HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Write the truth value of the following statement:

∀ n ∈ N, n + 6 > 8.

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

Find the vector equation of the line passing through points having position vector `3hati + 4hatj - 7hatk and 6hati - hatj + hatk`.

Find the vector equation of line passing through the point having position vector `5hat"i" + 4hat"j" + 3hat"k"` and having direction ratios  –3, 4, 2.

Find the vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k"  "and perpendicular to vectors"  hat"i" + hat"j" + hat"k" and 2hat"i" - hat"j" + hat"k"`.

Find the vector equation of the line passing through the point having position vector `-hat"i" - hat"j" + 2hat"k"  "and parallel to the line" bar"r" = (hat"i" + 2hat"j" + 3hat"k") + λ(3hat"i" + 2hat"j" + hat"k").`

Find the cartesian equations of the line passing through A(–1, 2, 1) and having direction ratios 2, 3, 1.

Find the Cartesian equations of the line passing through A(2, 2, 1) and B(1, 3, 0).

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.

Show that the lines given by `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1) and (x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` intersect. Also, find the coordinates of their point of intersection.

A line passes through (3, –1, 2) and is perpendicular to lines `bar"r" = (hat"i" + hat"j" - hat"k") + lambda(2hat"i" - 2hat"j" + hat"k") and bar"r" = (2hat"i" + hat"j" - 3hat"k") + mu(hat"i" - 2hat"j" + 2hat"k")`. Find its equation.

Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.

Find the principal solution of the following equation: 

cosθ = `(1)/(2)`

Find the principal solution of the following equation: 

Sec θ = `(2)/sqrt(3)`

Find the principal solution of the following equation :

cot θ = `sqrt(3)`

Find the principal solution of the following equation:

cot θ = 0

Find the principal solution of the following equation:

sin θ = `-1/2`

Find the principal solution of the following equation: 

tan θ = – 1

Find the principal solution of the following equation:

`sqrt(3)` cosecθ + 2 = 0 

Find the general solution of the following equation:

sinθ = `1/2`.