List of top Questions asked in JEE Main- 2013

A ring of mass \(M\) and radius \(R\) is rotating about its axis with angular velocity \(\omega\). Two identical bodies each of mass \(m\) are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be :

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is \(R_0=40\) m. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity \(v=20\) m/s on a horizontal surface? (\(g=10\ \mathrm{m/s^2}\))

Let ABC be a triangle with vertices at points \(A(2,3,5),\ B(-1,3,2)\) and \(C(\lambda,5,\mu)\) in three dimensional space. If the median through A is equally inclined with the axes, then \((\lambda,\mu)\) is equal to:

The equation of the curve passing through the origin and satisfying the differential equation \((1+x^2)\frac{dy}{dx}+2xy=4x^2\) is

Let \(\vec a=2\hat i+\hat j-2\hat k,\ \vec b=\hat i+\hat j\). If \(\vec c\) is a vector such that \(\vec a\cdot\vec c=|\vec c|,\ |\vec c-\vec a|=2\sqrt2\) and the angle between \(\vec a\times\vec b\) and \(\vec c\) is \(30^\circ\), then \(|(\vec a\times\vec b)\times\vec c|\) equals:

The equation of a plane through the line of intersection of the planes \(x+2y=3,\ y-2z+1=0\), and perpendicular to the first plane is :

Let \(f:[-2,3]\to[0,\infty)\) be a continuous function such that \(f(1-x)=f(x)\) for all \(x\in[-2,3]\). If \(R_1\) is the numerical value of the area of the region bounded by \(y=f(x),\ x=-2,\ x=3\) and the axis of \(x\) and \(R_2=\int_{-2}^{3}xf(x)\,dx\), then :

A spherical balloon is being inflated at the rate of 35cc/min. The rate of increase in the surface area (in \(\mathrm{cm^2}\)/min.) of the balloon when its diameter is 14 cm, is :

Let \(f(1) = -2\) and \(f'(x) \ge 4.2\) for \(1 \le x \le 6\). The possible value of \(f(6)\) lies in the interval :

If an equation of a tangent to the curve, \(y - \cos(x + y) = 0,\ -1 \le x \le 1 + \pi\), is \(x + 2y = k\) then \(k\) is equal to :

For \(0 \le x \le \frac{\pi}{2}\), the value of \(\displaystyle\int_0^{\sin^2 x}\sin^{-1}(\sqrt{t})\,dt+\int_0^{\cos^2 x}\cos^{-1}(\sqrt{t})\,dt\) equals :

Let \(p\) and \(q\) be any two logical statements and \(r : p \rightarrow (\sim p \vee q)\). If \(r\) has a truth value \(F\), then the truth values of \(p\) and \(q\) are respectively:

Let \(S=\left\{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}: a_{ij}\in\{0,1,2\},\ a_{11}=a_{22}\right\}\). Then the number of non-singular matrices in the set \(S\) is :

In a set of \(2n\) observations, half of them are equal to 'a' and the remaining half are equal to '\(-a\)'. If the standard deviation of all the observations is 2 ; then the value of \(|a|\) is :

Consider the function : \(f(x)=[x]+|1-x|,\ -1\le x\le3\) where \([x]\) is the greatest integer function. Statement 1 : \(f\) is not continuous at \(x=0,1,2\) and \(3\) Statement 2 : \(f(x)=\begin{cases}-x, & -1\le x\lt0\\ 1-x, & 0\le x\lt1\\ 1+x, & 1\le x\lt2\\ 2+x, & 2\le x\le3\end{cases}\)

Let \(A = \{\theta : \sin(\theta) = \tan(\theta)\}\) and \(B = \{\theta : \cos(\theta) = 1\}\) be two sets. Then:

If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle \(60^\circ\) on the circumference of the first circle, then the radius of the arc is:

A point on the ellipse, \(4x^2 + 9y^2 = 36\), where the normal is parallel to the line, \(4x - 2y - 5 = 0\), is :

Consider the system of equations : \(x + ay = 0,\ y + az = 0\) and \(z + ax = 0\). Then the set of all real values of 'a' for which the system has a unique solution is:

If for positive integers \(r \gt 1, n \gt 2\), the coefficients of the \((3r)^{\text{th}}\) and \((r+2)^{\text{th}}\) powers of \(x\) in the expansion of \((1 + x)^{2n}\) are equal, then \(n\) is equal to:

The value of \(1^2 + 3^2 + 5^2 + \ldots\ldots + 25^2\) is :

Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is :

5 - digit numbers are to be formed using \(2, 3, 5, 7, 9\) without repeating the digits. If \(p\) be the number of such numbers that exceed \(20000\) and \(q\) be the number of those that lie between \(30000\) and \(90000\), then \(p : q\) is:

Let \(z\) satisfy \(|z| = 1\) and \(z = 1 - \bar{z}\). Statement 1 : \(z\) is a real number. Statement 2 : Principal argument of \(z\) is \(\frac{\pi}{3}\).

If \(p\) and \(q\) are non-zero real numbers and \(\alpha^3 + \beta^3 = -p,\ \alpha\beta = q\), then a quadratic equation whose roots are \(\frac{\alpha^2}{\beta},\ \frac{\beta^2}{\alpha}\) is :