JEE Advanced 2010 · Paper 1

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression (a/c) sin 2C + (c/a) sin 2A is

1. A1/2
2. B√3/2
3. C1
4. D√3

Equation of the plane containing the straight line x/2 = y/3 = z/4 and perpendicular to the plane containing the straight lines x/3 = y/4 = z/2 and x/4 = y/2 = z/3 is

1. Ax + 2y − 2z = 0
2. B3x + 2y − 2z = 0
3. Cx − 2y + z = 0
4. D5x + 2y − 4z = 0

Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r₁, r₂ and r₃ are the numbers obtained on the die, then the probability that ωʳ¹ + ωʳ² + ωʳ³ = 0 is

1. A1/18
2. B1/9
3. C2/9
4. D1/36

Let P, Q, R and S be the points on the plane with position vectors −2î − ĵ, 4î, 3î + 3ĵ and −3î + 2ĵ respectively. The quadrilateral PQRS must be a

1. Aparallelogram, which is neither a rhombus nor a rectangle
2. Bsquare
3. Crectangle, but not a square
4. Drhombus, but not a square

The number of 3×3 matrices A whose entries are either 0 or 1 and for which the system A[x y z]ᵀ = [1 0 0]ᵀ has exactly two distinct solutions, is

1. A0
2. B2⁹ − 1
3. C168
4. D2

The value of limₓ→₀ (1/x³) ∫₀ˣ [t ln(1+t)/(t⁴+4)] dt is

1. A0
2. B1/12
3. C1/24
4. D1/64

Let p and q be real numbers such that p ≠ 0, p³ ≠ q and p³ ≠ −q. If α and β are nonzero complex numbers satisfying α + β = −p and α³ + β³ = q, then a quadratic equation having α/β and β/α as its roots is

1. A(p³ + q)x² − (p³ + 2q)x + (p³ + q) = 0
2. B(p³ + q)x² − (p³ − 2q)x + (p³ + q) = 0
3. C(p³ − q)x² − (5p³ − 2q)x + (p³ − q) = 0
4. D(p³ − q)x² − (5p³ + 2q)x + (p³ − q) = 0

Let f, g and h be real-valued functions defined on the interval [0, 1] by f(x) = eˣ² + e⁻ˣ², g(x) = xeˣ² + e⁻ˣ² and h(x) = x²eˣ² + e⁻ˣ². If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then

1. Aa = b and c ≠ b
2. Ba = c and a ≠ b
3. Ca ≠ b and c ≠ b
4. Da = b = c