Question Functions and are defined by i. Express in the form , where , and are constants. ii. State the range of . iii. State the domain of . iv. Sketch on the same diagram the graphs of , and , making clear the relationship between the graphs.. v. Find an expression for . Solution i. We […]
Question The diagram represents a metal plate , consisting of a sector of a circle with centre and radius , together with a triangle which is right-angled at C. Angle radians and is perpendicular to . i. Find an expression in terms of and for the perimeter of the plate. ii. For the case where and , find […]
Question The diagram shows the curvemeeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3. i. Find the coordinates of B and C. ii. Find the equation of the normal to the curve at C. iii. Find the volume obtained when the shaded region is rotated through about the y-axis. […]
Question A line has equation and a curve has equation , where is a constant. i. For the case where , the line and the curve intersect at points A and B. Find the distance AB and the coordinates of the mid-point of AB. ii. Find the two values of for which the line is a tangent to the curve. […]
Question Relative to an origin , the point A has position vector and the point B has position vector , where is a constant. i. Find . ii. Hence show that there are no real values of for which and are perpendicular to each other. iii. Find the values of for which angle AOB =. Solution i. A point has […]
Question The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48m. i. Find an expression for in terms of . ii. Given that the area of the garden is A m2, show that . iii. Given that can vary, find the maximum area of the garden, showing that this is a maximum […]
Question i. Sketch, on a single diagram, the graphs of and for . ii. Write down the number of roots of the equation in the interval . iii. Deduce the number of roots of the equation in the interval . Solution i. ii. If two lines (or a line and a curve) intersect each other at a […]
Question A curve has equation . Show that the gradient of the curve is never negative. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is: For the given case; Rule for differentiation of is: Therefore; Rule for differentiation of is: Rule for differentiation of is: Hence; […]
Question A function is defined for and is such that . The range of the function is given by . i. State the value of for which has a stationary value. ii. Find an expression for in terms of . Solution i. We have; The expression for represents derivative of . A stationary point on the curve is the point where gradient of the curve […]
Question Find the term independent of in the expansion of . Solution Expression for the Binomial expansion of is: In the given case: Hence; The term independent of in the expansion of is .
Question a) The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find the seventh term. b) A geometric progression has first term 1 and common ratio r. A second geometric progression has first term 4 and common ratio . The two progressions have the same sum to […]