Arts (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Find the direction cosines of a line which makes equal angles with the coordinate axes.

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).

If l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂ − m₂n₁, n₁l₂ − n₂l₁, l₁m₂ ­− l₂m₁.

If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.

Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`

Find the relationship between a and b so that the function f defined by f(x) = `{(ax + 1", if"  x<= 3),(bx + 3", if"  x > 3):}` is continuous at x = 3.

For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if"  x <= 0),(4x+ 1", if"  x > 0):}` continuous at x = 0? What about continuity at x = 1?

Is the function defined by f(x) = x² − sin x + 5 continuous at x = π?

Discuss the continuity of the following function:

f(x) = sin x × cos x

Find the value of k so that the function f is continuous at the indicated point.

f(x) = `{((kcosx)/(pi-2x)", if"  x != pi/2),(3", if"  x = pi/2):}` at x = `"pi/2`

Find the value of k so that the function f is continuous at the indicated point.

f(x) = `{(kx^2", if"  x<= 2),(3", if"  x > 2):}` at x = 2

Find the value of k so that the function f is continuous at the indicated point.

f(x) = `{(kx +1", if"  x<= pi),(cos x", if"  x > pi):}` at x = π

Find the value of k so that the function f is continuous at the indicated point.

f(x) = `{(kx + 1", if"  x <= 5),(3x - 5", if"  x > 5):}` at x = 5

Find the values of a and b such that the function defined by f(x) = `{(5", if"  x <= 2),(ax +b", if"  2 < x < 10),(21", if"  x >= 10):}` is a continuous function.