Complete the table below - Edexcel - A-Level Maths Pure - Question 4 - 2017 - Paper 2
Question 4
Complete the table below. The first one has been done for you.
For each statement you must state if it is always true, sometimes true or never true, giving a reason... show full transcript
Worked Solution & Example Answer:Complete the table below - Edexcel - A-Level Maths Pure - Question 4 - 2017 - Paper 2
Step 1
When a real value of x is substituted into $x^{2} - 6x + 10$ the result is positive.
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Answer
To determine if the expression x2−6x+10 is always positive, we can analyze its discriminant:
D=b2−4ac=(−6)2−4(1)(10)=36−40=−4.
Since the discriminant is negative, the quadratic does not intersect the x-axis and therefore does not have real roots. Consequently, the function is always positive for all real values of x. Thus, it is
Always True.
Step 2
If $ ax > b $ then $ x > \frac{b}{a} $.
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Answer
This statement is not always true. If a<0, the inequality flips when dividing by a, which indicates that x<ab. Thus, we can state that:
Sometimes True because the conclusion depends on the sign of a. If a>0, the statement holds; if a<0, it does not.
Step 3
The difference between consecutive square numbers is odd.
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Answer
The difference between consecutive square numbers can be expressed as:
(n+1)2−n2=(n2+2n+1)−n2=2n+1.
Since 2n+1 is always odd for any integer n, we can conclude that this statement is:
