# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined for the domain .      i.       Solve the equation .    ii.       Sketch the graph of .   iii.       Find the set of values of  for which the equation  has no solution. The function  is defined for the domain .   iv.       State the largest value of  for which g has an inverse.    v.        For […]

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

Question The diagram shows the parallelogram OABC. Given that  and , find i.       the unit vector in the direction of .    ii.       the acute angle between the diagonals of the parallelogram,   iii.       the perimeter of the parallelogram, correct to 1 decimal place. Solution i.   A unit vector in the direction of  is; Therefore, for the given case; It is evident […]

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

Question The diagram shows a triangle  in which  is  and  is . The gradients of ,  and  are ,  and  respectively, where  is a positive constant. i.       Find the gradient of  and deduce the value of m.    ii.       Find the coordinates of C. The perpendicular bisector of  meets  at .   iii.       Find the coordinates of D. Solution i.   Expression for […]

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

Question The diagram shows part of the curve  , which crosses the x-axis at A and the y-axis at B. The normal to the curve at A crosses the y-axis at C.     i.       Show that the equation of the line AC is .    ii.       Find the length of BC. Solution i.   To find the equation of the line either we […]

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

Question A curve is such that    and the point (9, 2) lies on the curve.     i.       Find the equation of the curve.    ii.       Find the x-coordinate of the stationary point on the curve and determine the nature of the stationary point. Solution i.   We can find equation of the curve from its derivative through integration; For the given case; […]

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

Question The diagram shows the curve  and the line . Find the area of the shaded region. Solution      i.   Consider the diagram below. It is evident that; Therefore, first we find area under the curve. To find the area of region under the curve , we need to integrate the curve from point  to  along x-axis. It is […]

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

Question The acute angle  radians is such that tan , where  is a positive constant. Express, in terms of ,      i.           ii.          iii.        Solution      i.   From the diagram below it is evident that if  is acute angle i.e.  lies in the first quadrant, then  will be in second quadrant. From the basic trigonometry […]

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

Question The function  is defined by  for .      i.       Express  in the form .    ii.       State the range of .   iii.        Find the set of values of  for which . The function g is defined by  for .   iv.       Find the value of the constant  for which the equation  has two equal roots. Solution i.   We have the expression; We use method […]

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

Question The function  is such that    for .      i.       Express  in the form , stating the values of  and .    ii.       State the greatest and least values of .   iii.       Solve the equation . Solution i.   We have the function; We have the trigonometric identity; We can rewrite it as; Therefore the function becomes; Hence; ii.   From (i), we can rewrite […]

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

Question i.       Find the first 3 terms in the expansion of  in descending powers of .    ii.       Hence find the coefficient of  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 value of   in the expansion of […]

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

Question The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49.     i.       Find the first term of the progression and the common difference.  The nth term of the progression is 46.    ii.       Find the value of n. Solution i.   From the given information, we can compile following data for Arithmetic Progression (A.P); […]