# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#6

Question In the diagram, ABC is a triangle in which AB=4 cm, BC=6 cm and angle ABC=150◦. The line CX is perpendicular to the line ABX.      i.       Find the exact length of BX and show that    ii.       Show that the exact length of AC is  cm. Solution i.   It is evident that BCX is a right-angled triangle. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#4

Question The position vectors of points  and  are  and   respectively, relative to an origin . i.       Calculate angle .    ii.       The point  is such that  . Find the unit vector in the direction of . Solution i.   We recognize that  is angle between  and  . Hence we use scalar/dot product of  and . The scalar or dot product of two vectors  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#2

Question Given that  , find the exact value of      i.           ii.        Solution i.   We are given that; We can rewrite it as; We have the trigonometric identity; We can also rewrite it as; For the given case; Substituting   in this equation; ii.   We have the trigonometric relation; Taking square of both sides; For the given case; Substituting values […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#10

Question The function  is defined by  for .      i.       Find the set of values of  for which .    ii.       Express  in the form , stating the values of  and .   iii.       Write down the range of .    iv.       State, with a reason, whether  has an inverse. The function  is defined by   for .    v.       Solve the equation . Solution i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#9

Question The diagram shows an open container constructed out of 200 cm2 of cardboard. The two vertical end pieces are isosceles triangles with sides  cm,  cm and  cm, and the two side pieces are rectangles of length  cm and width  cm, as shown. The open top is a horizontal rectangle.      i.       Show that      ii.       Show that the volume,  cm3, of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#3

Question In the diagram, AOB is a sector of a circle with centre O and radius 12 cm. The point A lies on the side CD of the rectangle OCDB. Angle  radians. Express the area of the shaded region in the form  , stating the values of the integers  and . Solution From the given information we can compile […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#8

Question The equation of a curve is  . i.       Calculate the gradient of the curve at the point where x = 1.    ii.       A point with coordinates (x, y) moves along the curve in such a way that the rate of increase of y has a constant value of 0.02 units per second. Find the rate of increase of x when […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#7

Question The diagram shows the curve y = x(x − 1)(x − 2), which crosses the x-axis at the points O(0, 0), A(1, 0) and B(2, 0).     i.       The tangents to the curve at the points A and B meet at the point C. Find the x-coordinate of C.    ii.       Show by integration that 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 2006 | Oct-Nov | (P1-9709/01) | Q#5

Question The three points A(1, 3), B(13, 11) and C(6, 15) are shown in the diagram. The perpendicular from C to AB meets AB at the point D. Find i.       the equation of CD,    ii.       the coordinates of D. Solution      i.   To find the equation of the line either we need coordinates of the two points on the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#1

Question Find the coefficient of  in the expansion of Solution Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the term with , we can  equate Finally substituting  in: Therefore the coefficient of  is .