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Range of the expression $f\left(x\right)=\frac{x^2-1}{x^2+1},x\in R$ is

Which of the following is the identity function?

Solution of the equation $xdy=(y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)})dx$

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that $Y\subseteq X,Z\subseteq XandY\cap Z$ is empty is:

For the given differential equation find the particular solution satisfying the given conditions.

$x^{2}dy+(xy+y^2)dx=0,y=1whenx=1$

If $x,y$ and $z$ are all positive, then the minimum value of $f(x,y,z)=x^3+12\left(\frac{yz}{x}\right)+16{\left(\frac{1}{yz}\right)}^{3/2}$ is

Integrate the function.
$\int \sqrt{1+3x-x^2}dx$

The cost of 4kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find cost of each item per kg by matrix method

An ester (X) on hydrolysis gives an acid 'A' and alcohol 'B'. The sodium salt of 'A' on electrolysis gives ethane. When 'B' is distilled with aqueous bleaching powder it forms chloroform. Therefore 'X' may be

If $I={\int }_0^{\frac{\pi }{2}}\frac{dx}{\sqrt{1+si{n}^{3}x}},$ then

The equation of the line passing through (1, 2, 3) and parallel to the planes x - y + 2z = 5 and 3x + y + z = 6, is
[DSSE 1986]

Which of the following are relations from the set $A=\{1,2,3,4\}$ to set $B=\{a,b,c\}$?

Let $P,Q,R,S$ be the four sets such that $P=\{3,5,7,9,11\},Q=\{9,11,13\},$R = \{3, 5, 9\}$and$S = \{13, 11\}$. Which of the following options is/are correct?

Sum to infinite terms of the series $co{t}^{-1}(1^2+\frac{3}{4})+co{t}^{-1}(2^2+\frac{3}{4})+co{t}^{-1}(3^2+\frac{3}{4})+.......$ is

If ${\theta }_1$ and ${\theta }_2$ be the angles which the lines $(x^2+y^2)({\cos }^2\theta {\sin }^2\alpha +{\sin }^2\theta )=(x\tan \alpha -y\sin \theta {)}^2$ make with the axis of $x$, then if $\theta =\frac{\pi }{6},\tan {\theta }_1+\tan {\theta }_2$ is equal to

Examine the following functions for continuity :

$f(\times )=\frac{{\times }^2-25}{\times +5},x\times not=-5$

Find all 3-digit natural numbers which are 12 times as large as the sum of their digits.

If $K+\frac{1}{2^{19}}=1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\cdots \text{upto}20\text{terms},$ then $K$ is divisible by

In the following diagram the area of the shaded portion is

If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ and $C=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, then $C=$

Area of the rectangle formed by the ends of latusrecta of the Ellipse $4x^2+9y^2$ = 144 is

Find the points on the curve x²
+ y² − 2x − 3 = 0 at which the
tangents are parallel to the x-axis.

Evaluate the integral $\int \frac{(x^2+2)}{x+1}dx$

If radius of sphere is measured as 5 cm with an error of .01 cm. Find the approximate error in the measure of volume of sphere

You are given a sequence of n elements to sort. The input sequence consists of n/k subsequences, each containing k elements. The elements in a given subsequence are all smaller than the elements in the succeeding subsequence and larger than the elements in the preceding subsequence. Thus, all that is needed to sort the whole sequence of length n is to sort the k elements in each of the n/k subsequences.

The lower bound on the number of comparisons needed to solve this variant of sorting problem is

Which of the following options are always true in their domain?

By using the method of contradiction verify that $\sqrt{5}$ is irrational.