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Let X₁, X₂, …. X₁₈ be eighteen observation such that ∑_i=1¹⁸ (X_i – ɑ) = 36 and

Answer: 4 Solution: Given, ∑i=118 (Xi - ɑ) = 36 ∑xi - 18ɑ = 36 ∑xi = 18 (ɑ + 2) ---- (1) ∑i=118 (Xi - β)2 = 90 ∑xi2 + 18β2 - 2β ∑xi = 90... View Article

Let ɑ and β be two real numbers such that ɑ + β = 1 and ɑ β = – 1.

Let ɑ and β be two real numbers such that ɑ + β = 1 and ɑ β = - 1. Let Pn = (ɑ)n + (β)n, Pn-1 = 11 and Pn+1 = 29 for some... View Article

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is

Answer: 1000 Since the required number has G.C.D with 18 as 3. It must be an odd multiple of ‘3’ but not a multiple of... View Article

The triangle of the maximum area that can be inscribed in a given circle of radius ‘r’ is:

a. A right-angle triangle having two of its sides of length 2r and r.b. An equilateral triangle of height 2r / 3.c. isosceles triangle with base... View Article

Let f (x) = ∫₀^x e^t f (t) dt + e^x be a differentiable function for all x ∈ R. Then f (x) equals

a. 2eex-1 - 1b. e(e^x-1)c. 2ee^x - 1d. ee^x - 1Solution:Answer: (a)Given, f (x) = ∫0x et f (t) dt + ex ---- (1)f’ (x) = ex . f (x) + ex... View Article

Let f (x) = sin^-1 x and g (x) = [x² – x – 2] / [2x² – x – 6]. If g (2) = lim_x→2g (x), then then the domain of the function fog is:

a. (- ∞, - 2] ⋃ [- 4 / 3, ∞]b. (- ∞, - 1] ⋃ [2, ∞)c. (- ∞, - 2] ⋃ [- 1, ∞]d. (- ∞, 2] ⋃ [- 3 / 2,... View Article

A natural number has prime factorization given by n = 2^x3^y5^z, where..

A natural number has prime factorization given by n = 2x3y5z, where y and z are such that y + z = 5 and y-1 + z-1 = 5 / 6, y > z. Then the... View Article

Find the number of possible functions g

Let A = {1, 2, 3 ……. , 10} and f : A → A be defined as . Then the number of possible functions g : A → A such that gof =... View Article

Let f : R → R be defined as

Let f : R → R be defined as If f (x) is continuous on R, then a + b equals:a. 3b. -1c. -3d. 1Solution:Answer: (b)If f is continuous at x = -... View Article

For x > 0, if f (x) = ∫₁^x log_e t / [1 + t] dt, then f (e) + f (1 / e) is equal to:

a. 1 / 2b. -1c. 1d. 0Solution:Answer: (a)... View Article

If vectors a₁ = xi – j + k and a₂ = i + yj + zk are collinear, then a possible unit vector parallel to the vector xi + yj + zk is

a. (1 / √2) (- j + k)b. (1 / √2) (i - j)c. (1 / √3) (i - j + k)d. (1 / √3) (i + j - k)Solution:Answer: (3)(x / 1) = - 1 /... View Article

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

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