function, then

The numerically greatest term in the expansion of $(3-5x{)}^{15}$ when $x=\frac{1}{5}$ is

The line $(3x-y+5)+\lambda (2x-3y-4)=0$ will be parallel to y-axis, if $\lambda$ =

Construct
a 2 ×
2 matrix,,
whose elements are given by:

(i)

(ii)

(iii)

The number of points at which $f\left(x\right)=\{\begin{array}{cc}min(x,x^2),& \text{if}-\infty <x<1\\ min(2x-1,x^2),& \text{if}x\ge 1\end{array}$ is not differentiable is

A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also find the number of unsuccessful attempts to open the lock.

A triangle has vertices A, $(x_1,y_1)$ for i=1,2,3. Then the determinant

$\Delta =\left|\begin{array}{ccc}{x}_2-x_3& y_2-y_3& y_1(y_2-y_3)+x_1(x_2-x_3)\\ x_3-x_1& y_3-y_1& y_2(y_3-y_1)+x_2(x_3-x_1)\\ x_1-x_2& y_1-y_2& y_3(y_1-y_2)+x_3(x_1-x_2)\end{array}\right|=0means$

Solve $|x+1|-|1-x|=2$