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

The function f(x) = x⁵ - 5x⁴ + 5x³ - 1 has ____________.

Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.

The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.

Find the area of the largest isosceles triangle having a perimeter of 18 meters.

The coordinates of the point on the parabola y² = 8x which is at minimum distance from the circle x² + (y + 6)² = 1 are ____________.

The distance of that point on y = x⁴ + 3x² + 2x which is nearest to the line y = 2x - 1 is ____________.

The function `"f"("x") = "x" + 4/"x"` has ____________.

The combined resistance R of two resistors R₁ and R₂ (R₁, R₂ > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R₁ + R₂ = C (a constant), then maximum resistance R is obtained if ____________.

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

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’

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.