ExamBazar Forum

[rahul]

IIT-JEE-Mathematics i.e conduct on 2008. This paper helps to prepare the IIT Mathematics paper.

1. A particle P starts from the point z0 = 1 + 2i, where i = √(-1). It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1 the particles moves √2 units in the direction of the unit vector i + j and then it moves through an angle Π/2 in anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by
(A) 6 + 7i
(B) -7 + 6i
(C) 7 + 6i
(D) -6 + 7i
2. Let the function g : (-∞, ∞) -->( -Π/2 , Π/2) be given by g(u) = 2tan-1(eu) -Π/2. Then, g is
(A) even and is strictly increasing in (0, ∞)
(B) odd and is strictly decreasing in (-∞, ∞)
(C) odd and is strictly increasing in (-∞, ∞)
(D) neither even or odd, but is strictly increasing in (-∞, ∞)
3. Consider a branch of the hyperbola
x2 - 2y2 - 2√2x - 4√2y - 6 = 0
with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is
(A) 1 - √(2/3)
(B) √(3/2) - 1
(C) 1 + √(2/3)
(D) √(3/2) + 1
4. The area of the region between the curves y = √( (1 + sin x)/cos x ) and y = √( (1 - sin x)/cos x ) bounded by the lines x = 0 and x = Π/4 is
area-under-the-curve
5. Consider three points P = (-sin(β-α), -cos β), Q = (cos(β-α), sin β) and R = (cos(β - α + θ), sin(β - θ), where 0 < α, β, θ < Π/4. Then,
(A) P lies on the line segment RQ
(B) Q lies on the line segment PR
(C) R lies on the line segment QP
(D) P, Q, R are non-collinear
6. An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is
(A) 2, 4 or 8
(B) 3, 6 or 9
(C) 4 or 8
(D) 5 or 10
7. Let two non-collinear unit vectors a and b form an acute angle. A point P moves so that at any tiem t the position vector OP (where O is the origin) is given by a cos t + b sin t. When P is farthest from origin O, let M be the length of vector OP and u be the unit vector along vector OP. Then, non-linear-unit-vector
8. Let equation1. Then, for an arbitrary constant C, the value of J - I equals.solutions
9. Let g(x)&nbs Let g(x) = log f(x) where f(x) is a twice differentiable positive function on (0, ∞) such that f(x + 1) = x f(x). Then, for N = 1, 2, 3, ......, differential-equation
10. Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let
b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and b4 = b3 + a4.
STATEMENT-1: The numbers b1, b2, b3, b4 are neither in A.P. nor in G.P. and
STATEMENT-2: The numbers b1, b2, b3, b4 are in H.P.
(A) Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.
(B) Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
11. Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x2 + 2px + q = 0 and α, 1/β are the roots of the equation ax2 + 2bx + c = 0, where β2 is not belongs to {-1, 0, 1}.
STATEMENT-1: (p2 - q)(b2 - ac) > 0 and
STATEMENT-2: b ¹ pa or c ¹ qa
(A) Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.
(B) Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
12. Consider
L1 : 2x + 3y + p - 3 = 0
L2 : 2x + 3y + p + 3 = 0
where p is a real number, and C : x2 + y2 + 6x + 10y + 30 = 0
STATEMENT-1: If line L1 is a chord of circle C, then line L2 is not always a diameter of circle C.
and
STATEMENT-2: If line L1 is a diameter of circle C, then line L2 is not a chord of circle C.
(A) Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.
(B) Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
13. Let a solution y = y(x) of the differential equation
x√(x2-1) dy-y√(y2-1) dx=0
satisfy y(2) = 2/√3.
STATEMENT-1: y(x) = sec(sec-1 x-π/6)
and
STATEMENT-2: y(x) is given by
1/y=(2√3)/x-√(1-1/x2 )
(A) Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for statement-1.
(B) Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for statement-1.
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
Paragraph Consider the function f : (–∞, ∞) --> (–∞, ∞) defined by
f(x) = (x2-ax+1)/(x2+ax+1), 0 < a < 2.
14. Which of the following is true?
(A) (2 + a)2 f”(1) + (2 – a)2 f”(–1) = 0
(B) (2 – a)2 f”(1) – (2 + a)2 f”(–1) = 0
(C) f’(1)f’(–1) = (2 – a)2
(D) f’(1)f’(–1) = (2 + a)2
15. Which of the following is true?
(A) f(x) is decreasing on (–1, 1) and has a local minimum at x = 1
(B) f(x) is increasing on (–1, 1) and has a local maximum at x = 1
(C) f(x) is increasing on (–1, 1) but has neither a local maximum nor a local minimum at x = 1.
(D) f(x) is decreasing on (–1, 1) but has nether a local maximum nor a local minimum at x = 1.
16. Let g(x) = ∫0ex(f'(t))/(1+t2 ) dt.
Which of the following is true?
(A) g’(x) is positive on (–∞, 0) and negative on (0, ∞)
(B) g’(x) is negative on (–∞, 0) and positive on (0, ∞)
(C) g’(x) is changes sign on both (–∞, 0) and (0, ∞)
(D) g’(x) is does not change sign on (–∞, ∞)
Paragraph Consider the lines
L1 : (x+1)/3=(y+2)/1=(z+1)/2
L2 : (x-2)/1=(y+2)/2=(z-3)/3
17. The unit vector perpendicular to both L1 and L2 is
(A) (-i ̂+7j ̂+7k ̂)/√99
(B) (-i ̂-7j ̂+5k ̂)/(5√3)
(C) (-i ̂+7j ̂+5k ̂)/(5√3)
(D) (7i ̂-7j ̂-k ̂)/√99
18. The shortest distance between L1 and L2 is
(A) 0
(B) 17/√3
(C) 41/(5√3)
(D) 17/(5√3)
19. The distance of the point (1, 1, 1) from the plane passing through the point (–1, –2, –1) and whose normal is perpendicular to both the lines L1 and L2 is
(A) 2/√75
(B) 7/√75
(C) 13/√75
(D) 23/√75