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IBMathsDPAnalysis & Approaches (AA)HLExam Questions1. Number & AlgebraProof by Induction & Contradiction

Proof by Induction & Contradiction (DP IB Analysis & Approaches (AA): HL): Exam Questions

2 hours20 questions
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15 marks

Use proof by contradiction to prove that there is no x element of straight real numbers such that  begin mathsize 16px style negative fraction numerator 2 over denominator x minus 2 end fraction equals x minus 3 end style.

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24 marks

Using the method of proof by contradiction, prove that square root of 7 is irrational.

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34 marks

Using mathematical induction, prove that 6 to the power of n minus 1 is divisible by 5 for n element of straight integer numbers comma space n greater or equal than 1.

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46 marks

Prove by mathematical induction 3 to the power of n greater or equal than 1 plus 2 n comma given n greater or equal than 0.

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56 marks

Prove by mathematical induction that if begin mathsize 16px style y equals fraction numerator 1 over denominator 1 minus x end fraction end style  then begin mathsize 16px style fraction numerator d to the power of n y over denominator d x to the power of n end fraction equals fraction numerator n factorial over denominator left parenthesis 1 minus x right parenthesis to the power of n plus 1. end exponent end fraction end style

How did you do?

66 marks

Prove by induction that

large capital sigma from r equals 1 to n ofr squared equals 1 over 6 n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses

for all values of n comma space n element of straight integer numbers to the power of plus.

How did you do?

14 marks

Prove by mathematical induction that  9 to the power of 2 n end exponent minus 1 comma space n element of straight integer numbers comma space n greater or equal than 1 is divisible by 16.

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24 marks

Prove by contradiction that square root of 10 is irrational.

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36 marks

Use mathematical induction to prove that the nth derivative of the function f open parentheses x close parentheses equals 5 over x is  given by

 fraction numerator 5 open parentheses negative 1 close parentheses to the power of n n factorial over denominator x to the power of open parentheses n plus 1 close parentheses end exponent end fraction 

for all integers, n, where n greater or equal than 1.

 

How did you do?

46 marks

Use proof by contradiction to prove that a squared minus 8 b minus 11 not equal to 0 if a comma space b element of straight integer numbers.

How did you do?

56 marks

Prove by mathematical induction, that for n element of straight integer numbers to the power of plus,

1 plus 2 open parentheses 1 half close parentheses plus 3 open parentheses 1 half close parentheses squared plus 4 open parentheses 1 half close parentheses cubed plus... plus n open parentheses 1 half close parentheses to the power of n minus 1 end exponent equals 4 minus fraction numerator n plus 2 over denominator 2 to the power of n minus 1 end exponent end fraction

How did you do?

66 marks

Use a contradiction to prove that the difference between a rational number and an irrational number is irrational.

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76 marks

Prove by mathematical induction that if f open parentheses x close parentheses equals x e to the power of 2 x end exponent then f to the power of open parentheses n close parentheses end exponent open parentheses x close parentheses equals open parentheses 2 to the power of n x plus n 2 to the power of n minus 1 end exponent close parentheses e to the power of 2 x end exponent .

How did you do?

86 marks

Prove by mathematical induction that

open parentheses cos space theta minus straight i space sin space theta close parentheses to the power of n equals cos open parentheses n theta close parentheses minus straight i space sin open parentheses n theta close parentheses comma space for space all space n element of straight integer numbers to the power of plus

How did you do?

16 marks

Prove that 2 to the power of n plus 2 end exponent plus 3 to the power of 3 n end exponent is divisible by 5 for n element of straight integer numbers comma space n greater or equal than 0.

How did you do?

24 marks

Prove that there are an infinite number of prime numbers.

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35 marks

Prove that the equation 5 x to the power of 4 plus 15 x cubed minus 20 x squared minus 4 equals 0 has no integer solutions.

How did you do?

4a3 marks

Show that the derivative of y equals x e to the power of negative x end exponent is

fraction numerator d y over denominator d x end fraction equals e to the power of negative x end exponent open parentheses 1 minus x close parentheses.

How did you do?

4b7 marks

Prove, by mathematical induction, that for n greater or equal than 1 comma

fraction numerator straight d to the power of n over denominator straight d x to the power of n end fraction equals e to the power of negative x end exponent open square brackets open parentheses negative 1 close parentheses to the power of n minus 1 end exponent n plus open parentheses negative 1 close parentheses to the power of n x close square brackets

How did you do?

57 marks

Prove that open square brackets r open parentheses cos space theta minus i sin theta close parentheses close square brackets to the power of n equals r to the power of n open square brackets cos open parentheses n theta close parentheses minus i sin open parentheses n theta close parentheses close square brackets,  for all n element of straight integer numbers to the power of plus  .

How did you do?

67 marks

Prove by Induction that

sum from r equals 1 to n of r open parentheses r squared minus 6 close parentheses equals 1 fourth n open parentheses n plus 1 close parentheses open parentheses n plus 4 close parentheses open parentheses n minus 3 close parentheses

for all positive integer values of n.

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