Questions C1 (1442 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI C1 Q15
Moderate -0.3
15
  1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
  2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
  3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
  4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.
OCR MEI C1 Q1
Moderate -0.8
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
OCR MEI C1 Q2
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. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\)
OCR MEI C1 Q3
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
Easy -1.8
4 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$
OCR MEI C1 Q5
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
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
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
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
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
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
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
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
Easy -1.8
6 Make \(b\) the subject of the following formula. $$a = \frac { 2 } { 3 } b ^ { 2 } c$$
OCR MEI C1 Q7
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
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
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
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
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
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
Easy -1.8
13 Make \(v\) the subject of the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\).
OCR MEI C1 Q14
Easy -1.8
14 Make \(t\) the subject of the formula \(s = \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 Q15
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
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
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.