Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2^nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3^rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.

State the order of the above given differential equation.

Write the sum of the order and the degree of the following differential equation:

`d/(dx) (dy/dx)` = 5

Find: `int (x + 1)/((x^2 + 1)x) dx`

The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",",  if x ≠ 0),(k",",  if x = 0):}` is continuous at x = 0 is ______.

If m and n, respectively, are the order and the degree of the differential equation `d/(dx) [((dy)/(dx))]^4` = 0, then m + n = ______.

P is a point on the line joining the points A(0, 5, −2) and B(3, −1, 2). If the x-coordinate of P is 6, then its z-coordinate is ______.

Define the relation R in the set N × N as follows:

For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.

Given a non-empty set X, define the relation R in P(X) as follows:

For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.

Find the general solution of the following differential equation:

`(dy)/(dx) = e^(x-y) + x^2e^-y`

Find the vector equation of a line passing through a point with position vector `2hati - hatj + hatk` and parallel to the line joining the points `-hati + 4hatj + hatk` and `-hati + 2hatj + 2hatk`.

Degree of the differential equation `sinx + cos(dy/dx)` = y² is ______.

Equation of a line passing through (1, 1, 1) and parallel to z-axis is ______.

Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.

The function f(x) = x |x| is ______.

Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, – 6), Q(5, – 3, 1), R(12, 4, 5) and S(11, 9, – 2). Use these equations to find the point of intersection of diagonals.

Read the following passage:

Based on the above information, answer the following questions:

1. Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
2. Prove that the function V(t) is an increasing function. (2)

Let A = {3, 5}. Then number of reflexive relations on A is ______.

The interval in which the function f(x) = 2x³ + 9x² + 12x – 1 is decreasing is ______.

The sum of the order and the degree of the differential equation `d/dx[(dy/dx)^3]` is ______.

Read the following passage:

Based on the above information, answer the following questions:

1. How many relations are possible from B to G? (1)
2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
3. Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
   OR
   A function f : B `rightarrow` G be defined by f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}. Check if f is bijective. Justify your answer. (2)