# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#4

Question      i. Solve the equation  for . ii. How many solutions has the equation  for ? Solution    i.   We are given; To solve this equation for , we can substitute . Hence, Since given interval is  , for  interval can be found as follows; Multiplying the entire inequality with 2; Since ; Hence the given interval for  is . To solve  equation for interval , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#11

Question The function  is such that , for .      i. Find the coordinates and the nature of the stationary point on the curve . The function g is such that , , where  is a constant.    ii. State the smallest value of  for which  has an inverse. For this value of k,   iii. Find an expression for .   iv. Sketch, on […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#1

Question i. Prove the identity ii. Use this result to explain why  for . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can rewrite the identity as; Therefore; Again using the relation , we can rewrite; ii.   It is evident that in equation  for  neither   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#10

Question The equation of a line is , where is a constant, and the equation of a curve is .     i.       In the case where, the line intersects the curve at the points A and B. Find the equation of the perpendicular bisector of the line AB.    ii.       Find the set of values of  for which the line  intersects the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#9

Question A curve is such that . The curve has a maximum point at (2, 12). i. Find the equation of the curve. A point P moves along the curve in such a way that the x-coordinate is increasing at 0.05 units per second. ii. Find the rate at which the y-coordinate is changing when x = 3, stating whether […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#8

Question In the diagram, AB is an arc of a circle with centre O and radius . The line XB is a tangent to the circle at B and A is the mid-point of OX. i. Show that angle  radians. Express each of the following in terms of,  and : ii. the perimeter of the shaded region, iii. the area of the shaded region. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#7

Question The curve  intersects the x-axis at A. The tangent to the curve at A intersects the y-axis at C.     i.       Show that the equation of AC is .    ii.     Find the distance AC. Solution i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form of Equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#5

Question The diagram shows part of the curve  , crossing the y-axis at the point A. The point B(6, 1) lies on the curve. The shaded region is bounded by the curve, the y-axis and the line y = 1. Find the exact volume obtained when this shaded region is rotated through about the y-axis. Solution i.   We are given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#2

Relative to an origin , the position vectors of the points ,  and  are given by and Find     i.      the unit vector in the direction of , ii.   the value of the constant p for which angle . Solution i.   A vector in the direction of  is; Therefore, for the given case; A unit vector in the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#3

Question The first three terms in th expansion of  , in the ascending powers of  , are  . Find the values of the constants   and . Solution Expression for the Binomial expansion of  is: In the given case: Hence; Now We know that first three terms in th expansion of  are ; Comparing the given and derived expansions. First […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#6

Question The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.      i.       Find the common difference of the progression. The first term, the ninth term and the nth term of this arithmetic progression are the first term, the  second term and the third term respectively of a geometric progression.    ii.       Find […]