Mathematics (Solved Paper 2024 – II) (51 to 100)

Total Questions: 50

31. Which one of the following is f(x) ?

Let int og(x) = cos²√x and g o f(x)= |cos x|.

Correct Answer: (c) cos² x
Solution:

32. Which one of the following is g(x) ?

Let int og(x) = cos²√x and g o f(x)= |cos x|.

Correct Answer: (a) √x
Solution:If f(x) = cos² x then g(x) = √x,

so that fog (x) = cos²√x

33. What is f(0.999) + f(1.001) equal to?

f(x) = [x]² - [x²].

Correct Answer: (b) 0
Solution:Given f(x) = [x]² - [x²]

34. Consider the following statements :

f(x) = [x]² - [x²].

A. f(x) is continuous at x = 0

B. f(x) is continuous at x = 1

Which of the statements given above is/are correct?

Correct Answer: (b) B only
Solution:Given f(x) = [x]² - [x²]

35. What is the greatest value of f(x) ?

Let f(x) = cos 2x + x on [ -π/2, π/2].

Correct Answer: (b)
Solution:f(x) = cos2x + x on [-π/2, π/2]

f ' (x) = -2 sin 2 x + 1

for maxima & minima f '(x) = 0

sin 2x = 1/2

x = π/12 or 5π/12 (Not in interval) of [-π/2 , π/2]

So, maximum value of f(x) occurs at x = π/12 ie

f(π/12) = cos π/6 + π/12 = √3/2 + π/12

36. What is the least value of f(x) ?

Let f(x) = cos 2x + x on [ -π/2, π/2].

Correct Answer: (b)
Solution:f(x) = cos2x + x on [-π/2, π/2]

Find the value of f(x) at boundary of interval i.e at

x = -π/2 and x = π/2

f(-π/2) = -1 + -π/2

f(π/2) = -1 + π/2

So, minimum value of f(x)² = -[1 + π/2]

37. What is the value of k?

The area bounded by the parabola y² = kx and the line x = k, where k > 0 is 4/3 square units.

Correct Answer: (b) 1
Solution:

38. What is the area of the parabola bounded by the latus rectum?

The area bounded by the parabola y² = kx and the line x = k, where k > 0 is 4/3 square units.

Correct Answer: (a) 1/6 square unit
Solution:

39. What are the order and degree respectively of the differential equation?

Let y dx + (x - y³) dy = 0 be a differential equation.

Correct Answer: (a) 1 and 1
Solution:

40. What is the solution of the differential equation?

Let y dx + (x - y³) dy = 0 be a differential equation.

Correct Answer: (d) 4xy - y⁴ = c
Solution: