JEE Advanced 2008 · Paper 1

Consider the two curves C₁ : y² = 4x, C₂ : x² + y² − 6x + 1 = 0. Then,

1. AC₁ and C₂ touch each other only at one point
2. BC₁ and C₂ touch each other exactly at two points
3. CC₁ and C₂ intersect (but do not touch) at exactly two points
4. DC₁ and C₂ neither intersect nor touch each other

If 0 < x < 1, then √(1 + x²) [{x cos (cot⁻¹ x) + sin (cot⁻¹ x)}² − 1]¹ᐟ² =

1. Ax/√(1 + x²)
2. Bx
3. Cx√(1 + x²)
4. D√(1 + x²)

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors â, b̂, ĉ such that â · b̂ = b̂ · ĉ = ĉ · â = 1/2. Then, the volume of the parallelepiped is

1. A1/√2
2. B1/(2√2)
3. C√3/2
4. D1/√3

Let a and b be non-zero real numbers. Then, the equation (a x² + b y² + c) (x² − 5xy + 6y²) = 0 represents

1. Afour straight lines, when c = 0 and a, b are of the same sign
2. Btwo straight lines and a circle, when a = b, and c is of sign opposite to that of a
3. Ctwo straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
4. Da circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Let g(x) = (x − 1)ⁿ / log cosᵐ (x − 1); 0 < x < 2, m and n are integers, m ≠ 0, n > 0, and let p be the left hand derivative of |x − 1| at x = 1. If limₓ→₁₊ g(x) = p, then

1. An = 1, m = 1
2. Bn = 1, m = −1
3. Cn = 2, m = 2
4. Dn > 2, m = n

The total number of local maxima and local minima of the function f(x) = {(2 + x)³, −3 < x ≤ −1; x²ᐟ³, −1 < x < 2} is

1. A0
2. B1
3. C2
4. D3

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 circumcircle, then

1. A1/PS + 1/ST < 2/√(QS × SR)
2. B1/PS + 1/ST > 2/√(QS × SR)
3. C1/PS + 1/ST < 4/QR
4. D1/PS + 1/ST > 4/QR

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

1. Ax² + 2√3 y = 3 + √3
2. Bx² − 2√3 y = 3 + √3
3. Cx² + 2√3 y = 3 − √3
4. Dx² − 2√3 y = 3 − √3