# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#10

Question The diagram shows part of the curve  . The curve intersects the y-axis at A . The  normal to the curve at A intersects the line  at the point B.      i.       Find the coordinates of B.    ii.       Show, with all necessary working, that the areas of the regions marked P and Q are equal. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#9

Question The equation of a curve is , where  is a positive constant.      i.       Show that the origin is a stationary point on the curve and find the coordinates of the other  stationary point in terms of .    ii.          Find the nature of each of the stationary points. Another curve has equation .   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#8

Question The function  is defined for .      i.       Solve the equation , giving your answer correct to 2 decimal places.    ii.       Sketch the graph of .   iii.       Explain why  has an inverse.   iv.       Obtain an expression for . Solution      i.   We are given; We are required to solve; Therefore; Using calculator;    ii.   Ware required to sketch […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#6

Question The line with gradient -2 passing through the point P(3t,2t) intersects the x-axis at A and the y-axis  at B.      i.       Find the area of triangle AOB in terms of t. The line through P perpendicular to AB intersects the x-axis at C.    ii.       Show that the mid-point of PC lies on the line y = x. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#5

Question A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius r cm.      i.       Show that the area of the sector, A cm2, is given by .    ii.       Express A in the form , where a and b are constants.   iii.       Given that r can vary, state […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#4

Question Relative to the origin O, the position vectors of points A and B are given by and i.       Find the cosine of angle AOB. The position vector of C is given by .      ii.       Given that AB and OC have the same length, find the possible values of k. Solution      i.   We recognize that angle […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#2

Question   The diagram shows the curve  and the points  and . The point Q lies on the  curve and PQ is parallel to the y-axis.   i. Express the area, A, of triangle XPQ in terms of .   The point P moves along the x-axis at a constant rate of 0.02 units per second and Q moves along  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#1

Question Given that  is an obtuse angle measured in radians and that , find, in terms of , an  expression for                     i.                          ii.                         iii.        Solution We are given that  is an obtuse angle. An obtuse angle  is such that . We are also given that; We know that for an obtuse angle  the sine of that obtuse angle equals the sine […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#3

Question      i.       Find the first three terms, in ascending powers of , in the expansion of  a.   b.      ii.       Hence find the coefficient of  in the expansion of . Solution i.   a)   Expression for the Binomial expansion of  is: First we rewrite the given equation in the standard form; In the given case: Hence; b)   Expression for the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#7

Question a)   The third and fourth terms of a geometric progression are  and  respectively. Find the sum to infinity of the progression. b)  A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic  progression. Given that the angle of the largest sector is 4 times the angle of […]