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The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point :

The shortest distance between the lines $\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$ and $x+y+z+1=0,2x-y+z+3=0$ is:

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :

A common tangent to $9x^2-16y^2=144$ and $x^2+y^2=9$, is

If $p=\underset{x\to \infty }{lim}(\sin \sqrt{x^2+1}-\sin |x|)$ and $q=\underset{x\to -\infty }{lim}\left[\sin \left(\cos {\left(\sqrt{|x|-x^3}\right)}^{-1}\right)\right]$

(Where [.] denotes greatest integer function), then which of the following is/are correct:

The shortest distance between the curves $y^2=x^3$ and $9x^2+9y^2–30y+16=0$ is

If $I=\underset{0}{\overset{1}{\int }}\frac{dx}{\sqrt{4-x^2-x^3}},$ then which of the following is/are TRUE?

A unit circle with the following P coordinate is given. The value of $\theta$ is:

Let $A(0,1),B(1,1),C(1,-1)$ and $D(-1,0)$ be four points. If $P$ is any other point, then the minimum value of $PA+PB+PC+PD$ is equal to

Let $f\left(x\right)=x\cdot \left[\frac{x}{2}\right],$ for $-10<x<10$, where $\left[t\right]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to

Let f:
{1, 3, 4} → {1, 2, 5} and g:
{1, 2, 5} → {1, 3} be given by f =
{(1, 2), (3, 5), (4, 1)} and g =
{(1, 3), (2, 3), (5, 1)}. Write down gof.

if a continuous function f(x) does not have a root in the interval $[a,b]$, then which one of the following statements is TRUE?

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.2² + 3.2² + … + n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2

For a standard hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Match the following.

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ 1.a^2>b^2& \text{P.Director circle is real}\\ 2.a^2=b^2& \text{Q.Director circle is imaginary}\\ 3.a^2<b^2& \text{R.Centre is the only point from which}\\ & \text{two perpendicular tangents can be drawn on the}\\ & \text{hyperbola}\end{array}$

If $k=\underset{0}{\overset{\pi }{\int }}x\sin xdx,$ then which of the following is/are correct ?

If the line $\frac{x-{\lambda }^2}{-3\lambda }=\frac{y+5}{\lambda }=\frac{z+3}{-1}$ lies on the plane $x+\lambda y-2z=0,$ then the value of $\lambda$ is equal to

What is the binomial conjugate of

$\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}+\frac{\sqrt{2}+1}{\sqrt{2}-1}$?

The integral $\underset{0}{\overset{\pi }{\int }}\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}dx$ equals to:

If two balanced dice are tossed once, the probability of the event, that the sum of the integers coming on the upper sides of the two dice is 9, is [MP PET 1987]

If at x = 1, y = 2x is tangent to the parabola $y=a{x}^2+bx+c$, then respective values of a, b, c possible are

Find the equations of the lines joining the vertex of the parabola $y^2=6x$ to the point on it which have abscissa 24.

In a group of $50$ people, $35$ speak Hindi and $25$ speak both English and Hindi. Then how many people speak only English ? (Assuming each person speaks at least one language)

Q. A bag contains 4 white, 5 red, and 6 blue balls. Three balls are drawn at random. What is the probability that one ball being red, the other being blue, and the last being white are drawn simultaneously?



Q. एक थैला में 4 सफेद, 5 लाल, और 6 नीली गेंदें हैं। 3 गेंदों को यादृच्छिक रूप से निकाल लिया गया है। क्या प्रायिकता है कि एक गेंद लाल, दूसरी नीली, और आखिरी सफेद एक साथ निकाली जा रही हो?

If $y=\frac{1}{\sqrt{a^2-x^2}},find,\frac{dy}{dx}$.

The solution of the differential equation
$x\frac{dy}{dx}+2y=x^2(x\ne 0)$ with $y\left(1\right)=1$ is

The length of the projection of the line joining (1, 2, 3) and (-1, 4, 2) on the line which has direction ratios (2, 3, -6) is .

Which among the following is the correct graphical representation of the quadratic polynomial $y=-x^2-2x+3?$

The product of the real roots of the equation $(x-1{)}^4+(x-5{)}^4=82$ is

Find the equation of
the plane through the line of intersection of the planes
and
which
is perpendicular to the plane