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If the variance of 10 natural numbers 1,1,1,…,1,k is less than 10, then the maximum possible value of k is

⇒ 10(9 + k2) - (81 + k2 + 18k) < 1000⇒ 90 + 10k2 - k2 - 18k - 81 < 1000⇒ 9k2- 18k + 9 < 1000⇒ (k - 1)2 <... View Article

The students S₁, S₂,… S₁₀ are to be divided into 3 groups A, B, and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is

The students S1, S2,... S10 are to be divided into 3 groups A, B, and C such that each group has at least one student and the group C has at most... View Article

The number of the real roots of the equation (x + 1)² + |x – 5| = 27/4 is

For x ≥ 5,(x + 1)2+ (x - 5) = 27/4⇒ x2 + 3x - 4 = 27/4⇒ x2 + 3x - 43/4 = 0⇒ 4x2 + 12x - 43 = 0x = (-12 ± √(144 +... View Article

The area of the region: R{(x, y): 5x² ≤ y ≤ 2x² + 9} is:

a. 9√3 square unitsb. 12√3 square unitsc. 11√3 square units/d. 6√3 square unitsAnswer: (b)Required area= = 12√3... View Article

If a curve y = f(x) passes through the point (1, 2) and satisfies xdy/dx + y = bx⁴, then for what value of b, [latex]int_{1}^{2}f(x)dx = frac{62}{5}[/latex]

a. 5b. 62/5c. 31/5d. 10Answer: (d)dy/dx + y/x = bx3I.F. = ⇒ (b/25) ×32 + c ln 2 - b/25 = 62/5⇒ c = 0 and b = 10... View Article

If n ≥ 2 is a positive integer, then the sum of the series ⁿ⁺¹C₂ + 2(²C₂ + ³C₂ + ⁴C₂ + …. + ⁿC₂) is

a. n(n + 1)2 (n + 2)/12b. n(n - 1)(2n + 1)/6c. n(n + 1)(2n + 1)/6d. n(2n + 1)(3n + 1)/6Answer: (c)2C2 = 3C3Let S = 3C3 + 3C2 + …….... View Article

Let A and B be 3×3 real matrices such that A is symmetric matrix and B is

Let A and B be 3×3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 −... View Article

For the statements p and q, consider the following compound statements:

(a) (~q∧(p→q)) → ~p(b) ((p∨q))∧~p) → qThen which of the following statements is correct?a. (a) is a tautology but not... View Article

The angle of elevation of a jet plane from a point A on the ground….

The angle of elevation of a jet plane from a point A on the ground is 600. After a flight of 20 seconds at the speed of 432 km/hour, the angle of... View Article

For which of the following curves, the line x + √3y = 2√3 is the tangent at the point (3√3/2, 1/2)?

a. x2 + 9y2 = 9b. 2x2 - 18y2 = 9c. y2 = x/(6√3)d. x2 + y2 = 7Answer: (a)Tangent to x2 + 9y2 = 9 at point (3√3/2, 1/2) is... View Article

For the system of linear equations:x − 2y = 1, x − y + kz = −2, ky + 4z = 6,k ∈ R

For the system of linear equations: x − 2y = 1, x − y + kz = −2, ky + 4z = 6,k ∈ R Consider the following... View Article

The negation of the statement ~p∧(p∨q) is∶

a. ~p∧qb. p∧~qc. ~p∨qd. p∨∼qAnswer: (d)Negation of ~p∧(p∨q) is∼[~p∧(p∨q)]≡ p∨ ∼(p∨q)≡... View Article

If the curve y = ax²+ bx + c,x ∈ R passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are:

a. a = 1, b = 1, c = 0b. a = -1, b = 1, c = 1c. a = 1, b = 0, c = 1d. a = 1/2, b = 1/2 ,c = 1Answer: (a)2 = a + b + cdy/dx = 2ax + b,... View Article

Let f: R → R be defined as

Let f: R → R be defined as Let A = {x ∈ R ∶ f is increasing}. Then A is equal to :a. (-5, -4) ∪ (4,∞)b. (-5,... View Article

The value of the integral, [latex]\int_{1}^{3}\left [ x^{2}-2x-2 \right ]dx[/latex] where [x] denotes the greatest integer less than or equal to x , is :

a. –4b. –5c. -√2 - √3 - 1d. -√2 - √3 + 1Answer: (c) I = -6 + (√2 - 1) + 2√3 - 2√2 + 6 -... View Article

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax²+ bx + 1 = 0, then the value of α² + β²– αβ is:

a. 71/256b. -69/256c. 69/256d. -71/256Answer: (d)2b = a + c(2a + 2)/3 = 10/3 and (2b + c)/3 = 7/3⇒ a = 42b + c = 7, 2b - c = 4 },solvingb =... View Article

The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is:

a. 65/27b. 135/29c. 65/28d. 35/27Answer: (b)Let A and B be two subsets.For each x∈{1,2,3,4,5} , there are four possibilities:x ∈... View Article

A possible value of tan (¼ sin^-1 √63/8) is:

a. 1/(2√2)b. 1/√7c. √7 - 1d. 2√2 - 1Answer: (b)tan (¼ sin-1 √63/8)Let sin-1(√63/8) = θsin... View Article

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x∈R. If |f(x) f'(x) f'(x) f”(x)| = 0, for all x∈R, then the value of f(1) lies in the interval:

a. (9, 12)b. (6, 9)c. (3, 6)d. (0, 3)Answer: (b)Given f(x) f''(x) - f'(x)2 =0Let h(x) = f(x)/f'(x)Then h'(x) = 0⇒ h(x) = k⇒ f(x)/f'(x)... View Article

The lanthanoid that does NOT shows +4 oxidation state is

1) Dy 2) Ce 3) Tb 4) Eu Answer: 4 Fact Eu does not show +4 oxidation state. Hence option 4 is the answer.... View Article