Questions — OCR MEI C1 (499 questions)

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OCR MEI C1 Q3
3 marks Easy -1.8
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
OCR MEI C1 Q4
2 marks Easy -1.8
4 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$
OCR MEI C1 Q5
3 marks Easy -1.8
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
OCR MEI C1 Q6
2 marks Easy -1.2
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.
OCR MEI C1 Q7
2 marks Moderate -0.5
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: \(y > 1\)
OCR MEI C1 Q1
5 marks Easy -1.2
1
  1. Expand and simplify \(( 3 + 4 \sqrt { 5 } ) ( 3 - 2 \sqrt { 5 } )\).
  2. Express \(\sqrt { 72 } + \frac { 32 } { \sqrt { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 Q2
5 marks Moderate -0.8
2
  1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 Q3
3 marks Moderate -0.8
3 Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac { 1 } { 3 } \pi r ^ { 2 } ( a + b )$$
OCR MEI C1 Q4
5 marks Moderate -0.8
4
  1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
  2. Simplify \(10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }\), giving your answer in the form \(a + b \sqrt { 5 }\).
OCR MEI C1 Q5
5 marks Easy -1.2
5
  1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
  2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 Q6
3 marks Easy -1.8
6 Make \(b\) the subject of the following formula. $$a = \frac { 2 } { 3 } b ^ { 2 } c$$
OCR MEI C1 Q7
5 marks Easy -1.2
7
  1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
  2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).
OCR MEI C1 Q8
4 marks Moderate -0.5
8 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.
OCR MEI C1 Q9
5 marks Easy -1.2
9
  1. Express \(\sqrt { 48 } + \sqrt { 27 }\) in the form \(a \sqrt { 3 }\).
  2. Simplify \(\frac { 5 \sqrt { 2 } } { 3 - \sqrt { 2 } }\). Give your answer in the form \(\frac { b + c \sqrt { 2 } } { d }\).
OCR MEI C1 Q10
5 marks Easy -1.2
10
  1. Simplify \(\frac { \sqrt { 48 } } { 2 \sqrt { 27 } }\).
  2. Expand and simplify \(( 5 - 3 \sqrt { 2 } ) ^ { 2 }\).
OCR MEI C1 Q11
5 marks Easy -1.2
11
  1. Express \(\sqrt { 75 } + \sqrt { 48 }\) in the form \(a \sqrt { 3 }\).
  2. Express \(\frac { 14 } { 3 - \sqrt { 2 } }\) in the form \(b + c \sqrt { d }\).
OCR MEI C1 Q12
5 marks Easy -1.2
12
  1. Express \(\frac { 1 } { 5 + \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
  2. Expand and simplify \(( 3 - 2 \sqrt { 7 } ) ^ { 2 }\).
OCR MEI C1 Q13
3 marks Easy -1.8
13 Make \(v\) the subject of the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\).
OCR MEI C1 Q14
3 marks Easy -1.8
14 Make \(t\) the subject of the formula \(s = \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 Q15
5 marks Easy -1.2
15
  1. Simplify \(\sqrt { 98 } \quad \sqrt { 50 }\).
  2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
OCR MEI C1 Q16
3 marks Easy -1.8
16 The volume of a cone is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). Make \(r\) the subject of this formula.
OCR MEI C1 Q17
5 marks Easy -1.3
17
  1. Simplify \(5 \sqrt { 8 } + 4 \sqrt { 50 }\). Express your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Express \(\frac { \sqrt { 3 } } { 6 \sqrt { 3 } }\) in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are rational.
OCR MEI C1 Q2
11 marks Easy -1.2
2 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{55e2d4f5-c84d-4577-988e-96071a220d60-2_689_811_430_662} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Write down the coordinates of B .
  2. Find the radius of the circle and hence write down the equation of the circle.
  3. AD is a diameter of the circle. Find the coordinates of D .
  4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\).
OCR MEI C1 Q3
12 marks Standard +0.3
3 The circle \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C .
  2. Find the coordinates of the intersections of the circle with the \(x\) - and \(y\)-axes.
  3. Show that the points \(\mathrm { A } ( 1,6 )\) and \(\mathrm { B } ( 7,4 )\) lie on the circle. Find the coordinates of the midpoint of AB . Find also the distance of the chord AB from the centre of the circle.
OCR MEI C1 Q4
3 marks Easy -1.8
4 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).