# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#8

Question The diagram shows a rhombus ABCD in which the point A is (−1, 2), the point C is (5, 4) and the point B lies on the y-axis. Find      i.       the equation of the perpendicular bisector of AC,    ii.       the coordinates of B and D,   iii.       the area of the rhombus. Solution i.   Consider the diagram below. It is evident that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#9

Question The diagram shows part of the curve  which has a minimum point at M. The line  intersects the curve at the points A and B.     i.        Find the coordinates of A, B and M.    ii.       Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. Solution i.   It is evident from […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#7

Question The diagram shows a metal plate ABCDEF which has been made by removing the two shaded regions from a circle of radius 10 cm and centre O. The parallel edges AB and ED are both of length 12 cm.     i.        Show that angle DOE is 1.287 radians, correct to 4 significant figures.    ii.       Find the perimeter of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#6

Question Relative to an origin , the position vectors of the points ,  and   are given by i.       Use a scalar product to find angle ABC.    ii.       Find the perimeter of triangle ABC, giving your answer correct to 2 decimal places. Solution i.   We recognize that angle ABC is between vectors  and  Therefore, first we find vectors  and . A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#5

Question The equation of a curve is such that . Given that the curve passes through the point , find     i.       the equation of the normal to the curve at P    ii.       the equation of the curve. Solution     i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#4

Question      i.       Show that the equation  can be expressed as .    ii.       Solve the equation   for . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can write it as; Therefore; ii.   To solve the equation  for , as demonstrated in (i), we can write the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#10

Question The function  is defined for .      i.       Find the values of the constant  for which the line  is a tangent to the curve .    ii.       Express  in the form of , where ,  and  are constants.   iii.       Find the range of . The function g is defined by  is defined for .   iv.       Find the smallest value of  for which g has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#3

Question The function  is defined for . Given that  and  find      i.       the values of  and ,    ii.       the range of .   iii.       the exact value of . Solution i.   We have the function; We are given that; Since Since Subtracting both equation; Substituting  in anyone of the above equations will yield value of . We choose; ii.   We have the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#2

Question i.    Find the first 3 terms, in descending powers of , in the expansion of  .    ii.       Find the coefficient of  in  in the expansion of . Solution i.   First rewrite the given expression in standard form. Expression for the Binomial expansion of  is: In the given case: Hence; ii.   To find the coefficient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#1

Question The first term of a geometric progression is 12 and the second term is −6. Find     i.       the tenth term of the progression,    ii.       the sum to infinity. Solution i.   From the given information, we can compile following data for Geometric Progression (G.P); To find the tenth term; Expression for the general term  in the Geometric Progression (G.P) is: […]