HSC Science (Electronics) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

Check whether the following matrix is invertible or not:

`[(cos theta, sin theta),(-sin theta, cos theta)]`

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

Select the correct option from the given alternatives:

In ΔABC if c² + a² – b² = ac, then ∠B = ____.

The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______

With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`

Find the principal solutions of tan x = `-sqrt(3)`

If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a², b², c² are in A.P.

Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`

Find the separate equations of the lines represented by the equation 3x² – 10xy – 8y² = 0.

Find k, if the sum of the slopes of the lines represented by x² + kxy − 3y² = 0 is twice their product. 

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0 

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

By vector method prove that the medians of a triangle are concurrent.

If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Prove that the volume of a parallelopiped with coterminal edges as  ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges  `bar i+barj, barj+bark `

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`

Prove by vector method, that the angle subtended on semicircle is a right angle.

If α, β, γ are direction angles of a line and α = 60°, β = 45°, then γ = ______.

If `bara, barb` and `barc` are position vectors of the points A, B, C respectively and `5bara - 3barb - 2barc = bar0`, then find the ratio in which the point C divides the line segement BA.