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Let $S_n={\cot }^{-1}(3x+\frac{2}{x})+{\cot }^{-1}(6x+\frac{2}{x})+{\cot }^{-1}(10x+\frac{2}{x})+\cdots +\text{upto}n\text{terms},$ where $x>0.$ If $\underset{n\to \infty }{lim}{S}_n=1,$ then $x$ equals to

The feasible solutions for a linear programming problem with constraints
x ≥0, y ≥ 0
3x+5y ≤ 15
5x+2y ≤ 10
is equal to

If P and Q are represented by the numbers $z_1$ and $z_2$ such that $|\frac{1}{z_2}+\frac{1}{z_1}|$ = $|\frac{1}{z_2}-\frac{1}{z_1}|$, then the circumcentre of $△OPQ$,(where O is the origin) is

If two normals to a parabola $y^2=4ax$ intersect at right angles, then the chord joining their feet passes through a fixed point whose coordinates are

The sum of the series 1+(1+2)+(1+2+3)+...........upto n terms,will be

The value of ${lim}_{x\to 0}\frac{sin\alpha x+bx}{ax+sinbx}$ is $(a,b,a+b\ne 0)$

${\int }_0^{\pi }\frac{x}{1+sinx}$

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for $n\ge 3.$ And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal. Further consider a matrix ${\cup }_{3\times 3}$ with its column as ${\cup }_1,{\cup }_2,{\cup }_3$ such that $A^{50}{\cup }_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],A^{50}{\cup }_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],A^{50}{\cup }_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ Then,

The value of $|A^{50}|$ equals

The value of $\underset{x\to \sqrt{2}}{lim}\frac{x^4-4}{x^2+3\sqrt{2}x-8}$ is

If $\theta$ is the angle which the straight line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ subtends at the origin, prove that $tan\theta =\frac{x_2{y}_1-x_1{y}_2}{x_1{x}_2+y_1{y}_2}$ and $cos\theta =\frac{x_1{x}_2+y_1{y}_2}{\sqrt{x_1^2+x_1^2\sqrt{x_2^2+y_2^2}}}$

Find the odd man out

If A
and B are
square matrices of the same order such that AB
= BA, then
prove by induction that.
Further, prove that
for
all n ∈
N

Find the value of ${\log }_7{\log }_7$ $\sqrt{7\left(\sqrt{7\sqrt{7}}\right)}$ .If ${\log }_{10}$ 7 = 0.8450 and ${\log }_{10}$ 2 = 0.3010

Find the values of a and b so that the function defined by f(x) = $\{\begin{array}{c}ax+1,ifx\le 3\\ bx+3,ifx>3\end{array}$ is continuous at x=3.

कृषि विभाग वालों ने मामले को हॉर्टीकल्चर विभाग को सौंपने के पीछे क्या तर्क दिए?

$A=\{x:x\in R,x^2=16\text{and}2x=6\}$ can be represented in the roster form as _________ .

The solution set of $\frac{x^2-7x+10}{14x-x^2-45}\ge 0$ is

The range of $t$ such that $2\sin t=\frac{1-2x+5x^2}{3x^2-2x-1}$ has a solution, where $t\in [-\frac{\pi }{2},\frac{\pi }{2}]$is

The coefficient of $x^4$ in ${(\frac{x}{2}-\frac{3}{x^2})}^{10}$ is:

The equation of the circle with the center (-3,4) and the radius 5 units is .

Let $\ast$ be the binary operation on N given by $a\ast b=LCM$ of a and b.
(i) Which elements of N are invertible for the operation $\ast$?

The number of ways in which $13$ non-distinguishable books can be distributed among $7$ students so that every student get at least one book and at least one student gets $4$ books but not more, is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

The differential equation of family of circles touching $x-$axis at origin is

Limit X tends to infinity

(X+C/X-C)^x=4

Then find C

If $y=x+4$ is a common tangent to the curves $y=e^x+3$ and $y=-a{x}^2,$ then the value of $a$ is