List of top Mathematics Questions asked in JEE Main

Let one root of the quadratic equation in $x$:
$(k^2 - 15k + 27)x^2 + 9(k - 1)x + 18 = 0$
be twice the other. Then the length of the latus rectum of the parabola $y^2 = 6kx$ is equal to:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}} \), then \( f \) is:

If \( \sum_{k=1}^{n} \varphi(k) = \frac{2\pi (n+1)}{} \), then \( \sum_{k=1}^{10} \frac{1}{\varphi(k)} \) is equal to:

The value of \( k \) for which each root of the equation \( x^2 - 6kx + 2k - 9k^2 = 0 \) is greater than 3, always satisfy the inequality:

If in the binomial expansion of \( (1 - x)^m(1 + x)^n \), the coefficients of \( x \) and \( x^2 \) are respectively 3 and -4, then the ratio \( m : n \) is equal to:

If you draw an object measuring 1 cm on the paper, what would it be equivalent to at a scale of 1:100?

If \( |z_1| = |z_2| = |z_3| = 1 \) and \( z_1 + z_2 + z_3 = \sqrt{2} + i \), then the number \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \) is:

In a car with seating capacity of exactly five persons, two persons can occupy the front seat and three persons can occupy the back seat. If amongst the seven persons, who wish to travel by this car, only two of them know driving, then the number of ways in which the car can be fully occupied and driven by them, is:

Number of non-negative integral solutions of the equation \[ a+b+2c=22 \] is

3 numbers are selected randomly from numbers \(1,2,3,\ldots,31\). The probability that they are in A.P. is

The shortest distance between the lines \[ \vec r=\frac13\hat i+2\hat j+\frac83\hat k+\lambda(2\hat i-5\hat j+6\hat k) \] \[ \vec r=\left(-\frac23\hat i-\frac13\hat k\right)+\mu(\hat j-\hat k), \quad \lambda,\mu\in\mathbb R \] is

Let \[ f(x) = \begin{cases} e^{x-1}, & x < 0 \\ x^2 - 5x + 6, & x \ge 0 \end{cases} \] and \( g(x) = f(|x|) + |f(x)| \). If \( \alpha \) = number of points of discontinuity of \( g(x) \) and \( \beta \) = number of points of non-differentiability of \( g(x) \), then \( \alpha + \beta = \)

In the expansion of \[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18}, \] if the coefficient of the term independent of \(x\) is \(221k\), then the value of \(k\) is:

The number of non-negative integer solutions of the equation \[ a + b + 2c = 22 \] is:

Evaluate \[ (0.2)^{\log_{\sqrt{5}}\alpha} + (0.04)^{\log_{5}\beta} \] if \[ \alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \] \[ \beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \]

P is the point of intersection of the two half-lines \[ x-\sqrt{3}y=\alpha, \quad \alpha>0 \] Points \(A\) and \(B\) lie on these lines at a distance \(\alpha\) from \(P\). If the length of perpendicular from \(P\) on \(AB\) is \(\frac{\alpha}{2}\) and the radius of the circumcircle of \(\triangle PAB\) is \(R\), then find \[ \frac{\alpha^2}{R} \]

P is a point on \[ \frac{x^2}{9}+\frac{y^2}{4}=1 \] as \(P(3\cos\alpha,2\sin\alpha)\). Q is a point on \[ x^2+y^2-14x+14y+82=0 \] R is a point on line \[ x+y=5 \] If the centroid of triangle \(PQR\) is \[ \left(\cos\alpha+2,\;\frac{2\sin\alpha}{3}+3\right) \] find the sum of possible ordinates of \(R\).

If \(\left(2\alpha+1,\;\alpha^2-3\alpha,\;\frac{\alpha-1}{2}\right)\) is the image of \((\alpha,2\alpha,1)\) in the line \[ \frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}, \] then the value of \(\alpha\) is:

If \(f(x)=(x-1)^4+1 \quad \forall x\in[1,\infty)\). Statement–1 : \(f(x)=f^{-1}(x)\) has only two solutions. Statement–2 : \(f^{-1}(x+1)=f(x)\) has no solution.

In the expansion of \[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18} \] if coefficient of the term independent of \(x\) is \(221k\), then the value of \(k\) is

If \(\hat u,\hat v\) are unit vectors and \[ |\hat u\times \hat v|=\frac{\sqrt3}{2} \] and \[ \vec A=\lambda\hat u+\hat v+\hat u\times \hat v \] then find \(\lambda\). (Angle between \(\hat u\) and \(\hat v\) is acute)

If \(z_1,z_2,z_3\) are roots of \[ x^3+ax^2+bx+c=0 \] Let \(z_1=1,\; z_2=1+i\sqrt2\) and \(a,b,c\in\mathbb{R}\). Then the value of \[ \int_{-1}^{1}(x^3+ax^2+bx+c)\,dx \] is