Page 44 - ISC-12

Numerical Valued Answer Type Jai Baba Ki5

sin θ π π
39. Let f(θ) = sin tan −1 √ , where − < θ < . Then the value of
cos 2θ 4 4
d
(f(θ))
d(tan θ)
is [JEE 2011]

d
0
0
0
0
40. Let y (x) + y(x)g (x) = g(x)g (x), y(0) = 0, x ∈ R, where f (x) denotes f(x) and g(x) is a given
dx
non-constant differentiable function on R with g(0) = g(2) = 0. Then the value of y(2) is
[JEE 2011]

41. Let M be a 3 × 3 matrix satisfying
           
0 −1 1 1 1 0
M 1 =  2  , M −1 =  1  , and M 1 =  0  .





 

 
 
0 3 0 −1 1 12
Then the sum of the diagonal entries of M is [JEE 2011]
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ~
ˆ
42. Let ~a = −i − k, b = −i + j and ~c = i + 2j + 3k be three given vectors. If ~r is a vector such that
~
~
~
~r × b = ~c × b and ~r · ~a = 0, then the value of ~r · b is [JEE 2011]
1
43. The maximum value of the expression is [JEE 2010]
2
2
sin θ + 3 sin θ cos θ + 5 cos θ
ˆ
ˆ
ˆ
ˆ ˆ 2i + j + 3k
i − 2j
~
~
44. If ~a and b are vectors in space given by ~a = √ and b = √ , then the value of
5 14
h i
~
~
~
2~a + b · ~a × b × ~a − 2b
is [JEE 2010]
45. If the distance between the plane Ax − 2y + z = d and the plane containing the lines
x − 1 y − 2 z − 3 x − 2 y − 3 z − 4
= = and = =
2 3 4 3 4 5
√
is 6, then |d| is [JEE 2010]
46. For any real number x, let bxc denote the largest integer less than or equal to x. Let f be a real-valued
function defined on the interval [−10, 10] by
(
x − bxc if bxc is odd
f(x) =
1 + bxc − x if bxc is even
Then the value of
π 2 Z 10
f(x) cos πx dx
10 −10
is [JEE 2010]