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The range of the function $f\left(x\right)={\sin }^{-1}\left(\frac{x^2}{1+x^2}\right),x\in R$ is equal to :

The coefficient of $x^{10}$ in the expansion of $[1+x^2(1-x){]}^8$ is

In a triangle $ABC,$ the minimum value if the sum of the squares of sides is ($\Delta$ is the area of the triangle $ABC$)

Let $a=2\hat{i}+\hat{j}-2\hat{k},b=\hat{i}+\hat{j}$ and $c$ be a vector such that $|c-a|=3,|(a\times b)\times c|=3$ and the angle between $c$ and $a\times b$ is ${30}^{\circ }$. Then, $a\cdot c$ is equal to

Let $p=\underset{x\to 0^{+}}{lim}(1+ta{n}^2\sqrt{x}{)}^{\frac{1}{2x}}$ then log p is equal to :

Of the 200 candidates who were interviewed for a position at a call center, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone. 40 of them had both two-wheeler and credit card, 30 had both credit card and mobile phone and 60 had both two wheeler and mobile phone and 10 had all three. How many candidates had none of the three?

The sides $AB,BC$ and $CA$ of $△ABC$ are marked with $3,4$ and $5$ interior points respectively. Number of triangles that can be constructed using these interior points as vertices is

If principal argument of $z_0$ satisfying $|z-3|\le \sqrt{2}$ and $\arg (z-5i)=-\frac{\pi }{4}$ simultaneously is $\theta ,$ then the CORRECT statement(s) is/are

${lim}_{x\to -\frac{1}{2}}\frac{8x^3+1}{2x+1}$

Value of limit $\underset{x\to 0^{+}}{lim}\frac{a^{\sqrt{x}}-a^{1/\sqrt{x}}}{a^{\sqrt{x}}+a^{1/\sqrt{x}}},a>1,$ is

Consider the following system of equations :

$ax+by+cz=0az+bx+cy=0ay+bz+cx=0$

$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If}a+b+c\ne 0\text{and}& \text{(P)}& \text{Planes meet only at one point}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0.& & \\ \text{(II)}& \text{If}a+b+c=0\text{and}& \left(Q\right)& \text{Equations represent the line}x=y=z\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(III)}& \text{If}a+b+c\ne 0\text{and}& \text{(R)}& \text{Equations represent identical planes}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(IV)}& \text{If}a+b+c=0\text{and}& \text{(S)}& \text{The solution of the system represents}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0& & \text{whole of the three dimensional space}\end{array}$





Which of the following is the "INCORRECT" option?

If the angle of intersection of the circles $x^2+y^2+x+y=0$ and $x^2+y^2+x-y=0$ is $\theta$, then equation of the line passing through $(1,2)$ and making an angle $\theta$ with the y − axis is

The value of $\underset{x\to 0}{lim}{(\tan \frac{x}{7}+\cos \frac{3x}{7})}^{14/x}$ is

Let $N$ be the set of natural numbers. Suppose $f:N\to N$ is a function satisfying the following conditions :

$\left(a\right)f(m+n)=f\left(m\right)+f\left(n\right)\left(b\right)f\left(2\right)=2$

Then the value of $\frac{1}{7}\sum _{k=1}^{20}f\left(k\right)$ is

How many of the following are consistent with the axiomatic definition of probability ___

1) The probability of sample space is 1

2) P(A) belongs to the set [0,1] for any event A

3) $P(A\cup B)=P\left(A\right)+P\left(B\right)$ if A and B are mutually exclusive events

For a linear plot of log $\left(\frac{x}{m}\right)$ versus log p in a Freundlich adsorption isotherm, which of the following statements is correct? (k and n are constants)

A bag contains $4$ white, $3$ black and $2$ red balls. Balls are drawn from the bag one by one without replacement.

The probability that the $4^{\text{th}}$ ball is red is

Which of the following is/are the term(s) of an AP whose first term is 2 and the common difference is 4?

If $A\equiv (9,-1)$ and $B\equiv (-9,5)$, then the locus of moving point $P$ such that $|\overline{AP}|:|\overline{BP}|=1:2$, where $|\overline{AP}|$ denotes the length of $AP$, is

1. 120

Let $f\left(x\right)={\cot }^{-1}(2x-x^2),x\in R$. Then the range of $f\left(x\right)$ is

Consider the process of inserting an element into a Max Heap, where the Max Heap is represented by an array. Suppose we perform a binary search on the path from the new leaf to the root to find the position for the newly inserted element, the number of comparisons performed is

Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin$ $\theta$ is equal to :

Find $\frac{d}{dx}\log 5x$

If the position vectors of the vertices $A,B$ and $C$ of a $△ABC$ are respectively $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$, then the position vector of the point, where the bisector of $\angle A$ meets $BC$ is :

$Avalueof\theta satisfyingcos\theta +\sqrt{3}sin\theta =2is$

In the expansion of ${(x+\frac{2}{x^2})}^{15}$, the term independent of x is

Solve the following system of inequalities graphically: 3x + 2y ≤ 12, x ≥ 1, y ≥ 2

The set of real values of $x$ satisfying $||x-1|-1|\le 2,$ is

Let $PS$ be the median of a triangle with vertices $P(2,2),$ $Q(6,-1)$ and $R(7,3).$ The equation of the line passing through $(1,-1)$ and parallel to $PS$ is

The number of solution(s) of $y=||x{|}^2-2|x|-2|$ and $y=1$ is

The value of given integral $\underset{0}{\overset{1}{\int }}\sin \left(\frac{\pi }{2}x\right)dx$ is