Page 45 - ISC-12

6Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal
2π 2π
47. Let ω be the complex number cos + i sin . Then the number of distinct complex numbers z
3 3
satisfying

z + 1 ω ω
2

ω z + ω 2 1 = 0

2
ω 1 z + ω

is equal to [JEE 2010]
48. Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If
the y-intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the
abscissa of P, then the value of f(−3) is equal to [JEE 2010]

49. Let f be a function defined on R (the set of all real numbers) such that
0
4
2
3
f (x) = 2010(x − 2009)(x − 2010) (x − 2011) (x − 2012) , for all x ∈ R.
If g is a function defined on R with values in the interval (0, ∞) such that
f(x) = ln (g(x)), for all x ∈ R,
then the number of points in R at which g has a local maximum is [JEE 2010]

50. Let k be a positive real number and let
√
 √   √ 
2k − 1 2 k 2 k 0 2k − 1 k
√ √
A =  2 k 1 −2k and B = 1 − 2k 0 2 k .




√ √ √
−2 k 2k −1 − k −2 k 0
6
If det(adj A) + det(adj B) = 10 , then bkc is equal to
[Note: adj M denotes the adjoint of a square matrix M and bkc denotes the largest integer less than
or equal to k.] [JEE 2010]
2
3
2
51. The maximum value of the function f(x) = 2x −15x +36x−48 on the set A = {x : x +20 ≤ 9x}
is [JEE 2009]
52. Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:
3x − y − z = 0
−3x + z = 0
−3x + 2y + z = 0

2
2
2
Then the number of such points for which x + y + z ≤ 100 is [JEE 2009]
√
0
0
53. Let ABC and ABC be two non-congruent triangles with sides AB = 4, AC = AC = 2 2 and
◦
angle B = 30 . The absolute value of the difference between the areas of these triangles is
[JEE 2009]
54. Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and

p(x)
lim 1 + = 2.
x→0 x 2
Then the value of p(2) is [JEE 2009]