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

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = cos x + C            y' + sin x = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(1+x^2)`                     `y'=(xy)/(1+x^2)`

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = x sin x              `xy'=y+xsqrt(x^2-y^2)`

Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) . 

Find the value of x, if tan `[sec^(-1) (1/x) ] = sin ( tan^(-1) 2) , x > 0 `.

Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation. 

If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3`, show that fof (x) = x for all `x ≠ 2/3`. Also, find the inverse of f.

Write the order and degree of the differential equation `((d^4"y")/(d"x"^4))^2 =  [ "x" + ((d"y")/(d"x"))^2]^3`.

Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.

Write the order and the degree of the following differential equation: `"x"^3 ((d^2"y")/(d"x"^2))^2 + "x" ((d"y")/(d"x"))^4 = 0`

If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.

Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.

if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and  hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.

Show that the relation R defined by (a, b)R(c,d) ⇒ a + d = b + c   on the A x A  , where A =  {1, 2,3,...,10}  is an equivalence relation. Hence write the equivalence class [(3, 4)]; a, b, c,d ∈ A.

For the matrix A = `[(2,3),(5,7)]`, find (A + A') and verify that it is a symmetric matrix.

A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.

Using matrices, solve the following system of linear equations :

x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11

Find the order and the degree of the differential equation `x^2 (d^2y)/(dx^2) = { 1 + (dy/dx)^2}^4`