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JEE Mathematics Solved Question Paper 2017

Multiple Choice Questions

31.
If $\int_{0}^{\infty} \frac{\log (1 + x^{2})}{1 + x^{2}} d x = k \int_{0}^{1} \frac{\log (1 + x)}{1 + x^{2}} dx$ , then k is equal to

4

8

$π$

2 $π$

We have,

$\int_{0}^{\infty} \frac{\log (1 + x^{2})}{1 + x^{2}} dx = k \int_{0}^{1} \frac{\log (1 + x)}{1 + x^{2}} dx \Rightarrow \int_{0}^{\frac{π}{2}} \log (\sec^{2} (θ)) dθ = k \int_{0}^{\frac{π}{4}} \log (1 + \tan (θ)) dθ where x = \tan (θ) \Rightarrow - 2 \int_{0}^{\frac{π}{2}} \log (\cos (θ)) dθ = k \int_{0}^{\frac{π}{4}} \log (1 + \tan (θ)) dθ \Rightarrow - 2 X - \frac{π}{2} \log_{e} (2) = k (\frac{π}{8} \log (2)) \Rightarrow k = 8$

32.

The degree of the differential equation satisfying $\sqrt{1 - x^{2}} + \sqrt{1 - y^{2}} = a (x - y)$

1
2
3
None

33.

If T = 2 $π$ $\sqrt{\frac{1}{g}}$ , then relative errors in T and l are in the ratio

1/2
2
1/2 $π$
None

34.

An unbiased coin is tossed n times. Let X denotes the number of times head occurs. If P(X = 4 ), P(X = 5) and PX = 6) are in AP, then the value of n can be

7, 14
10, 14
12, 7
None of these

35.

Let X and Y be two events such that P(X/Y) = 1/2, P(Y/X) = 1/3 and P(X ∩ Y) = 1/6. Which of the following is incorrect?

P(X $\cup$ Y) = 2/3
X and Y are independent
P(X^C ∩ Y) = 2/3
X and Y are not independent

36.

If a, b, c are non-coplanar unit vectors such that

$a \times (b \times c) = \frac{b + c}{\sqrt{2}}$ , then the angle between a and b is

$\frac{3 π}{4}$
$\frac{π}{4}$
$\frac{π}{2}$
$π$

37.

If a, b, c and a', b',c' form a reciprocal system of vectors, then a . a'+ b . b' + c · c' is equal to

38.

If a, b, c are three non - zero vectors which are pairwise non-collinear. Also, a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is