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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

31.

If a, b and c are three non-coplanar vectors, then (a + b - c) . [(a - b) x (b - c)] equals

0
a . b $\times$ c
a . c $\times$ b
3a . b $\times$ c

32.

The area of the region bounded by the curves x² + y² = 9 and x + y = 3 is

33.

For any three vectors a, b and c [a + b, b + c, c + a] is

[a, b, c]
3[a, b, c]
2[a, b, c]
0

34.
$\int_{0}^{\frac{π}{2}} \sin (2 x) . \log (\tan (x)) dx$

0

2

4

7

$Consider, I = \int_{0}^{\frac{π}{2}} (\log (\tan (x))) . \sin (2 x) dx . . . (i) I = \int_{0}^{\frac{π}{2}} (\log (\tan (\frac{π}{2} - x))) . \sin (2 (\frac{π}{2} - x)) dx$

$[∵ \int_{0}^{a} f (x) dx = \int_{0}^{a} f (a - x) dx]$

$\Rightarrow$ $I = \int_{0}^{\frac{π}{2}} \log (\cot (x)) \sin (2 x) dx$ ...(ii)

$[∵ \sin (π - 2 x) = \sin (2 x)]$

On adding Eqs. (i) and (ii), we get

$2 I = \int_{0}^{\frac{π}{2}} \log (\tan (x)) . \sin (2 x) dx + \int_{0}^{\frac{π}{2}} \log (\cot (x)) . \sin (2 x) dx = \int_{0}^{\frac{π}{2}} \sin (2 x) \log (\tan (x) . \cot (x)) dx [∵ \log (m) + \log (n) = \log (m . n)] = \int_{0}^{\frac{π}{2}} \sin (2 x) \log (1) dx \Rightarrow I = 0 [∵ \log (1) = 0] ∴ \int_{0}^{\frac{π}{2}} \sin (2 x) \log (\tan (x)) dx = 0$

35.

If a plane passing through the point (2, 2, 1) and is perpendicular to the planes 3x + 2y + 4z + 1 = 0 and 2x + y + 3z + 2 = 0. Then, the equation of the plane is

2x - y - z - 1 = 0
2x + 3y + z - 1 = 0
2x + y + z + 3 = 0
x - y + z - 1 = 0

36.

From a city population, the probability of selecting a male or smoker $\frac{7}{10}$ , a male smoker is $\frac{2}{5}$ and a male, if a smoker is already, selected, is $\frac{2}{3}$ Then, the probability of

selecting a male is $\frac{3}{2}$
selecting a smoker is $\frac{1}{5}$
selecting a non-smoker is $\frac{2}{5}$
selecting a smoker, if a male is first selected, is given by $\frac{8}{5}$

37.

The solution of $\frac{d^{2} x}{d y^{2}} - x = k$ where k is a non-zero constant, vanishes when y = 0 and tends of finite limit as y tends to infinity, is