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

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Answer the following using the above information.

• Let R = {(L₁, L₂ ): L₁ is parallel to L₂ and L₁: y = x – 4} then which of the following can be taken as L₂?

If A = {1,2,3}, B = {4,6,9} and R is a relation from A to B defined by ‘x is smaller than y’. The range of R is ____________.

The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} is ____________.

2x³ - 6x + 5 is an increasing function, if ____________.

If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:

The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.

The function f(x) = tan^-1 (sin x + cos x) is an increasing function in:

The function f(x) = x³ + 6x² + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.

The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.

`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.

The function `"f"("x") = "x"/"logx"` increases on the interval

The length of the longest interval, in which the function `3  "sin x" - 4  "sin"^3"x"` is increasing, is ____________.

Let h(x) = f(x) - [f(x)]² + [f(x)]³ for every real number x. Then ____________.

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.