Question In the diagram, the points A and C lie on the x- and y-axes respectively and the equation of AC is . The point B has coordinates (2, 2). The perpendicular from B to AC meets AC at the point X. i. Find the coordinates of X. The point D is such that the quadrilateral ABCD has AC as a […]
Question Relative to an origin , the position vectors of points and are and respectively. i. Find the value of for which and are perpendicular. ii. In the case where , use a scalar product to find angle , correct to the nearest degree. iii. Express the vector in terms of and hence find the values of for which […]
Question The diagram shows a curve for which where is a constant. The curve passes through the points (1,18) and (4,3). i. Show, by integration, that the equation of the curve is . The point P lies on the curve and has x-coordinate 1.6. ii. Find the area of the shaded region. Solution i. We can find equation of the […]
Question The diagram shows a circle with centre O and radius 5 cm. The point P lies on the circle, PT is a tangent to the circle and PT = 12 cm. The line OT cuts the circle at the point Q. i. Find the perimeter of the shaded region. ii. Find the area of the shaded region. Solution i. […]
Question i. Show that the equation can be written in the form . ii. Hence solve the equation , for . Solution i. We are given the equation; We can rewrite it as; We have the trigonometric relation; Substituting in the above equation; We have the trigonometric identity; We can rewrite this as; Therefore; ii. To solve equation for , we […]
Question Functions and are defined by , , where is a constant , i. Find the values of for which the equation has two equal roots. ii. Determine the roots of the equation for the values of found in part (i). Solution i. We have the functions; First we find ; We are given that; Hence; Standard form of quadratic equation is; Expression for discriminant […]
Question The function f is such that for . i. Obtain an expression for and hence explain why is an increasing function. ii. Obtain an expression for for and state the domain of . Solution i. We have the function; The expression for represents derivative of . Rule for differentiation of is: Rule for differentiation of is: […]
Question The equation of a curve C is and the equation of a line L is . i. Find the x-coordinates of the points of intersection of L and C. ii. Show that one of these points is also the stationary point of C. Solution i. If two lines (or a line and a curve) […]
Question In the triangle ABC, AB = 12 cm, angle BAC = 60o and angle ACB = 45o. Find the exact length of BC. Solution From the given data we can write the following; Expression for Law of Sines is; For the given case;
Question i. Find the first three terms in the expansion, in ascending powers of , of in ascending powers of . ii. Hence find the coefficient of in the expansion of . Solution i. Expression for the Binomial expansion of is: In the given case: Hence; ii. To find the coefficient of in the expansion of From (i) […]
Question The first term of an arithmetic progression is 6 and the fifth term is 12. The progression has n terms and the sum of all the terms is 90. Find the value of n. Solution From the given information, we can compile following data for Arithmetic Progression (A.P); Expression for the sum of number of terms in the […]
Question The first term of a geometric progression is 81 and the fourth term is 24. Find i. the common ratio of the progression, ii. the sum to infinity of the progression. The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression. iii. Find the sum of the first ten terms of […]