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The mean of n items is $\overline{x}.$ If the first term is increased by 1 second by 2 and so on, then new mean is

The coefficient of $t^8$ in ${(1+t)}^2{(1+t+t^2+....+t^9)}^3$ is

The value of k for which the system of equations:

2x + 3y - 2z = 0; 2x - y + 3z = 0 and 7x + ky - z = 0 has non-trivial solution, is .

Show that for any sets A and B, $A=(A\cap B)\cup (A-B)$ and $A\cup (B-A)=(A\cup B).$

If (1, 2), (4, y) (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x + y.

The straight line $x+2y=1$ meets the coordinate axes at $A$ and $B$. A circle is drawn through $A,B$ and the origin. Then the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is :

The solution set of ${\log }_3(x^2-2)<{\log }_3(\frac{3}{2}|x|-1)$ contains

If $P$ is a point on the rectangular hyperbola $x^2-y^2=a^2$, $C$ is its centre and $S$ and $S^{\prime }$ are foci, then $SP\cdot S^{\prime }P$ is equal to

Which of the following is an odd function in their domain ?

Using conradiction method, check the validity of the following statement if n is a real number with $n>3,then,n^2>9.$

Sum the series 5 + 55 + 555 + ... to n terms.

If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, then the value of $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ is

The locus of the point of intersection of the tangents to the parabola $y^2=4ax$ which makes angles ${\theta }_1$ and ${\theta }_2$ with its axis so that $\cot {\theta }_1+\cot {\theta }_2=k$ is

The mean and the variance of five observations are $4$ and $5.20$, respectively. If three of the observations are $3,4$ and $4$ ; then the absolute value of the difference of the other two observations, is :

If $\frac{3+5+7+........+nterms}{5+8+11+........+10terms}=7,$ the value of n is

The sum of three numbers in GP is 56. If we subtract 1, 7, 21 from these numbers in that order, we get an AP. Find the numbers.

Any ordinate $MP$ of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ meets the auxiliary circle at $Q$, then locus of the point of intersection of normals at $P$ and $Q$ to the respective curves is

For all $n\ge 1$ the sum of series of 1+4+7+..+(3n-2), is equal to

The range of the functions $f\left(x\right)={\int }_1^x|t|dt,xϵ[\frac{-1}{2},\frac{1}{2}]$ is

How many equivalence classes can be formed on a deck of cards, with respect to the relation "Belongs to the same suit"

If ${\log }_3{\left(\frac{2\sin 2x}{3}\right)}^2+1=0,x\in [0,2\pi ],$

then which among the following value(s) of $x$ satisfying the above equation

Let $A+B+C=\pi$ and $\alpha ={\sin }^3(B+C)\cdot \sin (2C+A),\beta ={\sin }^3(A+C)\cdot \sin (2A+B),\gamma ={\sin }^3(A+B)\cdot \sin (2B+C)$

are roots of the cubic equation $x^3+a{x}^2+bx+c=0$, then the value of $a$ is

In probability, the event ‘A or B’ can be associated with set :

The numerically greatest term in the expansion of $(3x+5y{)}^{24}$, when x=4 and y=2 is:

The range of $a$ for which $x^2-ax+1-2a^2$ is always positive for all real values of $x$, is

A circle touches the y-axis at the point (0, 4) and cuts the x-axis in a chord of length 6 units. The radius of the
circle is

${lim}_{x\to 1}(\frac{2}{1-x^2}+\frac{1}{x-1})=$___

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

A real-valued function $f\left(x\right)$ satisfies the functional equation $f(x-y)=f\left(x\right)f\left(y\right)-f(a-x)f(a+y)\forall x,y\in R$, where $a$ is a given constant and $f\left(0\right)=1$. Then, $f(2a-x)$ is equal to

Sum of common roots of the equations

$z^3$ + 2$z^2$ + 2z + 1 = 0 and $z^{100}$ + $z^{32}$ + 1 = 0 is equal to :

If n is an even natural number , then $\sum _{r=0}^n\frac{(-1{)}^r}{{}^n{C}_r}equals:$

If the values $1\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},......\frac{1}{n}$ occur at frequencies $1,2,3,4,5,\dots \dots ..,$ n in a distribution, then the mean is

If $p$ is the length of the perpendicular from origin to the line $\frac{x}{a}+\frac{y}{b}=1$, then the correct relation between $a,b$ and $p$ is

If the domain of the function $f\left(x\right)={\log }_e\left({\log }_{|\cos x|}\right(x^2-7x+26)-\frac{4}{{\log }_2|\cos x|})$ is set $A$, then $A$ contain(s) the interval(s)

$e^{|sinx|}+e^{-|sinx|}+4a=0$ will have exactly four different solutions in $[0,2\pi ]$ if

If ${}^{n^2-n}{C}_2$ = ${}^{n^2-n}{C}_{10}$, then n =

Three groups of children contain 3 girls and one boy, 2 girls and 2 boys, one girl and 3 boys. One child is selected at random form each group. What is the chance that the three selected consist of 1 girl and 2 boys?

Find the value of 'a' if one root of the quadratic equation ($a^2$ - 5a + 3)$x^2$ + (3a - 1) x + 2 = 0 is twice as large as the other.

The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p and q are the intercepts of the line L on the new axes, then $\frac{1}{a^2}-\frac{1}{p^2}+\frac{1}{b^2}-\frac{1}{q^2}$ is equal to

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The possible value of}a\text{if}\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})& \text{(P)}& -4\\ & \text{and}\overrightarrow{r}=(\hat{i}+2\hat{j})+\mu (-\hat{i}+\hat{j}+a\hat{k})\text{are not consistent,}& & \\ & \text{where}\lambda \text{and}\mu \text{are scalars, is}& & \\ \text{(B)}& \text{The angle between vectors}\overrightarrow{a}=\lambda \hat{i}-3\hat{j}-\hat{k}\text{and}& \text{(Q)}& -2\\ & \overrightarrow{b}=2\lambda \hat{i}+\lambda \hat{j}-\hat{k}\text{is acute, whereas vector}\overrightarrow{b}& & \\ & \text{makes an obtuse angle with the axes of coordinates.}& & \\ & \text{Then}\lambda \text{can be}& & \\ \text{(C)}& \text{The possible value of}a\text{such that}2\hat{i}-\hat{j}+\hat{k},& \text{(R)}& 1\\ & \hat{i}+2\hat{j}+(1+a)\hat{k}\text{and}3\hat{i}+a\hat{j}+5\hat{k}\text{are coplanar, is}& & \\ \text{(D)}& \text{If}\overrightarrow{A}=2\hat{i}+\lambda \hat{j}+3\hat{k},\overrightarrow{B}=2\hat{i}+\lambda \hat{j}+\hat{k},\overrightarrow{C}=3\hat{i}+\hat{j}& \text{(S)}& 2\\ & \text{and}\overrightarrow{A}+\lambda \overrightarrow{B}\text{is perpendicular to}\overrightarrow{C},\text{then}|2\lambda |\text{is}& & \\ & & \text{(T)}& 3\end{array}$



Which of the following is the only CORRECT combination?

Which of the following is an empty set?

Let $a,b$ and $c$ be the $7^{\text{th}},{11}^{\text{th}}$ and ${13}^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to :

Find the length of subtangent on the curve y = $\frac{x}{1+x}$ where the slope of the tangent is $\frac{1}{9}$

[ The point where the tangent is drawn is in first quadrant ]

1. 6

For all positive integral values of n, $3^{2n}$ - 2n + 1 is

divisible by