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The sum of mean and variance of a binomial distribution is 15 and their product is 54 then the distribution is

The root(s) of the quadratic equation $3x^2+x-10=0$ is/are

What should be the relation between range and codomain of a function, for a function to be an onto function -

The value of $\underset{x\to 1}{lim}\frac{(1-x)(1-x^2)\cdots (1-x^{2n})}{{\left\{\right(1-x)(1-x^2)\cdots (1-x^n)\}}^2},n\in N$ is

Applying the following transformations will result in on the matrix in order on the matrix in order $\left[\begin{array}{ccc}1& 5& 2\\ 3& 1& 4\\ 2& 2& 3\end{array}\right]$

$\sqrt{-3}\sqrt{-75}$

The maximum integral value of $a$ for which the equation $a\sin x+\cos 2x=2a-7$ has a solution is :

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,

Trace of $A^{50}$ equals

Find the sum to n
terms of the series whose n^th terms is given by n²
+ 2ⁿ

The top view of the area occupied by a famous airport is given. This airport has three runways (paths where the planes take off or land) and many taxiways (paths that planes use to get to the runways).



R1: 3x + 16y = 112 T1: 11y - 2x = 22



R2: 6x + 5y = 30 T2: 3x + 16y = -k



R3: 3x + 16y = 24



The three runways, R1, R2, and R3, are shown in pink, and two of the taxiways, T1 and T2 are shown in blue.





The airport security wants to constantly observe the intersection point of runways R2 and R3. What are the coordinates of this intersection point?



[1 mark]

If tan A = -

$\frac{1}{2}$ and tan B = -

$\frac{1}{3}$, then A + B =

[IIT 1967; MNR 1987; MP PET 1989]

A rectangle ABCD, A ≡ (0, 0), B ≡ (4, 0), C ≡ (4, 2), D ≡ (0,2) undergoes the following transformations successively
(i) $f_1(x,y)\to (y,x)$
(ii) $f_2(x,y)\to (x+3y,y)$
(iii) $f_3(x,y)\to (\frac{x-y}{2},\frac{x+y}{2})$.
The final figure will be

Let $\alpha$ and $\beta$ be the roots of $x^2-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6x+q=0$. If $\alpha ,\beta ,\gamma ,\delta$ form the geometric progression. Than ratio $(2q+p):(2q-p)$ is :

If the $X-$coordinate of a point ${}^{\prime }{P}^{\prime }$ on the segment joining $Q(2,2,1)\text{and}R(5,1,-2)$ is $4,$ then $Z-$ coordinate is:

If $m$ is choosen in the quadratic equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :

The area between the curve $y=x^2-3x$ and line $y=2x$ is

Solve the following systems of inequalities graphically:

$x+y\le 9my>x,x\ge 0$

Total number of values of 'a' so that $x^2$ - x - a = 0 has integral roots, $aϵN$ & 5 < a < 99 is

What is the point of contact between the hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and

the tangent $y=mx\pm \sqrt{a^2{m}^2-b^2}$.

Number of ways of creating a $5$ digit number with the help of $1,2$ and $3$ in such a way that sum of digits is $8$

$\frac{1}{1!(n-1)!}$ + $\frac{1}{3!(n-3)!}$ + $\frac{1}{5!(n-5)!}$ + ....... =

1. 60

If $27a+9b+3c+d=0$, then the equation $4a{x}^3+3b{x}^2+2cx+d=0$ has atleast one real root lying in

Prove that:

$tan(\frac{\pi }{4}+\theta )+tan(\frac{\pi }{4}-\theta )=2sec2\theta$

The value of the definite integral $\underset{-1}{\overset{1}{\int }}\frac{dx}{(1+e^x)(1+x^2)}$ is

Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $|z_1-z_2|$ is :