JEE Advanced 2020 Paper 2 Offline

Let \(\alpha\)2 + \(\beta\)2 + \(\gamma\)2 \(\ne\) 0 and \(\alpha\) + \(\gamma\) = 1. Suppose the point (3, 2, \(-\)1) is the mirror image of the point (1, 0, \(-\)1) with respect to the plane \(\alpha\)x + \(\beta\)y + \(\gamma\)...

Let O be the centre of the circle x2 + y2 = r2, where \(r > {{\sqrt 5 } \over 2}\). Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of the triangle ...

For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying \({z^4} - |z{|^4} = 4i{z^2}\), where i = \(\sqrt { - 1}\). Then the minimum possible value of |z1 \(-\) z2|2, where z1, z2$$...

Let \(f:R \to R\) be a differentiable function such that its derivative f' is continuous and f(\(\pi\)) = \(-\)6.If \(F:[0,\pi ] \to R\) is defined by \(F(x) = \int_0^x {f(t)dt}\), and if \(\int_0^\pi {(f'(x)} + F(x))\cos x\,dx\) = 2the...

Let b be a nonzero real number. Suppose f : R \(\to\) R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation \(f'(x) = {{f(x)} \over {{b^2} + {x^2}}}\)for all x\(\in\)R, then which of the fol...

Let the function f : [0, 1] \(\to\) R be defined by\(f(x) = {{{4^x}} \over {{4^x} + 2}}\)Then the value of $$f\left( {{1 \over {40}}} \right) + f\left( {{2 \over {40}}} \right) + f\left( {{3 \over {40}}} \right) + ... + f\left( {{{39} \ov...

Let the function \(f:(0,\pi ) \to R\) be defined by \(f(\theta ) = {(\sin \theta + \cos \theta )^2} + {(\sin \theta - \cos \theta )^4}\)Suppose the function f has a local minimum at \(\theta\) precisely when $$\theta \in \{ {\lambda _...

Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). Suppose the tangent to the hyperbola at P passes th...

Let f : R \(\to\) R and g : R \(\to\) R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y\(\in\)R. If \(\mathop {\lim }\limits_{x \to 0} g(x) = 1\), then which of the following statements is/are T...

The value of the limit$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{4\sqrt 2 (\sin 3x + \sin x)} \over {\left( {2\sin 2x\sin {{3x} \over 2} + \cos {{5x} \over 2}} \right) - \left( {\sqrt 2 + \sqrt 2 \cos 2x + \cos {{3x} \over 2}} \righ...

Let the functions \(f:( - 1,1) \to R\) and \(g:( - 1,1) \to ( - 1,1)\) be defined by \(f(x) = |2x - 1| + |2x + 1|\) and \(g(x) = x - [x]\), where [x] denotes the greatest integer less than or equal to x. Let \(f\,o\,g:( - 1,1) \to R\) be th...

For non-negative integers s and r, let\(\left( {\matrix{ s \cr r \cr } } \right) = \left\{ {\matrix{ {{{s!} \over {r!(s - r)!}}} & {if\,r \le \,s,} \cr 0 & {if\,r\, > \,s} \cr } } \right.\)For positive integers m and...

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 \(\times\) 2 matrix such that the trace of A is 3 and the trace of A3 is \(-\)18, then the value of the determinant of A is .............

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the...

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can...

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of ...

Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns ...

Let a and b be positive real numbers. Suppose \(PQ = a\widehat i + b\widehat j\) and \(PS = a\widehat i - b\widehat j\) are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of \(w = \widehat i + \widehat j\) alo...