# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#4

Question i.     Sketch and label, on the same diagram, the graphs of  and , for the interval . ii.       Hence state the number of solutions of the equation  in the interval . Solution i.   We sketch both graphs by considering multiple values of  within the given range . Now we can sketch both graphs. Blue graph is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#8

Question The points A and B have position vectors  and  respectively, relative to an origin O.      i.       Use a scalar product to calculate angle AOB, giving your answer in radians correct to 3 significant figures.    ii.       The point C is such that   . Find the unit vector in the direction of Solution i.   The angle AOB is the angle between  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#9

Question The function , where  is a constant, is defined for all real .      i.       In the case where , solve the equation . The function  is defined for all real .    ii.       Find the value of  for which the equation  has exactly one real solution. The function  is defined for the domain.   iii.       Express  in the form , where  and  are constants.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#6

Question The function  is defined for the domain .      i. Express  in the form , stating the values of  and .    ii. Hence find the values of  for which .   iii. State the range of . Solution i.   We have the function; We have the trigonometric identity; We can rewrite the identity as; Substituting the expression in the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#10

Question A curve has equation  .      i.       Write down expressions for  and    ii.      Find the coordinates of the stationary point on the curve and determine its nature.   iii.       Find the volume of the solid formed when the region enclosed by the curve, the x-axis and the lines x = 1 and x = 2 is rotated […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#7

Question A curve is such that  and  is a point on the curve.      i.       Find the equation of the normal to the curve at P, giving your answer in the form .    ii.       Find the equation of the curve. Solution i.   To find the equation of the normal to the curve at P; To find the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#5

Question The equation of a curve is  and the equation of a line is . The curve and the line intersect at the points A and B.      i.       The mid-point of AB is M. Show that the coordinates of M are .    ii.       Find the coordinates of the point Q on the curve at which the tangent is parallel […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#3

Question In the diagram, AC is an arc of a circle, centre O and radius 6 cm. The line BC is perpendicular to OC and OAB is a straight line. Angle radians. Find the area of the shaded region, giving your answer in terms of  and . Solution      i.   It is evident from the diagram that; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | Oct-Nov | (P1-9709/01) | Q#1

Question Find 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 term with  , 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 2004 | Oct-Nov | (P1-9709/01) | Q#2

Question Find      i.       the sum of the first ten terms of the geometric progression 81, 54, 36, . . .    ii.       the sum of all the terms in the arithmetic progression 180, 175, 170, . . . , 25. Solution i.   From the given information, we can compile following data for Geometric Progression (G.P) ; Expression for Common […]