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A rectangle of maximum area is inscribed in the circle $|z-3-4i|=1$. If one vertex of the rectangle is $4+4i,$ then another adjacent vertex of this rectangle can be

In the figure given below, If the areas of the two regions are equal then which of the following is true?

If $\left|\begin{array}{ccc}1+x& 1& 1\\ 1+y& 1+2y& 1\\ 1+z& 1+z& 1+3z\end{array}\right|=10kxyz(3+\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$ where $xyz\ne 0,3+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\ne 0$, then $k=$ _____

The quadratic equation $(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0$
(where $x\in R$) has real roots if $p$ lies in

The number of binary operations on the set $\{1,2,3\}$ is ______

A card is picked up from a deck of 52 playing cards

(i) What is the sample space of the experiment?

(ii) What is the event that the chosen card is black faced card?

If f(x) = |x - 2|, then at x = 2, f'(x) is

There are 6 books of physics, 3 of chemistry and 4 of biology. Number of ways in which these books can be placed on a shelf if the books of the same subject are to be together is

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

The number of $4$ letter words containing equal number of vowels and consonants, where repetition is allowed, is

In how many ways can 4 letters be posted in 5 letter boxes?

$\text{Let}f\left(x\right)=\{\begin{array}{cc}x,& x<0\\ 1,& x=0\\ x^2,& x>0\end{array}$ which of the following is not true?

Find the mean deviation from the mean for the following data :

(i) $\begin{array}{ccccccccc}\text{Classes}& 0-100& 10-200& 200-300& 300-400& 400-500& 500-600& 600-700& 700-800\\ \text{Frequencies}& 4& 8& 9& 10& 7& 5& 4& 3\end{array}$

(ii) $\begin{array}{ccccccc}\text{Classes}& 95-105& 105-115& 115-125& 125-135& 135-145& 145-155\\ \text{Frequencies}& 9& 13& 26& 26& 30& 12\end{array}$

(iii) $\begin{array}{ccccccc}\text{Classes}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60\\ \text{Frequencies}& 6& 8& 14& 16& 4& 2\end{array}$

A and B are two events such that $P\left(A\right)\ne 0$. Find$P\left(\frac{B}{A}\right)$if

A is a subset of B

$A\cap B=\varphi$

Prove that cos5A= 16cos⁵A -20cos³A +5cosA

Area of the region bounded by the curve xy – 3x – 2y – 10 = 0, x – axis, line x = 3 and x = 4

The area of the region bounded by y = x² - 2x and y = 4 - x² is.

A firm produces x type A and y type B bags. Demand for bag B is atmost one fourth of A. The corresponding constraint is

If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have a root in common and a, b, c $ϵ$ N, then minimum value of a + b + c =

If a, b, c, d are in G.P., prove that :

(i) $\frac{ab-cd}{b^2-c^2}=\frac{a+c}{b}$

(ii) $(a+b+c+d{)}^2=(a+b{)}^2+2(b+c{)}^2+(c+d{)}^2$

(iii) $(b+c)(b+d)=(c+d)(c+d)$

The value of $si{n}^{-1}\left[cos\left(\frac{33\pi }{5}\right)\right]$ is

(a) $\frac{3\pi }{5}$ (b) $\frac{-7\pi }{5}$ (c) $\frac{\pi }{10}$ (d) $\frac{-\pi }{10}$

If $x,y$ and $z$ be greater than $1$, then the value of $\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$ is

Prove that

$\left(i\right)\text{cos}(\frac{\pi }{4}+x)+\text{cos}(\frac{\pi }{4}-x)=\sqrt{2}\text{cos}x$

(ii)$\text{cos}(\frac{3\pi }{4}+x)-\text{cos}(\frac{3\pi }{4}-x)=-\sqrt{2}\text{sin}x$

If $\alpha$ and $\beta$ are roots of $x^2$ - x +2 =0, then ${\alpha }^2\beta +\alpha {\beta }^2=$

If the foci and vertices of an ellipse be (±1,0) and ( ±2,0) , then the minor axis of the ellipse os

Show that the relation R defined in the set A of all triangles as $R=\left\{\right(T_1,T_2):T_{1}issimilarto{T}_2\}$, is equivalence relation. Consider three right angle triangles $T_1$ with sides 3, 4, 5, T2 with sides 5, 12, 13 and $T_3$ with sides 6, 8, 10. Which triangles among $T_1,T_2$ and $T_3$ are related?

If $f\left(x\right)=8x^3,g\left(x\right)=x^{\frac{1}{3}},$ then $fog\left(x\right)$ is

Show that the given differential equation is homogeneous and then solve it.

$xdy-ydx=\sqrt{x^2+y^2}dx$

If $S_n$ denotes the sum of first 'n' terms of an A.P. and$\frac{S_{3n}-S_{n-1}}{S_{2n}-S_{2n-1}}=31$ then the value of n is

If $\alpha ,\beta$ are complex cube roots of unity, then the value of $\frac{a+b\alpha +c\beta }{a\alpha +b\beta +c}$ can be

The values of $k$ for which the point of minimum of the function $f\left(x\right)=4+k^{2}x-2x^3$ satisfies the inequality $\frac{x^2+2x+4}{x^2+6x+8}<0$ is

Total number of solutions for the equation ${\sin }^{4}x+{\cos }^{4}x=\sin x\cos x,x\in [0,2\pi ]$ is

The quadratic expression $a{x}^2+bx+c>0\forall xϵR$, then which of the following option(s) is/are correct ?

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :

A small sphere of mass ‘m’ is tied to the top of the smooth inclined plane with the help of a string of length ‘L’. The string and the inclined plane makes an angle ‘$\theta$’ with the horizontal. The inclined plane is rotated with a constant angular velocity ‘$\omega$’ about the vertical axis passing through the end of the string fixed to the plane as shown in the figure.

The maximum value of $\omega$ so that the sphere maintains contact with the inclined plane is:

A boats crew consists of 8 Men,3 of whom can only row on one side ,2 only on other .The number of ways the crew can be arranged is.

George is pulling Cameron on a toboggan and is exerting a force of 40N acting at an angle of ${60}^{\circ }$ to the ground. If Cameron is pulled by a distance of 100 m horizontally, then the work done by George is __ J

Sum of series ${\sum }_{r=1}^n(r^2+1)r!$ is