JEE Advanced 2018 Paper 2 Offline

Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let \(S\left( {{1 \over 2},{1 \over 2},{1 \over 2}} \right)\) be the centre of th...

Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P ...

Let s, t, r be non-zero complex numbers and L be the set of solutions \(z = x + iy(x,y \in R,\,i = \sqrt { - 1} )\) of the equation \(sz + t\overline z + r = 0\) where \(\overline z\) = x \(-\) iy. Then, which of the following statement(s...

The value of the integral\(\int_0^{1/2} {{{1 + \sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx}\) is ........

Let f : R \(\to\) R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation \({{dy} \over {dx}} = (2 + 5y)(5y - 2)\), then the value of \(\mathop {\lim }\limits_{n \to - \infty } f(x)\) is ...........

Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axi...

Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If \(\alpha\) is the number of one-one functions from X to Y and \(\beta\) is the number of onto functions from Y to X, then the value of $${1 \over {5!}}(\bet...

Let \({E_1} = \left\{ {x \in R:x \ne 1\,and\,{x \over {x - 1}} > 0} \right\}\) and $${E_2} = \left\{ \matrix{
x \in {E_1}:{\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right) \hfill \cr
is\,a\,real\,number \hfil...

Let \(H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\), where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60\(^\circ\) at one of its vertices N. Let the area of the \(\Delta\)LMN be $$4\...

Let T be the line passing through the points P(\(-\)2, 7) and Q(2, \(-\)5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point...

For any positive integer n, define \({f_n}:(0,\infty ) \to R\) as\({f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)\)for all x\(\in\)(0, \(\infty\)). (Here, the inverse trigonometric fu...

Let \({f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to R\) and \({f_4}:R \to R\) be functions defined by(i) \({f_1}(x) = \sin (\sqrt {1 - {e^{ - {x^2}}}} )\),(ii) $${f_2}(x) =...

Let f : (0, \(\pi\)) \(\to\) R be a twice differentiable function such that \(\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t - x}} = {\sin ^2}x\) for all x\(\in\) (0, \(\pi\)).If $$f\left( {{\pi \over 6}} \right...

Let \(X = {({}^{10}{C_1})^2} + 2{({}^{10}{C_2})^2} + 3{({}^{10}{C_3})^2} + ... + 10{({}^{10}{C_{10}})^2}\), where \({}^{10}{C_r}\), r \(\in\){1, 2, ..., 10} denote binomial coefficients. Then, the value of \({1 \over {1430}}X\) is ..........

Let S be the set of all column matrices \(\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]\) such that \({b_1},{b_2},{b_3} \in R\) and the system of equations (in real variables)$$\eqalign{
& - x + 2y + 5...

Let P be a matrix of order 3 \(\times\) 3 such that all the entries in P are from the set {\(-\)1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .

In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.(i) Let \(\alpha\)1 be the total number of ways in which the committee can be formed such that the committee has 5 ...

Let f : R \(\to\) R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y\(\in\) R.Then, the value of loge(f(4)) is ...........