Question i. Show that . ii. Using the identity in part (i), a. find the least possible value of as x varies. b. find the exact value of Solution i. […]
Question i. Express in the form , where a and b are integers. ii. Hence express in the form where R > 0 and . iii. Using the result of part (ii), solve the equation for . Solution i. We are given the expression; provided that provided that […]
Question i. Find ii. Without using a calculator, find the exact value of Solution i. We are required to find; We know that ; Therefore; Hence; Rule for integration of is: Rule for integration of is: Rule for integration of is: ii. We are required to find the exact […]
Question The diagram shows the curve with parametric equations for . i. Show that can be expressed in the form ii. Find the equation of the normal to the curve at the point where the curve crosses the positive y-axis. Give your answer in the form y = mx +c, where the constants […]
Question i. Show that ii. Solve the equation for . iii. Find the exact value of Solution i. We are required to show that; Since ; Therefore; provided that ii. We are required to solve the equation for . We are given that; As demonstrated in (i); Since ; […]
Question The polynomial is defined by where and are constants. It is given that is a factor of . It is also given that the remainder is 18 when is divided by . i. Find the values of a and b. ii. When a and b have these values […]
Question i. Show that ii. Hence find the exact value of; Solution i. We are required to show that; Since ; Therefore; ii. We are required to find the exact value of; As demonstrated in (i); Therefore; Rule for integration of is: Rule for integration of is: Rule for […]
Question A curve has equation y = 2 sin 2x − 5 cos 2x +6 and is defined for 0 ≤ x ≤ π. Find the x-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. Solution We are required to find the x-coordinates of stationary points of the […]
Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q. i. Show that . ii. Find the coordinates of M. iii. Find the gradient of the curve at P and at Q. Solution i. […]
Question Show that i. Hence a) find the exact value of , b) Solve the equation , for . Solution i. We are given that; We know that; Therefore; ii. a) We are required to find the exact value of From (i) we have found that; Therefore; b) We […]
Question i. Solve the equation . ii. Hence solve the equation for , giving your answer correct to 3 significant figures. Solution i. SOLVING EQUALITY: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases; OR We have the equation; We have to consider both moduli […]
Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q. i. Show that . ii. Find the coordinates of M. iii. Find the gradient of the curve at P and at Q. Solution i. […]
Question Show that i.Hence a) find the exact value of , b) Solve the equation , for . Solution i. We are given that; We know that; Therefore; ii. a) We are required to find the exact value of From (i) we have found that; Therefore; b) We […]
Question i. Solve the equation . ii. Hence solve the equation for , giving your answer correct to 3 significant figures. Solution i. SOLVING EQUALITY: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases; OR We have the equation; We have to consider both moduli […]
Question i. Find ii. Without using a calculator, find the exact value of giving your answer in the form , where a and b are integers. Solution i. We are required to find; provided that Rule for integration of is: We integrate both parts and one by […]
Question A curve is defined by the parametric equations for . i. Show that . ii. Find the coordinates of the stationary point. iii. Find the gradientof the curve at point . Solution i. We are required to show that for the parametric equations given below; If a curve […]
Question Solve the equation for . Solution We are given; except where or undefined Now we have two options. Using calculator; Now we find all solutions in the interval . Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity/symmetry property of to find other solutions (roots) of : Therefore; For ; […]
Question a. Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and −12r respectively. Find the value of the common sum to infinity. b. The first term of an arithmetic progression […]
Question Showing all necessary working, solve the equation for . Solution We are given equation; We know that ; therefore, Now we have two options. Using calculator we can find that; We have following properties of . Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity property of to find other solutions (roots) of . Therefore; For; […]
Question A function f is defined by for . i. Find the range of f. ii. Sketch the graph of iii. Solve the equation , giving answers in terms of . The function is defined by for , where k is a constant. iv. State the largest value of k for which g has an inverse. v. For this value of k, find […]