List of top Mathematics Questions asked in JEE Main

\(\lim\limits_{x\to0}\left(\tan\left(\frac{\pi}{4}+x\right)\right)^{\frac1x}\) is equal to :

A plane passing through the point \((3, 1, 1)\) contains two lines whose direction ratios are \(1, -2, 2\) and \(2, 3, -1\) respectively. If this plane also passes through the point \((\alpha, -3, 5)\), then \(\alpha\) is equal to:

Let \(E^C\) denote the complement of an event E. Let \(E_1\), \(E_2\) and \(E_3\) be any pairwise independent events with \(P(E_1) > 0\) and \(P(E_1 \cap E_2 \cap E_3) = 0\).
Then \(P\left(E_2^C \cap E_3^C / E_1\right)\) is equal to :

The imaginary part of \(\left(3+2\sqrt{-54}\right)^{\frac12}-\left(3-2\sqrt{-54}\right)^{\frac12}\) can be :

If the sum of first 11 terms of an A.P., \(a_1, a_2, a_3, \ldots\) is 0 \((a_1 \ne 0)\), then the sum of the A.P., \(a_1, a_3, a_5, \ldots, a_{23}\) is \(ka_1\), where k is equal to :

Let \(f : \mathbb{R} \to \mathbb{R}\) be a function which satisfies \(f(x+y) = f(x) + f(y)\ \forall\ x, y \in \mathbb{R}\). If \(f(1) = 2\) and \(g(n) = \sum\limits_{k=1}^{(n-1)} f(k)\), \(n \in \mathbb{N}\), then the value of n, for which \(g(n) = 20\), is :

If a curve \(y = f(x)\), passing through the point \((1, 2)\), is the solution of the differential equation, \(2x^2\,dy = (2xy + y^2)\,dx\), then \(f\left(\frac{1}{2}\right)\) is equal to :

Consider a region \(R = \{(x, y) \in \mathbb{R}^2 : x^2 \le y \le 2x\}\). If a line \(y = \alpha\) divides the area of region R into two equal parts, then which of the following is true?

Let \(A = \{X = (x, y, z)^T : PX = 0 \text{ and } x^2+y^2+z^2 = 1\}\) where \(P = \begin{bmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{bmatrix}\), then the set A :

If the equation \(\cos^4\theta + \sin^4\theta + \lambda = 0\) has real solutions for \(\theta\), then \(\lambda\) lies in the interval :

For some \(\theta \in \left(0, \frac{\pi}{2}\right)\), if the eccentricity of the hyperbola, \(x^2 - y^2\sec^2\theta = 10\) is \(\sqrt{5}\) times the eccentricity of the ellipse, \(x^2\sec^2\theta + y^2 = 5\), then the length of the latus rectum of the ellipse, is :

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ______.

The number of integral values of \(k\) for which the line, \(3x + 4y = k\) intersects the circle,
\(x^2 + y^2 - 2x - 4y + 4 = 0\) at two distinct points is ______.

If \(|x| \lt 1\), \(|y| \lt 1\) and \(x \ne y\), then the sum to infinity of the following series
\((x + y) + (x^2 + xy + y^2) + (x^3 + x^2y + xy^2 + y^3) + \ldots\)

Let \(P(h, k)\) be a point on the curve
\(y = x^2 + 7x + 2\), nearest to the line, \(y = 3x - 3\).
Then the equation of the normal to the curve at \(P\) is :

If
\(\lim_{x \to 1} \frac{x + x^2 + x^3 + \ldots + x^n - n}{x - 1} = 820\),
\((n \in N)\) then the value of \(n\) is equal to _______.

Let A be a \(2 \times 2\) real matrix with entries from \(\{0, 1\}\) and \(|A| \ne 0\). Consider the following two statements :

(P) If \(A \ne I_2\), then \(|A| = -1\)
(Q) If \(|A| = 1\), then \(\operatorname{tr}(A) = 2\),

where \(I_2\) denotes \(2 \times 2\) identity matrix and \(\operatorname{tr}(A)\) denotes the sum of the diagonal entries of A. Then :

Let S be the set of all \(\lambda \in R\) for which the system of linear equations

\(2x - y + 2z = 2\)
\(x - 2y + \lambda z = -4\)
\(x + \lambda y + z = 4\)

has no solution. Then the set S :

A line parallel to the straight line \(2x - y = 0\) is tangent to the hyperbola
\(\frac{x^2}{4} - \frac{y^2}{2} = 1\)
at the point \((x_1, y_1)\). Then \(x_1^2 + 5y_1^2\) is equal to :

The contrapositive of the statement
"If I reach the station in time, then I will catch the train" is :

The domain of the function
\(f(x) = \sin^{-1}\left(\frac{|x| + 5}{x^2 + 1}\right)\)
is \((-\infty, -a] \cup [a, \infty)\). Then a is equal to :

If p(x) be a polynomial of degree three that has a local maximum value 8 at \(x = 1\) and a local minimum value 4 at \(x = 2\); then p(0) is equal to :

Let \(\alpha \gt 0\), \(\beta \gt 0\) be such that
\(\alpha^3 + \beta^2 = 4\). If the maximum value of the term independent of x in
the binomial expansion of \(\left(\alpha x^{\frac19} + \beta x^{-\frac16}\right)^{10}\)
is \(10k\),
then k is equal to :

Let \(X = \{x \in N : 1 \le x \le 17\}\) and
\(Y = \{ax + b : x \in X \text{ and } a, b \in R,\ a \gt 0\}\). If mean
and variance of elements of Y are 17 and 216
respectively then \(a + b\) is equal to :

If \(R = \{(x, y) : x, y \in Z,\ x^2 + 3y^2 \le 8\}\) is a relation on the set of integers Z, then the domain of \(R^{-1}\) is :