# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#11

Question

i.      Solve the equation  for .

ii.      Find the set of values of  for which the equation has no solution.

iii.      For the equation , state the value of for which there are three solutions in the interval , and find these solutions.

Solution

i.

We have the equation;

Let ;

Now we have two options.

Since ;

 Using calculator;

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

We utilize the periodicity property of   to find other solutions (roots) of : .

Therefore;

For;

 for for

We are given that for .

Only following solutions (roots) are within the given interval ;

ii.

We are given the equation;

It is evident that given equation is a quadratic one.

Standard form of quadratic equation is;

Expression for discriminant of a quadratic equation is;

If  ; Quadratic equation has two real roots.

If  ; Quadratic equation has no real roots.

If  ; Quadratic equation has one real root/two equal roots.

Therefore, if the given equation has no solution, then for it;

As can be seen;

Hence;

iii.

We are given the equation;

It is evident that given equation is a quadratic one.

We are given that this equation has three solutions.

We know that only cubic equation can have three solutions but not the quadratic  equation.

The solutions to a quadratic equation are the places where it crosses the x-axis. Since the graph  of a quadratic equation is a parabola, it is impossible for it to cross the axis in more than two  places.

Therefore, we need to find a situation where this equation can yield three solutions.

By inspection it is evident that if , the equation will yield more than two  solutions.

Let ;

Now we have two solutions.

 Using calculator; Using calculator;

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

We utilize the periodicity property of   to find other solutions (roots) of : .

Therefore;

For;

 for for

We are given that for .

Only following solutions (roots) are within the given interval ;

Hence, the equation for has 03 solutions  when .