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Consider three observations a, b and c such that b = a+c. If the standard deviation of a+2, b+2, c+2 is d, then which of the following is true?

a. b2 = a2 + c2 + 3d2 b. b2 = 3 (a2 + c2) - 9d2 c. b2 = 3 (a2 + c2) + 9d2 d. b2 = 3 (a2 + c2 + d2) Answer: (b) For a, b, c mean = = [a + b + c]... View Article

Let e denote the base of the natural logarithm.

Let e denote the base of the natural logarithm. The value of the real number a for which the right-hand limit is equal to a nonzero real number,... View Article

For a polynomial g( ) with real coefficients

For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with... View Article

In a triangle PQR, let [latex]vec{a} = vec{QR}, vec{b} = vec{RP}: and : vec{c} = vec{PQ}[/latex].

In a triangle PQR, let . If Then the value of is _____ Answer: 108.00... View Article

Let f : [0, 2] ⟶ R be the function defined by

Let f : [0, 2] ⟶ R be the function defined by If , ∈ [0,2] are such that { ∈ [0, 2] ∶ f( ) ≥ 0} = [ , ], then the value... View Article

Let ₁, ₂, ₃,… be a sequence of positive integers in arithmetic progression

Let 1, 2, 3,… be a sequence of positive integers in arithmetic progression with common difference 2. Also, let 1, b2, 3,… be a... View Article

Let m be the minimum possible value of [latex]log_{3}(3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}})[/latex],

Let m be the minimum possible value of , where 1, 2, 3 are real numbers for which 1 + 2 + 3 = 9. Let M be the maximum possible value of... View Article

Which of the following inequalities is/are TRUE?

Which of the following inequalities is/are TRUE? a) b) c) d) Answer: a, b, d... View Article

Let ₁ and ₂ be the following straight lines.

Let 𝐿1 and 𝐿2 be the following straight lines. Suppose the straight line is Lies in the plane containing 𝐿1 and 𝐿2, and passes through the... View Article

Let x, and z be positive real numbers.

Let x, 𝑦 and z be positive real numbers. Suppose 𝑥, 𝑦 and z are the lengths of the sides of a triangle opposite to its angles 𝑋, 𝑌 and Z,... View Article

Let S be the set of all complex numbers z satisfying

Let S be the set of all complex numbers z satisfying | 2 + + 1| = 1. Then which of the following statements is/are TRUE? Answer: b, c... View Article

Let M be a 3 × 3 invertible matrix with real entries

Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If −1 = adj (adj ), then... View Article

Let the function : R → R be defined by ( ) = ³− ²+ ( − 1) sin

Let the function : R → R be defined by ( ) = 3 − 2 + ( − 1) sin and let : R → R be an arbitrary function. Let g : R... View Article

Consider all rectangles lying in the region

Consider all rectangles lying in the region and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among... View Article

Let C₁ and C₂ be two biased coins such that

Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are , respectively. Suppose α is the... View Article

Let , and λ be positive real numbers.

Let , and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola 2 = 4λ , and suppose the... View Article

Let the functions: R ⟶ R and g : R ⟶ R be defined by

Let the functions: R ⟶ R and g : R ⟶ R be defined by Then, the area of the region in the first quadrant bounded by the curves = ( ), = g( )... View Article

If the function : R ⟶ R is defined by ( ) = | | ( – sin ),

If the function : R ⟶ R is defined by ( ) = | | ( – sin ), then which of the following statements is TRUE? a) is one-one, but NOT onto... View Article

Suppose a, b denote the distinct real roots of the quadratic polynomial

Suppose , denote the distinct real roots of the quadratic polynomial 2 + 20 − 2020 and suppose , denote the distinct complex roots... View Article

Let the function f: (0, π) → R be defined by f(θ) = (sin θ + cos θ)² + (sinθ – cos θ)⁴ .

Let the function f: (0, π) → R be defined by f(θ) = (sin θ + cos θ)2 + (sinθ - cos θ)4 . Suppose, the... View Article

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