# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#11

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#10

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.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#8

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#7

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#4

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#2

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; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#9

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#3

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#6

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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#1

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 ;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#5

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; […]