HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

In ΔABC, prove that `("a - b")^2 cos^2  "C"/2 + ("a + b")^2 sin^2  "C"/2 = "c"^2`

In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.

If `(sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C"))`, then show that a², b², c² are in A.P.

State whether the following equation has a solution or not?

3 sin θ = 5

If 2 tan^-1(cos x) = tan^-1(2 cosec x), then find the value of x.

If sin^-1(1 - x) - 2 sin^-1x =  `pi/2`, then find the value of x.

If tan^-12x + tan^-13x = `pi/4`, then find the value of x.

Show that `tan^-1  1/2 - tan^-1  1/4 = tan^-1  2/9`.

Show that `cot^-1  1/3 - tan^-1  1/3 = cot^-1  3/4`.

Show that `tan^-1  1/2 = 1/3  tan^-1  11/2`

Show that `cos^-1  sqrt3/2 + 2 sin^-1  sqrt3/2 = (5pi)/6`.

Show that `2 cot^(-1)  3/2 +  sec^(-1)  13/12 = π/2`

Prove the following:

`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0

Prove the following:

`cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x")`, if x < 0

If | x | < 1, then prove that

`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`

If x, y, z are positive, then prove that

`tan^-1 (("x - y")/(1 + "xy")) + tan^-1 (("y - z")/(1 + "yz")) + tan^-1 (("z - x")/(1 + "zx")) = 0`

If `tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2,` then show that xy + yz + zx = 1

If cos^-1 x + cos^-1y + cos^-1z = 3π, then show that x² + y² + z² + 2xyz = 1.

Determine the order and degree of the following differential equation:

`("d"^2"y")/"dx"^2 + "x"("dy"/"dx")` + y = 2 sin x

Determine the order and degree of the following differential equation:

`root(3)(1 +("dy"/"dx")^2) = ("d"^2"y")/"dx"^2`