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Let the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be such that $|\overrightarrow{a}\times \overrightarrow{b}|$ is a unit vector and $|\overrightarrow{a}|=3,|\overrightarrow{b}|=\frac{\sqrt{2}}{3},$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

Compute area of region bounded by curves y=tanx and y=tan^2 x (-π/2<x<π/2)

Given the sequence $\left\{a_n\right\}$ is a $G.P.$ such that $\frac{a_4}{a_6}=\frac{1}{4}$ and $a_2+a_5=216$, then the first term is (common ratio is positive)

Let $n_1$ and $n_2$ be the number of red and black balls, respectively in box I. Let $n_3$ and $n_4$ be the number of red and black balls, respectively in box II.
A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$, then the correct option(s) with the possible values of $n_1$ and $n_2$ is/are

A sequence $a_1,a_2,a_3,\cdots$ is defined by letting $a_1=3$ and $a_k=7a_{k-1}$ for all natural numbers $k\ge 2.$ Then $a_n=$

(Solve using mathematical induction)

Let $\alpha$ and $\beta$ be the roots of $x^2+px+q=0$ and also of $x^{2008}+p^{1004}{x}^{1004}+q^{1004}=0$ where $q\ne 0.$ If $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ are the roots of $x^n+1+(x+1{)}^n=0,$ then value of $n$ must be

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$siny=xsin(\alpha +y),then\frac{dy}{dx}is$

If the equation $a{x}^2+2bx+c=0$ and $a{x}^2+2cx+b=0,b\ne c$, have a common root, then find the value of $\frac{a}{b+c}$.

If $I_1=\underset{0}{\overset{\pi }{\int }}xf({\sin }^{3}x+{\cos }^{2}x)dx$ and $I_2=\underset{0}{\overset{\pi /2}{\int }}f({\sin }^{3}x+{\cos }^{2}x)dx,$ then

Santosh painted a scenery on a square Canvas of side 2 dm. Shanthi also painted a scenery on a square Canvas of side twice that of Santosh's. How many times the size (the area) of Canvas used by Shanthi is greater than that used by Santosh.

Let $A_0{A}_1{A}_2{A}_3{A}_4{A}_5$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_0{A}_1,A_0{A}_{2}and{A}_0{A}_4$ is

If (2, 0) is the vertex and y-axis the directrix of a parabola, then its focus is

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively is received by station $A$ and transmitted to $B$. The probability each station receives correct signal is $\frac{3}{4}$. If the signal received by $B$ is green then the probability that original signal was green is:

Find the value of the trigonometric function

${lim}_{x\to 0}\frac{1-cos4x}{x^2}$

The equations of the asymptotes of the hyperbola $2x^2+5xy+2y^2-11x-7y-4=0$ are

If the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ at any point on it cuts the axes OX and OY at P and Q respectively, then OP +OQ is

If both the roots of $x^2$ - ax + a = 0 are greater than 2 then :

1. 140

Let $f\left(x\right)$ be a continuous and differentiable function on the interval $[2,10]$. It is known that $f\left(2\right)=8$ and the derivative on the given interval satisfies the condition $f^{\prime }\left(x\right)\le 4$ for all $x\in (2,10)$. Then the maximum possible value of the function at $x=10$ is

Let $f:R^{+}\to R^{+}$ be a differentiable function satisfying $f\left(xy\right)=\frac{f\left(x\right)}{y}+\frac{f\left(y\right)}{x}$ for all $x,yϵR^{+}$. Also $f\left(1\right)=0,f^{\prime }\left(1\right)=1$. If M is the greatest value of $f\left(x\right)$ then $[m+e]$ is ___ (where [.] represents Greatest Integer Function).

Let $z$ be a complex number satisfying $z^{117}=1.$ If $\omega$ is a cube root of unity, then the minimum value of $|z-\omega |$ is equal to

The below table shows that profit made by a group of shops in mall. Then the median is

$\begin{array}{ccccccc}\text{Profit per shop less than}\left(\%\right)& 10& 20& 30& 40& 50& 60\\ \text{No. of shops}& 12& 30& 57& 77& 94& 100\end{array}$

ABD : DGK : HMS :: MTB : SBL : ?

Let $f\left(x\right)=x+2|x+1|+2|x-1|$. If $f\left(x\right)=k$ has exactly one real solution, then the value of $k$ is

For the differential equation in given question find the general solution.​​​​

$(e^x+e^{-x})dy-(e^x-e^{-x})dx=0$

Question 2 (ii)

Write first four terms of the A.P. when the first term "a" and the common difference "d" are given as follows.

(ii) a = -2, d = 0

If the points $(\sqrt{3}\text{sin}\theta ,\sqrt{4}\text{cos}\theta )$ where $\theta \in R$, lies outside the hyperbola $\frac{x^2}{4}-\frac{y^2}{5}=1$, then the value of $\theta$ is:

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${\tan }^{-1}\left(\frac{1}{2}\right)$. Water is poured into it at a constant rate of $5$ cubic meter per minute. Then the rate (in m/min), at which the level of water is rising at the instant when the depth of water in the tank is $10m$, is :

The difference btw two no.s is 48, the difference btw the arithmetic mean and geometric mean of the no.s is 18. Find the no.

The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is