The value of non-zero scalars α and  β such that for all vectors $\vec{a}$  and $\vec{b}$ such that $\alpha (2\vec{a}-\vec{b})+\beta (\vec{a}+2\vec{b})=8\vec{b}-\vec{a}$ is

Number of real solutions of the equation $\sin\!\left(e^x\right) = 5^x + 5^{-x}$ is

The sum of infinite terms of a decreasing GP is equal to the greatest value of the function $f(x)=x^3+3x-9$ in the interval [-2,3] and the difference between the first two terms is $f'(0)$. Then the common ratio of GP is

Number of onto (surjective) functions from A to B if n(A)=6 and n(B)=3, is

A computer producing factory has only two plants T₁ and T₂ produces 20% and plant T₂ produces 80% of the total computers produced. 7% of the computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant T₁ 10P(computer turns out to be defective given that it is produced in plant T₂ ). A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T₂ is

If A > 0, B > 0 and A + B = $\frac{\pi}{6}$ , then the minimum value of $ \tan A + \tan B$

The tangent at the point (2,  -2) to the curve $x^2 y^2-2x=4(1-y)$ does not passes through the point

If all the words, with or without meaning, are written using the letters of the word QUEEN add are arranged as in  English Dictionary, then the position of the word QUEEN is

The curve satisfying the differential equation ydx-(x+3y²)dy=0 and passing through the point (1,1) also passes through the point __________

If S and S' are foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, B is the end of the minor axis and BSS' is an equilateral triangle, then the eccentricity of the ellipse is

The equation of the circle passing through the point (4,6) and whose diameters are along x + 2y - 5 =0 and 3x - y - 1=0 is

Let $X_i, i = 1,2,.. , n$ be n observations and $w_i = px_i +k, i = 1,2, ,n$ where p and k are constants. If the mean of $x_i 's$ is 48 and the standard deviation is 12, whereas the mean of $w_i 's$ is 55 and the standard deviation is 15, then the value of p and k should be

If , then the values of A₁, A₂, A₃, A₄ are

Equation of the tangent from the point (3,−1) to the ellipse 2x² + 9y² = 3 is

The $sin^2 x tanx + cos^2 x cot x-sin2x=1+tanx+cotx $, $x \in (0 , \pi)$, then x

Suppose A₁, A₂, ... ₃₀ are thirty sets, each with five elements and B₁, B₂, ...., B_n are n sets each with three elements. Let $\bigcup_{i=1}^{30} A_i= \bigcup_{j=1}^{n} Bj= S$. If each element of S belongs to exactly ten of the Ai' s and exactly nine of the Bj' s then n=

Let S be the set $\{a\in Z^+:a\leq100\}$.If the equation $[tan^2 x]-tan x - a = 0$ has real roots (where [ . ] is the greatest integer function), then the number of elements is S is

If a, b, c are in GP and log a - log 2b, log 2b - log 3c and log 3c - log a are in AP, then a, b, c are the lengths of the sides of a triangle which is

If (1 + x – 2x²)⁶ = 1 + a₁x + a₂x² + ... + a₁₂x¹², then the value a₂ + a₄ + a₆ + ... + a₁₂

If $x, y, z$ are distinct real numbers, then $$ \begin{vmatrix} x & x^{2} & 2 + x^{3} \\ y & y^{2} & 2 + y^{3} \\ z & z^{2} & 2 + z^{3} \end{vmatrix} = 0 $$ Then find $xyz$.

If $a, a, a_2, ., a_{2n-1},b$ are in AP, $a, b_1, b_2,...b_{2n-1}, b $are in GP and $a, c_1, c_2,... c_{2n-1}, b $ are in HP, where a, b are positive, then the equation $a_n x^2-b_n+c_n$ has its roots

Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x)=\begin{cases} x \sin\left(\frac{1}{x}\right), & x>0,\\ 0, & x \le 0. \end{cases}$ 

A particle P starts from the point z₀=1+2i, where i=√−1 . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z₁. From z₁ the particle moves √2 units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\dfrac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by

Two numbers $a$ and $b$ are chosen are random from a set of the first 30 natural numbers, then the probability that $a^2 - b^2$ is divisible by 3 is