HSC Science (Computer Science) 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)]`

Express the following equations in matrix form and solve them by the method of reduction:

x + 2y + z = 8, 2x + 3y – z = 11, 3x – y – 2z = 5.

In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot  C/2`.

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)`

In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.

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.

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

Find the value of k if the lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other. 

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

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 `

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

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

Find the vector equation of the line passing through the point having position vector `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Find the cartesian equation of the plane passing through the point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.