Maths

If a curve y=f(x), passing through the point (1,2), is the solution of the differential equation 2x²dy = (2xy+y²)dx, then f(1/2) is equal to

Let f: (-1, ∞) → R be defined by f(0) = 1 and f(x) = (1/x)log?(1+x), x≠0. Then the function f:

The imaginary part of (3 + 2√-54)¹/² - (3 - 2√-54)¹/² can be

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is

Consider the region R = {(x,y) ∈ R²: x² ≤ y ≤ 2x}. If a line y=α divides the area of region R into two equal parts, then which of the following is true?

For some θ ∈ (0, π/2), if the eccentricity of the hyperbola x² - y²sec²θ = 10 is √5 times the eccentricity of the ellipse x²sec²θ + y² = 5, then the length of the latus rectum of the ellipse is

lim(x→0) (tan(π/4 + x))¹/? is equal to

If the sum of first 11 terms of an A.P. a?, a?, a?, . is 0 (a? ≠ 0), then the sum of the A.P. a?, a?, a?, ., a? is ka?, where k is equal to

A plane passing through the point (3,1,1) contains two lines whose direction ratios are 1, -2, 2 and 2, 3, -1 respectively. If this plane also passes through the point (α, -3, 5), then α is equal to

Let f(x) be a quadratic polynomial such that f(-1) + f(2) = 0. If one of the roots of f(x)=0 is 3, then its other root lies in