List of top Mathematics Questions asked in JEE Main

If a line along a chord of the circle \(4x^2 + 4y^2 + 120x + 675 = 0\), passes through the point \((-30, 0)\) and is tangent to the parabola \(y^2 = 30x\), then the length of this chord is :
Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :
If \({}^{20}C_r\) is the co-efficient of \(x^r\) in the expansion of \((1+x)^{20\), then the value of \(\sum_{r=0}^{20} r^2 \cdot {}^{20}C_r\) is equal to :}
The sum of the series \(\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \dots + \frac{2^{100}}{x^{2^{100}}+1}\) when \(x = 2\) is :
The least positive integer n such that \(\frac{(2i)^n}{(1-i)^{n-2}}, i = \sqrt{-1}\), is a positive integer, is ___________.
Let A be a 3\(\times\)3 real matrix. If \(\det(2 \text{Adj}(2 \text{Adj}(\text{Adj} (2A)))) = 2^{41}\), then the value of \(\det (A^2)\) equals ___________.
Let \([t]\) denote the greatest integer less than or equal to t.
Let \(f(x)=x-[x]\), \(g(x)=1-x+[x]\), and \(h(x) = \min\{f(x), g(x)\}\), \(x \in [-2, 2]\).
Then h is :

A hall has a square floor of dimension 10 m \(\times\) 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is \(\cos^{-1}\frac{1}{5}\), then the height of the hall (in meters) is : 

 

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = \(\sqrt{5}\) inches and \(\angle PCB = \tan^{-1}(2)\). The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is : 

 

The local maximum value of the function \(f(x) = \left(\frac{2}{x}\right)^{x^2}, x>0\), is :
\(\lim_{x\to 2} \sum_{n=1}^{9} \frac{x}{n(n+1)x^2 + 2(2n+1)x + 4}\) is equal to :
Let \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{pmatrix} \). Then \( A^{2025} - A^{2020} \) is equal to :
Two fair dice are thrown. The numbers on them are taken as \(\lambda\) and \(\mu\), and a system of linear equations
\(x+y+z=5\)
\(x+2y+3z=\mu\)
\(x+3y+\lambda z=1\)
is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then :
If \((\sqrt{3} + i)^{100} = 2^{99}(p + iq)\), then p and q are roots of the equation :
Let a and b respectively be the points of local maximum and local minimum of the function \(f(x) = 2x^3 - 3x^2 - 12x\). If A is the total area of the region bounded by \(y=f(x)\), the x-axis and the lines \(x = a\) and \(x = b\), then 4A is equal to ___________.
Let Q be the foot of the perpendicular from the point P(7, -2, 13) on the plane containing the lines \(\frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8}\) and \(\frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}\). Then (PQ)\(^2\), is equal to ___________.
If the projection of the vector \(\hat{i} + 2\hat{j} + \hat{k}\) on the sum of the two vectors \(2\hat{i} + 4\hat{j} - 5\hat{k}\) and \(-\lambda\hat{i} + 2\hat{j} + 3\hat{k}\) is 1, then \(\lambda\) is equal to ___________.
Let the mean and variance of four numbers 3, 7, x and y (\(x>y\)) be 5 and 10 respectively. Then the mean of four numbers 3+2x, 7+2y, x+y and x-y is ___________.
Let \(a_1, a_2, ..., a_{10}\) be an AP with common difference -3 and \(b_1, b_2, ..., b_{10}\) be a GP with common ratio 2. Let \(c_k = a_k + b_k, k = 1, 2, ..., 10\). If \(c_2 = 12\) and \(c_3 = 13\), then \(\sum_{k=1}^{10} c_k\) is equal to ___________.
Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes \(\vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16\) and \(\vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6\). Then which of the following points does NOT lie on P ?
Let \(\binom{n}{k}\) denote \(^nC_k\).
If \(A_k = \sum_{i=0}^{9} \binom{9}{i} \binom{12}{12-k+i} + \sum_{i=0}^{8} \binom{8}{i} \binom{13}{13-k+i}\) and \(A_4 - A_3 = 190 p\), then p is equal to ___________.
The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is ___________.
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 :
Let \(\lambda \neq 0\) be in R. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - x + 2\lambda = 0\), and \(\alpha\) and \(\gamma\) are the roots of the equation \(3x^2 - 10x + 27\lambda = 0\), then \(\frac{\beta\gamma}{\lambda}\) is equal to ___________.
A fair die is tossed until six is obtained on it. Let X be the number of required tosses, then the conditional probability P(\(X \ge 5 | X>2\)) is :