June 2007 QP – C3 OCR.pdf

The exam is for Advanced GCE Unit 4723/01 Mathematics, Core Mathematics 3, and took place on Monday, June 11, 2007. The time allowed is 1 hour 30 minutes, and the total number of marks for the paper is 72. The exam consists of 9 questions, covering various topics in mathematics, including differentiation, inequalities, functions, integration, and trigonometry. Candidates are permitted to use a graphical calculator and are required to give non-exact numerical answers correct to 3 significant figures. The questions are:
1. Differentiate given functions with respect to x.
2. Solve the inequality |4x −3| < |2x + 1|.
3. Evaluate and find the inverse of a given function f(x) = 3 + √x.
4. Find the exact value of a given integral and approximate it using Simpson's rule.
5. Find the time taken for a substance to halve its mass and the value of t for which the mass is decreasing at a rate of 2.1 kg per year.
6. Show that a given equation holds and use an iterative formula to find the value of a correct to 3 decimal places.
7. Sketch the graph of y = secx, solve the equation sec x = 3, and solve the equation sec θ = 5 cosec θ.
8. Show that a given equation holds, find the exact value of the gradient of a curve, and find the exact volume of a solid produced when a shaded region is rotated about the x-axis.
9. Prove a trigonometric identity, solve a given equation, and show that another equation has two roots in a given interval.