IIT-JEE 2010 - Career Point

⎡1
1 ⎤ ⎡ ⎤
7. Let E = ⎥ + 1 2
⎢ + ⎢ + ⎥ + ... upto 50 terms, then -
⎣3
50 ⎦ ⎣3
50 ⎦
(A) E is divisible by exactly 2 primes
(B) E is prime
(C) E ≥ 30
(D) E ≤ 35
8. If m is a positive integer, then [( 3 + 1) ] + 1,
where [x] denotes greatest integer ≤ n, is divisible
by-
(A) 2 m (B) 2 m+1 (C) 2 m+2 (D) 2 2m
9. If A and B are acute angles such that sin A = sin 2 B,
2 cos 2 A = 3 cos 2 B; then -
(A) A = π/6 (B) A = π/2
(C) B = π/4 (D) B = π/3
This section contains 2 questions (Questions 10 to 11).
Each question contains statements given in two
columns which have to be matched. Statements (A, B,
C, D) in Column I have to be matched with statements
(P, Q, R, S, T) in Column II. The answers to these
questions have to be appropriately bubbled as
illustrated in the following example. If the correct
matches are A-P, A-S, A-T, B-Q, B-R, C-P, C-Q and
D-S, D-T then the correctly bubbled 4 × 5 matrix
should be as follows :
P Q R S T
A
B
C
D
P
P
P
P
Q
Q
Q
Q
R
R
R
R
S
S
S
S
T
T
T
T
10. For the circle x 2 + y 2 + 4x + 6y – 19 = 0
Column-I
Column-II
72 226
(A) Length of the tangent (P)
113
from (6, 4) to the circle
(B) Length of the chord (Q) 113
of contact from (6, 4)
to the circle
(C) Distance of (6, 4) (R) 113 – 32
from the centre of the
circle
(D) Shortest distance of (S) 9
(6, 4) from the circle
(T) None
11. Value of x when
Column-I
Column-II
(A) 5 2 5 4 5 6 ... 5 2x = (0.04) –28 (P) 3 log 3 5
⎛ 1 1 1 ⎞
(B) x 2 log 5 ⎜ + + + ... ⎟
=
⎝ 4 8 16
( 0.2)
⎠
(Q) 4
⎛ 1 1 1
log
⎞
2.5 ⎜ + + + ... ⎟
⎝ 3 2 3
3 3 ⎠
(C) x = ( 0.16)
(R) 2
2m
(D) 3 x–1 + 3 x–2 + 3 x–3 + ... (S) 7
⎛ 2 1 1 ⎞
= 2 ⎜5
+ 5 + 1+
+ + ... ⎟
⎝ 5
2
5 ⎠
(T) None
This section contains 8 questions (Questions 12 to 19).
The answer to each of the questions is a SINGLE-
DIGIT INTEGER, ranging from 0 to 9. The
appropriate bubbles below the respective question
numbers in the OMR have to be darkened. For
example, if the correct answers to question numbers X,
Y, Z and W (say) are 6, 0, 9 and 2, respectively, then
the correct darkening of bubbles will look like the
following :
X Y Z W
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
12. Fifteen persons, among whom are A and B, sit down at
random at a round table. if p is The probability that
there are exactly 4 persons between A and B find 14 p.
13. If l is the length of the intercept made by a common
tangent to the circle x 2 + y 2 = 16 and the ellipse
x 2 /25 + y 2 /4 = 1, on the coordinate axes, then
81l
2 + 3
is equal to
1059
14. If x + y = k is a normal to the parabola y 2 = 12x, p is
the length of the perpendicular from the focus of the
3k
3 2
+ 2p
parabola on this normal; then
is equal to
741
15. The volume of the tetrahedron whose vertices are
(0, 1, 2) (3, 0, 1) (4, 3, 6) (2, 3, 2) is equal to
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
16. Let a 1 = 2
1 , ak+1 = a k 2 + a k ∀ k ≥ 1 and
1 1 1
x n = + + ... +
a1 + 1 a 2 + 1 a n + 1
Find [x 100 ] where [x] denotes the greatest integer ≤ x.
17. Find the value of x which satisfy the equation
log 2 (x 2 – 3) – log 2 (6x – 10) + 1 = 0
18. Find the coefficient of x 2009 in the expansion of
(1 – x) 2008 (1 + x + x 2 ) 2007
1/ log x
2
19. Find the value of x satisfying 4 = 2.
XtraEdge for IIT-JEE 68 DECEMBER 2009