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Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then the probability of event A is equal to

a. 4 / 9 b. 9 / 56 c. 3 / 7 d. 11 / 27 Answer: (a) Total case = 6∟6 Fav. case = (0, 1, 2, 3, 4, 5) + (0, 1, 2, 4, 5, 6) + (1, 2, 3, 4, 5,... View Article

Let the lengths of intercepts on x-axis and y-axis made by the circle x² + y² + ax + 2ay + c = 0, (a < 0) be 2 √2 and 2 √5, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to:

a. √10 b. √6 c. √11 d. √7 Answer: (b) 2 √(a2 / 4 - c) = 2√2 √(a2 - 4c) = 2√2 a2 - 4c = 8 --- (1) 2√(a2 - c) = 2√5 a2 - c = 5 --- (2)... View Article

Let f be a real-valued function, defined on R – {–1, 1} and given by f(x) = 3 log_e |(x – 1) / (x + 1)| – 2 / (x – 1). Then in which of the following intervals, function f (x) is increasing?

a. (- ∞, - 1) ⋃ ([1 / 2, ∞) - {1}) b. (- 1, 1 / 2] c. (–∞, ∞) – {–1, 1} d. (- ∞, 1 / 2] - {- 1} Answer: (a) f’ (x) = [(x + 1) / (x - 1) * (x... View Article

Let A (–1, 1), B (3, 4) and C(2, 0) be given three points. A line y = mx,

Let A (–1, 1), B (3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let... View Article

Consider the integral I = ∫₀¹⁰ {[x] e^[x]} / [e^x-1] dx, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

a. 45 (e – 1) b. 45 (e + 1) c. 9(e – 1) d. 9(e + 1) Answer: (a) I = ∫010 {[x] e[x]} / [ex-1] dx = ∫010 [x] e[x]+1-xdx = ∫12 e2-x dx + ∫23... View Article

If (x, y, z) be an arbitrary point lying on a plane P which passes through the points

If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of... View Article

If the foot of the perpendicular from point (4, 3, 8) on the line

If the foot of the perpendicular from point (4, 3, 8) on the line L1 : [x - a] / l = [y - 2] / 3 = [z - b] / 4, l ≠ 0 is (3, 5, 7), then the... View Article

If y = y (x) is the solution of the differential equation (dy / dx) + (tan x) y = sinx, 0 ≤ x ≤ π / 3, with y(0) = 0, then y (π / 4) equal to:

a. loge 2 b. (1 / 2) loge 2 c. (1 / 2 √2) loge 2 d. (1 / 4) loge 2 Answer: (c) I.F. = e∫tanx dx = eln secx = sec x Solution of the... View Article

Let f : S → S where S = (0, ∞) be a twice differentiable function such that f (x + 1) = x f (x). If g : S → R be defined as g (x) = log_ef (x), then the value of |g”(5) – g”(1)| is equal to:

a. 197 / 144 b. 187 / 144 c. 205 / 144 d. 1 Answer: (c) f(x+1) = xf(x) g(x+1) = loge(f(x+1)) g(x+1) = loge x + loge f(x) g(x+1) – g(x) =... View Article

The least value of |z| where z is a complex number which satisfies the inequality exp [{(|z| + 3) (|z| – 1)} / {||z| + 1}] log_e 2 ≥ log_√2 |5 √7 + 9i|, i = √-1, is equal to:

a. 2 b. 3 c. 8 d. √5 Answer: (b) 2[{(|z| + 3) (|z| - 1)} / {|z| + 1}] ≥ 23 ⇒ [{(|z| + 3) (|z| - 1)} / {||z| + 1}] ≥ 3 |z|2 + 2 |z| - 3 ≥ 3 |z| +... View Article

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Archives

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then the probability of event A is equal to

Let the lengths of intercepts on x-axis and y-axis made by the circle x² + y² + ax + 2ay + c = 0, (a < 0) be 2 √2 and 2 √5, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to:

Let f be a real-valued function, defined on R – {–1, 1} and given by f(x) = 3 log_e |(x – 1) / (x + 1)| – 2 / (x – 1). Then in which of the following intervals, function f (x) is increasing?

Let A (–1, 1), B (3, 4) and C(2, 0) be given three points. A line y = mx,

Consider the integral I = ∫₀¹⁰ {[x] e^[x]} / [e^x-1] dx, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

If (x, y, z) be an arbitrary point lying on a plane P which passes through the points

If the foot of the perpendicular from point (4, 3, 8) on the line

If y = y (x) is the solution of the differential equation (dy / dx) + (tan x) y = sinx, 0 ≤ x ≤ π / 3, with y(0) = 0, then y (π / 4) equal to:

Let f : S → S where S = (0, ∞) be a twice differentiable function such that f (x + 1) = x f (x). If g : S → R be defined as g (x) = log_ef (x), then the value of |g”(5) – g”(1)| is equal to:

The least value of |z| where z is a complex number which satisfies the inequality exp [{(|z| + 3) (|z| – 1)} / {||z| + 1}] log_e 2 ≥ log_√2 |5 √7 + 9i|, i = √-1, is equal to:

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