List of top Questions asked in JEE Main

Let S = {1, 2, 3, …, 2022}. Then the probability that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is

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 statement (p∧q) ⇒ (p∧r) is equivalent 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

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

The sum 1 + 2 ⋅ 3 + 3 ⋅ 3² + …. + 10 ⋅ 3⁹ is equal to

The equation of a common tangent to the parabolas y = x² and y = –(x – 2)² is

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

Let f be a real valued continuous function on [0, 1] and
\(f(x) = x + \int_{0}^{1} (x - t) f(t) \,dt\)
Then, which of the following points (x, y) lies on the curve y = f(x)?

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

Let the solution curve y = f(x) of the differential equation
\(\frac{dy}{dx} + \frac{xy}{x^2 - 1} = + \frac{ x^4+2x}{\sqrt{1 - x^2}}, \quad x \in (-1, 1)\) pass through the origin. Then
\(\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) \,dx\)is

\(f(x) = \begin{cases} \frac{sin(x-|x|)}{x-|x|} & \quad {x \in(-2,-1) } \\ max{2x,3[|x|]}, & \quad \text{|x|<1}\\1 & \quad \text{,otherwise} \end{cases}\) 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

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 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

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

Let the system of linear equations
x + y + az = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α is

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

The area of the smaller region enclosed by the curves y² = 8x + 4 and
x²+y²+4√3x-4=0
is equal to

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

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

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

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