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.
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Since ;
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Using calculator; |
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Properties of |
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Domain |
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Range |
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Periodicity |
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Odd/Even |
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Translation/ Symmetry |
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We utilize the periodicity property of to find other solutions (roots) of
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Therefore;
For;
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We are given that for
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Only following solutions (roots) are within the given interval ;
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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.
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Using calculator; |
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Using calculator; |
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Properties of |
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Domain |
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Range |
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Periodicity |
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Odd/Even |
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Translation/ Symmetry |
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We utilize the periodicity property of to find other solutions (roots) of
: .
Therefore;
For;
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We are given that for
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Only following solutions (roots) are within the given interval ;
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Hence, the equation for
has 03 solutions when
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