OCR MEI C1 (Core Mathematics 1)

1 Explain why each of the following statements is false. State in each case which of the symbols ⟹, ⟸ or ⇔ would make the statement true.

1. ABCD is a square ⇔ the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer

2 Complete each of the following by putting the best connecting symbol ⟵, ⟸ or ⇒) in the box. Explain your choice, giving full reasons.

1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\)

3 Select the best statement from $$\begin{aligned} & \mathrm { P } \Rightarrow \mathrm { Q }
& \mathrm { P } \Leftarrow \mathrm { Q }
& \mathrm { P } \Leftrightarrow \mathrm { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.

1. P: WXYZ is a quadrilateral with 4 equal sides
   \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
3. P : \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer

4 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$

5 Given that \(n\) is a positive integer, write down whether the following statements are always true (T), always false (F) or could be either true or false (E).

1. \(2 n + 1\) is an odd integer
2. \(3 n + 1\) is an even integer
3. \(n\) is odd \(\Rightarrow n ^ { 2 }\) is odd
4. \(n ^ { 2 }\) is odd \(\Rightarrow n ^ { 3 }\) is even

6 The converse of the statement ' \(\mathrm { P } \Rightarrow \mathrm { Q }\) ' is ' \(\mathrm { Q } \Rightarrow P\) '.
Write down the converse of the following statement. $$\text { ' } n \text { is an odd integer } \Rightarrow 2 n \text { is an even integer.' }$$ Show that this converse is false.

7 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .

1. P: \(x ^ { 2 } + x - 2 = 0\) Q: \(x = 1\)
2. P: \(y ^ { 3 } > 1\) Q: \(y > 1\)