Figure it Out - Mock-Test 1: Practice Problems
1. Which is greater: (a + b)² or a² + b²? Justify your answer.
2. Express 84 as the difference of two squares.
3. Find 295², 88², 198², 1003², and 315² using the identities you have learnt so far.
4. Does the identity a² − b² = (a + b)(a − b) hold for decimal numbers? What about irrational numbers like √2 and √3? Justify your answer with examples.
Bonus Questions
5. Which is greater: (2a)² or 2a²? Support your answer with reasoning.
6. Express 96 as the difference of two squares in at least two different ways.
7. Without actually multiplying, find the values of:
- 103 × 97
- 52 × 48
- 205 × 195
(Hint: Use the identity for difference of squares)
8. Can every positive integer be expressed as the difference of two squares? Test this for numbers 1 through 10 and identify any pattern.
9. Using algebraic identities, simplify:
- (x + 5)² − (x − 5)²
- (2m + 3)² − (2m − 3)²
10. If a + b = 10 and ab = 21, find the value of a² + b² without finding a and b separately.
Figure it Out - Mock-Test 2: Practice Problems
Which is greater: (a + b)² or (a − b)²? Justify your answer.
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Express 225 as the difference of two squares.
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Compute the following squares using algebraic identities (no long multiplication): 398², 88², 155², 1203², and 176².
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Do the identities a² − b² = (a − b)(a + b) and (a ± b)² = a² ± 2ab + b² hold for irrational numbers (like √2) and complex numbers (like 3 + 2i)? Justify your answer.
Bonus (optional, for extra practice):
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If x − y = 9 and x + y = 31, find x² − y² without finding x and y individually. Explain your method.
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Show that any even integer N can be expressed as a difference of two squares if and only if N is divisible by 4. Give examples and a brief proof sketch.
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