HSC Science (General) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

In any Δ ABC, prove the following:

`"cos 2A"/"a"^2 - "cos 2B"/"b"^2 = 1/"a"^2 - 1/"b"^2`

Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`

Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`

In any Δ ABC, prove the following:

`("b" - "c")/"a" = (tan  "B"/2 - tan  "C"/2)/(tan  "B"/2 +tan  "C"/2)`

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`

Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`

In Δ ABC, if a, b, c are in A.P., then show that cot `"A"/2, cot  "B"/2, cot  "C"/2` are also in A.P.

Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`

Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`

In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`

Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`

In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.

Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`

In Δ ABC, if sin² A + sin² B = sin² C, then show that the triangle is a right-angled triangle.

Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`

In Δ ABC, prove that a² (cos² B - cos² C) + b² (cos² C - cos² A) + c² (cos² A - cos² B) = 0.

Evaluate the following integrals:

`int (2x + 1)/(x^2 + 4x - 5).dx`

Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`

Evaluate the following integrals :  `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`

Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`