PRACTICE PAPER For IIT
( COMPETITIVE EXAMINATION)
1) Three dice are thrown simultaneously. The probability of getting a sum of 15, is
a) 1/72 b) 5/36 c) 5/72 d) none
2) Two dice are thrown simultaneously to get the coordinates of a point on x - y plane. Then the probability that this point lies inside or on the region bounded by | x |+| y | =3, is
a) 3/14 b) 2/3 c) 1/12 d) 4/14.
3) Let (x) denotes the probability of the occurence of event x. Then all those points (x, y) = P(A), P(B), in a plane which satisfy the cinditiins, P(A) ≥ 3/4 and 1/8 ≤ P(A∩B) ≤ 3/8 imply.
a) P(A)+P(B) < 11/8
b) P(A)+P(B)>11/8
c) 7/8 ≤ P(A)+ P(B) < 11/8
d) none of these.
4) Two points are taken at random on the given straight line segment of length a. Then the probability for the distance between them to exceed a given length c,
where 0 < c < a, is
a) (1+c/a)². b) (1 - c/a)².
c)(1+a/c)². d) None
5) If a ∈ [20,0], then the probability that the Equation
16x² +8(a+5)x - 7a - 5 =0 has imaginary roots, is
a)13/20. b) 20/13.
c) 13/24. d) 15/20.
6) If the sides of a triangle are decided by the row of a single die thrice, the probability that triangle is of maximum area given that its an isosceles triangle, is
a) 1/7. b) 1/27. c) 1/14. d) none.
7) Four cards are drawn from a pack of 52 playing cards. Then the probability of drawing at least one pair is
a) 1 - {(4⁴ x ¹³C₄)}/⁵²C₄
b) 1 - ¹³C₄/⁵²C₄
c) 1 - {(4⁴x 13!)}/52!
d) none
8) If P(A/B)=P(B/A). A and B are two non- mutually exclusive events, then
a) A and B are necessarily same events.
b) P(A) = P(B)
c) P(A∩B) = P(A) P(B)
d) All if the above.
9) The probability that
sin⁻¹(sin x) + cos⁻¹ (cosy) is an Integer x, y ∈ {1,2,3,4}, is
a) 1/16 b) 3/16 c) 15/16 d) none
10) A and B throw a dice each. The probability that A's throw is not greater than B's throw, is
a) 7/12 b) 5/12 c) 1/12 d)1/2
11) If A and B are square metrices of order ' n' such that
A² - B²=(A - B)(A+B), then which of the following will be true ?
a) Either of A or B is zero metrix.
b) A=B. c) AB=BA d) Either of A or B is an identity metrix.
12)If Metrix A= x 2. And | A³ |=125
2 x. Then x= ?
a) ±1 b) ±2 c) ±3. d) ±5
13) If Determinants A= x 1 1
B = x 1 1 x 1
1 x 1 1 x
Then d A/d x =
a) 3B+1. b) 3B. c) - 3B. d) 1-3B
14) If the Determinant of the adjoint of a (real) matrix of order 3 is 25, then the Determinant of the inverse of the metrix is -
a) 0.2. b) ±5. c) 1/⁵√625. d) ±0.2
15) A tangent is drawn to the circle to the circle 2x²+2y²-3x+4y=0 at the point 'A' and it meets the line x+y=3 at B(2,1), then AB = ?
a)√10. b) 2. c)2√2. d) 0
16) The sum of the squares of the eccentricities of the conics
x²/2+ y²/3=1 and x²/4 - y²/3=1 is
a) 2. b) √(7/8) c) √7. d) √3
17) If x+y=tan⁻¹ and
d²y/dx² = f(y)dy/dx, then f(y) = ?
a) -2/y³ b) 2/y³ c) 1/y d) -1/y
18) The coefficient of x⁴ in the expansion of (1+x+x²+x³)ⁿ is
a) ⁿC₄ b) ⁿC₄ + ⁿC₂
c) ⁿC₄ + ⁿC₂+ⁿC₄ . ⁿC₂
d) ⁿC₄ + ⁿC₂+ⁿC₁. ⁿC₂
19) Range of the function
f(x)=(x²+x+2)/(x²+x+1) , x ∈ R is -
a) (1,∞) b) (1,11/7)
c) (1,7/3) d) (1,7/5)
20) If α, β are the roots of 8x²-3x+27=0, then the value of
{(α²/β)¹/³ + (β²/α)¹/³} is
a) 1/3 b) 1/4 c) 1/5 d) 1/6
21) Let A={a,b,c}. which of the following is not an equivalence relation in A .
a) R₁ ={(a,b),(b,c),(a,c),(a,a)}
b) R₂= {(c,b),(c,a),(c,c),(b,b)}
c) R₃= {(a,a),(b,b),(c,c),(a,b)}
d) None of these.
22) If ∫ x dx/(x²-4x+8) =
k(x²-4x+8) + tan⁻¹{(x-2)/2} +c,
find k.
a) 1/2 b) 1 c) 2 d) 3.
23) Given two events A and B. if odds against A are as 2/1 and those in favour of A∪B are as 3/1, then .
a) 1/2< P(B)≤3/4
b) 5/12≤P(B)≤3/4
c) 1/4≤P(B)≤3/5 d) None.
24) If sinα,sin²α,sin⁴α,sin⁵α,
(-π<α <π) are in A.P.,
then α lies in the interval.
a) (-π/2, π/2) b) (-π/3,π/3)
c) (-π/6, π/6) d) None .
25) The point on the curve y=(x-3)², where the tangent is parallel to the chord joining (3,0) and (4,1) is-
a) cosx b) tan x c) x² d) | x- 1|
26) cos[2cos⁻¹1/5 + sin⁻¹ 1/5] =
a) 1/5 b)-2√6/5 c) -1/5 d)√6/5.
27) ∫ cosⁿ⁻¹x/sinⁿ⁺¹x dx, n≠ 0
a) cotⁿx/n + C b) -cotⁿ⁻¹x/(n-1)+C
c) -cotⁿx/ n + C d) cotⁿ⁻¹x/(n-1) +C
28) The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is-
a) 8π sq.units. b) 4 sq.units
c) 5 sq.units. d) 8 sq units.
29) The 100th term of the sequence 1,2,2,3,3,3,4,4,4,4....is
a) 12. b) 13. c) 14 d) 15. e) 16.
30) Five Numbers are in H.P The middle term is 1 and the ratio of the second and the fourth terms is 2:1. Then the sum of the first three terms is-
a) 11/2 b) 5. c) 2. d) 14/3
31) Find x
sin[2cos⁻¹{cot(2tan⁻¹x)}] =0
a) 1 b) -1 c) 1± √(2) d) -(1±√2)
32) Cos4x + (10tanx)/(1+tan²x) =3
in the interval (-π/2 , π/2) has
a) no solution b) one solution
c) two solution d) three solution
33) The largest interval for which
x²² - x¹⁹ + x¹⁴ - x⁵ + 1 > 1 is
a) (0,1) b) (-∞,1] c)[1,∞) d) (-∞,∞)
34) 100 students appeared for a test comprising 5 subjects: Hindi, English, Mathematics, Physics and Chemistry. The number of them passed in these subjects are 99,85,91,89,87 respectively. The number of students who passed in all the 5 subjects can be
a) 45. b) 56. c) 67. d) 78
Continue.........
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