List of top Questions asked in JEE Main- 2014

A Carnot engine absorbs \(1000\ \mathrm{J}\) of heat energy from a reservoir at \(127^\circ\mathrm{C}\) and rejects \(600\ \mathrm{J}\) of heat energy during each cycle. The efficiency of engine and temperature of sink will be:

A bullet of mass 4 g is fired horizontally with a speed of 300 m/s into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately?

If the distance between planes, \(4x-2y-4z+1=0\) and \(4x-2y-4z+d=0\) is 7, then \(d\) is:

A particle is moving in a circular path of radius \(a\), with a constant velocity \(v\) as shown in the figure. The centre of circle is marked by '\(C\)'. The angular momentum from the origin \(O\) can be written as:

A symmetrical form of the line of intersection of the planes \(x=ay+b\) and \(z=cy+d\) is

Three masses \(m\), \(2m\) and \(3m\) are moving in \(x-y\) plane with speed \(3u\), \(2u\) and \(u\) respectively as shown in figure. The three masses collide at the same point at P and stick together. The velocity of resulting mass will be:

The integral \(\int\frac{\sin^2x\cos^2x}{(\sin^3x+\cos^3x)^2}\,dx\) is equal to:

Let \(f\) and \(g\) be two differentiable functions on \(R\) such that \(f'(x)>0\) and \(g'(x)<0\) for all \(x\in R\). Then for all \(x\):

If for a continuous function \(f(x)\), \(\int_{-\pi}^{t}(f(x)+x)\,dx=\pi^2-t^2\), for all \(t\ge-\pi\), then \(f\left(-\frac{\pi}{3}\right)\) is equal to:

The general solution of the differential equation, \(\sin2x\left(\frac{dy}{dx}-\sqrt{\tan x}\right)-y=0\), is:

If [ ] denotes the greatest integer function, then the integral \(\int_0^{\pi}[\cos x]\,dx\) is equal to:

If \(\hat{x},\hat{y}\) and \(\hat{z}\) are three unit vectors in three-dimensional space, then the minimum value of \(|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2\) is:

Let \(f,g:\mathbb{R}\to\mathbb{R}\) be two functions defined by \(f(x)=\begin{cases}x\sin\left(\frac{1}{x}\right), & x\ne0\\ 0, & x=0\end{cases}\) and \(g(x)=xf(x)\).
Statement I: \(f\) is a continuous function at \(x=0\).
Statement II: \(g\) is a differentiable function at \(x=0\).

If \(f(x)=x^2-x+5,\ x>\frac{1}{2}\), and \(g(x)\) is its inverse function, then \(g'(7)\) equals:

If \(\begin{vmatrix} a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2 \end{vmatrix} = k\lambda \begin{vmatrix} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}, \lambda \ne 0\), then \(k\) is equal to:

If \(f(\theta) = \begin{vmatrix} 1 & \cos\theta & 1 \\ -\sin\theta & 1 & -\cos\theta \\ -1 & \sin\theta & 1 \end{vmatrix}\) and \(A\) and \(B\) are respectively the maximum and the minimum values of \(f(\theta)\), then \((A, B)\) is equal to:

Statement I: The equation \((\sin^{-1}x)^3+(\cos^{-1}x)^3-a\pi^3=0\) has a solution for all \(a\ge\frac{1}{32}\).
Statement II: For any \(x\in\mathbb{R}\), \(\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\) and \(0\le\left(\sin^{-1}x-\frac{\pi}{4}\right)^2\le\frac{9\pi^2}{16}\).

Let \(\bar{X}\) and M.D. be the mean and the mean deviation about \(\bar{X}\) of \(n\) observations \(x_i, i = 1, 2, n\). If each of the observations is increased by 5, then the new mean and the mean deviation about the new mean, respectively, are:

Let \(p, q, r\) denote arbitrary statements. Then the logically equivalent of the statement \(p \Rightarrow (q \vee r)\) is:

If the three distinct lines \(x + 2ay + a = 0\), \(x + 3by + b = 0\) and \(x + 4ay + a = 0\) are concurrent, then the point \((a, b)\) lies on a:

If \(A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}\) be such that \(AB = \begin{bmatrix} 6 \\ 8 \end{bmatrix}\), then:

For the two circles \(x^2 + y^2 = 16\) and \(x^2 + y^2 - 2y = 0\), there is/are

A relation on the set \(A = \{x : |x| \lt 3, x \in Z\}\), where \(Z\) is the set of integers is defined by \(R = \{(x, y) : y = |x|, x \ne -1\}\). Then the number of elements in the power set of \(R\) is:

8-digit numbers are formed using the digits \(1, 1, 2, 2, 2, 3, 4, 4\). The number of such numbers in which the odd digits do not occupy odd places, is:

Let \(G\) be the geometric mean of two positive numbers \(a\) and \(b\), and \(M\) be the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\). If \(\frac{1}{M} : G\) is \(4 : 5\), then \(a : b\) can be: