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If P and Q are two statements, then which of the following compound statement is a tautology?

a. ((P ⇒ Q) ^ ~Q) ⇒ Pb. ((P ⇒ Q) ^ ~ Q) ⇒ ~ Pc. ((P ⇒ Q) ^ ~ Q)d. ((P ⇒ Q) ^ ~ Q) ⇒ QSolution:Answer: (b)(P... View Article

Define a relation R over a class of n × n real matrices A and B as “ARB iff there exists a non-singular matrix P such that PAP^-1 = B”. Then which of the following is true?

a. R is reflexive, symmetric but not transitiveb. R is symmetric, transitive but not reflexivec. R is an equivalence relationd. R is reflexive,... View Article

The area bounded by the curve 4y² = x² (4 – x) (x – 2) is equal to:

a. 3π/2b. π/16c. π / 8d. 3π/8Solution:Answer: (a)The domain of 4y2 = x2 (4 - x) (x - 2) is [2, 4] ⋃ {0}Area of loop = 2 * (1/2)... View Article

Let a complex number be w = 1 – √3i. Let another complex number z be such that |zw| = 1 and arg (z) – arg (w) = π / 2. Then the area of the triangle with vertices origin, z and w is equal to:

a. 1/2b. 4c. 2d. 1/4Solution:Answer: (a)w = 1 - √3i|w| = 2|zw| = 1 ⇒ |z| = 1/|w| = 1/2arg (z) - arg (w) = π / 2Area of... View Article

Let a and b be two non-zero vectors perpendicular to each other and |a| = |b|. If |a x b| = |a|, then the angle between the vectors (a + b + (a x b)) and a is equal to

a. sin-1 (1/√3) b. cos-1 (1/√3) c. sin-1 (1/√6) d. cos-1 (1/√2) Solution: Answer: (b)... View Article

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to:

a. 80/243b. 32/625c. 128/625d. 40/243Solution:Answer: (b)5C1 p1 q4 = 0.4096 --- (1)5C2 p2 q3 = 0.2048 --- (2)(1)/(2) ⇒ q/2p = 2 ⇒ q =... View Article

In a triangle ABC, if vector BC = 8, CA = 7, AB = 10, then the projection of the vector AB on AC is equal to:

a. 25/4b. 85/14c. 127/20d. 115/16Solution:Answer: (b)Projection of AB on AC is = AB cos A = 10 cos ABy cosine rulecos A = [102 + 72 - 82]/[2 . 10... View Article

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to

a. 2√2b. 3√2c. 7√2d. 9/√2Solution:Answer: (d)r = |(0 + 0 - 3)/√2| = 3/√2sin 30° = r/R = 1/2R = 2rSo, r +... View Article

Let S₁ = x² + y² = 9 and S₂ = (x – 2)² + y² = 1

Let S1 = x2 + y2 = 9 and S2 = (x - 2)2 + y2 = 1. Then the locus of the centre of a variable circle S which touches S1 internally and S2... View Article

Let S₁ be the sum of the first 2n terms of an arithmetic progression

Let S1 be the sum of the first 2n terms of an arithmetic progression. Let S2 be the sum of the first 4n terms of the same arithmetic progression.... View Article

Let f : R – {3} → R – {1] be defined by f (x) = (x – 2) / (x – 3). Let g : R – R be given as g (x) = 2x – 3. Then, the sum of all the values of x for which f^-1 (x) + g^-1 (x) = 13 / 2 is equal to

a. 7 b. 5 c. 2 d. 3 Solution: Answer: (b) f-1 (x) + g-1 (x) = 13/2 ⇒ [3x - 2]/[x - 1] + [x + 3]/2 = 13/2 ⇒ 2 (3x - 2) + (x - 1) (x + 3) =... View Article

If 15 sin⁴ θ + 10 cos⁴ θ = 6, for some θ ∈ R, then the value of 27 sec⁶ θ + 8 cosec⁶ θ is equal to:

a. 250b. 500c. 400d. 350Solution:Answer: (a)15 sin4 θ + 10 cos4 θ = 6⇒ 15 sin4 θ + 10 (1 - sin2 θ)2 = 6⇒ 25... View Article

Let g (x) =∫ ₀^t f (t) dt where f is a continuous function

Let g (x) =∫ 0t f (t) dt where f is a continuous function in [0, 3] such that (1/3) ≤ f (t) ≤ 1 for all t ∈ [0, 1] and 0... View Article

Let in a series of 2n observations, half of them are equal to a and the remaining half are equal to -a.

Let in a series of 2n observations, half of them are equal to a and the remaining half are equal to -a. Also by adding a constant b in each of... View Article

A pole stands vertically inside a triangular park ABC. Let the angle of elevation

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be... View Article

Let the system of linear equations has a non-trivial solution. Then which of the following is true

Let the system of linear equations 4x + λy + 2z = 0 2x - y + z = 0 μx + 2y + 3z = 0, λ, μ ∈ R Has a non-trivial solution. Then which... View Article

The sum of fraction is equal to

Solution: , , , = 504 Answer: (504)... View Article

The number of 4 letter words (with or without meaning) that can be made from the eleven letters of the word “EXAMINATION” is

Solution: Word “EXAMINATION” consists of 2 , 2 , 2 , , , , , Case I: All different letters are selected Number of words formed = 8 4 × 4! =... View Article

tan( + 2 ) is equal to

, α, β ∈ (0, π/2 ), then tan(α + 2β) is equal to _________ . Solution: => tan α = 1/7 => (√2 sinβ) /√2 = 1 √10 => sinβ = 1... View Article

Let f( ) be a polynomial of degree 3 such that (−1) = 10, (1) = −6, ( ) has a critical point at = −1 and ′( ) has a critical point at = 1. Then the local minima at =

Solution: Let the polynomial be f( ) = 3 + 2 + + ⇒ ′ ( ) = 3 2 + 2 + ⇒ ′′( ) = 6 + 2 ′′(1) = 0 ⇒ 6 + 2 = 0 ⇒ = −3... View Article