March 2011 - Career Point

81. If in the expansion of (1 + x) m (1 – x) n , the coefficient
of x and x 2 are 3 and – 6 respectively, then m is
(A) 6 (B) 9 (C) 12 (D) 24
82. ~ (p ∨ q) ∨ (~ p ∧ q) is logically equivalent to
(A) ~ p (B) p (C) q (D) ~ q
83. The position vectors of the vertices of a quadrilateral
ABCD are a r , b r , c r , d r respectively. Area of the
quadrilateral formed by joining the middle points of
its sides is
(A) 4
1 | b
r
× c
r + c
r × d
r
+ d
r
× b
r
|
(B) 4
1 | a
r × b
r
+ b
r
× d
r
+ d
r
× a
r |
(C)
( a + b + c + d)!
a!
b!
c!
(D) None of these
87. A bag contains (2n + 1) coins. It is known that n of
these coins have a head on both side, whereas the
remaining n + 1 coins are fairs. A coin is picked up at
random from the bag and tossed. If the probability
that the toss results in a head is 31/42, then n is equal
to
(A) 13 (B) 12 (C) 11 (D) 10
−1⎛
1 ⎞
88. tan ⎜ ⎟ =
2
⎝ 2i
⎠
∑ ∞
i=
1
(A) 2
π
−π
(B) 4
(C) 4
1 | a
r × b
r
+ b
r
× c
r + c
r × d
r
+ d
r
× a
r |
(C) 4
π
(D) None of these
(D) 4
1 | b
r
× c
r + c
r × d
r
+ a
r × d
r
+ b
r
× a
r |
84. In the triangle with vertices at A(6, 3), B(–6, 3) and
C(–6, –3), the median through A meets BC at P, the
line AC meets the x-axis at Q, while R and S
respectively denote the orthocentre and centroid of
the triangle. Then the correct matching of the
coordinates of points in List-I to List -II is
List- I List-II
(i) P (a) (0, 0)
(ii) Q (b) (6, 0)
(iii) R (c) (–2, 1)
(iv) S (d) (–6, 0)
(e) (–6, –3)
(f) (–6, 3)
(i) (ii) (iii) (iv)
(A) d a e c
(B) d b e c
(C) d a f c
(D) b a f c
85. The value of definite integral
I =
∫
π / 2
e x
0
{cos(sin
x)cos
2
x
+ sin (sin x)sin
2
2
x
}
2
1
(A) [e π/2 e π / 2
(cos 1 + sin 1)–1] (B) (cos 1 + sin 1)
2 2
1
(C) (e π/20 e π / 2
cos 1 –1) (D) [cos 1+sin 1–1]
2 2
86. A library has a copies of one book, b copies of each
of the two books, c copies of each of three books and
single copy of d book. The total number of ways in
which these books can be arranged is
( a + 2b
+ 3c
+ d)!
( a + 2b
+ 3c
+ d)!
(A)
(B)
2 3
a!(
b!)
( c!)
a!
b!
c!
dx
is
Passage Based Question (Q. 89 to 90)
Consider the quadratic equation
x 2 – mx + (m – 1) = 0. Where m is real number .
On the basis of above information, answer the
following questions
89. Find the no. of integral values of m such that given
quadratic equation has equal roots–
(A) 1 (B) 2
(C) 0
(D) None of these
90. Find the set of values of m such that given quadratic
equation has one root positive and other root
negative–
(A) m∈ R (B) m∈ (–∞, 1)
(C) m∈ (∞, 0)
(D) None of these
1. The storm cock or male mistlethrush sings as
a thunderstorm approaches.
2. A typical double mattress contains as many
as two million house dust mites.
3. The average human will grow 590 miles
(949.5 km) of hair in their lifetime.
4. About 51% of incoming solar radiation is
absorbed by the earth's surface and 14%
absorbed by the atmosphere.
5. Only 2% of male red deer are seriously
injured in their antler-rattling contests.
6. The United States recycles 25 percent of its
annual 180 million tons of household
rubbish.
7. Venus is the hottest planet with a
temperature of 480 °C.
XtraEdge for IIT-JEE 78
MARCH 2011