Maths

The value of ((1+sin(2π/9)+icos(2π/9))/(1+sin(2π/9)-icos(2π/9)))³ is

Let P(h,k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x - 3. Then the equation of the normal to the curve at P is

Let α > 0, β > 0 be such that α³ + β² = 4. If the maximum value of the term independent of x in the binomial expansion of (αx¹/⁹ + βx⁻¹/⁶)¹⁰ is 10k, then k is equal to

If |x| < 1, |y| < 1 and x ≠ y, then the sum to infinity of the following series (x+y) + (x²+xy+y²) + (x³+x²y+xy²+y³) + . is

If |x| < 1, |y| < 1 and x ≠ y, then the sum to infinity of the following series (x+y) + (x²+xy+y²) + (x³+x²y+xy²+y³) + . is

If |x| < 1, |y| < 1 and x ≠ y, then the sum to infinity of the following series (x+y) + (x²+xy+y²) + (x³+x²y+xy²+y³) + . is

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be 45°. After walking a distance of 80 meters towards the top, up a slope inclined at an angle of 30° to the horizontal plane, the angle of elevation of the top of the hill becomes 75°. Then the height of the hill (in metres) is [numerical value].

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD=8m, BC=11m and AB=10m; then the distance (in meters) of a point M on AB from the point A such that MD²+MC² is minimum is [numerical value].

If a and b are unit vectors, then the greatest value of √3|a+b|+|a-b| is [numerical value].

Let f:R→R be defined as f(x)={x⁵sin(1/x)+5x², x<0; 0, x=0; xcos(1/x)+x, x>0}. The value of λ for which f"(0) exists, is [numerical value].