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If the roots of the equation $a{x}^2+bx+c=0$ are reciprocal of the roots of the equation $p{x}^2+qx+r=0$, then which of the following options is always correct?

Prove by induction the inequality $(1+x{)}^n\ge 1+nx,$ whenever x is positive and n is a positive integer.

A point is randomly selected with uniform probability in the x-y plane with in the rectangle with corners at (0, 0), (1, 0), (1, 2) and (0, 2). If p is the length of the position vector of the point, then the expected value of $p^2$ is

1. 5

The solution of differential equation $(1+e^{2y})e^{{\tan }^{-1}x}dx-(1+x^2)(e^y+{(e^y-1)}^2)dy=0$ is

$($Here, $C$ is a constant of integration$)$

A plane P passes through a point P (3, -2, 1) and is perpendicular to the vector

$\overrightarrow{V}=4\hat{i}+7\hat{j}-4\hat{k}.$ The distance between the plane P and the plane. $\overrightarrow{r}.(4\hat{i}+7\hat{j}-4\hat{k})+33=0$

If $a$ and $b$ are unit vectors and $\theta$ is the angle between them, then $|a+b|<1$, if

The angle between the tangents to the curve $y^2=2ax$ at the points where $x=\frac{a}{2}$, is

The value of $\underset{x\to 2}{lim}\frac{2^x+2^{3-x}-6}{\sqrt{2^{-x}}-2^{1-x}}$ is

The minimum value of $\frac{(6+x)(11+x)}{(2+x)},x\ge 0$ is

Side of an equilateral triangle expands at the rate of 2 cm/s. The rate of increase of its area when each side is 10 cm, is

In a triangle $ABC$, with usual notation, $CD$ is the bisector of the angle $C.$ If $\cos \frac{C}{2}$ has the value $\frac{1}{2}$ and the length of $CD$ is $6$, then $(\frac{1}{a}+\frac{1}{b})$ has the value equal to

If $\lambda$ be the ratio of the roots of the quadratic equation in $x,3m^2{x}^2+m(m-4)x+2=0,$ then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :

f(x) and g(x) are continuous functions such that ${lim}_{x\to a}\left[3f\right(x)+g(x\left)\right]=6and{lim}_{x\to a}\left[2f\right(x)-g(x\left)\right]=4$. Given that the function h(x).g(x) is continuous at x = a and h(a)=4, which of the following must be true?

If an angle $A$ of a $△ABC$ satisfies $5\cos A+3=0,$ then the roots of the quadratic equation, $9x^2+27x+20=0$ are:

The equation of the tangent to the parabola $y^2=8x$ inclined at ${30}^{\circ }$ to the x axis is

Which of the following functions are strictly decreasing on?

(A) cos x (B) cos 2x (C) cos 3x (D) tan x

Write ${\sum }_{r=0}^m{}^{n+r}{C}_r$ in the simplified form.

Find the equation of the hyperbola satisfying the given conditions,

Vertices $(0,\pm 5)$ foci $(0,\pm 8)$

Cardinal number of the set A = { x: x is prime number and x < 9} is ____.

Solve the following system of inequalities graphically: 2x + y≥ 4, x + y ≤ 3, 2x – 3y ≤ 6

The coordinates of two consecutive vertices $A$ and $B$ of a regular hexagon $ABCDEF$ are $(1,0)$ and $(2,0),$ respectively. Then the equation of the diagonal $CE$ is

Find the values of other five trigonometric functions if , x lies in fourth quadrant.

Let $f:[-10,10]\to R,$ where $f\left(x\right)=\sin x+\left[\frac{x^2}{a}\right]$ be an odd function. Then the set of values of parameter $a$ is (where $[.]$ denotes greatest integer function)

Pointing to a photograph, a man said, “I have no brother or sister but that man’s father is myfather’s son”. Whose photograph was it:

Find the particular solution of differential equation : $\frac{dy}{dx}=-\frac{x+ycosx}{1+sinx}$ given that y = 1 when x = 0.

Let $\cos A+\cos B=x$; $\cos 2A+\cos 2B=y$; $\cos 3A+\cos 3B=z$, then which of the following is true?

There are 3 dice of different colours namely red, blue and green. What is the probability that when 3 of them are thrown together sum of three numbers is 15 with the least number occuring on red die?

If the equation of normal to the circle $x^2+y^2-16x-12y+99=0$, which is also a tangent to $x^2+y^2=4$ is ax + by + c = 0, find the value of $\left[|\right(\frac{a}{b}\left)|\right]$, where [x] is the greatest integer function

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)=\underset{0}{\overset{g\left(x\right)}{\int }}\frac{dt}{\sqrt{1+t^3}}\text{where}g\left(x\right)=\underset{0}{\overset{\cos x}{\int }}(1+\sin t^2)dt,\text{then the value of}{f}^{\prime }\left(\pi /2\right)\text{is equal to}& \text{(P)}& 3\\ \text{(B)}& \text{If}f\left(x\right)\text{is a non-zero differentiable function such that}\underset{0}{\overset{x}{\int }}f\left(t\right)dt=\left(f\right(x){)}^2\text{for all}x,\text{then}f(2)\text{equals}& \text{(Q)}& 2\\ \text{(C)}& \text{If}\underset{a}{\overset{b}{\int }}(2+x-x^2)dx,(a<b)\text{is maximum, then the value of}(a+b)\text{is equal to}& \text{(R)}& 1\\ \text{(D)}& \text{If}\underset{x\to 0}{lim}(\frac{\sin 2x}{x^3}+a+\frac{b}{x^2})=0,\text{then the value of}(3a+b)\text{is equal to}& \text{(S)}& -1\end{array}$



Which of the following is a CORRECT combination?

If four points P (1,2,3) , Q (2,3,4), R(3,4,5), S(4,5, k) are coplanar then the value of k is / are -

For the given differential equation find the general solution.

$(1+x^2)dy+2xydx=cotxdx$

For natural number n, $(n!{)}^2$ > $n^n$, if

If $x^y=e^{x-y},then\frac{dy}{dx}=$

It is given that at x = 1, the
function x⁴− 62x² + ax
+ 9 attains its maximum value, on the interval [0, 2]. Find the value
of a.

Let $P(3,3)$ and $Q(1,2)$ be two points. If $R$ is a point such that the straight lines $PQ$ and $QR$ are equally inclined to the tangent of the circle $x^2+y^2=5$ at $Q,$ then equation of the line $QR$ is

Tap on the bubbles with equations using distributive property.