Questions C1 (1562 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
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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 \(k\).
OCR MEI C1 Q5
12 marks Moderate -0.3
5 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).
  1. Write down the radius of the circle and the coordinates of its centre.
  2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
  3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0$$
  4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle.
OCR MEI C1 Q3
11 marks Moderate -0.3
3 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_520_873_478_675} \captionsetup{labelformat=empty} \caption{Fig. 12.1}
\end{figure}
  1. State the value of \(x\) at A , where the arch meets the road.
  2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
  3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_528_870_1558_717} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
    \end{figure}
  4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.
OCR MEI C1 Q4
11 marks Moderate -0.3
4 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).
  1. State the coordinates of the centre and the radius of this circle.
  2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
  3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
  4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.
OCR MEI C1 Q2
13 marks Moderate -0.8
2 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).
  1. Show that the centre of this circle is \(C ( 4,2 )\) and find the radius of the circle.
  2. Show that the origin lies inside the circle.
  3. Show that AB is a diameter of the circle, where A has coordinates ( 2,7 ) and B has coordinates \(( 6 , - 3 )\).
  4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).
OCR MEI C1 Q3
12 marks Moderate -0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50cfc73d-850e-4a9b-b088-cc9741b66ffb-2_445_617_1008_741} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Not to scale A circle has centre \(\mathrm { C } ( 1,3 )\) and passes through the point \(\mathrm { A } ( 3,7 )\) as shown in Fig. 11.
  1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
  2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
  3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.
OCR MEI C1 Q1
11 marks Moderate -0.5
1 Point A has coordinates ( 4,7 ) and point B has coordinates ( 2,1 ).
  1. Find the equation of the line through A and B .
  2. Point C has coordinates \(( - 1,2 )\). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
  3. Find the coordinates of \(D\), the midpoint of AC. Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.
OCR MEI C1 Q2
2 marks Easy -1.2
2 A line has gradient 3 and passes through the point \(( 1 , - 5 )\). The point \(( 5 , k )\) is on this line. Find the value of \(k\).