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If $A_j=\sum _{r=0}^j{}^j{C}_r{7}^r$, then the value of $\sum _{j=1}^{21}{A}_j$ is

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Four of the following five are alike in a certain way based on their positions in the above arrangement and so form a group. Which is the one that does not belong to the group?

Let $z$ be a unit complex number having the argument $\theta$, $0<\theta <\frac{\pi }{2}$ and satisfying the relation $|z-3i|=3,$ then $\arg (\cot \theta -\frac{6}{z})$

A box contains $15$ red and $10$ blue balls. If $10$ balls are randomly drawn one by one with replacement then the variance of the number of red balls is

In how many ways can the letters of the words 'ARRANGE' be arranged so that the two R's are never together?

Question 75.Consider the following matrix-

Select a suitable figure from the following given four alternatives that would complete the figure-matrix?

General solution of the equation ${\sin }^{2}x+{\text{cosec}}^{2}x-2(\sin x+\text{cosec}x)+2=0$ is

The point which is equidistant from the points $(-1,1,3),(2,1,2),(0,5,6)$ and $(3,2,2)$ is

Let E and F be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If $P\left(T\right)$ denotes the probability of occurrence of the event T, then

Let $A=\{1,2,3\},B=\{4,5,6,7,8\},$ $C=\{4,6,8,10,12\},$ then $n\left[\right(A\times B)-(A\times C\left)\right]=$

Evaluate the determinants.

$\left|\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right|$

The function is defined by f(x) =$\{\begin{array}{c}Kx=1,ifx\le 5\\ 3x-5,ifx>5\end{array}$ at x = 5.

For the LP problem, maximize z = 2x + 3y, the coordinates of the corner points of the bounded feasible region are $A(3,3),B(20,3),C(20,10),D(18,12)$ and $E(12,12)$ ,the maximum value of z is

Let a line $L:2x+y=k,k>0$ be a tangent to the hyperbola $x^2-y^2=3.$ If $L$ is also a tangent to the parabola $y^2=\alpha x,$ then $\alpha$ is equal to:

The contra-positive of the statement p $\to$ q is

In how many ways a commitee of six members be formed from 7 men and 5 women if the commitee contains 4 men and 2 women?

Let $A(6,-1),B(1,3)$ and $C(x,8)$ be three points such that $AB=BC.$ Then the value of $x$ can be

Find the 20^th term of the
series 2 × 4 + 4 ×
6 + 6 × 8 + … + n
terms.

Let $C_1$ be the curve obtained by solution of differential equation $2xy\frac{dy}{dx}=y^2-x^2,$ $x>0.$ Let the curve $C_2$ be the solution of $\frac{2xy}{x^2-y^2}=\frac{dy}{dx}.$ If both the curves pass through $(1,1),$ then the area enclosed by the curves $C_1$ and $C_2$ is equal to :

The equation of line passing through point $(1,-2,-1)$ and parallel to the line $2x+y=3$ and $y+z=1,$ is

The strength of 20 volume of $H_2{O}_2$ is :

How many ways are there to arrange the letters of the word EDUCATION so that all the following three conditions hold?

– the vowels occur in the same order (EUAIO);

– the consonants occur in the same order(DCTN);

– no two consonants are next to each other.

The expression $co{s}^2\phi +co{s}^2(\alpha +\phi )-2cos\alpha cos\phi cos(\alpha +\phi )$ is independent of

Consider the experiment of throwing a die, if a multiple of 3 comes up throw the die again and if any other number comes, toss a coin. Find the conditional probability fo the event the coin shows a tail, given that atleast one die shows a 3.

Solve the equation 27x² – 10x + 1 = 0

What does the "c" in the equation y = mx + c, stand for? Where x, y are variables and m, c are constants.

${lim}_{x\to 0}\frac{e^{2x}-e^x}{sin2x}$

If a and b are two arbitrary constants, then the straight line (a-2b)x + (a+3b)y + 3a+4b = 0 will pass through