Maths

. In ?ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of ?ABC is 30 cm² and R and r are respectively the radii of circumcircle and incircle of ?ABC, then the value of 2R + r (in cm) is equal to.

Let 1/16, a and b be in G.P. and 1/a, 1/b, 6 be in A.P., where a, b > 0. Then 72(a+b) is equal to.

Let n be a positive integer. Let A = ∑_{k=0 to n} (-1)ᵏ ⁿCₖ [ (1/2)ᵏ + (3/4)ᵏ + (7/8)ᵏ + (15/16)ᵏ + (31/32)ᵏ ]. If 63A = 1 - 1/2³⁰, then n is equal to.

If the distance of the point (1,-2,3) from the plane x+2y-3z+10=0 measured parallel to the line, (x-1)/3 = (2-y)/m = (z+3)/1 is √(7/2), then the value of |m| is equal to.

Let A = [a₁; a₂] and B = [b₁; b₂] be two 2x1 matrices with real entries such that A = XB, where X = (1/√3)[1 -1; 1 k] and k ∈ R. If a₁²+a₂² = (2/3)(b₁²+b₂²) and (k²+1)b₂² ≠ -2b₁b₂, then the value of k is.

For real numbers α, β, γ and δ, if ∫ [ (x²-1) + tan⁻¹((x²+1)/x) ] / [ (x⁴+3x²+1)tan⁻¹((x²-1)/x) ] dx = αlogₑ|tan⁻¹((x²+1)/x)| + βtan⁻¹((γ(x²-1))/x) + δtan⁻¹((x²+1)/x) + C where C is an arbitrary constant, then the value of 10(α + βγ + δ) is equal to.

Let Sₙ(x) = logₐ^(1/2) x + logₐ^(1/3) x + . up to n-terms. Where a > 1. If S₂₄(x) = 1093 and S₁₂(2x) = 265, then the value of a is equal to.

If the points of intersections of the ellipse x²/16 + y²/b² = 1 and the circle x² + y² = 4b, b > 4 lie on the curve y² = 3x², then b is equal to :

Let A(-1,1), B(3, 4) and C(2,0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A? and A? be the area of ?ABC and ?PQC respectively, such that A? = 3A?, then the value of m is equal to:

Let a = i + 2j - 3k and b = 2i - 3j + 5k. If r x a = b x r, r.(αi+2j+k) = 3 and r.(2i + 5j - αk) = -1, α ∈ R, then the value of α + |r|² is equal to :