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The slope of normal to the curve $y=2x^2+3sinx$ at x = 0 is

Set of all real values of $x$ satisfying the inequation $\frac{{\log }_2(x^2-5x+4)}{{\log }_2(x^2+1)}>1$ is

The circle $x^2+y^2=4x+8y+5$ interesects the line 3x - 4y = m at two distinct points if

(2010)

Let $f\left(x\right)=\frac{1+x}{1-x}.$ If $A$ is matrix for which $A^3=O,$ then $f\left(A\right)$ is

Which one of the following Boolean expressions is NOT a tautology?

For given binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(vi) On R-{-1},define $a\ast b=\frac{a}{b+1}$

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)

Number of tangents to $y^2=2x$ through $(1,2)$ is

If the graph of $y=x^2$ is given by


then graph of $y-2=3(x-5{)}^2$ will be given by:

The shortest distance of the point (a, b, c) from the x-axis is
[MP PET 1999; DCE 1999]

A variable line $L$ is drawn through $O(0,0)$ to meet $L_1:x-y-8=0$ and $L_2:x-y-16=0$ at points $A$ and $B$ respectively. A point $P$ is taken on $L$ such that $\frac{1}{4OP}=\frac{1}{OA}+\frac{1}{OB}$. Then the locus of $P$ is

The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chance of happening of each event are

(Where, $p_1,p_2$ be the chances of happening of the first and second events, respectively)

A letter is chosen at random from the word $“STATISTICS”$. The probability of getting a vowel is

Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$. The roots of $a(x-2{)}^2-b(x-2\left)\right(x-3)+c(x-3{)}^2=0$, where $a\ne 0$ are

If the equation $2^x+4^y=2^y+4^x$ is solved for $y$ in terms of $x$, where $x<0$, then the sum of the solutions is

Find the vector
equation of the plane passing through (1, 2, 3) and perpendicular to
the plane

The foot of perpendicular of the point $M(1,-2,1)$ on the plane $P:2x-y+2z+3=0$ is

Let 'f' be a function such that $f\left(xy\right)=f\left(x\right)f\left(y\right)\forall x,yϵR^{+}$ and $f(1+x)=1+x(1+g(x\left)\right)$ where ${lim}_{x\to 0}g\left(x\right)=0$. Then ${\int }_1^2\frac{f\left(x\right)}{f^{\prime }\left(x\right)}.\frac{1}{1+x^2}dx=$

If two tangents drawn from a point $P$ to the parabola $y^2=16(x–3)$ are at right angles, then the locus of point $P$ is

For two finite disjoint sets $A$ and $B$, if the number of elements in power set of $A$ is $224$ more than the number of elements in power set of $B$, then the number of elements present in either of the sets is

A rod $AB$ of the length $30$ units slips on the coordinate axes such that $A$ lies on $x-$axis and $B$ lies on $y-$axis. Then, the locus of $C$ if $BC=2AC$ is

Let $f\left(x\right)=x\sin x-\frac{1}{2}{\sin }^{2}x$ $\forall x\in (0,\frac{\pi }{2}),$ then which of the following option(s) is/are CORRECT?

Does the point (–2.5,
3.5) lie inside, outside or on the circle x² + y²
= 25?

The minimum value of $(\sin x{)}^{\sin x},$ where $0<x<\frac{\pi }{2}$ is

The ratio in which the sphere $x^2+y^2+z^2=504$ divides the line segment AB joining the points A(12,-4,8) and (27,-9,18) is given by

If the power of point $(2,1)$ with respect to the circle $2x^2+2y^2-8x-6y+k=0$ is positive, then

Find the equation of a line which is perpendicular to the line $\sqrt{3}x-y+5=0$ and which cuts off an intercept of 4 units with the negative direction of y-axis.

State the first principle of mathematical induction.

The sum of the divisors of $2^5\cdot 3^4\cdot 5^2$ is

If a $ϵ$ R- and a $\ne$ - 2 then the equaiton $x^2+a|x|+1=0:$

The shortest distance $\left(\text{in units}\right)$ between the lines whose equations are $\overrightarrow{r}=3\hat{i}+5\hat{j}+7\hat{k}+\lambda (\hat{i}+2\hat{j}+\hat{k})$ and $\overrightarrow{r}=-\hat{i}-\hat{j}+\hat{k}+s(7\hat{i}-6\hat{j}+\hat{k})$ is:

Consider a hyperbola whose centre is at origin. If line $x+y=2$ touches this hyperbola at $P(1,1)$ and intersects the asymtotes at $A$ and $B$ such that $AB=6\sqrt{2}$ units, then the equation of the tangent to the hyperbola at $(-1,\frac{7}{2})$ is