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If variance of first n natural numbers is 10 and variance of first m even natural numbers is 16, m+n is equal to

Solution: For natural number variance is given by σ2 = (n2-1)/12 = 10 n = 11 Variance of (2, 4, 6…) =... View Article

[latex]\lim_{x\to 2}\frac{3^{x}+3^{3-x}-12}{3^{-\frac{x}{2}}-3^{1-x}}[/latex] is equal to

Solution: Put 3x/2 = t The answer is 36.... View Article

If the system of linear equations

If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c ∈ R are non-zero and distinct;... View Article

Five numbers are in A.P., whose sum is 25 and product is 2520.

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is -1/2, then the greatest number amongst them is a)... View Article

If y = y(x) is the solution of the differential equation e^y(dy/dx)-1 = e^x such that y(0) = 0, then y(1) is equal to

a) loge 2 b) 2e c) 2 + loge 2 d) 1+loge 2 Solution: (d) y( ′ − 1) = ⇒ dy/dx = − + 1 Let x-y = t 1 - dy/dx = dt/dx... View Article

Let the function, f:[-7, 0] →R be continuous on [−7, 0] and differentiable on (−7, 0).

Let the function, f:[-7, 0] →R be continuous on [−7, 0] and differentiable on (−7, 0). If f(-7)= −3 and ′( ) ≤... View Article

Let x^k + y^k = a^k, (a,k>0) and (dy/dx) + (y/x)^1/3 = 0, then k is

a) 1/3 b) 3/2 c) 2/3 d) 4/3 Answer: (c) xk + yk = ak kxk-1 + kyk-1 (dy/dx) = 0 dy/dx = -(x/y)k-1 dy/dx + (y/x)1-k = 0 1-k = 1/3 k = 2/3... View Article

Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is:

a) (6, 5, 2) b) (6, 5, −2) c) (4, 3, 2) d) (3, 4, −2) Answer: (b) Points A(2, 1, 0), B(4, 1, 1) C(5, 0, 1) Equation of... View Article

Let and are two real roots of the equation

Let and are two real roots of the equation ( + 1) tan2 − √2 tan = 1 − , where ( ≠ −1) and are real... View Article

If g(x) = x² + x – 1 and (gof)(x) = 4x² – 10x + 5, then f(5/4) is equal to

a) -3/2 b) -1/2 c) 1/2 d) 3/2 Answer: (b) g(x) = x2 + x - 1 gof(x) = 4x2 - 10x + 5 g(f(x) = 4x2 - 10x + 5 f2(x) + f(x) - 1 = 4x2 - 10x + 5... View Article

Let be a root of the equation x²+ x+ 1= 0

Let be a root of the equation x2 + x+ 1= 0 and the matrix A = then the matrix A31 is equal to a) A b) A2 c) A3 d) I3 Answer: (c) The... View Article

If y = mx + 4 is a tangent to both the parabolas

If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to a) -64 b) 128 c) -128 d) -32 Answer: (c) Any... View Article

If y(α) = [latex]sqrt{2(frac{tan alpha + cot alpha }{1+

a) -1/4 b) 4/3 c) 4 d) -4 Answer: (c) y(α) = y(α) = √(1+cotα )2 y(α) = -1 - cotα dy/dα... View Article

A vector ⃗ = ̂ + 2 ̂ + ̂ ( , ∈ ) lies in the plane of the vectors

A vector 𝑎⃗ = 𝛼𝑖̂ + 2𝑗̂ + 𝛽𝑘̂ (𝛼, 𝛽 ∈ 𝑹) lies in the plane of the vectors, ⃗𝑏 = 𝑖̂ + 𝑗̂ and 𝑐⃗ = 𝑖̂ − 𝑗̂ + 4𝑘̂. If 𝑎⃗ bisects the... View Article

The greatest positive integer , for which 49 + 1 is a factor of the sum 49¹²⁵ + 49¹²⁴ + ⋯ + 49² + 49 + 1, is

a) 32 b) 60 c) 65 d) 63 Answer: (d) 1 + 49 + 492 + …. + 49125 = (49126 - 1)/(49 - 1) = (4963 + 1)( 4963 - 1)/48 = (4963 + 1)((1 +... View Article

The logical statement (p⇒q) ∧ (q ⇒∼ p) is equivalent to

a) ∼ p b) p c) q d) ∼ q Answer: (a) 𝒑 𝒒 𝒑 ⇒ 𝒒 ∼ 𝒑 𝒒 ⇒∼ 𝒑 (𝒑 ⇒ 𝒒) ∧ (𝒒 ⇒∼ 𝒑) T T... View Article

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is

a) 2√3 b) √3 c) 3/√2 d) 3√2 Answer: (d) Let the equation of ellipse be x2/a2 + y2/b2 = 1 a>b Now 2ae = 6 2a/e... View Article

If f(a + b + 1 – x) = f(x) ∀x, where a and b are fixed positive real numbers

If f(a + b + 1 - x) = f(x) ∀x, where a and b are fixed positive real numbers, then is equal to a) b) c) d) Answer: (c) f(a... View Article

If Re (z-1)/(2z + i) = 1, where z = x+iy, then the point (x,y) lies on a

a) circle whose centre is at (-1/2, -3/2) b) straight line whose slope is 3/2. c) circle whose diameter is √5/2 d) straight line whose... View Article

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