# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/22) | Q#3

Question A curve has equation Find the equation of the normal to the curve at the point (1, 2). Give your answer in the form ax + by  + c = 0, where a, b and c are integers. Solution We are given equation of the curve as; We are required to find the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/22) | Q#2

Question i.       Find where a is a positive constant.    ii.       Deduce the value of Solution      i.  We are given that; Rule for integration of  is: Rule for integration of , or ;      ii.   We are required to deduce the value of. From (i) we have demonstrated; Therefore replacing  with;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/23) | Q#5

Question i.       Given that  and  are factors of    ii.       When a and b have these values, factorise completely, and hence solve the equation giving any answers correct to 3 significant figures. Solution      i.   We are given that;   We are also given that and are factors of . We can write the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/23) | Q#3

Question A.         Find B.         Find the exact value of Solution A.   We are required to find; We know that; It can be written for  as; It can be rearranged as; Hence, integral can be written as; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:   B.  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/21) | Q#5

Question i.       Given that  and  are factors of    ii.       When a and b have these values, factorise completely, and hence solve the equation giving any answers correct to 3 significant figures. Solution      i.   We are given that;    We are also given that and are factors of . We can write […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/21) | Q#3

Question A.         Find B.         Find the exact value of Solution A.   We are required to find; We know that; It can be written for  as; It can be rearranged as; Hence, integral can be written as; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:   B.  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#4

Question The parametric equations of a curve are Find the equation of the tangent to the curve at the point for which t = 0. Give your answer in the  form ax + by + c = 0, where a, b and c are integers. Solution We are required to find the equation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#4

Question The parametric equations of a curve are Find the equation of the tangent to the curve at the point for which t = 0. Give your answer in the  form ax + by + c = 0, where a, b and c are integers. Solution We are required to find the equation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#6

Question i.       Show that ii.       Use the trapezium rule with four intervals to find an approximation to giving your answer correct to 3 significant figures. Solution i.   We are required to show that; Rule for integration of  is: This integral is valid only when . Division Rule; Power Rule; ii.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#4

Question      i.       By sketching a suitable pair of graphs, show that the equation has exactly one real root.    ii.       Show by calculation that the root lies between 2.0 and 2.5.   iii.       Use the iterative formula to find the root correct to 3 decimal places. Give  the result of each iteration to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#2

Question Find the gradient of each of the following curves at the point for which x = 0.     i.           ii.         Solution      i.   We are required to find the gradient of the curve at the point for which x = 0. Therefore first we need to find the expression for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/23) | Q#2

Question The variables x and y satisfy the equation y = a(bx), where a and b are constants. The graph of      ln y against x is a straight line passing through the points (0.75,1.70) and (1.53,2.18), as shown in  the diagram. Find the values of a and b, correct to 2 decimal […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/22) | Q#4

Question a.   Find the value of x satisfying the equation 2 ln (x – 4) − ln x = ln 2. b.   Use logarithms to find the smallest integer satisfying the inequality 1.4y > 1010 Solution a.   We are given; Power Rule; Division Rule; Taking anti-logarithm of both sides;  for any We have […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/21) | Q#2

Question The variables x and y satisfy the equation y = a(bx), where a and b are constants. The graph of      ln y against x is a straight line passing through the points (0.75,1.70) and (1.53,2.18), as shown in  the diagram. Find the values of a and b, correct to 2 decimal […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#5

Question The variables x and y satisfy the equation y = k(2px), where k and p are constants. The graph of     ln y  against x is a straight line passing through the points (1.35,1.87) and (3.35, 3.81), as shown in  the diagram. Find the values of k and p, correct to 2 decimal […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#5

Question The variables x and y satisfy the equation y = k(2px), where k and p are constants. The graph of     ln y  against x is a straight line passing through the points (1.35,1.87) and (3.35, 3.81), as shown in  the diagram. Find the values of k and p, correct to 2 decimal […]