Exam Questions & Step-by-Step Markschemes (172 Problems)
Question 1 • 2023
1 Marks
The value of k for which f(x) = \begin{cases} 3x + 5, & x \geq 2 \\ kx^2, & x < 2 \end{cases} is a continuous function, is :
Answer: (D) \frac{11}{4}
Question 2 • 2023
2 Marks
If y = (x + \sqrt{x^2 - 1})^2 , then show that (x^2 - 1)\left(\frac{dy}{dx}\right)^2 = 4y^2 .
Answer: Proved.
Question 3 • 2023
4 Marks
The equation of the path traced by a roller-coaster is given by the polynomial f(x) = a(x + 9)(x + 1)(x - 3) . If the roller-coaster crosses y-axis at a point (0, -1) , answer the following : (i) Find the value of ‘a’. (2) (ii) Find f''(x) at x = 1 . (2)
Answer: (i) a = \frac{1}{27} (ii) f''(1) = \frac{20}{27}
Question 4 • 2023
1 Marks
For what value of k may the function f(x) = \begin{cases} k(3x^2 - 5x), & x \leq 0 \\ \cos x, & x > 0 \end{cases} become continuous ?
Answer: (D) No value
Question 5 • 2023
1 Marks
The function f(x) = |x| - x is :
Answer: (A) continuous but not differentiable at x = 0 .
Question 6 • 2023
2 Marks
If x = \sqrt{a^{\tan^{-1} t}} , y = \sqrt{a^{\cot^{-1} t}} , then show that x\frac{dy}{dx} + y = 0 .
Answer: Proved.