JEE Advanced 2015 Paper 2 Offline

Let \(f, g :\) \(\left[ { - 1,2} \right] \to R\) be continuous functions which are twice differentiable on the interval \((-1, 2)\). Let the values of f and g at the points \(-1, 0\) and \(2\) be as given in the following table:

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Let \(F:R \to R\) be a thrice differentiable function. Suppose that
\(F\left( 1 \right) = 0,F\left( 3 \right) = - 4\) and \(F\left( x \right) < 0\) for all \(x \in \left( {{1 \over 2},3} \right).\) Let $$f\left( x \right) = xF\left( x \ri...

Let \(f:R \to R\) be a continuous odd function, which vanishes exactly at one point and \(f\left( 1 \right) = {1 \over {2.}}\) Suppose that \(F\left( x \right) = \int\limits_{ - 1}^x {f\left( t \right)dt}\) for all $$x \in \,\,\left[ { - 1...

For any integer k, let \({a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)\), where \(i = \sqrt { - 1} \,\). The value of the expression $${{\sum\limits_{k = 1}^{12} {\left| {{\alpha _{k + 1}} -...

Let \(f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}\) for all \(x \in R\,\,\) with \(f\left( {{1 \over 2}} \right) = 0\).
If \(m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,}\) then the possible values of \(m\) an...

If \(\alpha = \int\limits_0^1 {\left( {{e^{9x + 3{{\tan }^{ - 1}}x}}} \right)\left( {{{12 + 9{x^2}} \over {1 + {x^2}}}} \right)} dx\) where \({\tan ^{ - 1}}x\) takes only principal values, then the value of $$\left( {{{\log }_e}\left| {1 ...

The option(s) with the values of a and \(L\) that satisfy the following equation is (are)
$$${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }...

Let \(f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x\) for all \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).\)
Then the correct expression(s) is (are)

Let \(F:R \to R\) be a thrice differentiable function. Suppose that
\(F\left( 1 \right) = 0,F\left( 3 \right) = - 4\) and \(F'\left( x \right) < 0\) for all \(x \in \left( {{1 \over 2},3} \right).\) Let $$f\left( x \right) = xF\left( x \r...

Let \({E_1}\) and \({E_2}\) be two ellipses whose centres are at the origin. The major axes of \({E_1}\) and \({E_2}\) lie along the \(x\)-axis and the \(y\)-axis, respectively. Let \(S\) be the circle $${x^2} + {\left( {y - 1} \right)^2} ...

Consider the hyperbola \(H:{x^2} - {y^2} = 1\) and a circle \(S\) with center \(N\left( {{x_2},0} \right)\). Suppose that \(H\) and \(S\) touch each other at a point \(P\left( {{x_1},{y_1}} \right)\) with \({{x_1} > 1}\) and \({{y_1} > 0}\)...

If \(\alpha\) \(= 3{\sin ^{ - 1}}\left( {{6 \over {11}}} \right)\) and \(\beta = 3{\cos ^{ - 1}}\left( {{4 \over 9}} \right),\) where the inverse trigonimetric functions take only the principal values, then the correct options(s) is (are...

Let m and n be two positive integers greater than 1. If
\($\mathop {\lim }\limits_{\alpha \to 0} \left( {{{{e^{\cos \left( {{\alpha ^n}} \right)}} - e} \over {{\alpha ^m}}}} \right) = - \left( {{e \over 2}} \right)\)$
then the value of $$...

Suppose that the foci of the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 5} = 1\) are \(\left( {{f_1},0} \right)\) and \(\left( {{f_2},0} \right)\) where \({{f_1} > 0}\) and \({{f_2} < 0}\). Let \({P_1}\) and \({P_2}\) be two parabolas wit...

Let \({n_1}\) and \({n_2}\) be the number of red and black balls, respectively, in box \({\rm I}\). Let \({n_3}\) and \({n_4}\) be the number of red and black balls, respectively, in box \({\rm I}{\rm I}.\)
One of the two boxes, box $${\rm...

Let \({n_1}\) and \({n_2}\) be the number of red and black balls, respectively, in box \({\rm I}\). Let \({n_3}\) and \({n_4}\) be the number of red and black balls, respectively, in box \({\rm I}{\rm I}.\)
A ball is drawn at random from b...

Let \(S\) be the set of all non-zero real numbers \(\alpha\) such that the quadratic equation \(\alpha {x^2} - x + \alpha = 0\) has two distinct real roots \({x_1}\) and \({x_2}\) satisfying the inequality $$\left| {{x_1} - {x_2}} \right|...

Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the co...

The coefficient of \({x^9}\) in the expansion of (1 + x) (1 + \({x^2)}\) (1 + \({x^3}\)) ....\((1 + {x^{100}})\) is

Suppose that \(\overrightarrow p ,\overrightarrow q\) and \(\overrightarrow r\) are three non-coplanar vectors in \({R^3}\). Let the components of a vector \(\overrightarrow s\) along \(\overrightarrow p ,\) \(\overrightarrow q\) and $$...