Question The diagram shows the curve with equation . The minimum point on the curve has coordinates and the x-coordinate of the maximum point is , here and are constants. i. State the value of . ii. Find the value of . iii. Find the area of the shaded region. iv. The gradient, , of the curve has […]
Question A straight line has equation , where is a constant, and a curve has equation . i. Show that the x-coordinates of any points of intersection of the line and curve are given by the equation . ii. Find the two values of for which the line is a tangent to the curve. […]
Question A curve is such that i. Find ii. Verify that the curve has a stationary point when and determine its nature. iii. It is now given that the stationary point on the curve has coordinates (−1, 5). Find the equation of the curve. Solution i. Second derivative is the derivative of the derivative. If we have […]
Question i. The diagram shows part of the curve and part of the straight line meeting at the point , where and are positive constants. Find the values of and . ii. The function f is defined for the domain by Express in a similar way. Solution i. It is evident from the diagram that […]
Question In the diagram, D lies on the side AB of triangle ABC and CD is an arc of a circle with centre A and radius 2 cm. The line BC is of length 2√3 cm and is perpendicular to AC. Find the area of the shaded region BDC, giving your answer in terms of and . Solution […]
Question It is given that , for . Show that is a decreasing function. Solution To test whether a function is increasing or decreasing at a particular point , we take derivative of a function at that point. If , the function is increasing. If , the function is decreasing. If , the test is inconclusive. We are given that; […]
Question The position vectors of points A and B relative to an origin O are given by and where is a constant. i. In the case where OAB is a straight line, state the value of and find the unit vector in the direction of . ii. In the case where OA […]
Question Solve the equation , for . Solution We have the equation; We have the trigonometric identity; We can rewrite the identity as; Therefore; To solve this equation for , we can substitute . Hence, Now we have two options; Since; NOT POSSIBLE Using calculator we can find the values of . We utilize the symmetry property of to find other […]
Question The functions and are defined for by Solve the following equations for . i. , giving your answer in terms of . ii. , giving your answers correct to 2 decimal places. Solution i. We have the functions; We write as; We are given that; Therefore; ii. We have the functions; We write as; We are given that; Therefore; We utilize the […]
Question Find the coefficient of in the expansion of . Solution First rewrite the given expression in standard form. Expression for the general term in the Binomial expansion of is: In the given case: Hence; Since we are looking for the term of : we can equate Subsequently substituting in: Since we are interested in the coefficient of ;
Question The first term of a geometric progression is and the fourth term is . Find i. the common ratio, ii. the sum to infinity. Solution From the given information, we can compile following data for Geometric Progression (G.P); i. Expression for the general term in the Geometric Progression (G.P) is: Expressions for 4th term can be written as; […]