Question A curve passes through the point (4, −6) and has an equation for which . Find the equation of the curve. Solution We are required to find the equation of the curve from the given derivative. We can find equation of the curve from its derivative through integration; We are given; Therefore; Rule for […]
Question i. Express in the form , where and , giving the value of correct to 2 decimal places. ii. Hence solve the equation for . Solution i. We are given the expression; We are required to write it in the form; If and are positive, then; can be written in the […]
Question A curve is defined by the parametric equations for . i. Find the exact gradient of the curve at the point for which . ii. Find the value of at the point where the gradient of the curve is 2, giving the value correct to 3 significant figures. Solution i. […]
Question a. Show that b. Find the exact value of Show all necessary working. Solution a. We are required to show; Rule for integration of is: Division Rule; Power Rule; b. We are required to find; Since , we can rearrange to write; Rule for integration of is: Rule for integration of is: […]
Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is equal to zero; Therefore, we find the expression for […]
Question The sequence of values converges to the value . i. Use the iterative formula to find the value of correct to 4 significant figures. Give the result of each iteration to 6 significant figures. ii. State an equation satisfied by and hence determine the exact value of . Solution i. […]
Question The variables x and y satisfy the equation y = Kxa, where K and a are constants. The graph of ln y against ln x is a straight line passing through the points (0.22, 3.96) and (1.32, 2.43), as shown in the diagram. Find the values of K and a, correct to 3 significant […]
Question i. Solve the equation ii. Hence, using logarithms, solve the equation giving the answer correct to 3 significant figures. Solution SOLVING EQUALITION: PIECEWISE i. 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 polynomial f(x) is defined by Find the quotient and remainder when f(x) is divided by . Solution Hence quotient is and remainder is .
Question i. Show that ii. Hence, show that iii. Solve the equation For . Show all the necessary working. Solution i. First we are required to show that; provided that provided that provided that ii. We are required to show that; As shown in (i); Therefore; […]
Question The diagram shows the curve with equation The maximum point is denoted by M. i. Find an expression for and determine the gradient of the curve at the point where the curve crosses the x-axis. ii. Show that the x-coordinate of the point M satisfies the equation iii. Use an […]
Question The polynomial is defined by where and are constants. It is given that is a factor of and that remainder is 27 when is divided by . i. Find the values of a and b. ii. Hence factorise completely. iii. State the number of roots of the equation p(2y) = 0, […]
Question a. Find b. Find the exact value of Show all necessary working. Solution a. We are given that; Rule for integration of is: Rule for integration of is: Rule for integration of is: b. We are required to find the exact value of; Rule for integration of is: Rule for integration of […]
Question Find the equation of the normal to the curve at the point (3, 1). Solution We are given equation of the curve as; We are required to find the equation of normal to the curve at the point (3, 1). To find the equation of the line either we need coordinates of the two […]
Question i. Solve the inequality . ii. Hence find the greatest integer satisfying the inequality Solution SOLVING INEQUALITY: PIECEWISE i. Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with […]
Question Show that Solution We are given; Division Rule;
Question The equation of a curve is i. Find an expression for and show that the gradient of the curve at the point (−1, 2) is . ii. Show that the curve has no stationary points. iii. Find the x-coordinate of each of the points on the curve at which the […]
Question a. Showing all necessary working, solve the equation for . b. Showing all necessary working, solve the equation for . Solution a. We are given; provided that provided that Let , then; We are given that ; interval for can be found as follows. Multiplying entire inequality with 2; Hence, […]
Question It is given that Where is a constant. i. Show that ii. Using the equation in part (i), show by calculation that 0.5 < a < 0.75. iii. Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 significant figures. Give […]
Question The polynomial is defined by where is a constant. It is given that is a factor of i. Find the value of a . ii. Using this value of a, factorise completely. iii. Hence solve the equation , giving the answer correct to 2 significant figures. Solution i. We are […]