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

Let f(x) = 1 + 2x² + 2²x⁴ + …… + 2¹⁰x²⁰. Then f (x) has ____________.

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.

The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.

One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following case is the events E and F independent?

E : ‘the card drawn is a spade’

F : ‘the card drawn is an ace’

One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following case is the events E and F independent?

E : ‘the card drawn is black’

F : ‘the card drawn is a king’

Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.

`(dy)/(dx) + ycotx = 2/(1 + sinx)`

The solution set of the inequality 3x + 5y < 4 is ______.

The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at ______.

Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6 and P(A' ∩ B') is ______.

Assertion (A): The acute angle between the line `barr = hati + hatj + 2hatk  + λ(hati - hatj)` and the x-axis is `π/4`

Reason(R): The acute angle 𝜃 between the lines `barr = x_1hati + y_1hatj + z_1hatk  + λ(a_1hati + b_1hatj + c_1hatk)` and  `barr = x_2hati + y_2hatj + z_2hatk  + μ(a_2hati + b_2hatj + c_2hatk)` is given by cosθ = `(|a_1a_2 + b_1b_2 + c_1c_2|)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)`

If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`

Solve the following Linear Programming Problem graphically:

Maximize Z = 400x + 300y subject to x + y ≤ 200, x ≤ 40, x ≥ 20, y ≥ 0

Read the following passage and answer the questions given below.

1. Is the function differentiable in the interval (0, 12)? Justify your answer.
2. If 6 is the critical point of the function, then find the value of the constant m.
3. Find the intervals in which the function is strictly increasing/strictly decreasing.
   OR
   Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.

Read the following passage and answer the questions given below.

1. If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
2. Find the critical point of the function.
3. Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
   OR
   Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

Events A and Bare such that P(A) = `1/2`, P(B) = `7/12` and `P(barA ∪ barB) = 1/4`. Find whether the events A and B are independent or not.

The probability that A hits the target is `1/3` and the probability that B hits it, is `2/5`. If both try to hit the target independently, find the probability that the target is hit.

Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.

Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.

Find the general solution of the differential equation:

`log((dy)/(dx)) = ax + by`.

Find the general solution of the differential equation:

`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`

If `[(1, 2, 1),(2, 3, 1),(3, a, 1)]` is non-singular matrix and a ∈ A, then the set A is ______.