Hits: 192

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#6

Hits: 192 Question i.       Express  in the form , where  and , giving the exact  value of R and the value of  correct to 2 decimal places.    ii.       Hence solve the equation Giving all solutions in the interval . Solution      i.   We are given the expression; We are required to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#4

Hits: 183   Question a.   Find . b.   Express  in terms of  and hence find . Solution a.     We are required to find; Rule for integration of , or ; b.     We know that , therefore; Hence; Therefore; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#8

Hits: 182   Question The diagram shows the curve , for . The point  lies on the curve. i.       Show that the normal to the curve at Q passes through the point .    ii.       Find .   iii.       Hence evaluate Solution      i.   If a point P(x,y) lies on a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#5

Hits: 66   Question Solve the equation , giving all solutions in the interval . Solution We are given; We know that ; Let , then we can write; Now we have two options. Since ; We know that except where  or undefined, therefore; Using calculator we can find that; Properties of Domain Range Periodicity […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/21) | Q#8

Hits: 109   Question The diagram shows the curve , for . The point  lies on the curve.     i.       Show that the normal to the curve at Q passes through the point .    ii.       Find .   iii.       Hence evaluate Solution      i.   If a point P(x,y) lies on […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/21) | Q#5

Hits: 71   Question Solve the equation , giving all solutions in the interval . Solution We are given; We know that ; Let , then we can write; Now we have two options. Since ; We know that except where  or undefined, therefore; Using calculator we can find that;   Properties of Domain Range […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#8

Hits: 76   Question i.       Prove the identity    ii.       Hence solve the equation For . Solution      i.   We are given that; We utilize following two addition formulae;    ii.   We are required to solve; As demonstrated in (i); Therefore; Since   provided that ; Since ; Therefore, we solve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#4

Hits: 73   Question a.   Show that b.   By using an appropriate trigonometrical identity, find the exact value of Solution a.     We are required to show that; Rule for integration of  is: b.     We are required to find exact value of; We know that , therefore; Hence; Rule for integration of  is: Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/22) | Q#8

Hits: 115 Question i.       Prove the identity    ii.       Hence solve the equation For . Solution      i.   We are given that; We utilize following two addition formulae;    ii.   We are required to solve; As demonstrated in (i); Therefore; Since   provided that ; Since ; Therefore, we solve  for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/22) | Q#4

Hits: 106     Question a.   Show that b.   By using an appropriate trigonometrical identity, find the exact value of Solution a.     We are required to show that; Rule for integration of  is:   b.     We are required to find exact value of; We know that , therefore; Hence; Rule for integration of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/21) | Q#8

Hits: 144 Question      i.       By differentiating , show that if y = cot x then    ii.       By expressing in terms of and using the result of part (i), show that   iii.       Express cos 2x in terms of sin2 x and hence show that can be expressed as .  Hence […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/21) | Q#3

Hits: 97     Question      i.       Show that the equation can be written in the form    ii.       Hence solve the equation  for . Solution      i.   We are given; We apply following addition formula on left side of given equation. Therefore; Since;      ii.   We are required to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#4

Hits: 445   Question i.       Sketch the curve  for . ii.       By adding a suitable straight line to your sketch, determine the number of real roots of the equation State the equation of the straight line. Solution i.   We are required to sketch  for ; We can sketch the graph of  for  as follows. We can find the points of the graph as follows. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 52   Question Solve the equation  for   . Solution We can rearrange the given equation as; We have the trigonometric identity; This identity can be rearranged as; Substituting  expression in the given equation; Let ; We can solve this quadratic equation through factorization. Now we have two options. Since ; Using calculator we can find the values of . We utilize the symmetry […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#7

Hits: 807   Question A function  is defined by   for .      i.       State the range of .    ii.       State the exact value of .   iii.       Sketch the graph of . The function  is defined for the domain .   iv.       State the largest value of  for which g has an inverse.    v.       Obtain an expression, in terms of , for  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#11

Hits: 1242   Question The function  is defined for the domain .      i.       Solve the equation .    ii.       Sketch the graph of .   iii.       Find the set of values of  for which the equation  has no solution. The function  is defined for the domain .   iv.       State the largest value of  for which g has an inverse.    v.    […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 322 Question Solve the equation  for . Solution We have the equation; We have the trigonometric identity; We can write it as; Therefore; To solve this equation for , we can substitute . Hence, Now we have two options; Since; Using calculator we can find the values of . We utilize the symmetry property of   to find other […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#2

Hits: 467 Question Prove the identity Solution We have the equation; We have the relation , therefore, We have the trigonometric identity; We can write it as; Therefore; We have the relation , therefore,

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#4

Hits: 1231 Question      i.       Prove the identity    ii.      Hence solve the equation  for . . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can write it as; Therefore; Using the formula; ii.   To solve the equation  for , as demonstrated in (i), […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#4

Hits: 322 Question      i.       Show that the equation  can be expressed as .    ii.       Solve the equation   for . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can write it as; Therefore; ii.   To solve the equation  for , as demonstrated in (i), we can […]