Maths

For real numbers α and β ≠ 0, if the point of intersection of the straight lines (x-α)/1 = (y-1)/2 = (z-1)/3 and (x-4)/β = (y-6)/3 = (z-7)/3 lies on the plane x+2y-z = 8, then α - β is equal to:

The value of limₓ→₀ x/(√1-sinx - √1+sinx) is equal to :

A possible value of 'x', for which the ninth term in the expansion of {3ˡᵒᵍ₃√⁽²⁵⁽ˣ⁻¹⁾⁺⁷⁾ + 3⁻¹/ˡᵒᵍ₃⁽⁵⁽ˣ⁻¹⁾⁺¹⁾}¹⁰ in the increasing powers of 3⁻¹/ˡᵒᵍ₃⁽⁵⁽ˣ⁻¹⁾⁺¹⁾ is equal to 180, is :

Let C be the set of all complex numbers. Let S₁ = {z∈ C:|z-2|≤1} and S₂ = {z∈ C: z(1+i)+z̄(1-i) ≥ 4}. Then, the maximum value of |z - 5/2|² for z ∈ S₁ ∩ S₂ is equal to:

Let α = maxₓ∈ᵣ{8²sin³ˣ 4⁴cos³ˣ} and β = minₓ∈ᵣ{8²sin³ˣ 4⁴cos³ˣ}. If 8x² + bx + c = 0 is a quadratic equation whose roots are α¹/⁵ and β¹/⁵, then the value of c – b is equal to :

Let A and B be two 3 x 3 real matrices such that (A² - B²) is invertible matrix. If A⁵ = B⁵ and A³B² = A²B³, then the value of the determinant of the matrix A³ + B³ is equal to :

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept 6√5 on the x-axis. Then the radius of the circle C is equal to :

The point P(a,b) undergoes the following three transformations successively :
(a) reflection about the line y = x
(b) translation through 2 units along the positive direction of x-axis.
(c) rotation through angle π/4 about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point P are (-1/√2, 7/√2), then the value of 2a + b is equal to :

Kindly consider the following 

Let f: R → R be defined as f(x+y)+f(x-y) = 2f(x)f(y), f(1/2) = -1. Then, the value of Σ₂₀_(k=_1) 1/(sin(k)sin(k+f(k))) is equal to :