HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find the measure of the acute angle between the lines given by x² − 4xy + y² = 0 

Find the value of h, if the measure of the angle between the lines 3x² + 2hxy + 2y² = 0 is 45°. 

If θ is the acute angle between the lines given by ax² + 2hxy + by² = 0 then prove that tan θ = `|(2sqrt("h"^2) - "ab")/("a" + "b")|`. Hence find acute angle between the lines 2x² + 7xy + 3y² = 0 

Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”

If the angle between the lines represented by ax² + 2hxy + by² = 0 is equal to the angle between the lines 2x² − 5xy + 3y² = 0, then show that 100(h² − ab) = (a + b)²

Write the dual of the following

(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q)

The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______

System of equations x + y = 2, 2x + 2y = 3 has ______

If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A⁻¹X = B.

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

Solve the following by inversion method 2x + y = 5, 3x + 5y = −3

Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices

Solve the following equations by using inversion method.

x + y + z = −1, x − y + z = 2 and x + y − z = 3

If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A

Solve the following system of equations by using inversion method

x + y = 1, y + z = `5/3`, z + x = `4/3`

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices

sin⁻¹x − cos⁻¹x = `pi/6`, then x = ______

The principal value of sin⁻¹`(1/2)` is ______

The principal value of cos⁻¹`(-1/2)` is ______

`tan^-1(tan  (7pi)/6)` = ______