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The vector having magnitude equal to $5$ and perpendicular to the vector $\overrightarrow{B}=2\hat{i}-3\hat{j}+2\hat{k}$ and making equal angle with the positive $X$-axis and the positive $Y$-axis is

If $z_1,z_2,z_3$ and $z_4$ are the roots of the equation $z^4+z^3+z^2+z+1=0$, then





$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \left|\sum _{i=1}^4(z_i{)}^4\right|\text{is equal to}& \text{(p)}& 0\\ \text{(B)}& \sum _{i=1}^4(z_i{)}^5\text{is equal to}& \text{(q)}& 4\\ \text{(C)}& \prod _{i=1}^4(z_i+2)\text{is equal to}& \text{(r)}& 1\\ \text{(D)}& \text{Least value of}[|z_1+z_2|]\text{is}& \text{(s)}& 11\end{array}$



$\text{where}\left[\right]$ represents greatest integer function



Which of the following is the correct combination

The angle between the minute hand and hour hand of the clock when the time is $09:30$ hours is

What are the conditions under which the logarithm ${\log }_{2x}\left(x^2-1\right)$ is defined ?

The letter's of the word $LABOUR$ are permuted in all possible ways and the words thus formed are arranged as in a dictionary. The rank of the word $LABOUR$ will be

Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1(b<5)$ and the hyperbola, $\frac{x^2}{16}-\frac{y^2}{b^2}=1$ respectively satisfying $e_1{e}_2=1$. If $\alpha$ and $\beta$ are the distance between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha ,\beta )$ is equal to:

If $3x+5y+17=0$ is polar for the circle $x^2+y^2+4x+6y+9=0$, then the pole is

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x^2+(3-\lambda )x+2=\lambda$ has the least value is:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

The equation(s) of the circle(s) having radius $5$, centre on the line $y=x$ and touching both the coordinate axes is(are)

The coordinates of the orthocenter of the triangle, having vertices (0, 0), (2, –1) and (–1, 3), are

The principal value of ${\tan }^{-1}(-\sqrt{3})$ is

${lim}_{x\to \frac{\pi }{2}}\frac{e^{cosx}-1}{cosx}$

Find the equation of a normal to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$ at the point (4, 3).

Solve the given inequality graphically in two-dimensional plane: 2x – 3y > 6

Write the set of values of x satisfying the inequation $(x^3-2x+1)(x-4)\ge 0$.

The inverse of $A=\left[\begin{array}{ccc}3& 0& 2\\ 2& 0& -2\\ 0& 1& 1\end{array}\right]$ is

The value of $k$ for which the equation $|x-2|+|x-6|-|x+1|=k$ has atleast one solution

If A + C = B, then tan A tan B tan C =

How many number of four digits can be formed with the digits 1,3,3,0 ?

A rifleman is firing at a distance target and has only $10\%$ chance of hitting it. Then the number of rounds, he must fire in order to have more than $50\%$ chance of hitting it at least once is:

The value of ${\sec }^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ is

The roster form of the set $\{x:x$ is a real number and $x^3=3x^2-2x\}$

If tan $\alpha =x+1,tan\beta =x-1$. show that 2 cot $(\alpha -\beta )=x^2$.

If l₁,
m₁, n₁ and l₂,
m₂, n₂ are the direction cosines
of two mutually perpendicular lines, show that the direction cosines
of the line perpendicular to both of these are m₁n₂
− m₂n₁, n₁l₂
− n₂l₁, l₁m₂
­− l₂m₁.