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The greatest value of the function $\varphi \left(x\right)=\frac{\sin 2x}{\sin (x+\frac{x}{4})}$ on the interval $[0,\frac{\pi }{2}]$ is

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): (ax + b)ⁿ

The equation of plane whose perpendicular distance from origin is $\sqrt{17}\text{units}$ and normal vector to plane is $2\hat{i}-3\hat{j}-2\hat{k},$ is

$\int \frac{2x-1}{(x-1)(x+2)(x-3)}dx$

List all events associated with the andom experiment of tossing of two coins. How many of them arc elementary events ?

If $A=\left[\begin{array}{cc}-3& 2\\ -1& 0\end{array}\right]$, then $A^9$ equals

If ${\sin }^{-1}\left(\frac{x}{5}\right)+{\text{cosec}}^{-1}\left(\frac{5}{4}\right)=\frac{\pi }{2},$ then the value of $x$ is

If $(1+x{)}^n=C_0+C_{1}x+\dots ..+C_n{x}^n$, then the value of $\underset{0\le r<s\le n}{\sum \sum }{C}_r{C}_s$ is equal to

41ⁿ – 14ⁿis a multiple
of 27.

The solution of the equation $x\underset{0}{\overset{x}{\int }}y\left(t\right)dt=(x+1)\underset{0}{\overset{x}{\int }}ty\left(t\right)dt,x>0$ is

(where $c$ is integration constant)

Let L be the set of all lines in
XY plane and R be the relation in L defined as R = {(L₁,
L₂): L₁ is parallel
to L₂}. Show that R is an equivalence relation.
Find the set of all lines related to the line y = 2x +
4.

If a + ib =,
prove that a² + b² =

The arithmetic mean of two positive numbers is $2$ more than its geometric mean. If their difference is $8,$ then the numbers are

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of following is true ?

Let S be the sum, P the
product and R the sum of reciprocals of n terms in a G.P.
Prove that P²Rⁿ = Sⁿ

There are two works each of 3 volumes and two works each of 2 volumes ; In how many ways can the 10 books be placed on a shelf so that the volumes of the same work are not separated?

A salesman sold 25 items, the number of sales increased at the rate of 20 items per day. Which of the given equations represent the correct number of sales the salesman made along with its domain?

If the slope of one of the lines represented by 4ax²+xy+4y²=0 is the square of the other then a equals :

If $A={}^{2n}{C}_0\cdot {}^{2n}{C}_1+{}^{2n}{C}_1{}^{2n-1}{C}_1+{}^{2n}{C}_2{}^{2n-2}{C}_1+\dots {+}^{2n}{C}_{2n-1}\cdot {}^1{C}_1$
then $A$ is

$\pi$/24 radians corresponds to ………………... degrees.

If truth value for $p\to (q\vee r)$ is false, then the truth values of $p,q,r$ are respectively :

If R is a relation on the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} given by $xRy\iff y=3x.$ then R =

If A=$\left[\begin{array}{cc}lo{g}_{b}a& lo{g}_{d}c\\ lo{g}_{c}d& lo{g}_{a}b\end{array}\right]$ then $A^{-1}$

Turn the graph of $y=\frac{1}{x}$ by ${45}^{\circ }$ counter-clockwise and consider the bowl-like top part of the curve (the part above $y=0$). We let a $2D$ fluid accumulate in this $2D$ bowl until the maximum depth of the fluid is $\frac{2\sqrt{2}}{3}.$ What is the area of the fluid used?

If $y\left(\alpha \right)=\sqrt{2\left(\frac{\tan \alpha +\cot \alpha }{1+{\tan }^2\alpha }\right)+\frac{1}{{\sin }^2\alpha }}$ where $\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

If f' $(x^2-4x+3)>0,\forall xϵ(2,3)$; then f (sin x) is increasing on

• You read that in Changthang the temperature drops below 0°C. Look at newspapers on the TV to find cities in India or abroad where the temperature drops below 0°C. In which months do you expect to see this?