List of top Mathematics Questions asked in JEE Main

If \(\int_0^{\pi/4} \left[ \cot(x - \frac{\pi}{3}) - \cot(x + \frac{\pi}{3}) + 1 \right] dx = \alpha \log(\sqrt{3} - 1)\) then \(9\alpha\) is

The mean & variance of \(x_1, x_2, x_3, x_4\) is 1 and 13 respectively and the mean and variance of \(y_1, y_2, \dots, y_6\) be 2 and 1 respectively, the variance of \(x_1, x_2, \dots, x_4, y_1, y_2, \dots, y_6\) will be

The number of values of \(z \in \mathbb{C}\) satisfying \(|z-4-8i| = \sqrt{10}\) and \(|z-3-5i| + |z-5-11i| = 4\sqrt{5}\) is

Locus of the mid-point of chord of circle \(x^2 + y^2 - 6x - 8y - 11 = 0\), subtending a right angle at the center is

Let \( \frac{x^2}{f(a^2 + 2a + 7)} + \frac{y^2}{f(3a + 14)} = 1 \) represent an ellipse. The major axis of the given ellipse is the y-axis and \( f \) is a decreasing function. If the range of \( a \) is \( R - [\alpha, \beta] \), then \( \alpha + \beta \) is:

A person has 3 different bags & 4 different books. The number of ways in which he can put these books in the bags so that no bag is empty, is

If \( \alpha = 3 + 4 + 8 + 9 + 13 + \dots \) up to 40 terms and \( (\tan \beta)^{\frac{\alpha}{1020}} \) is the root of the equation \( x^2 - x - 2 = 0 \), then the value of \( \sin^2 \beta + 3\cos^2 \beta \) is:

The line passing through point of intersection of \(3x + 4y = 1\) and \(4x + 3y = 1\) intersects axes at P and Q, then locus of midpoint of PQ is

Let \(\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}\), \(\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}\) and a vector \(\vec{c}\) be such that \(2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = 0\). If \(\vec{a} \cdot \vec{c} = 15\), then the value of \(\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})\) is

A person goes to college either by bus, scooter or car. The probability that he goes by bus is \(\frac{2}{5}\), by scooter is \(\frac{1}{5}\) and by car is \(\frac{3}{5}\). The probability that he entered late in college if he goes by bus is \(\frac{1}{7}\), by scooter is \(\frac{3}{7}\) and by car is \(\frac{1}{7}\). If it is given that he entered late in college, then the probability that he goes to college by car is

Statement 1: \( f(x) = e^{|\sin x| - |x|} \) is differentiable for all \( x \in \mathbb{R} \).
Statement 2: \( f(x) \) is increasing in \( x \in \left( -\pi, -\frac{\pi}{2} \right) \).

If \((x\sqrt{1-x^2}) \, dy - (y\sqrt{1-x^2} - x^2 \cos^{-1} x) \, dx = 0\) and \(\lim_{x \to 1^-} y(x) = 1\), then \(y\left(\frac{1}{2}\right)\) is

If \(\alpha = 3\sin^{-1}\left(\frac{6}{11}\right)\) and \(\beta = 3\cos^{-1}\left(\frac{4}{9}\right)\), consider statements:
Statement 1: \(\cos(\alpha + \beta) > 0\)
Statement 2: \(\cos\alpha < 0\)
Then which of the following is true?

Let \(A = \{-2, -1, 0, 1, 2\}\). A relation \(R\) is defined on set \(A\) such that \(aRb \Rightarrow 1 + ab > 0\).
Statement-1: It is an equivalence relation.
Statement-2: Number of elements in \(R\) is 17.

Consider the system of equations
\(x + y + z = 6\)
\(x + 2y + 5z = 18\)
\(2x + 2y + \lambda z = \mu\)
If the system of equations has infinitely many solutions, then the value of \((\lambda + \mu)\) is equal to

Let \( f(x) = \dfrac{x-1}{x+1} \), \( f^{(1)}(x) = f(x) \), \( f^{(2)}(x) = f(f(x)) \), and \( g(x) + f^{(2)}(x) = 0 \). The area of the region enclosed by the curves \( y = g(x) \), \( y = 0 \), \( x = 4 \), and \( 2y = 2x - 3 \) is:

If \( \lim_{x \to \frac{\pi}{2}} \dfrac{b(1 - \sin x)(\pi - 2x)^2}{1} = \frac{1}{3} \), then \( \int_0^{3b-6} |x^2 + 2x - 3| \, dx \) is equal to:

Let \(A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \\ 0 & 0 \\ 4 & 5 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -5\alpha & 0 & 0 & 4\alpha \\ -2\alpha & 0 & 0 & 0 \end{bmatrix} + \operatorname{adj}(A)\). If \(\det(B) = 66\), then \(\det(\operatorname{adj}(A))\) equals:

Let a line \(L_1\) pass through the origin and be perpendicular to the lines \(L_2: \vec{r} = (3 + t)\hat{i} + (2t - 1)\hat{j} + (2t + 4)\hat{k}\) and \(L_3: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}\). If \((a, b, c)\), \(a \in \mathbb{Z}\), is the point on \(L_3\) at a distance of \(\sqrt{17}\) from the point of intersection of \(L_1\) and \(L_2\), then \((a + b + c)^2\) is equal to ________.

Let the foot of perpendicular from the point \((\lambda, 2, 3)\) on the line \(\frac{x-4}{1} = \frac{y-9}{2} = \frac{z-5}{1}\) be the point \((1, \mu, 2)\). Then the distance between the lines \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+4}{6}\) and \(\frac{x-\lambda}{2} = \frac{y-\mu}{3} = \frac{z+5}{6}\) is equal to:

Let \(f\) be a polynomial function such that \(\log_2(f(x)) = \left(\log_2\left(2 + \frac{2}{3} + \frac{2}{9} + \dots \infty\right)\right) \cdot \log_3\left(1 + \frac{f(x)}{f(1/x)}\right)\), \(x>0\) and \(f(6) = 37\). Then \(\sum_{n=1}^{10} f(n)\) is equal to ________.

Consider the circle \(C: x^2 + y^2 - 6x - 8y - 11 = 0\). Let a variable chord AB of the circle \(C\) subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle \(x^2 + y^2 - \alpha x - \beta y - \gamma = 0\), then \(\alpha + \beta + 2\gamma\) is equal to ________.

If \(\int_{\pi/6}^{\pi/4} \left( \cot\left(x - \frac{\pi}{3}\right) \cot\left(x + \frac{\pi}{3}\right) + 1 \right) dx = \alpha \log_e(\sqrt{3}-1)\), then \(9\alpha^2\) is equal to ________.

Let \(\frac{x^2}{f(a^2 + 7a + 3)} + \frac{y^2}{f(3a + 15)} = 1\) represent an ellipse with major axis along y-axis, where \(f\) is a strictly decreasing positive function on \(\mathbb{R}\). If the set of all possible values of \(a\) is \(\mathbb{R} - [\alpha, \beta]\), then \(\alpha^2 + \beta^2\) is equal to: