Question The diagram shows part of the curve . The curve intersects the x-axis at A. The normal to the curve at A intersects the y-axis at B. i. Obtain expressions for and ii. Find the coordinates of B. iii. Find, showing all necessary working, the area of the shaded region. Solution i. […]
Question Functions f and g are defined by; for for Where is a constant. i. Find the value of for which the line is a tangent to the curve . ii. In the case where , find the set of values of for which . iii. In the case where , […]
Question a) Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km. On the first day she runs 13 km. (i) Find the distance she runs on the last day of the 21-day period. (ii) Find the total distance […]
Question The diagram shows a three-dimensional shape OABCDEFG. The base OABC and the upper surface DEFG are identical horizontal rectangles. The parallelograms OAED and CBFG both lie in vertical planes. Points P and Q are the mid-points of OD and GF respectively. Unit vectors and are parallel to and respectively and the unit vector is […]
Question a) Given that x > 0, find the two smallest values of x, in radians, for which . Show all necessary working. b) The function is defined for . i. Express f(x) in the form , where a and b are constants. ii. Find the range of f. Solution a) We […]
Question The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of h cm. i.Show that the volume, V cm3, of the cone is given by . [The volume of a cone of radius and vertical height is .] ii.Given that can vary, find the value of […]
Question The diagram shows a circle with centre O and radius cm. Points A and B lie on the circle and angle radians. The tangents to the circle at A and B meet at T. i. Express the perimeter of the shaded region in terms of and . ii. In the case where […]
Question A curve is such that , where is a constant. The points P(1, −1) and Q(4, 4) lie on the curve. Find the equation of the curve. Solution We are given that; We can find equation of the curve from its derivative through integration; Therefore; Rule for integration of is: If a point lies […]
Question The point M is the mid-point of the line joining the points (3, 7) and (−1, 1). Find the equation of the line through M which is parallel to the line . Solution We are required to find the equation of the line which passes through the point M and is parallel to […]
Question The coefficient of in the expansion of is 3. Find the value of the constant a. Solution i. First, we expand . Expression for the Binomial expansion of is: In the given case: Hence; Now can we find the coefficient of in the expansion of . We write only terms […]