# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#7

Question     i.      Show that the equation  can be expressed as Where   ii.      Hence solve the equation   for . Solution i.   We are given the equation; We have the trigonometric identity; From this we can substitute in above equation; Let ;      ii.   We are required to solve the equation  for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#6

Question a)   Given that x > 0, find the two smallest values of x, in radians, for which . Show  all necessary working. b)  The function is defined for . i.     Express f(x) in the form , where a and b are constants. ii.      Find the range of f. Solution a)     We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#8

Question The diagram shows a sector OAC of a circle with centre O. Tangents AB and CB to the circle meet  at B. The arc AC is of length 6 cm and angle  radians. i.       Find the length of OA correct to 4 significant figures.    ii.       Find the perimeter of the shaded region.  iii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#5

Question     i.      Given that , show, without using a calculator, that .    ii.       Hence, showing all necessary working, solve the equation  for Solution i.   We are given that; Since ; We have the trigonometric identity; From this we can write; Therefore; Let ; Now we have two options. Since ; We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#9

Question The functions is defined for , where and are positive constants. The diagram shows the graph of y = f(x).     i.      In terms of and state the range of .    ii.       State the number of solutions of the following equations.                  a)        […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#3

Question The diagram shows triangle ABC which is right-angled at A. Angle radians and AC = 8  cm. The points D and E lie on BC and BA respectively. The sector ADE is part of a circle with centre  A and is such that BDC is the tangent to the arc DE at D. i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#6

Question The equation of a curve is and the equation of a line is . i.State the smallest and largest values of y for both the curve and the line for . ii.Sketch, on the same diagram, the graphs of and for . iii.State the number of solutions of the equation for . Solution i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#4

Question Angle x is such that sin x = a + b and cos x = a − b, where a and b are constants. A curve is such that . The point P (2,9) lies on the curve.     i.       Show that a2 +b2 has a constant value for all values of x.    […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#9

Question The functions f is defined by  for  ,     i.      State the range of .   ii.     Sketch the graph of . The functions g is defined by for , where is a constant.  iii.     State the largest value of for which g has an inverse.  iv.     For this value of , find an […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#6

Question      i.       Prove the identity    ii.       Hence solve the equation  for Solution i.   First, we are required to show that; Since ; We have the trigonometric identity; From this we can write; Therefore; We have the algebraic formula;      ii.   We are required to solve the equation;   For . […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#7

Question a)  Solve the equation   for . b)    The diagram shows part of the graph of , where is measured in radians and and are constants. The curve intersects the x-axis at  and the y-axis at . Find the values of and . Solution a)   We are required to solve the equation for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#8

Question     i.       Express  in the form , where and  , giving the value of  correct to 2 decimal places.    ii.       Hence solve the equation  for . Solution      i.   We are given the expression; We are required to write it in the form; If  and are positive, then; can be written in the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#7

Question A curve is defined by the parametric equations for .       i.       Find the exact gradient of the curve at the point for which .    ii.       Find the value of at the point where the gradient of the curve is 2, giving the value correct to 3 significant figures. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#6

Question a.   Show that b.   Find the exact value of Show all necessary working. Solution a.     We are required to show; Rule for integration of  is: Division Rule; Power Rule; b.     We are required to find; Since , we can rearrange to write; Rule for integration of  is: Rule for integration of  is: […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#7

Question      i.       Show that    ii.       Hence, show that   iii.       Solve the equation For . Show all the necessary working. Solution i.   First we are required to show that;   provided that   provided that   provided that ii.   We are required to show that; As shown in (i); Therefore; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#4

Question a.   Find b.   Find the exact value of Show all necessary working. Solution a.   We are given that; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is: b.   We are required to find the exact value of; Rule for integration of  is: Rule for integration of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#6

Question a.   Showing all necessary working, solve the equation for . b.   Showing all necessary working, solve the equation for . Solution a.     We are given;   provided that   provided that Let , then; We are given that ; interval for  can be found as follows. Multiplying entire inequality with 2; Hence, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#5

Question It is given that Where  is a constant.     i.       Show that    ii.       Using the equation in part (i), show by calculation that 0.5 < a < 0.75.   iii.       Use an iterative formula, based on the equation in part (i), to find the value of a  correct to 3 significant figures. Give […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#6

Question a.   Showing all necessary working, solve the equation for . b.   Showing all necessary working, solve the equation for . Solution a.     We are given;   provided that   provided that Let , then; We are given that ; interval for  can be found as follows. Multiplying entire inequality with 2; Hence, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#5

Question It is given that Where  is a constant.     i.       Show that    ii.       Using the equation in part (i), show by calculation that 0.5 < a < 0.75.   iii.       Use an iterative formula, based on the equation in part (i), to find the value of a  correct to 3 significant figures. Give […]