JEE Advanced 2016 Paper 1 Offline

Consider a pyramid \(OPQRS\) located in the first octant \(\left( {x \ge 0,y \ge 0,z \ge 0} \right)\) with \(O\) as origin, and \(OP\) and \(OR\) along the \(x\)-axis and the \(y\)-axis, respectively. The base \(OPQR\) of the pyramid is a s...

The least value of a \(\in R\) for which \(4a{x^2} + {1 \over x} \ge 1,\), for all \(x>0\). is

Let RS be the diameter of the circle \({x^2}\, + \,{y^2} = 1\), where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at...

The total number of distinct \(x \in \left[ {0,1} \right]\) for which \(\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1\)

Let \(f:(0,\infty ) \to R\) be a differentiable function such that \(f'(x) = 2 - {{f(x)} \over x}\) for all \(x \in (0,\infty )\) and \(f(1) \ne 1\). Then

A solution curve of the differential equation
\(\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,\) \(x>0,\) passes through the
point \((1,3)\). Then the solution curve

Let \(f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}\) and \(h:\mathbb{R} \to \mathbb{R}\) be differentiable functions such that \(f\left( x \right)= {x^3} + 3x + 2,\) \(g\left( {f\left( x \right)} \right) = x\) and $$h\left( {g\...

Let \(\alpha\), \(\beta\) \(\in\) R be such that \(\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha x - \sin x}} = 1\). Then 6(\(\alpha\) + \(\beta\)) equals _________.

Let \(m\) be the smallest positive integer such that the coefficient of \({x^2}\) in the expansion of $${\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,$...

Let \(P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]\), where \(\alpha\) \(\in\) R. Suppose \(Q = [{q_{ij}}]\) is a matrix such that PQ = kl, where k \(\in\) R, k \(\ne\) 0 a...

The total number of distinct x \(\in\) R for which
\(\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10\) is ______________.

Let \(z = {{ - 1 + \sqrt 3 i} \over 2}\), where \(i = \sqrt { - 1}\), and r, s \(\in\) {1, 2, 3}. Let \(P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]\) and I be the identity matrix of...

The circle \({C_1}:{x^2} + {y^2} = 3,\) with centre at \(O\), intersects the parabola \({x^2} = 2y\) at the point \(P\) in the first quadrant, Let the tangent to the circle \({C_1}\), at \(P\) touches other two circles \({C_2}\) and $${C_3}...

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of...

A computer producing factory has only two plants \({T_1}\) and \({T_2}.\) Plant \({T_1}\) produces \(20\)% and plant \({T_2}\) produces \(80\)% of the total computers produced. \(7\)% of computers produced in the factory turn out to be defe...

In a triangle \(\Delta\)\(XYZ\), let \(x, y, z\) be the lengths of sides opposite to the angles \(X, Y, Z\) respectively, and \(2s = x + y + z\).
If \({{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}\) and area of incircle of the ...

Let \(- {\pi \over 6} < \theta < - {\pi \over {12}}.\) Suppose \({\alpha _1}\) and \({\beta_1}\) are the roots of the equation \({x^2} - 2x\sec \theta + 1 = 0\) and \({\alpha _2}\) and \({\beta _2}\) are the roots of the equation $${...

Let \(S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.\) The sum of all distinct solutions of the equation \(\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0\) in the set S is equa...