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Solutions to IIT–JEE 2008 (PAPER 1)

PART I

MATHEMATICS

Section - I

Straight objective type

This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct

1. Consider the tow curves

C₁ : y² = 4x

C₂ : x² + y² - 6x + 1 = 0

Then,

(A) C₁ and C₂ touch each other only at one point

(B) C₁ and C₂ touch each other exactly at two points

(D) C₁ and C₂ neither nor touch each other

1. (B)

C₁ : y² = 4x

C₂ : x² + y² - 6x + 1 = 0

For intersection point C₁ and C₂

x² + 4x - 6x + 1 = 0

x² - 2x + 1 = 0

(x - 1)² = 0 Þ x = (1, 1)

\ y² = 4

Þ y = ± 2

Intersection point (1, 2) and (1 -2)

2. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector such that

Then, the volume of the parallelopiped is

(A) (B)

2. (A)

3. If 0 < x < 1, then

(A) (B) x

3. (C)

4. Let , m and n are integers, m ¹ 0, n > 0 and let p be the left hand derivative of |x -1| at x = 1.

If , then

(A) n = 1, m = 1 (B) n = 1, m = -1

4. (C)

LHD of |x - 1| = 1

\ p = -1

Given of m and n are integer m ¹ 0, n > 0.

Þ Þ

Þ n = 2, m = 2.

5. Let a and b be non-zero real numbers. Then, the equation

(ax² + by² + c) (x² - 5xy + 6y²) = 0

represents

(A) four straight lines, when c = 0 and a, b are of the same sign

(B) two straight lines and a circle, when a = b, and c is of sign opposite to that of a

(D) a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

5. (B)

(ax² + by² + c) (x - 2y) (x - 3y) = 0

two lines x = 2y and x = 3y

a = b and c is of opposite sign of a and b

ax² + cos² + c = 0

If c/a < 0

So, this represents circle centred at origin.

6. The total number of local maxima and local minima of the function

(A) 0 (B) 1

6. (C)

SECTION-II

Multiple correct answers type

This section contains 4 multiple choice correct answers(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct

7. Let P(x₁, y₁) and Q(x₂, y₂), y₁ < 0, y₂ < 0, be the end points of the latus rectum of the ellipse x² + 4y² = 4. The equations of parabolas with latus rectum PQ are

(A) x² + 2 y = 3 + (B) x² - 2 y = 3 +

7. (B), (C)

1 = 4(1 - e²)

e² = 1 - =

P º

Q º

Length of latus rectum of parabola = and equation of latus rectum of parabola
y = -

Only (B) and (C) satisfy equation latus rectum of parabola i.e., y = -.

8. Let

and , for n = 1, 2, 3, …. Then,

(A) (B)

8. (A), (C)

we know that

9. Let f(x) be a non-constant twice differentiable function defined on (-¥, ¥)l such that f(x) = f(1 - x) and . Then

(A) f²(x) vanishes at least twice on [0, 1] (B)

9. (A), (B), (C), (D)

10. A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the cricumcricle, then

(A) (B)

10. (B), (D)

PS.ST = QS. SR

Now, QS.SR £ OQ.OR

QS.SR £ OQ²

QS.SR £

\ .

SECTION-III

Reasoning type

This section contains 4 reasoning type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct

11. Consider the system of equations

x - 2y + 3z = -1

-x + y - 2z = k

x - 3y + 4z = 1

STATEMENT - 1 : The system of equations has no solution for k ¹ 3.

and

STATEMENT-2: The determinant ¹ 0, for k ¹ 3.

(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

(D) STATEMENT - 1 is False, STATEMENT - 2 is True

11. (B)

12. Let f and g be real valued functions defined on interval (-1, 1) such that g² (x) is continuous, g(0) ¹ 0, g¢(0) = 0, g²(0) ¹ 0, and f(x) = g(x) sinx

TATEMENT - 1 : .

and

STATEMENT-2: f¢(0) = g(0).

(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