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${\sum }_{m-1}^{\pi }ta{n}^{-1}\left(\frac{2m}{m^4+m^2+2}\right)$ is equal to

The sums of n terms of three A.P.'s whose first term is 1 and common differences are 1, 2, 3 are $s_1,s_2,s_3$ respectively. The true relation is

A license plate is 3 capital letters of english alphabets and 3 digits.If all cases are equally likely possibility that the plate has either a letter or digit palindrome or both is:

Consider the numbers 1,2,34,5,67,8,9,10. If 1 is added to each numbers, the variance of the numbers so obtained is

How many of the following statements are correct ___

1. A conic with eccentricity equal to one is called a parabola

2. If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a parabola, then

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0and{h}^2=ab$

If ${\int }_0^b\frac{dx}{1+x^2}={\int }_b^{\infty }\frac{dx}{1+x^2}$,then b=

The perpendicular distance from the origin to the plane containing the two lines,
$\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$ , is:

The general solution of the differential equation $\frac{dy}{dx}=\frac{1+y^2}{1+x^2}$ is

If $f\left(x\right)=x^2$, find $\frac{f\left(1.1\right)-f\left(1\right)}{(1.1-1)}.$

Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that $A\cup B$ can have.

Find the equation of the line, which passes through P (1, -7) and meets the axes at A and B respectively so that 4 AP - 3 BP = 0.

The equation $si{n}^2\theta =\frac{x^2+y^2}{2xy},x,y\ne 0$ is possible if

The tangents to the curve $y=\sin (x+y),-2\pi \le x\le 2\pi$ that are parallel to the line $x+2y=0$ is

A geometric progression with common ratio $r,$ consists of an even number of terms. If the sum of all terms is $5$ times the sum of the terms occupying the odd places, then $\sum _{i=1}^4(ir{)}^2$ is

Let A and B be two sets such that n(A) = p and n(B) = q, write the number of functions from A to B.

Which of the following binary operations defined on the set of real numbers is not associative?

Evaluate the following definite integrals as limit of sums.
${\int }_2^3{x}^{2}dx.$

If $2{\sin }^{2}x+3\sin x-2>0$ and $x^2-x-2<0$, then $x$ lies in the interval

The value of $C_0^2+3C_1^2+5C_2^2+\cdots$ up to $51$ terms is equal to
( where $C_r={}^{50}{C}_r$)

If the lines joining the points $(K,2),(3,6)$ and $(2,3),(K,7)$ are perpendicular to each other, then the possible value of $K$ is

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for $n\ge 3.$ And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal. Further consider a matrix ${\cup }_{3\times 3}$ with its column as ${\cup }_1,{\cup }_2,{\cup }_3$ such that $A^{50}{\cup }_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],A^{50}{\cup }_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],A^{50}{\cup }_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ Then,

The value of $|\cup |$ equals

Question 4 (d)
Find
d) $75\%of1kg$

If $z$ is a complex number then $(z+5)(\overline{z}+5)$ is

Let $a,b,c,d$ be real numbers such that $\{\begin{array}{c}{a}^2+b^2+2a-4b+4=0\\ c^2+d^2-4c+4d+4=0\end{array}.$ Let $m$ and $M$ be the minimum and the maximum values of $(a-c{)}^2+(b-d{)}^2,$ respectively. The value of $m\times M$ is
(correct answer + 3, wrong answer 0)

The integrating factor of the differential equation $x\frac{dy}{dx}+2y=x^2$ is $(x\ne 0)$

Integrate the rational functions.
$\int \frac{2x}{(x^2+1)(x^2+3)}dx$.

Determine wherther or not each of the defintion of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this.
(ii) ON $Z^{+}$, defined $\ast$ by $a\ast b=ab$

Let $A=\{1,2,3,4\}.$ Examine whether the statements given below are true or false.

(i) $\exists x\in A$ such that $x+3=8.$

(ii) $\forall x\in A,x+2<7.$

(iii) $\exists x\leftrightarrow A$ such that $x+1<3.$

(iv) $\forall x\in A,x+3\ge 5.$

The middle term in the expansion of $(x+\frac{1}{x}{)}^{2n}$ is:

Question:- (1/x ^a-b)^1/(a-c) . (1/x ^b-c)^1/(b-a) . (1/x ^c-a) ^1/(c-b) = ?

A) 0

B) 1

C) (a+b+c)

D) (a-b+c) ²

The area bounded by the curve y² = 16x, x = 1, x = 3 and X-axis is:

Anil introduces Rohit as the son of the only brother of his father's wife. How is Rohit related to Anil?

If the ratio of sum of m terms and n terms of an A.P. is $m^2$:$n^2$, then the ratio of its $m^{th}$ and $n^{th}$ terms will be

If the relation f is defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 3\le x\le 10\end{array}$
and the relation g is defined by $g\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 2\le x\le 10\end{array}$
then,

From a well-shuffled deck of $52$ cards, $9$ cards are taken out one by one with replacement. Then the probability that out of $9$ cards, $5$ cards are spades, is

If $a{\sin }^{-1}x-b{\cos }^{-1}x=c,$ then $a{\sin }^{-1}x+b{\cos }^{-1}x$ is equal to

If $z_1,z_2,z_3$ are complex numbers such that $|z_1|=|z_2|=|z_3|=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1,\text{then find the value of}|z_1+z_2+z_3|.$

A line passes through $(x_1,y_1)$ and (h, k). If slope of the line is m, show that $k-y_1=m(h-x_1)$.

If $C_r={}^n{C}_r$, then $C_0{C}_4-C_1{C}_3+C_2{C}_2-C_3{C}_1+C_4{C}_0=$

If two roots of biquadratic equation $x^4+12x-5=0$ are -1 + $\sqrt{2}and1+\sqrt{2}i$ . Find the sum of the square of all the roots.

If y = (1 + tan A)(1 - tan B) where A - B = $\frac{\pi }{4}$, then

$(y+1{)}^{y+1}$ is equal to

Work done in splitting a drop of water of 1 mm radius into ${10}^6$ droplets is (Surface tension of water =$72\times {10}^{-3}J/m^2)$