JEE MAIN Questions

A square lamina $OABC$ of side $10\text{cm}$ is pivoted at $O$. Forces act on the lamina as shown in the figure. If the lamina remains stationary, then the magnitude of $F$ is:

Let $A=\{2,3,4,5,6\}$. Let $R$ be a relation on the set $A\times A$ given by $(x,y)R(z,w)$ if and only if $x$ divides $z$ and $y=w$. Then the number of elements in $R$ is ______.

If $\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)=K$, then the value of $\sin \left(\frac{10K\pi }{3}\right)$ is:

Let the parabola $y=x^2+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2+q^2$ is:

If the domain of the function $f(x)=\sqrt{{\log }_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty ,a]\cup \{b\}\cup [c,d)\cup (e,\infty )$, then the value of $a+b+c+d+e$ is ____.

The number of seven-digit numbers that can be formed by using the digits $1,2,3,5,7$ such that each digit is used at least once, is:

The figure given below shows an LCR series circuit with two switches S1 and S2. When switch S1 is closed keeping S2 open, the phase difference ($\varphi$) between the current and source voltage is $30^{\circ }$ and phase difference is $60^{\circ }$ when S2 is closed keeping S1 open. The value of $(3L_1-L_2)$ is ________ H. The capacitor $C=100\mu \text{F}$ and the source is $v=V_0\sin (300t)$.

Let an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with $a<b$, pass through the point $(4,3)$ and have eccentricity $\frac{\sqrt{5}}{3}$. Then the length of its latus rectum is:

The equation of motion of a particle is given by $x=a\sin (50t+\frac{\pi }{3})$ cm. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are ________.

Identify the correct pair having amino acid (A) and the hormone (B) that is iodinated derivative of the amino acid (A). (T and Y represent one-letter codes for amino acids)

Amino acid (A) ------ Hormone (B)

Let $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy=12$ such that the mid-point of the line segment $PQ$ is $\left(\frac{1}{2},-\frac{1}{2}\right)$. Then the area of the triangle $OPQ$ equals:

Let the circles $C_1:|z|=r$ and $C_2:|z-3-4i|=5$, $z\in \mathbb{C}$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min |z_1-z_2|=2$, then $\max |z_1-z_2|$ is equal to:

Two blocks of masses $2\text{kg}$ and $1\text{kg}$ respectively are tied to the ends of a string which passes over a light frictionless pulley. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in $2\text{s}$ is _______ m. (Take $g=10{\text{m/s}}^2$)

A solution is made by mixing one mole of volatile liquid A with 3 moles of volatile liquid B. The vapour pressure of pure A is 200 mm Hg and that of the solution is 500 mm Hg. The vapour pressure of pure B and the least volatile component of the solution, respectively, are:

Let $A=\left[\begin{array}{ccc}1& 21& \alpha \end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 3\beta & 2\end{array}\right]$. If $A^2-4A+I=O$ and $B^2-5B-6I=O$, then among the two statements:
(S1): ${\left[(B-A)(B+A)\right]}^T=\left[\begin{array}{ccc}13& 157& 10\end{array}\right]$ (S2): $\det (\mathrm{adj}(A+B))=-5$
which one is correct?

Among $F{e}^{2+}$, $F{e}^{3+}$, $C{r}^{2+}$ and $Z{n}^{2+}$, the ion that shows positive borax bead test and with highest ionisation enthalpy is :

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x+(k-1)y+3=0$ and $2x+k^{2}y-4=0$. If the line $x-y+2=0$ intersects the circle at the points $A$ and $B$, then $(AB{)}^2$ is equal to:

If a body of mass $1\text{kg}$ falls on the earth from infinity, it attains velocity $(v)$ and kinetic energy $(k)$ on reaching the surface of earth. The values of $v$ and $k$ respectively are ________. (Take radius of earth to be $6400\text{km}$ and $g=9.8{\text{m/s}}^2$.)

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are 100 divisions in circular scale and pitch of screw gauge is $0.1\text{mm}$. When diameter of a sphere is measured, the reading of main scale is $5\text{mm}$ and $50^{th}$ division of circular scale coincides with the reference line of main scale. The diameter of sphere is ______ mm.

Dimensions of universal gravitational constant $G$ in terms of Planck's constant ($h$), distance ($L$), mass ($M$) and time ($T$) are _______.