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The
general solution of the differential equation

is

A. xe^y
+ x² = C

B. xe^y
+ y² = C

C. ye^x
+ x² = C

D. ye^y
+ x² = C

O is the origin and A is the point (3,4). If a point P moves so that the line segment OP is always parallel to the line segment OA, then the equation to the locus of P is

Let P = {(x,y) $x^2+y^2=1,x,y\in R\}$. Then P is.

Suppose that p,q and r are three non-coplanar vectors in $R^3$. Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$, $y$ and $z$ respectively, then the value of $2x+y+z$ is ___

Two boys raising a load pull at an angle to each other. If they exert forces of 30 N and 60 N respectively and their effective pull is at right angles to the directions of the pull of the first boy, what is the angle between their arms? What is the effective pull.

If $y=f\left(x\right),$ where $f:A\to B$ is a function in $x,$ then which of the following statements is true?

(i) Every element of $A$ needs to have an image.

(ii) $x\in A$ must be related to one and only one value of $y$ of $B.$

One Indian and four American men and their wives are to be seate randomly around a circular table. Then, the conditional probability that Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

A person draws a card from a pack, replaces it shuffles the pack, again draws a card, replace it and draws again. This he does unitl he draw a heart. The probability that he will have to make at least four draws is:

Question 1

Make up as many expressions with numbers (no variables) as you can from three numbers 5, 7 and 8. Every number should be used not more than once. Use only addition, subtraction and multiplication.

The value of $\underset{x\to \infty }{lim}(\frac{x^3}{3x^2-4}-\frac{x^2}{3x+2})$ is:

If $g\left(x\right)=\ln \left[f\right(x\left)\right]$ and $f(x+1)=xf\left(x\right),$ then $g^{\prime \prime }\left(\frac{5}{2}\right)-g^{\prime \prime }\left(\frac{1}{2}\right)$ is equal to

There are $10$ points in a plane of which no $3$ points are collinear and $4$ points are concyclic. Number of different circles that can be drawn through at least $3$ points is

Let $\overrightarrow{p}$ and $\overrightarrow{q}$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|\overrightarrow{p}|=p,|\overrightarrow{q}|=q.$ The points $R$ and $S$ divides $PQ$ internally and externally in the ratio $3:2$ respectively. If $\overrightarrow{OR}$ and $\overrightarrow{OS}$ are perpendicular, then

Compute the magnitude of the following vectors:
$a=\hat{i}+\hat{j}+\hat{k};b=2\hat{i}-7\hat{j}-3\hat{k};c=\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}-\frac{1}{\sqrt{3}}\hat{k}.$

The graph of $\sqrt{x}+\sqrt{y}=\sqrt{2}$ is

The equation of tangent to the curve $x{y}^3+x^{2}y+2x^2+3y-2x=0$ at origin, is

Let $f\left(x\right)=\{\begin{array}{cc}\left(\right[x]+\sqrt{\{x}\}),& x\ne 0\\ \lambda ,& x=0\end{array}$. If $f\left(x\right)$ is continuous at $x=0$, then the value of $\lambda$ is

(where $[.]$ represents the greatest integer function, {.} represents the fractional part function and $\lambda \in R$)

The image of a point $(3t+1,1-4t),\forall t\in R-\left\{0\right\}$ in a line lies on $3x-4y+1=0.$ Then the slope of line(s) is/are

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}.$ The probability that the missile hits the target, given that it is not intercepted is $\frac{3}{4}.$ If three missiles are fired independently from the ship, then the probability that all three hit the target, is

In $△ABC$, if $\sin 2A=\sin 2B$ but $\angle A\ne \angle B$ and $3\tan A-4=0$, then value of expression $\frac{1}{1+{\tan }^{2}B}+\frac{\sin (B-A)}{2}+\cot C(\cos B\sqrt{1+{\sin }^{2}A}-\cos A\sqrt{1+{\sin }^{2}B})$ is

If $f\left(x\right)=\underset{0}{\overset{\pi }{\int }}\frac{t\sin t}{\sqrt{1+{\tan }^{2}x{\sin }^{2}t}}dt$ for $0<x<\frac{\pi }{2},$ then

The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\text{and}\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$is :

If $a_1,a_2,a_3,\cdots$ are in A.P. such that $a_1+a_7+a_{16}=40$, then the sum of the first $15$ terms of this A.P. is :

$\frac{3(x-2)}{5}\le \frac{5(2-x)}{3}$

Let $y=y\left(x\right)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$, is equal to :

The distance of the point (-1, -5, -10) from the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane x - y + z = 5, is
[AISSE 1985; DSSE 1984; MP PET 2002]