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SAMPLE QUESTIONS

Q1. Let 𝛼, 𝛽, 𝛾, 𝛿 be real numbers such that 𝛼 2 + 𝛽 2 + 𝛾 2 ≠ 0 and 𝛼 + 𝛾 = 1. Suppose the point (3, 2, −1) is the mirror image of the point (1, 0, −1) with respect to the plane 𝛼𝑥 + 𝛽𝑦 + 𝛾𝑧 = 𝛿. Then which of the following statements is/are TRUE?

(A) 𝛼 + 𝛽 = 2

(B) 𝛿 − 𝛾 = 3

(C) 𝛿 + 𝛽 = 4

(D) 𝛼 + 𝛽 + 𝛾 = 𝛿

Q2. Suppose 𝑎, 𝑏 denote the distinct real roots of the quadratic polynomial 𝑥2 + 20𝑥 − 2020 and suppose 𝑐, 𝑑 denote the distinct complex roots of the quadratic polynomial 𝑥2 − 20𝑥 + 2020. Then the value of

𝑎𝑐(𝑎 − 𝑐) + 𝑎𝑑(𝑎 − 𝑑) + 𝑏𝑐(𝑏 − 𝑐) + 𝑏𝑑(𝑏 − 𝑑) is

(A) 0

(B) 8000

(C) 8080

(D) 16000

Q3. If the function 𝑓: is defined by 𝑓(𝑥) = |𝑥|(𝑥 − sin 𝑥), then which of the following statements is TRUE?

(A) 𝑓 is one-one, but NOT onto

(B) 𝑓 is onto, but NOT one-one

(C) 𝑓 is BOTH one-one and onto

(D) 𝑓 is NEITHER one-one NOR onto

Q4. Let the function 𝑓: be defined by 𝑓(𝑥) = 𝑥3𝑥2 + (𝑥 − 1) sin 𝑥 and let 𝑔: be an arbitrary function. Let 𝑓𝑔: be the product function defined by (𝑓𝑔)(𝑥) = 𝑓(𝑥)𝑔(𝑥). Then which of the following statements is/are TRUE?

(A) If 𝑔 is continuous at 𝑥 = 1, then 𝑓𝑔 is differentiable at 𝑥 = 1

(B) If 𝑓𝑔 is differentiable at 𝑥 = 1, then 𝑔 is continuous at 𝑥 = 1

(C) If 𝑔 is differentiable at 𝑥 = 1, then 𝑓𝑔 is differentiable at 𝑥 = 1

(D) If 𝑓𝑔 is differentiable at 𝑥 = 1, then 𝑔 is differentiable at 𝑥 = 1

Q5. 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 statements is/are ALWAYS TRUE?

(A) 𝑀 = 𝐼

(B) det 𝑀 = 1

(C) 𝑀2 = 𝐼 Mathematics Paper I Mathematics Paper II Final Answer Keys