HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Important Questions

In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled

In ∆ABC, prove that `sin  ((A - B)/2) = ((a - b)/c) cos  C/2` 

Prove that cot⁻¹(7) + 2 cot⁻¹(3) = `pi/4`

In ΔABC, prove that `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0

In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0

If f(x) = x⁵ + 2x – 3, then (f^–1)¹ (–3) = ______.

Find the principal value of `cot^-1 ((-1)/sqrt(3))`

If f'(x) = x^–1, then find f(x)

Find the principal solutions of cot θ = 0

Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.

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

Find the general solution of sin θ + sin 3θ + sin 5θ = 0

If –1 ≤ x ≤ 1, the prove that sin^–1 x + cos^–1 x = `π/2`

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

The Cartestation equation of  line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Find the value of p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l₁.

Find p and q if the equation px² – 8xy + 3y² + 14x + 2y + q = 0 represents a pair of prependicular lines.