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$Ify=500e^{7x}+600e^{-7x},showthat\frac{d^{2}y}{d{x}^2}=49y.$

The value of $\underset{3}{\overset{4}{\int }}|x^2-7x+12|dx$ is

If $S=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right],A=\frac{1}{2}\left[\begin{array}{ccc}b+c& c-a& b-a\\ c-b& c+a& a-b\\ b-c& a-c& a+b\end{array}\right],$ then $SA{S}^{-1}=$

The number of natural numbers less than $7,000$ which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to :

Sketch the graph of
and
evaluate

The values of $x$ for which the logarithmic function $f\left(x\right)={\log }_{|-x^2+2x-4|}\left(x\right(3-x\left)\right)$ is defined, is

The sum of the series $(\sqrt{2}+1)+1+(\sqrt{2}-1)+\cdots$ up to infinite term is

A polynomial in $x$ of degree greater than $3,$ leaves remainders $2,1$ and $-1$ when divided by $(x-1),(x+2)\text{and}(x+1)$ respectively. What will be the remainder when it is divided by $(x-1)(x+2)(x+1).$

The equation of angular bisector between the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{4-z}{1}$ and $\frac{1-x}{1}=\frac{y-4}{2}=\frac{z-5}{1}$ is given by

The greatest positive integer $k$, for which ${49}^k+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\cdots +{49}^2+49+1,$ is

Given $\int \sin \left(x\right)dx=-\cos \left(x\right)$ and $\int \cos \left(x\right)dx=\sin \left(x\right)$. If $f\left(x\right)=36\int \left(\sin \left(2x\right)+\cos \left(3x\right)\right]dx$, then find the value of $-f\left(\pi \right)$ [ take constant of integration equal to zero]

The solution of differential equation $\frac{xdy}{dx}+y=xlogx$ is

If the first term of a G.P., $a_1,a_2,a_3,..$ is unity, then the value of $4a_2+5a_3$ will be minimum when the common ratio is

In $\Delta ABC,\frac{a}{b}=2+\sqrt{3}$ and $\angle C={60}^0.$ Then the ordered pair $(\angle A,\angle B)$ is equal to

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin, respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at $S$ and $R$, then the area (in sq. units) of $\Delta SOR$ is equal to

If the points (0,4) and (0,2) are respectively the vertex and focus of the parabola, then find the equation of the parabola.

In how many ways can 12 balls be divided between 2 boys, one receiving 5 and the other 7 balls?

If $2\hat{i}+3\hat{j}+4\hat{k}$ and $\hat{i}-4\hat{j}+7\hat{k}$ are the sides of a parallelogram, then is the vector representing its diagonal.

Find the sum total of 3, 5 and 9.

$f\left(x\right)=\frac{1}{2}-tan\left(\frac{\pi x}{2}\right)[-1<x<1]$

and g(x) =$\sqrt{(3+4x-4x^2)}$

Find domain of (f + g).

If $\cos x\frac{dy}{dx}-y\sin x=6x,(0<x<\frac{\pi }{2})$ and $y\left(\frac{\pi }{3}\right)=0$, then $y\left(\frac{\pi }{6}\right)$ is equal to :

Equation of the auxiliary circle of the ellipse
$\frac{x^2}{12}+\frac{y^2}{18}=1$ is

If the function $f\left(x\right)=x^4+b{x}^2+8x+1$ has a horizontal tangent and a point of inflection for the same value of $x,$ then the value of $b$ is equal to

Simplify
and solve the linear equation

Can i complete complex number in 1 day if i were studying it first time........

In which one of the following intervals, Rolle's theorem holds good for $f\left(x\right)=x^2\sin \frac{1}{x}+x^3\cos \frac{1}{2x}$?

The vertex of a parabola is the point (a,b) and latus rectum is of length /. If the axis of the parabola is along the positive direction of y-axis, then its equation is

The ratio of sum of first three terms of a G.P. to the sum of first six terms is 64:91, the common ratio of G.P. is

The pth, qth and rth terms of an A.P. are a, b, c, respectively. Show that $(q-r)a+(r-p)b+(p-q)c=0$.

If $\alpha ,\beta ,\gamma$ be the angles which a line makes with the positive direction of co-ordinate axes, then $si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma =$

[RPET 2000; AMU 2002; MP PET 1989, 98, 2000, 03, Pb. CET 2001]

The value of integration $\int (\sin x{)}^{\ln x}\left\{\frac{\left(x\ln x\right)\cot x+\ln \sin x}{x}\right\}dx$ is

If $X$ and $Y$ are two sets, then $X\cap (Y\cup X{)}^C$ equals to

The point of inflection for the function $f\left(x\right)={\sin }^{-1}x$ is: