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If $f\left(x\right)=x+\frac{1}{x},x\ne 0;\mathrm{then}{\left(f\left(x\right)\right]}^3=$

If $z^3+(3+2i)z+(-1+ia)=0$ has one real root, then the value of ${}^{\prime }{a}^{\prime }$ lies in the interval
$(a\in R)$

How many different signals can be made from 4 red, 2 white and 3 green flags by arranging all of them vertically on a flagstaff?

Let $\ast$ be the binary operation on N defined by $a\ast b=HCF$ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on N?

1. 2
2. 3
3. 4
4. 5

If ${}^n{P}_r=336,{}^n{C}_r=56$, then find n and r and hence find ${}^{n-1}{C}_{r-1}$

The solution of the differential equation $\frac{d^{2}y}{d{x}^2}=e^{-2x},$ where $c$ and $d$ are constants of integration, is

Find
the inverse of each of the matrices, if it exists.

$\left|z_1\right|=1,\left|z_2\right|=2,\left|z_3\right|=3and|9z_1{z}_2+4z_1{z}_3+z_2{z}_3|=12$, then the value of $|z_1+z_2+z_3|$ is equal to

The value of $sinco{t}^{-1}tanco{s}^{-1}x$ is equal to
[Bihar CEE 1974]

If the expansion of $\frac{1}{(1-ax)(1-bx)}=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n+\cdots$, then $a_n$ is (where $a\ne b,|ax|,|bx|<1$)

Question 10

The distribution of heights (in cm) of 96 children is given below.

$\begin{array}{cc}Height\left(incm\right)& Numberofchildren\\ 124-128& 5\\ 128-132& 8\\ 132-136& 17\\ 136-140& 24\\ 140-144& 16\\ 144-148& 12\\ 148-152& 6\\ 152-156& 4\\ 156-160& 3\\ 160-164& 1\end{array}$

Draw a less than type cumulative frequency curve for this data and use it to compute median height of the children.

If x, y, z are three real numbers of the same sign then the value of x/y +y/z+z/x lies in the interval

The letters of the word SACHIN are permuted and are arranged in an alphabetical order as in an English dictionary. Then, the rank of the word SACHIN is ___.

Find the length of the line segment joining the vertex of the parabola $y^2=4ax$ and a point on the parabola where the line-segment makes an angle $\theta$ to the x-axis.

The value of $\frac{29{\int }_0^1{(1-x^4)}^{7}dx}{10{\int }_0^1{(1-x^4)}^{6}dx}$ is

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant when α varies?

(i) Show
that the matrix
is
a symmetric matrix

(ii) Show
that the matrix
is
a skew symmetric matrix

Cards are drawn from a pack of 52 cards one by one. The probability that exactly 10 cards will be drawn before the first ace cards is

If $\int \frac{1}{1+{\cos }^{2}x+2\cos x\sin x}dx={\tan }^{-1}\left[f\right(x\left)\right]+C,$ then $f(\frac{\pi }{4}-x)$ is equal to

(where $C$ is integration constant)

The condition for which the quadratic equation $(p^2-3p+2)x^2+(p^2-4)x+(p-1)=0$ will have a graph which is an upward opening parabola is

Consider the system of linear equations.

x + 2y + z =1

2x + 3y + z = 2

2x + 2y + 4z =1. The sum of values in solution of (x,y,z) will be __

The value of $\underset{0}{\overset{2\pi }{\int }}\left[\sin 2x\right(1+\cos 3x\left)\right]dx$, where $\left[t\right]$ denotes the greatest integer function, is :

If x, y and z are positive real numbers such that x+y+z=a then

Consider the following operation along with Enqueue and Dequeue operations on queue where k is a global parameter.



Multi - Dequeue (Q)

{

m = k ;

while ((Q is not empty) and (m > 0))

{

Dequeue (Q);

m = m - 1 ;

}

}



What is the worst case time complexity of a sequence of n queue operations on an initially empty queue?

The solution to the recurrence equation $T\left(2^k\right)=3T\left(2^{k-1}\right)+1,T\left(1\right)=1is$

The equation of the line which passes through the point (1, 1, 1) and intersecting the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ is

Ajay writes five letters to his five friends and addresses the corresponding .The number of ways can the letters be placed in the envelopes so that at least 2 of them are in the wrong envelopes