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A variable straight line is drawn through the point of intersection of the straight lines $\frac{x}{2}+\frac{y}{3}=1$ and $\frac{x}{3}+\frac{y}{2}=1$ and meets the coordinate axes at $A$ and $B$. Then the locus of the mid-point of $AB$ is

If x,y,z are in A.P., then $3^{-x}$, $3^{-y}$, $3^{-z}$are in :

Which of the following numbers can never be an outcome for the single roll of an unbiased die?

The transformed equation of $9x^2+2\sqrt{3}xy+7y^2=10$ when the axes are rotated through an angle of $\frac{\pi }{6}$ (in the anti clockwise direction) is

The straight lines $L_1,L_2,L_3$ are parallel and lie in the same plane. A total

number of m points are taken on $L_1$,n points on $L_2$, k points on $L_3$. The

maximum number of triangles formed with vertices at these points are

$\begin{array}{ccc}\text{Function}& \text{Domain}& \text{Range}\\ \text{(i) sinx}& \text{(P) R}& \text{(L) [-1, 1]}\\ \text{(ii) cos x}& \text{(Q) R-n}\pi & \text{(M) R}\\ \text{(iii) tan x}& \text{(R) R- {(2n+1)}}& \text{(N)}(-\infty ,\text{-1]}\cup \text{[1,}\infty )\\ \text{(iv) cosec x}& & \\ \text{(v) sec x}& & \\ \text{(vi) cot x}& & \end{array}$

How many of the following are matched correct?

(i) -P-L, (ii) -P-L, (iii)-R-M, (iv) -Q-N, (v) -R-N, (vi) -Q-M

Prove by the principle of mathematical induction that $(2n+7)<(n+3{)}^2$ for all natural numbers n.
Or
Prove by the principle of mathematical induction that n (n + 1) (2n + 1) is divisible by 6 for all $nϵN.$

The general solution of $\tan x+\tan 2x+\sqrt{3}\tan x\cdot \tan 2x=\sqrt{3}$ is

$(\text{where}n\in Z)$

If degree sequence of a simple graph G is {3, 2, 2 , 1 , 0} then degree sequence of

$\overline{G}is$ ______

The general solution of $(\sqrt{3}-1)\sin \theta +(\sqrt{3}+1)\cos \theta =2$ is

(where $n\in Z$)

Let $S_1,S_2,S_3$ and $S_4$ be four sets defined as

$S_1=\{y:y\in Z\text{and}y=\frac{x^2+4x+3}{x^2+7x+14}\text{for}x\in R\}$

$S_2=\{x:x\in Z\text{and}|1-\frac{|x|}{1+|x|}|\ge \frac{1}{3}\}$

$S_3=\left\{x:x^2-3x+2\text{sgn}\left(x\right)=0\right\},$ where $\text{sgn}\left(x\right)$ represents the signum function.

$S_4=\left\{\right(x,y):x,y\in Z,x^2+y^2\le 4\}$.



$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be correctly matched with a unique entry of $\text{List II}.$



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& n\left(S_1\Delta S_2\right)& \text{(P)}& 9\\ \text{(B)}& n((S_1\times S_2)\cap (S_2\times S_1))& \text{(Q)}& 12\\ \text{(C)}& n(S_1\cap S_2\cap S_3^{\prime })& \text{(R)}& 36\\ \text{(D)}& n(S_4\times S_3)& \text{(S)}& 2\\ & & \text{(T)}& 0\end{array}$



Which of the following is the only CORRECT combination?

Find a G.P for which sum of the first two terms is -4 and the fifth term is 4 times the third term.

The solution set for the inequality $\sin \alpha {\cos }^3\alpha >{\sin }^3\alpha \cos \alpha$ where $\alpha \in (0,\pi )$ is

If $\sin \theta ,\cos \theta$ are the roots of the equation $a{x}^2-bx+c=0,$ then the relation among a,b,c is

Three sets $A,B\&C$ such that $A\subset B,$ then $A\cup (B\cup C)=$

If the sum of the squares of the distance of a point from the three co-ordinate axes be 36,then its distance from origin

The statement $(p\Rightarrow \sim p)\wedge (\sim p\Rightarrow p)$ is a:

$\int \frac{1}{2+3sinx}dx$

The value of the integral ${\int }_a^{a+\frac{\pi }{2}}(|sinx|+|cosx|)dxis$

The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$, and having centre at $(0,3)$ is

Tangents are drawn from any point on the circle $x^2+y^2=R^2$ to the circle $x^2+y^2=r^2$. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals

Statement 1: For every natural number $n\ge 2.$

$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}>\sqrt{n}.$

Statement 2: For every natural number $n\ge 2,$

$\sqrt{n(n+1)}<n+1.$

The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.

Plot the graph of $y=($x-3${)}^2+7$

If $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a chord of hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ which passes through $(4,0)$, then the value of $\tan \frac{\theta }{2}\cdot \tan \frac{\varphi }{2}$is

Let $\overrightarrow{a}=2\hat{i}+{\lambda }_1\hat{j}+3\hat{k},$$\overrightarrow{b}=4\hat{i}+(3-{\lambda }_2)\hat{j}+6\hat{k}$ and $\overrightarrow{c}=3\hat{i}+6\hat{j}+({\lambda }_3-1)\hat{k}$ be three vectors such that $\overrightarrow{b}=2\overrightarrow{a}$ and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}$. Then the possible value of $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ is :

1. 32

In a non-zero G.P., if $T_{p-1}+T_{p+1}=3T_p,$ where $T_n$ denotes the $n^{th}$ term of the G.P., then the common ratio of the G.P. can be

The coordinate planes divide the space into ___ octants.

$1+i^2+i^4+i^6+.....i^{2n}$ is:

Which of the following is/are not an identity relation on the set $A=\{a,b,c\}$

If for a real number $y,\left[y\right]$ is the greatest integer less than or equal to $y$, then value of the integral $\underset{\frac{\pi }{2}}{\overset{\frac{3\pi }{2}}{\int }}\left[2\sin x\right]dx$, is

$y=\frac{sinx}{x+cosx},then\frac{dy}{dx}=?$

All possible values of expression $x^2-4x+9$ is

If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$ , then $\overline{z}$ is equal to :

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that.
(a) You both enter the same section?
(b) You both enter the different sections?

The sum of the intercepts on the coordinate axes of the plane passing through the point $(–2,–2,2)$ and containing the line joining the points $(1,–1,2)$ and $(1,1,1)$, is :

The points of extremum of the function $F\left(x\right)={\int }_1^x{e}^{-t^2/2}(1-t^2)$ dt are

The fundamental period of the function $f\left(x\right)=2cos\frac{1}{3}(x-\pi )$ is

Which of the following are the conditions to be satisfied in the axiomatic approach to probability?

Associative Property for three sets $A,B,\&C$ states $A\cap (B\cap C)=$

The value of ${2000}_{C_2}+{2000}_{C_5}+{2000}_{C_8}+...+{2000}_{C_{2000}}=?$

A. $\frac{2^{1999}+1}{3}$

B. $\frac{2^{1999}-1}{3}$

C. $\frac{2^{2000}+1}{3}$

D. $\frac{2^{2000}-1}{3}$

The number of permutations of the word $AUROBIND$ in which vowels appear in the alphabetical order, is

The discriminant of the quadratic equation $3x^2-4\sqrt{3}x+4=0$ is

Statement 1: If a line $L=0$ is tangent to the circle $S=0$, then it will also be a tangent to the circle $S+\lambda L=0$

Statement 2: If a line touches a circle, then perpendicular distance of the line from the centre of the circle is equal to the radius of the circle.

A fresh radioactive mixture has short lived species $A$ and $B$. Both emit $\alpha$ − particles initially at 8000 particles per minute. 20 minutes later, they emit 3500 $\alpha$ − particles per minute. If the half – lives of the species $A$ and $B$ are 10 minutes and 500 hours respectively, then the ratio of activities of A : B in the initial mixture was:

If tan $\theta$ = $\frac{-4}{3}$ then sin$\theta$ is