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

Question The diagram shows the curve  and the point A(1, 2) which lies on the curve. The tangent to the curve at A cuts the y-axis at B and the normal to the curve at A cuts the x-axis at C. i.           Find the equation of the tangent AB and the equation of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#9

Question The position vectors of points A and B relative to an origin O are  and  respectively. The position vectors of points C and D relative to O are  and  respectively. It is given that and      i.       Find the unit vector in the direction of  .    ii.       The point E is the mid-point of CD. Find angle EOD. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#8

Question The diagram shows the curve  and the straight line . The curve and straight line intersect at  and , where  is a constant. i.      Show that . ii.   Find, showing all necessary working, the area of the shaded region. Solution i.   It is evident from the diagram that line and the curve intersect at two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#7

Question i.       Solve the equation  , for .          ii.    The smallest positive solution of the equation , where  is a positive integer, is . State the value of  and hence find the largest solution of this equation in the interval . Solution i.   We have the equation; We have the trigonometric identity; We can rewrite the identity as; Therefore; To […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#6

Question The diagram shows a sector OAB of a circle with centre O and radius . Angle AOB is  radians. The point C on OA is such that BC is perpendicular to OA. The point D is on BC and the circular arc AD has centre C.     i.       Find AC in terms of  and .    ii.    Find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#5

Question A curve has equation  . Verify that the curve has a stationary point at  and determine its nature. Solution i.   A stationary point  on the curve  is the point where gradient of the curve is equal to zero; Therefore first we need gradient of the given curve. Gradient (slope) of the curve is the derivative of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#3

Question An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#2

Question A curve is such that  and the point (2, 4) lies on the curve. Find the equation of the curve. Solution i.   We can find equation of the curve from its derivative through integration; We are given that; Rule for integration of  is: Rule for integration of  is: If a point   lies on the curve , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#10

Question The function  is defined by  for .      i.       Express  in the form , and hence state the coordinates of the vertex of the graph of  . The function  is defined by , for .    ii.       State the range of .   iii.       Find an expression for  and state the domain of . Solution i.   We have the function; We have the expression; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#4

Question      i.       Find the first 3 terms in the expansion of , in ascending 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.   First we know from (i) that Therefore: […]

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

Question The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first n  terms is n. Find the value of the positive integer n. Solution   From the given information, we can compile following data for Arithmetic Progression (A.P); The expression for difference  in Arithmetic Progression (A.P) is: For the given case; […]