Page 43 - ISC-12

4Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal
−1
28. Let f : [0, 4π] → [0, π] be defined by f(x) = cos (cos x). The number of points x ∈ [0, 4π]
satisfying the equation
10 − x
f(x) =
10
is [JEE 2014]

29. For a point P in the plane, let d 1 (P) and d 2 (P) be the distances of the point P from the lines x−y = 0

and x+y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant
of the plane and satisfying 2 ≤ d 1 (P) + d 2 (P) ≤ 4 is [JEE 2014]

π
~
30. Let ~a, b and ~c be three non-coplanar unit vectors such that the angle between every pair of them is .
3
2
2
p + 2q + r 2
~
~
~
If ~a × b + b × ~c = p~a + qb + r~c, where p, q and r are scalars, then the value of is
q 2
[JEE 2014]
31. A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from
the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on
the removed cards is k, then k − 20 = [JEE 2013]

32. Of the three independent events E 1 , E 2 and E 3 , the probability that only E 1 occurs is α, only E 2
occurs is β and only E 3 occurs is γ. Let the probability p that one of the events E 1 , E 2 or E 3 occurs
satisfy the equations
(α − 2β)p = αβ and (β − 3γ)p = 2βγ.

All the given probabilities are assumed to lie in the interval (0, 1). Then
Probability of occurrence of E 1
=
Probability of occurrence of E 3
[JEE 2013]
n o
ˆ
ˆ
ˆ
33. Consider the set of eight vectors V = ai + bj + ck : a, b, c ∈ {−1, 1} . Three non-coplanar vectors
p
can be chosen from V in 2 ways. Then p is [JEE 2013]
2
34. Let f : R → R be defined as f(x) = |x| + |x − 1|. The total number of points at which f attains
either a local maximum or a local minimum is [JEE 2012]

v
 
s
u
r
1 u 1 1 1
35. The value of 6 + log 3/2  √ t 4 − √ 4 − √ 4 − √ · · ·  is [JEE 2012]
3 2 3 2 3 2 3 2
36. Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local
0
minimum at x = 3. If p(1) = 6 and p(3) = 2, then p (0) is [JEE 2012]
~
~
~
2
2
~ 2
37. If ~a, b and ~c are unit vectors satisfying |~a − b| + |b − ~c| + |~c − ~a| = 9, then |2~a + 5b + 5~c| is
[JEE 2012]
−5
8
−3
−4
38. The minimum value of the sum of real numbers a , a , 3a , 1, a and a 10 with a > 0 is
[JEE 2011]