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JEE Mathematics Solved Question Paper 2016

Multiple Choice Questions

31.

The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal

is linear
is homogeneous of second degree
has separable variables
is of second order

32.

The solution of the equation $\frac{d y}{d x} = \cos (x - y)$ is

$x + \cot (\frac{x - y}{2}) = C$
$y + \cot (\frac{x - y}{2}) = C$
$x + \tan (\frac{x - y}{2}) = C$
None of these

33.

The differential equation of all parabolas each of which has a latusrectum 4a and whose axes are parallel to the Y-axis is

of order 1 and degree 2
of order 2 and degree 3
of order 2 and degree 1
of order 2 and degree 2

34.

The value of the parameter a such that the area bounded by y = a²x² + ax + 1, coordinate axes and the line x = 1 attains its least value is equal to

- 1
$- \frac{1}{4}$
$- \frac{3}{4}$
$- \frac{1}{2}$

35.

$\int_{0}^{x} |\sin (t)| dt, where x \in [2 nπ, (2 n + 1) π], n \in N$ , is equal to

4n - 1 - cos(x)
4n - sin(x)
4n - cos(x)
4n + 1 - cos(x)

36.

Let I₁ = $\int_{0}^{1} \frac{e^{x} dx}{1 + x}$ and I₂ = $\int_{0}^{1} \frac{x^{2} dx}{e^{x^{3}} (2 - x^{3})}$ Then, $\frac{I_{1}}{I_{2}}$ is equal to

$\frac{1}{3 e}$
3e
$\frac{e}{3}$
$\frac{3}{e}$

37.
$\int_{\frac{5}{2}}^{5} \frac{\sqrt{{(25 - x^{2})}^{3}}}{x^{4}} dx$ is equal to

$\frac{π}{3}$

$\frac{2 π}{3}$

$\frac{π}{6}$

$\frac{5 π}{6}$

$\frac{π}{3}$

$I = \int_{\frac{5}{2}}^{5} \frac{\sqrt{{(25 - x^{2})}^{3}}}{x^{4}} dx Let x = 5 \sin (θ) \Rightarrow dx = 5 \cos (θ) dθ ∴ I = \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{\sqrt{{(25 - 25 \sin^{2} (θ))}^{3}}}{x^{4} \sin^{4} (θ)} . 5 \cos (θ) dθ$

$= \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{5^{3} \cos^{3} (θ) . 5 \cos (θ)}{5^{4} \sin^{4} (θ)} = \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) (\csc^{2} (θ) - 1) dθ = \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) \csc^{2} (θ) dθ - \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) = \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) \csc^{2} (θ) dθ - \int_{\frac{π}{6}}^{\frac{π}{2}} (\csc^{2} (θ) - 1) dθ = {[- \frac{\cot^{3} (θ)}{3} + \cot (θ) + θ]}_{\frac{π}{6}}^{\frac{π}{2}} = (- 0 + 0 + \frac{π}{2}) - (- \frac{3 \sqrt{3}}{3} + \sqrt{3} + \frac{π}{6}) = \frac{π}{3}$

38.

If n integers taken at random are multiplied together, then the probability that the last digit of the product is 1, 3, 7 or 9 is

$\frac{4^{n}}{5^{n}}$
$\frac{2^{n}}{5^{n}}$
4 - $\frac{2^{n}}{5^{n}}$
None

39.

A bag contains (n + 1) coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is $\frac{7}{12}$ , then the value of n is