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A seven-digit number is formed using digit $3,3,4,4,4,5,5.$ The probability, that number so formed is divisible by $2$, is:

Which among the following would be obtained on simplifying the expression $(A^{T}BA{)}^T$.

The focus of the parabols $x^2=-16y$ is

If a tangent is drawn to the parabola $y^2=4x$ through the point $(-2,1)$ then the points of contact is/are

The projectile is thrown at $25\sqrt{2}m/s$ at an angle of ${45}^{\circ }$, if the objective is just to clear both the posts, each with a height of $30m$, find the distance between the posts?

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three unit vectors such that $\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{a}\cdot \overrightarrow{c}=0$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3},$ then the value of $|\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{a}\times \overrightarrow{c}|$ is

The contrapositive of the statement “if I am not feeling well, then I will go to the doctor” is

Find the value of if
sin⁻¹
x = y,
then

(A) (B)

(C) (D)

At each point of curve $y=x^3-3a{x}^2+6x+b,$ its tangent is inclined at an acute angle, then which of the following is/are true

The number of solutons of the equation tan x +sec x = 2cos x lying in the interval $[0,2\pi ]$ is

The value of $\underset{0}{\overset{1}{\int }}\frac{x^a-1}{\ln x}dx,$ (where $a$ is parameter) is equal to

If x and y are acute angles such that cos x + cos y = 3/2 and sinx + siny = 3/4 then the value of sin ( x + y)

For any two sets A and B, prove the following :

(i) $A\cap (A^{\prime }\cup B)=A\cap B$

(ii) A - (A - B) = $A\cap B.$

(iii) $A\cap (A\cup B{)}^{\prime }=\varphi$

(iv) A - B =$A\Delta (A\cup B)$.

If $z_1$ and $z_2$ lie on a circle having centre at the origin, then point of intersection of the tangents at $z_1$ and $z_2$ is

The remainder when the determinant $\left|\begin{array}{ccc}{2014}^{2014}& {2015}^{2015}& {2016}^{2016}\\ {2017}^{2017}& {2018}^{2018}& {2019}^{2019}\\ {2020}^{2020}& {2021}^{2021}& {2022}^{2022}\end{array}\right|$ is divided by $5$ is

Solution of the differential equation $(x{e}^{\frac{y}{x}}-y\sin \frac{y}{x})dx+x\sin \frac{y}{x}dy=0;x>0$ is:

(where $c$ is integration constant and $\log$ is given with base ${}^{\prime }{e}^{\prime }$)

What are the limitations of dimensional analysis?

The projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ if $\overrightarrow{a}\cdot \overrightarrow{b}=8$ and $\overrightarrow{b}=2\hat{i}+6\hat{j}+3\hat{k}$ will be

If the length of the chord of the circle, $x^2+y^2=r^2(r>0)$ along the line,$y-2x=3$ is $r,$ then $r^2$ is equal to

If $\overrightarrow{a},\overrightarrow{b}$ lie in a plane normal to the plane containing $\overrightarrow{c}$ and $\overrightarrow{d},$ then $\left|\right(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{c}\times \overrightarrow{d}\left)\right|$ is equal to

In an equilateral $△ABC$ coordinates of the centroid of the triangle is ($1,2,3)$. Find the coordinates of the orthocenter of the triangle.

Find the equation of an ellipse whose eccentricity is $\frac{2}{3}$, the latus-rectum is 5 and the centre is at the origin.

The value of $\sin \frac{\pi }{5}\sin \frac{2\pi }{5}\sin \frac{3\pi }{5}\sin \frac{4\pi }{5}$ is

The quadriatic approximation of $f\left(x\right)=x^3-3x^2-5$ at the point $x=0$ is

If $\left[2\sin x\right]+\left[\cos x\right]=-3$, then the range of the function $f\left(x\right)=\sin x+\sqrt{3}\cos x$ in $[0,2\pi ]$ is (where $[.]$ denotes the greatest integer function)

A circle passes through $(-2,4)$ and touches the $y-$axis at $(0,2).$ Then which of the following equations can represent a diameter of the circle?

Which of the following point lie inside the circle $x^2+y^2-2x+3y-11=0$