Question Functions and are defined by , , , , i. Solve the equation . ii. Express and in terms of . iii. Show that the equation has no solutions. iv. Sketch in a single diagram the graphs of and , making clear the relationship between the graphs. Solution i. We have the functions; We can write these functions […]
Question The equation of a curve is . The equation of a line is . On the same diagram, sketch the curve and the line for . Solution First we sketch for . We can find the points of the graph as follows. Now we sketch the line for . We can write it as; Slope-Intercept form of the […]
Question Relative to an origin , the position vectors of the points , and are given by and i. Find angle AOB. ii. Find the vector which is in the same direction as and has magnitude 30. iii. Find the value of the constant for which is perpendicular to . Solution i. We recognize that is angle between and . […]
Question The equation of a curve is . i. Obtain an expression for . ii. Find the equation of the normal to the curve at the point P(1, 3). iii. A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of […]
Question A curve is such that , where k is a constant. i. Given that the tangents to the curve at the points where and are perpendicular, find the value of . ii. Given also that the curve passes through the point (4, 9), find the equation of the curve. Solution i. If two lines are perpendicular […]
Question The equation of a curve is . i. Find the coordinates of the stationary point on the curve and determine its nature. ii. Find the area of the region enclosed by the curve, the x-axis and the lines and . Solution i. Coordinates of stationary point on the curve can be found from the derivative of equation […]
Question The diagram shows a semicircle ABC with centre O and radius 6 cm. The point B is such that angle BOA is 90o and BD is an arc of a circle with centre A. Find i. the length of the arc BD, ii. the area of the shaded region. Solution i. Expression for length of a […]
Question Solve the equation for . Solution To solve this equation for , we can substitute . Hence, Since given interval is , for interval can be found as follows; Multiplying the entire inequality with 2; Adding to entire inequality; Since ; Hence the given interval for is . To solve equation for interval , Using calculator we can find the value of . To find all […]
Question i. Find the first three terms in the expansion of in ascending powers of . ii. Given that the coefficient of in the expansion of is , find the value of the constant . Solution i. First rewrite the given expression in standard form. Expression for the Binomial expansion of is: In the given […]
Question The first term of an arithmetic progression is 8 and the common difference is d, where d ≠ 0. The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is r. i. Write down two equations connecting […]