Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.

The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.

P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.

If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.

A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.

If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.

If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin²α + sin²β + sin²γ is ______.

If a line makes an angle of `pi/4` with each of y and z-axis, then the angle which it makes with x-axis is ______.

The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`

Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.

If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ² = δl² + δm² + δn² 

O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.

If the directions cosines of a line are k,k,k, then ______.

Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.

The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.

The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.

The line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.

A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?

Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:

The point(s), at which the function f given by f(x) = `{("x"/|"x"|","  "x" < 0),(-1","  "x" ≥ 0):}` is continuous, is/are: