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Let I be an identity matrix of order 2 times 2 and P is a matrix. Then the value of n belongs to N for which Pn = 5I – 8P is equal to

Solution: Answer: (6) P6 = 5I - 8P Thus, n = 6... View Article

If f (x) and g (x) are two polynomials such that the polynomial P (x) = f (x³) + x g (x³) is divisible by x² + x + 1, then P (1) is equal to

Solution:Answer: 0Roots of x2 + x + 1 are ω and ω2 nowQ (ω) = f (1) + ω g (1) = 0 …(1)Q (ω2) = f (1) +... View Article

Let the mirror image of the point (1, 3, a) with respect to the plane r . (2i – j + k) – b = 0 be (-3, 5, 2). Then, the value of |a + b| is equal to

Solution:Answer: (1)Plane : 2x -y + z = bR ≡ [- 1, 4, (a + 2) / 2] on plane- 2 - 4 + (a + 2) / 2 = b⇒ a + 2 = 2b + 12 ⇒ a = 2b +... View Article

Let P (x) be a real polynomial of degree 3 which vanishes at x = – 3. Let P(x) have local minima at x = 1, local maxima at x = -1 and ∫_-1¹ P (x) dx = 18, then the sum of all the coefficients of the polynomial P (x) is equal to

Solution:Answer: (8)P’ (x) = a (x + 1) (x - 1)P (x) = (ax3 / 3) - ax + cP (-3) = 0 (given)⇒ a(- 9 + 3) + C = 0⇒ 6a = C... View Article

Let ⁿC_r denote the binomial coefficient of x^r in the expansion of (1 + x)ⁿ. If ∑_k=0¹⁰ [2² + 3k] ⁿC_k = ɑ . 3¹⁰ + β . 2¹⁰ then ɑ + β is equal to

Solution:n must be equal to 10∑k=010 [22 + 3k] nCk= ∑k=010 [4 + 3k] nCk= 4 ∑k=010nCk + 3 ∑k=010 k nCk= 4 (210) + 3 * 10 * 29= 19... View Article

The term independent of x in the expansion of [(x + 1) / (x^2/3 – x^1/3 + 1) – (x – 1) / (x – x^1/2)]¹⁰, x ≠ 1, is equal to

Solution:Answer: (210)[(x1/3 + 1) - [√x + 1] / [√x]]10 = (x1/3 - x-½)10General term, Tr+1 = 10Cr (x1/3)10-r (- x-½)rFor... View Article

∑_r=1¹⁰ r! (r³ + 6r² + 2r + 5) = α (11!). Then the value of α is equal to

Solution:Answer: (160)Tr = r! ((r + 1) (r + 2) (r + 3) - 9r - 1)= (r + 3) ! - 9r . r! - r!= (r + 3) ! - 9 (r - 1 + 1)) r! - r!= (r + 3) ! - 9 (r... View Article

Let P be a plane containing the line [x – 1] / 3 = [y + 6] / 4 = [z + 5] / 2 and parallel to the line [x – 3] / 4 = [y – 2] / – 3 = [z + 5] / 7. If the point (1, -1, α) lies on the plane P, then the value of |5α| is equal to

Solution:Answer: (38)DR’s of normal (34, -13, -25)P ≡ 34 (x - 1) - 13 (y + 6) - 25 (z + 5)Q (1, -1, α) lies on P.⇒ 3 (1 -... View Article

Let a tangent be drawn to the ellipse (x² / 27) + y² = 1 at (3√3 cos θ, sin θ) where θ ∈ (0, π / 2). Then the value of θ such that the sum of intercepts on axes made by a tangent is minimum is equal to:

a. π/8b. π/6c. π/3d. π/4Solution:Answer: (b)Equation of tangent [x/3√3] cos θ + y sin θ = 1A [(3√3/cos... View Article

Let y = y (x) be the solution of the differential equation dy / dx = (y + 1) [(y + 1) e^{x^2/2} – x], 0 < x < 2.1, with y (2) = 0. Then the value of dy / dx at x = 1 is equal to:

a. [e5/2]/[1 + e2]2b. [5e1/2]/[e2 + 1]2c. - 2e2/(1 + e2)2d. [-e3/2]/[e2 + 1]2Solution:Answer: (d)dy/dx = (y + 1) [(y + 1) ex^2/2 - x]⇒ [-... View Article

Let f : R → R be a function defined as

Let f : R → R be a function defined as . If f is continuous at x = 0, then the value of a + b is equal toa. - 2b. - 2/5c. - 3/2d. -... View Article

Consider a hyperbola H : x² – 2y² = 4……

Consider a hyperbola H : x2 - 2y2 = 4. Let the tangent at a point P (4, √6) meet the x-axis at Q and latus rectum at R (x1, y1), x1 > 0.... View Article

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