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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

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: 0 Roots of x2 + x + 1 are ω and ω2 now Q (ω) = 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 = b R ≡ [- 1, 4, (a + 2) /... 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) –... 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... 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)10 General term, Tr+1 =... 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) =... 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\(\begin{array}{l}\\\vec{n}\equiv \vec{b_{1}}\times\vec{b_{2}}\\ \vec{n}=\begin{vmatrix} i &j &k \\ 3 &4 &2 \\ 4&-3 &7 \end{vmatrix}\end{array} \)... 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/8 b. π/6 c. π/3 d. π/4 Solution: Answer: (b) Equation of tangent [x/3√3] cos θ + y sin... 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]2 b. [5e1/2]/[e2 + 1]2 c. – 2e2/(1 + e2)2 d. [-e3/2]/[e2 + 1]2 Solution: Answer: (d)... View Article

Let f : R → R be a function defined as

Let f : R → R be a function defined as \(\begin{array}{l}f(x)=\left\{\begin{matrix} [sin(a+1)x+sin2x]/2x, &x<0 & \\ b,& x=0 & \\... 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... View Article

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

a. ((P ⇒ Q) ^ ~Q) ⇒ P b. ((P ⇒ Q) ^ ~ Q) ⇒ ~ P c. ((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 transitive b. R is symmetric, transitive but not reflexive c. R is an... View Article

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

a. 3π/2 b. π/16 c. π / 8 d. 3π/8 Solution: Answer: (a) The domain of 4y2 = x2 (4 –... 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. 1/2 b. 4 c. 2 d. 1/4 Solution: Answer: (a) w = 1 – √3i |w| = 2 |zw|... 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) \(\begin{array}{l}\\|\vec{a}\times\vec{b}|=|\vec{a}|=|\vec{b}|\\ cos\Theta =\frac{\vec{a}\times(\vec{a}+\vec{b}+(\vec{a}\times\vec{b}))}{|\vec{a}|.|\vec{a}+\vec{b}+\vec{a}\times\vec{b}|}\\ |\vec{a}|=a\\ cos\Theta... 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. 80/243 b. 32/625 c. 128/625 d. 40/243 Solution: Answer: (b) 5C1 p1 q4 = 0.4096 — (1) 5C2 p2... 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. 25/4 b. 85/14 c. 127/20 d. 115/16 Solution: Answer: (b) Projection of AB on AC is = AB cos... 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√2 b. 3√2 c. 7√2 d. 9/√2 Solution: Answer: (d) r = |(0 + 0 – 3)/√2| = 3/√2... View Article

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