# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#7

Question A function is defined by   for .    i.     State the range of .  ii.     State the exact value of .  iii.      Sketch the graph of .  iv.      Obtain an expression, in terms of , for Solution i.   We have the function;  for This can be written as; We know that range […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#11

Question The equation of a curve is  .     i.        Find an expression for  and determine, with a reason, whether the curve has any stationary points.    ii.       Find the volume obtained when the region bounded by the curve, the coordinate axes and the line  is rotated through 360o about the x-axis.   iii.       Find the set of values of  for which […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#10

Question The equation of a curve is .     i.       Show that the equation of the normal to the curve at the point (3, 6) is .    ii.       Given that the normal meets the coordinate axes at points A and B, find the coordinates of the mid-point of AB.   iii.       Find the coordinates of the point at which the normal meets […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#9

Question The diagram shows two circles, C1 and C2, touching at the point T. Circle C1 has centre P and radius 8 cm; circle C2 has centre Q and radius 2 cm. Points R and S lie on C1 and C2 respectively, and RS is a tangent to both circles. i.       Show that RS=8 cm.    ii.       Find angle RPQ in radians […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#8

Question The diagram shows a metal plate consisting of a rectangle with sides  cm and  cm and a quarter-circle of radius  cm. The perimeter of the plate is 60 cm. i.       Express  in terms of .    ii.       Show that the area of the plate,  cm2, is given by . Given that  can vary,   iii.       find the value of  at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#5

Question The diagram shows a pyramid OABC with a horizontal base OAB where OA = 6 cm, OB = 8 cm and angle . The point C is vertically above O and OC = 10 cm. Unit vectors ,  and  are parallel to ,  and  as shown. Use a scalar product to find angle ACB. Solution i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#4

Question      i.       Prove the identity    ii.      Hence 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; Using the formula; ii.   To solve the equation  for , as demonstrated in (i), we can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#1

Question Find   Solution Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#3

Question Functions  and  are defined for  by  Express  in the form , where a, b and c are constants. Solution i.   We have the functions; For ; We have the expression; We use method of “completing square” to obtain the desired form. We take out factor ‘4’ from the terms which involve ; Next we complete the square for the terms […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#2

Question In the expansion of  , where  is constant, the coefficient of  is . Find the coefficient of . Solution Expression for the Binomial expansion of  is: In the given case: Hence; We know that coefficient of   is . i.e. The coefficient of   is ;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#6

Question a)   The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75. Find the  first term and the common difference. b)  The first term of a geometric progression is 16 and the fourth term is  . Find the sum to infinity  of the progression. Solution a)   From […]