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A seven-digit number is formed using the digit 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is:

a. 6 / 7b. 4 / 7c. 3 / 7d. 1 / 7Solution:Answer: (3)n (S) = 7! / [2! 3! 2!]n (E) = 6! / [2! 2! 2!]P (E) = n (E) / n (S) = [6! / 7!] * [[2! 3! 2!]... View Article

Consider the following system of equations:

Question 10: Consider the following system of equations: x + 2y - 3z = a2x + 6y - 11 z = bx - 2y + 7z = c,Where a, b and c are real constants.... View Article

Let F₁ (A, B, C) = (A ∧ ~B) ∨ [~C ∧….

Question 9: Let F1 (A, B, C) = (A ∧ ~B) ∨ [~C ∧ (A ∨ B)] ∨ ~A and F2 (A, B) = (A ∨ B) ∨ (B → ~A) be two logical... View Article

If 0 < a, b < 1, and tan^-1 a + tan^-1 b = π / 4, then the value of (a + b) – [a² + b²] / 2 + [a³ + b³ ] / 3 – [a⁴ + b⁴] / 4 + …. is

a. loge2b. loge (e / 2)c. ed. e2 - 1Solution:Answer: (a)tan-1 {[a + b] / [1 - ab]} = (π / 4)a + b = 1 - ab(1 + a) (1 + b) = 2(a + b) - [a2 +... View Article

Let A₁ be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant.

Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the... View Article

Let the slope of the tangent line to a curve at any point P (x, y) be given by [xy² + y] / x. If the curve intersects the line x + 2y = 4 at x = – 2, then the value of y, for which the point (3, y) lies on the curve, is:

a. - 18 / 11b. - 18 / 19c. - 4 / 3d. 18 / 35Solution:Answer: (b)dy / dx = [xy2 + y] / x[x dy - y dx] / y2 = x dx- d (x / y) = d (x2 / 2)(- x / y)... View Article

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x²+ y²= 1 is a circle of the radius r, then r is equal to

a. 14b. 12c. 1d. 13Solution:Answer: (b)P ≡ (2h - 3), (2k - 2) → on circle[h - (3 / 2)]2 + (k - 1)2 = 1 / 4Radius = 1 / 2... View Article

Let A (1, 4) and B (1, – 5) be two points. Let P be a point on the circle (x – 1)² + (y – 1)² = 1 such that (PA)² + (PB)² have maximum value, then the points P, A and B lie on

a. a parabolab. a straight linec. a hyperbolad. an ellipseSolution:Answer: (b)PA2 = cos2 θ + (sin θ - 3)2 = 10 - 6 sin θPB2 =... View Article

Let f(x) be a differentiable function at x = a with f’ (a) = 2 and f (a) = 4. Then lim_x→a [x f (a) – a f (x)] / [x – a] equals:

a. 2a + 4b. 2a - 4c. 4 - 2ad. a + 4Solution:Answer: (c)By L-H ruleL = limx→a [f (a) - a f’ (x)] / [1]L = 4 - 2a... View Article

The sum of the series ∑_n=1^∞∞[n²+ 6n + 10] / (2n + 1)! is equal to:

a. (41/ 8) e + (19 / 8) e-1 - 10b. (- 41/ 8) e + (19 / 8) e-1 - 10c. (41/ 8) e - (19 / 8) e-1 - 10d. (41/ 8) e + (19 / 8) e-1 +... View Article

Let L be a line obtained from the intersection of two planes

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P (ɑ, β, γ) is the foot of the... View Article

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is

a) 3√(5/2)b) 3√5/2c) √10d) 2√10Solution:As we know centroid divides line joining circumcentre and orthocentre internally 1:2.So, C(6,2)AC =... View Article

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series.

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series. 12 +2.22+32+2.42+52 + 2.62+..If B-2A = 100λ,... View Article

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is

a) at least 500 but less than 750b) at least 750 but less than 1000c) at least 1000d) less than 500Solution:We have 6 novels and 3 dictionaries.... View Article

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR…

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at... View Article

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x+y+z = 7 is

a) 1/3b) √(2/3)c) 2/√3d) 2/3Solution:Direction ratios of AB = (1,0,1)Let be angle between line AB and normal of plane.So cos = 1×1+0×1+1×1)/√2√3=... View Article

Let u be a vector coplanar with the vectors ….

Let u be a vector coplanar with the vectors Answer: (c)... View Article

Let y = y(x) be the solution of the differential equation sin x(dy/dx)+y cosx = 4x, x∈(0,π). If y(π/2) = 0, then y(π/6) is equal to

a) (-8/9)π2b) (-4/9)π2c) (4/9√3)π2d) (-8/9√3)π2Solution:(dy/dx)+y cotx = 4x/sin xIF = e∫cot xdx = elog|sin x| = sin x as x∈(0,π)y... View Article

Let S = {t∈R: f(x) = |x-π|(e^|x|-1)sin x is not differentiable at t}. Then the set S is equal to:

a) {π}b) {0,π}c) (an empty set)d) {0}Solution:f(x) = |x-π|(e|x|-1) sinxObviously differentiable at x = 0Check at x = π So,... View Article

Solve the integral of fraction – Step by Step Solution

a) 1/(1+cot3x) + C b) -1/(1+cot3x) + C c) 1/3(1+tan3x) + C d) -1/3(1+tan3x) + C where C is the constant of integration. Solution: Answer: (d)... View Article

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