List of top Questions asked in JEE Main

Let \( Z \) be the set of all integers,
\( A = \{(x, y) \in Z \times Z : (x - 2)^2 + y^2 \leq 4\} \),
\( B = \{(x, y) \in Z \times Z : x^2 + y^2 \leq 4\} \) and
\( C = \{(x, y) \in Z \times Z : (x - 2)^2 + (y - 2)^2 \leq 4\} \)
If the total number of relations from \( A \cap B \) to \( A \cap C \) is \( 2^p \), then the value of \( p \) is :
Let M and m respectively be the maximum and minimum values of the function \(f(x) = \tan^{-1}(\sin x + \cos x)\) in \([0, \pi/2]\). Then the value of \(\tan(M-m)\) is equal to :
Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|2\vec{a} + 3\vec{b}| = |3\vec{a} + \vec{b}|\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(60^\circ\). If \(\frac{1}{8} \vec{a}\) is a unit vector, then \(|\vec{b}|\) is equal to :
Let A + 2B = [1 2 0; 6 -3 3; -5 3 1] and 2A - B = [2 -1 5; 2 -1 6; 0 1 2]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) - Tr(B) has value equal to :
A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, -1, 1), then the projection of $\vec{OP}$ on this plane is of length :
Let us consider a curve, $y = f(x)$ passing through the point $(-2, 2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x) + x f'(x) = x^2$. Then :
A function f(x) is given by f(x) = $\frac{5^x}{5^x + 5}$, then the sum of the series $f(\frac{1}{20}) + f(\frac{2}{20}) + \dots + f(\frac{39}{20})$ is equal to:
Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :
Let $f:[0, \infty) \to [0, \infty)$ be defined as $f(x) = \int_0^x [y] dy$, where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true ?
If the curve $x^2+2y^2 = 2$ intersects the line $x+y=1$ at two points P and Q, then the angle subtended by the line segment PQ at the origin is :
Let f be a twice differentiable function defined on ℝ such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x ∈ ℝ. If $| f(x) \ f'(x) | \ | f'(x) \ f''(x) | = 0$, then the value of f(1) lies in the interval :
If \( \frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}, y(0) = 0 \), then for \( y = 1 \), the value of \( x \) lies in the interval :
The equation of the plane passing through the line of intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 \) and parallel to the x-axis is :
If \((\sqrt{3} + i)^{100} = 2^{99}(p + iq)\), then p and q are roots of the equation :
If \( y = f(x) \) passes through \( (1, 2) \) and \( x \frac{dy}{dx} + y = b x^4 \), then for what value of \( b \), \( \displaystyle \int_{1}^{2} f(x)\,dx = \frac{62}{5} \) ?
The value of \(2 \sin(\frac{\pi}{8}) \sin(\frac{2\pi}{8}) \sin(\frac{3\pi}{8}) \sin(\frac{5\pi}{8}) \sin(\frac{6\pi}{8}) \sin(\frac{7\pi}{8})\) is :
A hyperbola passes through the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
Three numbers are in an increasing geometric progression with common ratio \( r \). If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference \( d \). If the fourth term of GP is \( 3 r^2 \), then \( r^2 - d \) is equal to :
The value of the integral \( \int_{0}^{1} \frac{\sqrt{x} dx}{(1+x)(1+3x)(3+x)} \) is :
The function \( f(x) = \dfrac{4x^3 - 3x^2}{6} - 2 \sin x + (2x - 1)\cos x \) :
The equation of one of the straight lines which passes through the point (1, 3) and makes an angle tan⁻¹(√2) with the straight line, y + 1 = 3√2 x is :
Let the foot of perpendicular from a point $P(1, 2, -1)$ to the straight line $L: \frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the plane $x + y + 2z = 0$ which meets $L$ at point $Q$. If $\alpha$ is the acute angle between the lines $PN$ and $PQ$, then $\cos \alpha$ is equal to ________.
The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m - n = 0 and mn + nl + lm = 0, is :
Which one of the following lanthanides exhibits +2 oxidation state with diamagnetic nature ? (Given Z for Nd=60, Yb=70, La=57, Ce=58)
Which one of the following is correct for the adsorption of a gas at a given temperature on a solid surface ?