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The solution of the differential equation $x^{2}dy+y(x+y)dx=0$ is:

(where $c$ is integration constant)

If $f(r+s)=f\left(r\right)+f\left(s\right)\forall r,s\in R$. Let $m$ and $n$ be integers. Then $f\left(\frac{n}{m}\right)$ is equal to

Maximize:Z=x+2y
Subject to the constraints:$x+2y\ge 100,2x-y\le 0,2x+y\le 200,x\ge 0,y\ge 0$
Solve the L.P.P. graphically.

Points (–2, 4, 7), (3, –6, –8) and (1, –2, –2) are
[AI CBSE 1982]

If both $\theta$ and $\varphi$ are acute angles such that $\sin \theta =\frac{1}{2}$ and $\cos \varphi =\frac{1}{3}$, then $(\theta +\varphi )$ belongs to

The equation of the planes passing through the line of intersection of the planes 3x-y-4z=0 and x+3y+6=0 whose distance from the origin is 1, are

The mean of the values $0,1,2,\dots \dots \dots ,n$ having corresponding weight ${}^n{C}_0{,}^n{C}_1{,}^n{C}_2,\dots \dots \dots {\dots }^n{C}_n,$ respectively is

A planar structure of length L and width W is made of two different optical media of refractive indices $n_1=1.5$ and $n_2=1.44$ as shown in figure. If $L>>W$, a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the structure. For $L=9.6m$, if the incident angle $\theta$ is varied, the maximum time taken by a ray to exit the plane CD is $t\times {10}^{-9}s$, where $t$ is. [Speed of light $c=3\times {10}^{8}m/s$]

Let $f\left(x\right)=2x^2+5,g\left(x\right)=\sin x+\cos x,$ then the interval of $x$ for which $f\left(g\right(x\left)\right)$ is increasing, is

If the roots of $x^2-ax+b=0$ are real and differ by a quantity which is less than $c(c>0)$ then $b\in$

In triangle $ABC,CD$ is the bisector of the angle $C.$ If $\cos \frac{C}{2}=\frac{1}{3}$ and $CD=6$. If $(\frac{1}{a}+\frac{1}{b})=\frac{1}{\lambda },$ then $\lambda$ is:

Let $\alpha =\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}$ and $\beta ={\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$. Then the value of $\alpha \beta$ is

The p^th,
q^th and r^th terms of an A.P. are
a, b, c respectively. Show that

Represent to solution set of each of the following in equations graphically in two dimensional plane :

$x+2y\ge 0$

If the system of equations $2x+3y-z=0,x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z)$, then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to :

If $f\left(x\right)=\{\begin{array}{cc}\frac{\ln (1+3x)-\ln (1-2x)}{x},& x\ne 0\\ a,& x=0\end{array}$ is continuous at $x=0$, then the value of $a$ is

If $1+(1+x)+(1+x{)}^2+(1+x{)}^3+......+(1+x{)}^n={\sum }_{k=0}^n{a}_k{x}^k,$ then which of the following is true?

sin²
6x
– sin²
4x
= sin 2x
sin 10x

Derivative of $f\left(x\right)=\frac{\sin x-x\cos x}{(x\sin x+\cos x)}$ w.r.t. $x$ is

P is a moving point, S is the focus and L is the directrix as shown in figure.

Which of the following represents the equation of a conic?

In $\Delta ABC,a,b$ and $c$ are the lengths of the sides opposite to the angles $A,B$ and $C$ respectively. The bisector of the $\angle A$ meets the side $BC$ at $D$ and the circumscribed circle at $E$. Then $DE$ equals

Let two events $A$ and $B$ are such that, probability of occurrence of only $A$ is $\frac{3}{10}$ and probability of occurrence of $B$ is $\frac{3}{5}.$ Then the probability of occurrence of any of two events is

If a double ordinate of the parabola $y^2=4ax$ be of length $8a$, then the angle between the lines joining the vertex of the parabola to the ends of this double ordinate is

Slope of a normal at x = a on the graph of f(x) is -1/2. If the rate of rate of change of f(x) at the point a is $\alpha$ , find the value of $4{\alpha }^2.$

The equation of the reflection of the hyperbola $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{9}=1$ about the line $x+y-2=0$ is

The complete set of values of $k$, for which the quadratic equation $x^2-kx+k+2=0$ has equal roots, consists of

If a and b are the coefficients of xn in the expansion of $(1+x{)}^{2n}$ and $(1+x{)}^{2n-1}$ respectively, find $\frac{a}{b}$

Choose the correct answer in the following question:
If a matrix A is both symmetric and skew-symmetric matrix, then
(a)A is a diagonal matrix
(b)A is zero matrix
(c)A is a square matrix
(d)None of these

A man running round a race course notes that the sum of the distances of two flag posts from him is 8 meters.The area of the path he encloses in square meters if the distance between flag posts is 4 is

Let $f^{\prime }\left(x\right)=\frac{192x^3}{2+{\sin }^4\pi x},\forall x\in R$ and $f\left(\frac{1}{2}\right)=0$. If $m\le \underset{1/2}{\overset{1}{\int }}f\left(x\right)dx\le M$, then the values of $m$ and $M$ are

Find the following integrals.
$\int \frac{se{c}^{2}x}{cose{c}^{2}x}dx.$