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Let f: R → R and g: R → R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y)

Let f: R → R and g: R → R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) For all x, y ∈ R. If... View Article

Let a and b be positive real numbers such that a > 1 and b < a.

Let a and b be positive real numbers such that a > 1 and b < . Let P be a point in the first quadrant that lies on the hyperbola .... View Article

Let b be a nonzero real number.

Let b be a nonzero real number. Suppose f = R → R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the... View Article

The value of the limit

The value of the limit Answer: 8 Solution: Using transformation and submultiple angle formula in denominator... View Article

Let the functions f: (−1,1) → R and g: (−1, 1) → (−1, 1) be defined by f(x) = |2x − 1| + |2x + 1|and g (x) = x − [x]

Let the functions f: (−1,1) → R and g: (−1, 1) → (−1, 1) be defined by f(x) = |2x − 1| + |2x + 1|and g (x) = x... View Article

The trace of a square matrix is defined to be the sum of its diagonal entries.

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 x 2 matrix such that the trace of A is 3 and the trace... View Article

Let O be the centre of the circle x² + y² = r², where r > √5/2 .

Let O be the centre of the circle x2 + y2 = r2, where r > √5/2 . Suppose PQ is a chord of this circle and the equation of the line... View Article

The probability that a missile hits a target successfully is 0.75.

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are... View Article

For a complex number z, let Re(z) denote the real part of z.

For a complex number z, let Re(𝑧) denote the real part of z. Let S be the set of all complex numbers z satisfying 𝑧4 − |𝑧|4 = 4 𝑖𝑧2, where... View Article

Let A(1,0), B(6,2), C(3/2, 6) be the vertices of a triangle ABC.

Let A(1,0), B(6,2), C(3/2, 6) be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangle APC, APB and BPC... View Article

Let S be the set of points where the function

Let S be the set of points where the function, (𝑥) = |2 − |𝑥 − 3||, 𝑥 ∈ 𝐑, is not differentiable. Then, the value of... View Article

If the sum of the coefficients of all even powers of x

If the sum of the coefficients of all even powers of in the product (1 + x + x2 + x3 … . +x2 )(1 − x + x2 − x3 … . +... View Article

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

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