Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions

For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A² + aA + bI = 0.

For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A³ − 6A² + 5A + 11 I = 0. Hence, find A⁻¹.

If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A³ − 6A² + 9A − 4I = 0 and hence find A⁻¹.

If A is an invertible matrix of order 2, then det (A⁻¹) is equal to ______.

If A⁻¹ = `[(3,-1,1),(-15,6,-5),(5,-2,2)]` and B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`, find (AB)⁻¹.

Let A = `[(1,2,1),(2,3,1),(1,1,5)]` verify that

1. [adj A]^–1 = adj(A^–1)
2. (A^–1)^–1 = A

If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.

Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ ≤ 2π, then ______.

Show that the three lines with direction cosines `12/13, (-3)/13, (-4)/13;  4/13, 12/13, 3/13;  3/13, (-4)/13, 12/13 ` are mutually perpendicular.

Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector `3hati+2hatj-2hatk`.

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector `2hati -hatj+4hatk`  and is in the direction `hati + 2hatj - hatk`.

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by `(x+3)/3 = (y-4)/5 = (z+8)/6`.

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

Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).

Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).

Show that the lines `(x-5)/7 = (y + 2)/(-5) = z/1` and `x/1 = y/2 = z/3` are perpendicular to each other.

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, – 1), (4, 3, – 1).