List of top Mathematics Questions asked in JEE Main

Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}\). If \(\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})\), then \(\lambda\) is equal to:

The shortest distance between the lines
\(\vec{r} = (\frac{1}{3}\hat{i} + \frac{8}{3}\hat{j} - \frac{1}{3}\hat{k}) + \lambda(2\hat{i} - 5\hat{j} + 6\hat{k})\)
and \(\vec{r} = (-\frac{2}{3}\hat{i} - \frac{1}{3}\hat{k}) + \mu(\hat{j} - \hat{k}), \lambda, \mu \in \mathbb{R}\), is:

If \((2\alpha + 1, \alpha^2 - 3\alpha, \frac{\alpha - 1}{2})\) is the image of \((\alpha, 2\alpha, 1)\) in the line \(\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z}{1}\), then the possible value(s) of \(\alpha\) is (are):

In the expansion of \(\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x>0\), if the term independent of \(x\) is \((221)k\), then \(k\) is equal to:

Let \(A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}\) and \(\det(A - \alpha I) = 0\), where \(\alpha\) is a real number. If the largest possible value of \(\alpha\) is \(p\), then the circle \((x - p)^2 + (y - 2p)^2 = 320\), intersects the co-ordinate axes at:

If the quadratic equation \((\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \lambda \neq -2\), has two positive roots, then the number of possible integral values of \(\lambda\) is:

For 10 observations \(x_1, x_2, \dots, x_{10}\), if \(\sum_{i=1}^{10} (x_i + 2)^2 = 180\) and \(\sum_{i=1}^{10} (x_i - 1)^2 = 90\), then their standard deviation is:

If \(\alpha = 1\) and \(\beta = 1 + i\sqrt{2}\), where \(i = \sqrt{-1}\) are two roots of the equation
\(x^3 + ax^2 + bx + c = 0, a, b, c \in \mathbb{R}\), then \(\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx\) is equal to:

If the system of equations:
\(x + y + z = 5\)
\(x + 2y + 3z = 9\)
\(x + 3y + \lambda z = \mu\)
has infinitely many solutions, then the value of \(\lambda + \mu\) is:

Let \(S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}\). Then \(\sum_{z \in S} |z + \sqrt{3}i|^2\) is equal to:

For the function \(f: [1, \infty) \rightarrow [1, \infty)\) defined by \(f(x) = (x - 1)^4 + 1\), among the two statements:
(I) The set \(S = \{x \in [1, \infty) : f(x) = f^{-1}(x)\}\) contains exactly two elements, and
(II) The set \(S = \{x \in [1, \infty) : f(x) = f^{-1}(x + 1)\}\) is an empty set,
Options:

The number of points, at which the function \(f(x) = \max\{6x, 2 + 3x^2\} + |x - 1| \cos|x^2 - \frac{1}{4}|\), \(x \in (-\pi, \pi)\), is not differentiable, is ____.

Consider the parabola \(P : y^2 = 4kx\) and the ellipse \(E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Let the line segment joining the points of intersection of \(P\) and \(E\), be their latus rectum. If the eccentricity of \(E\) is \(e\), then \(e^2 + 2\sqrt{2}\) is equal to ____.

Let \(\vec{a}_k = (\tan \theta_k) \hat{i} + \hat{j}\) and \(\vec{b}_k = \hat{i} - (\cot \theta_k) \hat{j}\), where \(\theta_k = \frac{2^{k-1}\pi}{2^n+1}\), for some \(n \in \mathbb{N}\), \(n>5\). Then the value of \(\frac{\sum_{k=1}^{n} |\vec{a}_k|^2}{\sum_{k=1}^{n} |\vec{b}_k|^2}\) is ____.

If \(A = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}\) and \(B = \tan 81^\circ - \tan 3^\circ\), then \(\frac{B}{A}\) is equal to ____.

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is \(p\), then \(96p\) is equal to ____.

If \( y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right) \), then \(\frac{dy}{dx}\) at \(x = \frac{\sqrt{5}}{2}\) is equal to:

The area of the region \(\{(x, y): y \le \pi - |x|, y \le |x \sin x|, y \ge 0\}\) is:

Let \(\int_{-2}^{2} (|\sin x| + |\cos x|) \, dx = 2(3 - \cos 2) + \beta\). Then \(\beta \sin \left( \frac{\beta}{2} \right)\) equals:

Let \(y = y(x)\) be the solution of the differential equation \(\frac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)\), \(y(0) = \frac{1}{2}\). Then \((2y(1) - 1)\) is equal to:

Let \(f\) be a real polynomial of degree \(n\) such that \(f(x) = f'(x)f''(x)\), for all \(x \in \mathbb{R}\). If \(f(0) = 0\), then \(36(f''(2) + f''(2) + \int_0^2 f(x)\,dx)\) is equal to:

Let the vertex A of a triangle ABC be (1, 2), and the mid-point of the side AB be (5, -1). If the centroid of this triangle is (3, 4) and its circumcenter is \((\alpha, \beta)\), then \(2(10\alpha + \beta)\) is equal to:

Let the line \(L_1 : x + 3 = 0\) intersect the lines \(L_2 : x - y = 0\) and \(L_3 : 3x + y = 0\) at the points A and B, respectively. Let the bisector of the obtuse angle between the lines \(L_2\) and \(L_3\) intersect the line \(L_1\) at the point C. Then \(BC^2 : AC^2\) is equal to:

The square of the distance of the point (-2, -8, 6) from the line \(\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{1}\) along the line \(\frac{x+5}{1} = \frac{y+5}{1} = \frac{z}{2}\) is equal to:

Let the smallest value of \(k \in \mathbb{N}\), for which the coefficient of \(x^3\) in \((1+x)^3 + (1+x)^4 + (1+x)^5 + \dots + (1+x)^{99} + (1 + kx)^{100}\), \(x \neq 0\), is \((43n + \frac{101}{4}) \binom{100}{3}\) for some \(n \in \mathbb{N}\), be \(p\). Then the value of \(p + n\) is: