List of top Questions asked in JEE Main

Let a triangle ABC be inscribed in the circle
$x² - \sqrt2(x+y)+y² = 0$
such that ∠BAC= π/2. If the length of side AB is √2, then the area of the ΔABC is equal to :

Let A be a matrix of order 3×3 and det (A) = 2. Then det (det (A) adj (5 adj (A³))) is equal to ______.

Negation of the Boolean statement (p ∨ q) ⇒ ((~ r) ∨ p) is equivalent to

The area enclosed by y² = 8x and y = √2x that lies outside the triangle formed by $y=√2x,x=1,y=2√2$, is equal to

Let
$\vec{a}=α\hat{i}+\hat{j}−\hat{k}\ and\ \vec{b}=2\hat{i}+\hat{j}−α\hat{k},α>0$.
If the projection of $\vec{a}×\vec{b}$ on the vector $−\hat{i}+2\hat{j}−2\hat{k}$
is 30, then α is equal to

If m and n respectively are the number of local maximum and local minimum points of the function
$f(x)=∫_0 ^{x^2} \frac{t^2–5t+4}{2+et}dt$, then the ordered pair (m, n) is equal to

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with

$f(x) = \begin{cases} \frac{sin(x-|x|)}{x-|x|} & \quad {x \in(-2,-1) } \\ max{2x,3[|x|]}, & \quad \text{|x|<1}\\1 & \quad \text{,otherwise} \end{cases}$ Where [t] denotes greatest integer t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is

The number of distinct real roots of the equation $x^7 – 7x – 2 = 0$ is

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{4} = 1, a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6\sqrt3$. Then the eccentricity of the ellipse is

Let $\hat a$ and $\hat b$ be two unit vectors such that $| (a+b)+2(a × b)|$ =$ 2$. If $θ ∈ (0, π)$ is the angle between a and b, then among the statements: $(S1)$: $2|a×b|=|a-b|$ $(S2)$: The projection of a on $(\hat a+\hat b)$ is $\frac{1}{2}$

Let the hyperbola
$H:\frac{x^2}{a^2}−y^2=1$
and the ellipse
$E:3x^2+4y^2=12$
be such that the length of latus rectum of H is equal to the length of latus rectum of E. If eH and eE are the eccentricities of H and E respectively, then the value of
$12 (e^{2}_H+e^{2}_E)$ is equal to _____ .

Let $\overrightarrow a$ and $\overrightarrow b$ be the vectors along the diagonals of a parallelogram having area $2\sqrt2$. Let the angle between $\overrightarrow a$ and $\overrightarrow b$ be acute,$ |\overrightarrow a|=1$, and $|\overrightarrow a⋅\overrightarrow b|=|\overrightarrow a×\overrightarrow b|$ If $\overrightarrow c=2\sqrt2(\overrightarrow a×\overrightarrow b→)−2\overrightarrow b$ then an angle between$ \overrightarrow b$ and $\overrightarrow c$ is

Consider the following statements:
A : Rishi is a judge.
B : Rishi is honest.
C : Rishi is not arrogant.
The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is

The area of the region
$\left\{(x,y) : y² ≤ 8x, y ≥ \sqrt2x, x ≥ 1 \right\}$
is

Let y = y(x) be the solution of the differential equation
$x ( 1 - x² ) \frac{dy}{dx} + ( 3x²y - y - 4x³ ) = 0, x > 1$
with y(2) = –2. Then y(3) is equal to

Let the solution curve y = y(x) of the differential equation
$[ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ] x \frac{dy}{dx} = x + [ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ]y$
pass through the points (1, 0) and (2α, α), α> 0. Then α is equal to

If y = y (x) is the solution of the differential equation
$(1 + e^{2x})\frac{dy}{dx} + 2(1 + y^2)e^x = 0$
and y(0) = 0, then
$6 \left( y'(0) + \left( \log_e\left(\sqrt{3}\right) \right)^2 \right)$
is equal to

Let
$\vec{a}=\hat{i} - 2\hat{j} + 3\hat{k}, \vec{b}=\hat{i} - \hat{j} + \hat{k} $ and $\vec{c}$
be a vector such that
$\vec{a} + (\vec{b}×\vec{c}) = \vec{0}$ and $\vec{b}.\vec{c} = 5.$
Then the value of 3($\vec{c}.\vec{a}$) is equal to

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R=\{(x, y): 3 x+\alpha y$ is a multiple of 7$\}$ The relation $R$ is an equivalence relation if and only if :

The area of the region enclosed between the parabolas y² = 2x – 1 and y² = 4x – 3 is

While estimating the nitrogen present in an organic compound by Kjeldahl’s method, the ammonia evolved from 0.25 g of the compound neutralized 2.5 mL of 2 M H₂SO₄. The percentage of nitrogen present in organic compound is ________

Let the mirror image of the point (a, b, c) with respect to the plane 3x – 4y + 12z + 19 = 0 be (a – 6, β, γ). If a + b + c = 5, then 7β – 9γ is equal to __________.

Choose the correct answer :

1. The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be $\frac{24}{5} $ and $\frac{194}{25}$, respectively. If the mean and variance of the first 4 observations are $\frac{7}{2}$ and a, respectively, then (4a + x₅) is equal to