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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

11.

System software : Utility software : :

Operating system· Anti-virus
Anti-virus Operating system
Anti-virus MS office
MS office Anti-virus

12.

If f(x) = $\{\frac{\sin [x]}{[x]}, [x] \neq 0 0, [x] = 0\}$ , where [x] denotes the gretest function less than or equal to x, then $\lim_{x \to 0} f (x)$ is equal to

1
0
- 1
Does not exist

13.

The number of common tangents to two circles x² + y² = 4 and x² + y² - 8x + 12 = 0 is

14.

If for all x, y $\in$ N, there exists a function f(x) satisfying f(x + y) = f(x) · f(y) such that f(1) = 3 and $\sum_{x = 1}^{n} f (x) = 120$ , then value of n will be

4
5
6
None of these

15.

If sin(a) and cos(a) are the roots of the equation ax² + bx + c = 0, then

a² - b² + 2ac = 0
(a - c)² = b² + c²
a² + b² - 2ac = 0
a² + b² + 2ac = 0

16.
In $∆ ABC$ , if cot(A), cot(B) and cot(C) are in AP, then a², b² and c² are in

HP

AP

GP

None of the above

$It is given, \cot (A), \cot (B) and \cot (C) are in AP . \Rightarrow 2 \cot (B) = \cot (A) + \cot (C) \Rightarrow 2 \frac{\cos (B)}{\sin (B)} = \frac{\cos (A)}{\sin (A)} + \frac{\cos (C)}{\sin (C)} ∴ \frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c} = k \Rightarrow \frac{2 \cos (B)}{kb} = \frac{\cos (A)}{ak} + \frac{\cos (C)}{ck} \Rightarrow \frac{2 \cos (B)}{b} = \frac{\cos (A)}{a} + \frac{\cos (C)}{c} \Rightarrow \frac{2}{b} [\frac{a^{2} + c^{2} - b^{2}}{2 ac}] = \frac{1}{a} [\frac{b^{2} + c^{2} - a^{2}}{2 bc}] + \frac{1}{c} [\frac{a^{2} + b^{2} - c^{2}}{2 ab}] \Rightarrow \frac{2 (a^{2} + c^{2} - b^{2})}{2 abc} = \frac{1}{2 abc} [(b^{2} + c^{2} - a^{2}) + (a^{2} + b^{2} - c^{2})] \Rightarrow 2 (a^{2} + c^{2} - b^{2}) = 2 b^{2} \Rightarrow a^{2} + c^{2} = 2 b^{2} ∴ a^{2}, b^{2} and c^{2} are in AP .$