# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#7

Question The function  is defined for the domain .      i.       Find the range of .    ii.       Sketch the graph of . A function g is defined by , for , where A is a constant.   iii.     State the largest value of  for which  has an inverse.   iv.       When  has this value, obtain an expression, in terms of , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#11

Question Relative to an origin , the position vectors of the points , and   are given by i.       Use a scalar product to find angle AOB, correct to the nearest degree.    ii.       Find the unit vector in the direction of  .   iii.       The point C is such that  , where  is a constant. Given that the lengths of  and  are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#3

Question      i.       Show that the equation  can be expressed as .    ii.       Hence solve the equation , for . Solution i.   First we need to manipulate the given equation to write it in a single trigonometric ratio i.e. ; Dividing the entire equation by ; Using the relation ; ii.   To solve this equation for , Using calculator we can find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#10

Question The equation of a curve is . i.       Show that the whole of the curve lies above the x-axis.    ii.       Find the set of values of x for which  is a decreasing function of x. The equation of a line is , where k is a constant.   iii.       In the case where k = 6, find the coordinates of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#9

Question A curve has equation  . i.       The normal to the curve at the point (4, 2) meets the x-axis at P and the y-axis at Q. Find the length of PQ, correct to 3 significant figures.    ii.       Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#8

Question In the diagram, ABC is a semicircle, centre O and radius 9 cm. The line BD is perpendicular to the diameter AC and angle AOB = 2.4 radians. i.       Show that BD = 6.08 cm, correct to 3 significant figures.    ii.       Find the perimeter of the shaded region.   iii.       Find the area of the shaded region. Solution i.   It is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#5

Question The diagram shows a rhombus ABCD. The points B and D have coordinates (2, 10) and (6, 2) respectively, and A lies on the x-axis. The mid-point of BD is M. Find, by calculation, the coordinates of each of M, A and C. Solution M is the mid-point of BD. To find the mid-point of a line we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#2

Question Find the gradient of the curve  at the point where x = 3. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: In the given case: We can rewrite the equation as; Rule for differentiation of  is: Rule for differentiation of  is: Rule for differentiation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#1

Question A curve is such that  . Given that the point (3, 8) lies on the curve, find the equation of the curve. Solution We can find equation of the curve from its derivative through integration; In the given case: Therefore; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is: If a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#4

Question i.     Find the first three terms in the expansion of  in ascending powers of . ii.       Find the value of  for which there no term in  in the expansion of  Solution i.   Expression for the Binomial expansion of  is: First rewrite the given expression in standard form. In the given case: Hence; ii.   To find the value of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#6

Question A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An  arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression,  find the first term and the […]