GCE Advanced Level (Sri Lanka) · Combined Mathematics

Pure Maths: Differentiation, Integration and Limits

69 practice questions, each with the answer and an explanation.

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All 69 questions

The curve $y = x^{3} + 3 x^{2} - 9 x + 7$ has two stationary points. Find the difference between their $y$ -coordinates (larger minus smaller).
Find the volume of the solid formed by rotating $y = 3 x$ about the $x$ -axis from $x = 0$ to $x = 1$ , in terms of $π$ .
Evaluate $x \to 0 lim \frac{sin 6 x}{2 x}$ .
If $y = ln (4 x + 5)$ , find $\frac{d y}{d x}$ at $x = 1$ .
Find the local minimum value of $y = x^{3} + 6 x^{2} + 9 x + 7$ .
Find the area under the curve $y = 5 x^{2}$ between $x = 0$ and $x = 3$ .
Find the local minimum value of $y = x^{3} - 12 x + 7$ .
If $y = (x + 2) (x^{2} + 2)$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the local minimum value of $y = x^{3} - 6 x^{2} + 7$ .
Evaluate $\int_{2}^{3} (3 x^{2} - 4 x + 4) d x$ .
Evaluate $x \to \infty lim \frac{10 x ^{2} + 5 x}{5 x ^{2} + 2}$ .
If $y = (4 x + 1)^{2}$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the area under the curve $y = 3 x^{2}$ between $x = 0$ and $x = 2$ .
Evaluate $x \to \infty lim \frac{10 x ^{2} + 2 x}{2 x ^{2} + 5}$ .
If $f (x) = 2 x^{3} - x^{2} - 4 x + 9$ , find $f^{'} (- 1)$ .
Find the local minimum value of $y = x^{3} - 3 x^{2} + 7$ .
Evaluate $\int_{2}^{3} (3 x^{2} + 4 x - 3) d x$ .
Evaluate $x \to \infty lim \frac{20 x ^{2} + 5 x}{5 x ^{2} + 4}$ .
If $y = (5 x + 1)^{4}$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the local minimum value of $y = x^{3} - 3 x + 7$ .
Evaluate $x \to \infty lim \frac{30 x ^{2} + 5 x}{5 x ^{2} + 6}$ .
If $y = (x + 1) (x^{2} + 1)$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the area under the curve $y = 2 x^{2}$ between $x = 0$ and $x = 2$ .
Evaluate $x \to \infty lim \frac{15 x ^{2} + 5 x}{5 x ^{2} + 3}$ .
If $f (x) = 4 x^{3} - 2 x^{2} - 3 x + 9$ , find $f^{'} (- 3)$ .
Evaluate $\int_{2}^{3} (2 x^{2} + 2 x + 1) d x$ .
Evaluate $x \to 3 lim \frac{x ^{2} - 9}{x - 3}$ .
If $y = ln (2 x + 5)$ , find $\frac{d y}{d x}$ at $x = 0$ .
Find the local minimum value of $y = x^{3} + 3 x^{2} + 7$ .
Find the volume of the solid formed by rotating $y = 3 x$ about the $x$ -axis from $x = 0$ to $x = 2$ , in terms of $π$ .
Evaluate $x \to 6 lim \frac{x ^{2} - 36}{x - 6}$ .
If $y = (2 x + 1)^{4}$ , find $\frac{d y}{d x}$ when $x = 1$ .
The curve $y = x^{3} - 6 x^{2} + 7$ has two stationary points. Find the difference between their $y$ -coordinates (larger minus smaller).
Evaluate $\int_{1}^{2} (3 x^{2} + 2 x + 4) d x$ .
Evaluate $x \to 5 lim \frac{x ^{2} - 25}{x - 5}$ .
Find the local minimum value of $y = x^{3} + 3 x^{2} - 9 x + 7$ .
If $y = (4 x + 1)^{4}$ , find $\frac{d y}{d x}$ when $x = 1$ .
Evaluate $x \to 4 lim \frac{x ^{2} - 16}{x - 4}$ .
If $f (x) = 4 x^{3} - 3 x^{2} - 2$ , find $f^{'} (1)$ .
The curve $y = x^{3} + 6 x^{2} + 9 x + 7$ has two stationary points. Find the difference between their $y$ -coordinates (larger minus smaller).
Evaluate $x \to 0 lim \frac{sin 3 x}{6 x}$ .
If $y = (3 x + 1)^{2}$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the area under the curve $y = 5 x^{2}$ between $x = 0$ and $x = 1$ .
If $y = (x + 4) (x^{2} + 3)$ , find $\frac{d y}{d x}$ when $x = 2$ .
Find the area under the curve $y = 4 x^{2}$ between $x = 0$ and $x = 2$ .
If $f (x) = x^{3} + 2 x^{2} - 6 x + 8$ , find $f^{'} (- 1)$ .
Find the area enclosed between the curve $y = 2 x^{2}$ and the line $y = 4 x$ .
Evaluate $x \to 0 lim \frac{sin 4 x}{3 x}$ .
If $y = (x + 3) (x^{2} + 2)$ , find $\frac{d y}{d x}$ when $x = 3$ .
The curve $y = x^{3} - 3 x^{2} - 9 x + 7$ has two stationary points. Find the difference between their $y$ -coordinates (larger minus smaller).
If $f (x) = 2 x^{3} - 3 x^{2} - x + 2$ , find $f^{'} (- 2)$ .
If $f (x) = 3 x^{3} - 2 x^{2} - x - 9$ , find $f^{'} (1)$ .
Evaluate $x \to 8 lim \frac{x ^{2} - 64}{x - 8}$ .
If $f (x) = 3 x^{3} - 5 x^{2} + 6 x + 8$ , find $f^{'} (3)$ .
Find the area under the curve $y = 6 x^{2}$ between $x = 0$ and $x = 2$ .
Evaluate $x \to 0 lim \frac{sin 2 x}{3 x}$ .
If $y = (x + 2) (x^{2} + 1)$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the area under the curve $y = 3 x^{2}$ between $x = 0$ and $x = 1$ .
Find the local minimum value of $y = x^{3} - 3 x^{2} - 9 x + 7$ .
Evaluate $\int_{2}^{3} (2 x^{2} - 2 x + 2) d x$ .
Evaluate $x \to 0 lim \frac{sin 2 x}{7 x}$ .
If $y = ln (4 x + 6)$ , find $\frac{d y}{d x}$ at $x = 3$ .
Find the area enclosed between the curve $y = 2 x^{2}$ and the line $y = 2 x$ .
Evaluate $x \to \infty lim \frac{6 x ^{2} + 2 x}{2 x ^{2} + 3}$ .
If $f (x) = 4 x^{3} - 3 x^{2} - 5 x - 9$ , find $f^{'} (1)$ .
Evaluate $\int_{1}^{4} (3 x^{2} - 4 x - 5) d x$ .
Evaluate $x \to \infty lim \frac{6 x ^{2} + 3 x}{3 x ^{2} + 2}$ .
If $y = ln (5 x + 5)$ , find $\frac{d y}{d x}$ at $x = 1$ .
The curve $y = x^{3} - 3 x^{2} + 7$ has two stationary points. Find the difference between their $y$ -coordinates (larger minus smaller).

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