September 08, 2021

JEE Advanced Mock Test 6

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## Section - (I)

a. This section contains Six questions.

b. Each question has FOUR options (A), (B), (C) and (D). Only one of these four options is correct.

1. The number of points in (-∞, ∞), for which x² - xSinx - Cosx = 0 is/are.

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2. If the line (x - 2)/a = (3 - y)/5 = z/2 completely lies in the plane x + 3y - 2z + b = 0, then the value of (a + b) is.

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3. If a chord which is not a tangent of the parabola y² = 16x has the equation 2x + y = p and midpoint (h, k), then which of the following are possible value of values of p, h, and k?

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4. Tangents drawn from the point P(1, 8) to the circle x² + y² - 6x - 4y - 11 = 0 touch the circle at the point A and B. The equation of the circumcircle of the triangle PAB is.

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5. Let Arg((z + 1)/(z - 1)) = π/4. Then the locus of z be a circle whose radius and center respectively are.

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6. A dice is rolled till 6 appears, then the probability P((x ≥ 5)/(x > 2)) is {where x = r means 6 appear in rth throw}.

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## SECTION – (II)

a. This section contains Four Questions.

b. Each question has FOUR options (A), (B), (C) and (D) ONE OR MORE of these four options is/are correct.

1. Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M⁻¹ = adj(adj M), then which of the following statement is/are ALWAYS TRUE ?

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2. Let f: R → R, g: R → R and h: R → R be differentiable functions such that f(x) = x³ + 3x + 2, g(f(x)) = x and h(g(g(x))) = x for all x in R. Then.

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3. A vertical line passing through the point T(h, 0) intersect the ellipse x²/4 + y²/3 = 1 at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. if Δ(h) = area of the triangle PQR, Δ₁ = maximum Δ(h) (where 1/2 ≤ h ≤ 1) and Δ₂ = minimum Δ(h) (where 1/2 ≤ h ≤ 1) then the value of (8Δ₁/√5) - 8Δ₂ is/are.

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4. A circle is given by x² + (y - 1)² = 1, another circle C touches it externally and also the x-axis, then the locus of its center is.

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## SECTION - (III)

a. This section contains Four questions.

b. The answer to each question is a Whole Number.

1. Let f : [0, 2] → R be the function defined by f(x) = (3 - Sin(2πx)).Sin(πx - π/4) - Sin(3πx + π/4). If a, b ∈ [0, 2] are such that {x ∈ [0, 2] : f (x) ≥ 0} = [a, b], then the value of (b – a) is.

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2. Suppose that the focii of the ellipse x²/9 + y²/5 = 1 are f1 and f2, where f1 > 0 and f2 < 0. Let P1 and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f1, 0) and (2f2, 0) respectively. Let T1 be the tangent to P1 which passes through (2f2, 0) and T2 be a tangent to P2 which passes through (f1, 0). The m is the slope of T1 and n is the slope of T2, then the value of

(m⁻² + n²) is.

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3. Let f: R → R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x)f'(y) + f(y)f'(x) for all x and y in R. Then the value of lnf(8) is.

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4. If a = i + j + k, b = i - k, a x c = b, and a . c = 3, then the value of |[a b c]| is.

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