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The value of ${\tan }^{-1}\cot \frac{12\pi }{7}$ is

Total number of solutions of equation sin x tan 4x = cos x belonging to $(0,\pi )$ are:

The equation of the line whose x intercept is 3 and y intercept is -2 is

The equation of the plane containing the lines



$\overrightarrow{r}={\overrightarrow{a}}_1+\lambda {\overrightarrow{a}}_{2}and\overrightarrow{r}={\overrightarrow{a}}_2+\mu {\overrightarrow{a}}_1$ is

If the mean of 1, 2, 3,..........,n is $\frac{6n}{11}$, then n is

A box $B_1$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B_2$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B_3$ contains 3 white ball, 4 red balls and 5 black balls. If one ball is drawn from each of the boxes $B_1,B_2$ and $B_3$, the probability that all three drawn balls are of the same colour is

The number of 5-digit numbers which are divisible by 3 that can be formed by using the digits 1,2,3,4,5,6,7,8 and 9, when repetition of digits is allowed, is

If $g\left(x\right)=x^2+x-1$ and $(g\circ f)\left(x\right)=4x^2-10x+5,$ then $f\left(\frac{5}{4}\right)$ is equal to

If sin x + cos x = t, then sin x cos x is equal to

How many trivial relations can be formed on a set which has 64 elements?

Let $L_1$ and $L_2$ are two intersecting lines.

If the image of $L_1$ w.r.t. $L_2$ and $L_2$ w.r.t. $L_1$

coincide, then angle between $L_1$ and $L_2$ is-

The numerically greatest term in the expansion of $(2+3x{)}^{12}$ when $x=\frac{5}{6}$ is:

Let the line $\frac{x}{a}+\frac{y}{b}=1$ where $ab>0,$ passes through $P(\alpha ,\beta ),$ where $\alpha \beta >0.$ If the area formed by the line and the coordinate axes is $S,$ then the least value of $S$ is

If $A=\{5,6,7\},B=\{1,2,3,4\}$, then number of elements in set $(A-B)\times B$ is

If p and q are chosen randomly from the set {1,2,3,4,5,6,7,8,9,10} with replacement, determine the probability that the roots of the equation $x^2+px+q=0$ are real.___

The vertex of the conic represented by $25(x^2+y^2)=(3x-4y+12{)}^2$ is

If a rectangular hyperbola of latus rectum $4$ units passing through $(0,0)$ have $(2,0)$ as its one focus, then equation of locus of the other focus is

$\mathrm{If}{D}_r=\left(\begin{array}{ccc}{2}^{r-1}& 2.3^{r-1}& 4.5^{r-1}\\ \alpha & \beta & \gamma \\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}{\sum }_{r=1}^n{D}_r\mathrm{is}$

If $f\left(x\right)=\frac{\tan (x+\frac{\pi }{6})}{\tan x}$ attains local minimum at $x=a\pi$ in the interval $(\frac{\pi }{3},\pi )$ and the local minimum value is $b,$ then the value of $a+b$ is

If the intercepts of the variable circle on the $x$ and $y$-axis are $2$ units and $4$ units respectively, then the locus of the centre of the variable circle is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are vectors such that $\overrightarrow{a}.\overrightarrow{b}=0$ and $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$, then

If $A=\{x\in N:{\log }_{2}x+\frac{3}{{\log }_{2}x}<4,x>1\}$ and $B=\{x\in Z:(4-x^2\left)\right(x^2-8x+15)\ge 0\},$ then the number of subsets of the set $(A\cup B)-(A\cap B)$ is

The equation of the axis of symmetry of the quadratic polynomial $y=3x^2+6x-1$ is

If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6}$, is :

If $g\left(x\right)$ is shifted $1$ unit left of $f\left(x\right)=x^2-6,$ then $g\left(x\right)$ is

If $\sum _{r=0}^{44}{}^{49-r}{C}_5={}^{50}{C}_x$, then the value(s) of $x$ is/are

If the average rate of change of f(x) with respect to x over the interval [a, b] is m, then find the value of $\frac{1}{m^2}$

$\underset{x\to \infty }{lim}{\left(\frac{x^5+5x+3}{x^2+x+2}\right)}^x\text{equals}$

If two sides of a triangle are roots of the equation $x^2-7x+8=0$ and the angle between these sides is ${60}^{\circ }$ then the product of inradius and circumradius of the triangle is

The value of m for which y = mx + 6 is a tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{49}=1$ is .

Equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane $5x+3y-6z+8=0$ is

The coefficient of $x^{203}$ in $(1-x)(2-x^2)(3-x^3)\cdots (20-x^{20})$ is

(correct answer + 1, wrong answer - 0.25)

Let $\hat{a},\hat{b}and\hat{c}$ be three unit vectors such that

$\hat{a}\times (\hat{b}\times \hat{c})=\frac{\sqrt{3}}{2}(\hat{b}+\hat{c}).$ If

$\hat{b}$ is not parallel to $\hat{c}$, then the angle between $\hat{a}$ and $\hat{b}$ is

A, B, C in order cut a pack of cards, replacing them after each cut, on the condition that the first who cuts a spade shall win a prize; find their respective chances.

If the sum of two of the roots of $x^3+p{x}^2+qx+r=0$ is zero, then pq =

Consider a family of straight lines $(x+y)+\lambda (2x-y+1)=0$. Then the equation of the straight line belonging to this family that is farthest from $(1,-3)$ is:

The value of $c$ in the Lagrange's mean value theorem for the function $f\left(x\right)=x^3-4x^2+8x+11$, where $x\in [0,1]$ is :

If the $n^{\text{th}}$ term and the sum of $n$ terms of the series $2,12,36,80,150,252,.....$ is $T_n$ and $S_n$ repectively then

The two numbers between $\frac{1}{16},\frac{1}{6}$ such that first three may be in G.P. and the last three in H.P. are respectively.

If $(\lambda -2)x^2+(\lambda -1)x+3<0,\forall x\in R,$ then the range of $\lambda$ is

The set of values of 'c' so that the equations $y=\left|x\right|+cand{x}^2+y^2-8\left|x\right|-9=0$ have no solution is

If $f\left(x\right)=\int \frac{5x^8+7x^6}{(x^2+1+2x^7{)}^2}dx$ , if f(0) = 0, then the value of f(1) is

If the angle θ between the vectors $a=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $b=7\hat{i}-2\hat{j}+x\hat{k}$ is such that ${90}^{\circ }$ < $\theta$ < ${180}^{\circ }$

then x lies in the interval:

If $\overrightarrow{p}=\frac{\overrightarrow{b}\times \overrightarrow{c}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{q}=\frac{\overrightarrow{c}\times \overrightarrow{a}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$, where $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then the value of the expression $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot (\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$ is

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.

$|OX\times OY|=$