JEE Advanced 2016 · Paper 2

Let P = [[1, 0, 0], [4, 1, 0], [16, 4, 1]] and I be the identity matrix of order 3. If Q = [qᵢⱼ] is a matrix such that P⁵⁰ − Q = I, then (q₃₁ + q₃₂)/q₂₁ equals

1. A52
2. B103
3. C201
4. D205

Area of the region {(x, y) ∈ ℝ² : y ≥ √|x + 3|, 5y ≤ x + 9 ≤ 15} is equal to

1. A1/6
2. B4/3
3. C3/2
4. D5/3

Let P be the image of the point (3, 1, 7) with respect to the plane x − y + z = 3. Then the equation of the plane passing through P and containing the straight line x/1 = y/2 = z/1 is

1. Ax + y − 3z = 0
2. B3x + z = 0
3. Cx − 4y + 7z = 0
4. D2x − y = 0

Let a, b ∈ ℝ and a² + b² ≠ 0. Suppose S = {z ∈ ℂ : z = 1/(a + ibt), t ∈ ℝ, t ≠ 0}, where i = √−1. If z = x + iy and z ∈ S, then (x, y) lies on

1. Athe circle with radius 1/(2a) and centre (1/(2a), 0) for a > 0, b ≠ 0
2. Bthe circle with radius −1/(2a) and centre (−1/(2a), 0) for a < 0, b ≠ 0
3. Cthe x-axis for a ≠ 0, b = 0
4. Dthe y-axis for a = 0, b ≠ 0

Let P be the point on the parabola y² = 4x which is at the shortest distance from the center S of the circle x² + y² − 4x − 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then

1. ASP = 2√5
2. BSQ : QP = (√5 + 1) : 2
3. Cthe x-intercept of the normal to the parabola at P is 6
4. Dthe slope of the tangent to the circle at Q is 1/2

Let α, λ, μ ∈ ℝ. Consider the system of linear equations αx + 2y = λ; 3x − 2y = μ. Which of the following statement(s) is(are) correct?

1. AIf α = −3, then the system has infinitely many solutions for all values of λ and μ
2. BIf α ≠ −3, then the system has a unique solution for all values of λ and μ
3. CIf λ + μ = 0, then the system has infinitely many solutions for α = −3
4. DIf λ + μ ≠ 0, then the system has no solution for α = −3

Let û = u₁î + u₂ĵ + u₃k̂ be a unit vector in ℝ³ and ŵ = 1/√6(î + ĵ + 2k̂). Given that there exists a vector v⃗ in ℝ³ such that |û × v⃗| = 1 and ŵ · (û × v⃗) = 1. Which of the following statement(s) is(are) correct?

1. AThere is exactly one choice for such v⃗
2. BThere are infinitely many choices for such v⃗
3. CIf û lies in the xy-plane then |u₁| = |u₂|
4. DIf û lies in the xz-plane then 2|u₁| = |u₃|

Let F₁(x₁, 0) and F₂(x₂, 0), for x₁ < 0 and x₂ > 0, be the foci of the ellipse x²/9 + y²/8 = 1. Suppose a parabola having vertex at the origin and focus at F₂ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. The orthocentre of the triangle F₁MN is

1. A(−9/10, 0)
2. B(2/3, 0)
3. C(9/10, 0)
4. D(2/3, √6)