Question The function f is defined by , i. Sketch, in a single diagram, the graphs of and , making clear the relationship between the two graphs. The function g is defined by , ii. Express in terms of , and hence show that the maximum value of is 9. The function h is defined by , […]
Question The function f is such that for , where and are positive constants. The maximum value of is 10 and the minimum value is −2. i. Find the values of and . ii. Solve the equation . iii. Sketch the graph of . Solution i. We have the function; We know that; For ; For ; […]
Question The diagram shows the curve and the points P(0,1) and Q(1,2) on the curve. The shaded region is bounded by the curve, the y-axis and the line y = 2. i. Find the area of the shaded region. ii. Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. Tangents are drawn to the […]
Question The equation of a curve is . i. Show that the equation of the normal to the curve at the point is This normal meets the curve again at the point Q. ii. Find the coordinates of Q. iii. Find the length of PQ. Solution i. We are required to find the equation of the normal to the curve. To […]
Question A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side x cm and the other piece is bent to form a circle of radius r cm (see diagram). The total area of the square and the circle is Acm2. i. Show that ii. Given that x and r […]
Question In the diagram, the circle has centre O and radius 5 cm. The points P and Q lie on the circle, and the arc length PQ is 9 cm. The tangents to the circle at P and Q meet at the point T. Calculate i. angle POQ in radians, ii. the length of PT, iii. the area of the shaded region. […]
Question The diagram shows a semicircular prism with a horizontal rectangular base . The vertical ends and are semicircles of radius 6 cm. The length of the prism is 20 cm. The mid-point of is the origin , the mid-point of is and the mid-point of is . The points and are the highest points of the […]
Question Prove the identity Solution We are given the equation; We can rewrite it as; We have the trigonometric identity; We can rewrite it as; Therefore;
Question Find the value of the coefficient of in the expansion of Solution Expression for the general term in the Binomial expansion of is: In the given case: Hence; Since we are looking for the term with , we can equate Finally substituting in: Therefore the coefficient of is .