# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#6

Question      i. Sketch the graph of the curve , for . The straight line , where  is a constant, passes through the maximum point of this curve for .    ii. Find the value of k in terms of .   iii. State the coordinates of the other point, apart from the origin, where the line and the curve intersect. Solution i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#8

Question The points , ,  and  have position vectors , ,   and  respectively. i.       Use a scalar product to show that  and  are perpendicular.    ii.       Show that  and  are parallel and find the ratio of the length of  to the length of . Solution We have the position vectors for points , ,  and ; i.   First of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#2

Question Find all the values of  in the interval  which satisfy the equation; Solution First we need to manipulate the given equation to write it in a single trigonometric ratio; Dividing the entire equation by ; Using the relation  we can rewrite the equation as, To solve this equation for , we can substitute . Hence, Since given interval is  , for  interval can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#11

Question The equation of a curve is .      i. Express  in the form , stating the numerical values of  and .    ii. Hence, or otherwise, find the coordinates of the stationary point of the curve.   iii. Find the set of values of  for which . The function g is defined by  for .   iv. State the domain and range […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#5

Question The function  is defined by , for  , where a and b are constants. It is given that  and .      i. Find the values of a and b.    ii. Solve the equation . Solution i.   We are given that; We utilize the given data. Subtracting both equations; We can substitute  in any of the above two equations to find the value […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#10

Question The equation of a curve is      i.      Calculate the gradient of the curve at the point where .    ii.       A point with coordinates  moves along the curve in such a way that the rate of increase of  has the constant value 0.03 units per second.  Find the rate of increase of   at the instant when […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#9

Question The diagram shows a semicircle ABC with centre O and radius 8 cm. Angle  radians. i.       In the case where , calculate the area of the sector BOC.    ii.       Find the value of  for which the perimeter of sector AOB is one half of the perimeter of sector BOC.   iii.       In the case where , show that the exact length […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#7

Question The line  has equation . The line  passes through the point  and is perpendicular to . i.       Find the equation of .    ii.       Given that the lines  and  intersect at the point B, find the length of AB. Solution      i.   To find the equation of the line either we need coordinates of the two points on […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#3

Question a)   Differentiate  with respect to . b)  Find Solution a)   First write the expression in equation form. We can rewrite the equation without fractions: In the given case: Rule for differentiation of  is: Rule for differentiation of  is: b)   The given case is: We can rewrite the expression without fractions: Rule for integration of  is: Rule for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#1

Question Find the value of the coefficient of  in the expansion of . Solution First rewrite the given expression in standard form. Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the value of the coefficient of   , i.e.  we can equate Finally substituting  in: Therefore the coefficient of  is .

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#4

Question In an arithmetic progression, the 1st term is −10, the 15th term is 11 and the last term is 41. Find the  sum of all the terms in the progression. Solution From the given information, we can compile following data for Arithmetic Progression (A.P) ; Expression for the sum of  number of terms in the Arithmetic Progression (A.P) […]