Question The function is defined for . i. Find the exact value of for which . ii. State the range of . iii. Sketch the graph of . iv. Find an expression for . Solution i. We are given that; We can write it as; We are given that , therefore; Using calculator; ii. […]
Question A curve is such that . The curve has a stationary point at where . i. State, with a reason, the nature of this stationary point. ii. Find an expression for . iii. Given that the curve passes through the point , find the coordinates of the […]
Question The diagram shows a trapezium ABCD in which AB is parallel to DC and angle BAD is . The coordinates of A, B and C are , and respectively. i. Find the equation of AD. ii. Find, by calculation, the coordinates of D. The point E is such that ABCE is a parallelogram. iii. Find the length […]
Question The diagram shows a pyramid OABCX. The horizontal square base OABC has side 8 units and the centre of the base is D. The top of the pyramid, X, is vertically above D and XD = 10 units. The mid- point of OX is M. The unit vectors and are parallel to and respectively and the unit vector is vertically upwards. […]
Question The equation of a curve is , where and are constants. i. In the case where the curve has no stationary point, show that . ii. In the case where and , find the set of values of for which is a decreasing function of . Solution i. We are given; Gradient […]
Question i. Show that the equation can be expressed as ii. Hence solve the equation for . Solution i. We are given; We know that; We can write the given equation as; We have the trigonometric identity; We can rearrange this identity as; Substituting in above equation; ii. From (i) we know that we can write the given equation […]
Question A curve has equation . i. Find . A point moves along this curve. As the point passes through A, the x-coordinate is increasing at a rate of 0.15 units per second and the y-coordinate is increasing at a rate of 0.4 units per second. ii. Find […]
Question The diagram shows a triangle AOB in which OA is 12 cm, OB is 5 cm and angle AOB is a right angle. Point P lies on AB and OP is an arc of a circle with centre A. Point Q lies on AB and OQ is an arc of a circle with centre B. […]
Question The diagram shows part of the curve . Find the volume obtained when the shaded region is rotated through about the y-axis. Solution Expression for the volume of the solid formed when the shaded region under the curve is rotated completely about the y-axis is; We are given; We can rearrange it to change the subject; We can see that curve […]
Question i. Find the first 3 terms, in ascending powers of , in the expansion of . The coefficient of in the expansion of is 95. ii. Use the answer to part (i) to find the value of the positive constant . Solution i. Expression for the Binomial expansion of is: In the given case: Hence; ii. To evaluate […]
Question a) The sum, , of the first terms of an arithmetic progression is given by . Find the first term and the common difference. b) A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression. Solution […]