JEE Advanced 2014 · Paper 2 · Mathematics Q58

MathematicsList match+3 / -1Question 58 of 60

Match List I with List II and select the correct answer using the code given below the lists.

List I

P
The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\le2$ , satisfying $f(0)=0$ and $\int_0^1f(x)\,dx=1$ , is
Q
The number of points in the interval $[-\sqrt{13},\sqrt{13}]$ at which $f(x)=\sin(x^2)+\cos(x^2)$ attains its maximum value, is
R
$\displaystyle\int_{-2}^{2}\frac{3x^2}{(1+e^x)}\,dx$ equals
S
$\dfrac{\left(\int_{-1/2}^{1/2}\cos2x\,\log\left(\frac{1+x}{1-x}\right)dx\right)}{\left(\int_0^{1/2}\cos2x\,\log\left(\frac{1+x}{1-x}\right)dx\right)}$ equals

List II

P → 3; Q → 2; R → 4; S → 1

P → 2; Q → 3; R → 4; S → 1

P → 3; Q → 2; R → 1; S → 4

P → 2; Q → 3; R → 1; S → 4

Answer and worked solution

Official answer: D

P: $f(x)=bx+cx^2$ with $\frac b2+\frac c3=1$ , i.e. $3b+2c=6$ with $b,c\ge0$ integers: $(2,0)$ and $(0,3)$ . Two polynomials → 2.
Q: $\sin u+\cos u$ has maximum $\sqrt2$ at $u=\frac\pi4+2n\pi$ . With $u=x^2\le13$ : $u=\frac\pi4$ and $\frac{9\pi}4\approx7.07$ ( $\frac{17\pi}4\approx13.35$ is too big). Each gives $x=\pm\sqrt u$ : 4 points → 3.
R: Adding the integrand at $x$ and $-x$ : $\frac{3x^2}{1+e^x}+\frac{3x^2}{1+e^{-x}}=3x^2$ . So the integral is $\int_0^23x^2dx=8$ → 1.
S: $\cos2x\log\frac{1+x}{1-x}$ is odd, so the numerator is 0 (the denominator is non-zero) → 4.

(D) P → 2, Q → 3, R → 1, S → 4

JEE Advanced 2014 · Paper 2, Mathematics question 58