Questions — OCR C1 (324 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 C1 Q5
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.
  1. Calculate the radius of the circle and the coordinates of the centre.
  2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
  3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).
OCR C1 2005 January Q1
1
  1. Express \(11 ^ { - 2 }\) as a fraction.
  2. Evaluate \(100 ^ { \frac { 3 } { 2 } }\).
  3. Express \(\sqrt { 50 } + \frac { 6 } { \sqrt { 3 } }\) in the form \(a \sqrt { } 2 + b \sqrt { } 3\), where \(a\) and \(b\) are integers.
OCR C1 2005 January Q2
2 Given that \(2 x ^ { 2 } - 12 x + p = q ( x - r ) ^ { 2 } + 10\) for all values of \(x\), find the constants \(p , q\) and \(r\).
OCR C1 2005 January Q3
3
  1. The curve \(y = 5 \sqrt { } x\) is transformed by a stretch, scale factor \(\frac { 1 } { 2 }\), parallel to the \(x\)-axis. Find the equation of the curve after it has been transformed.
  2. Describe the single transformation which transforms the curve \(y = 5 \sqrt { } x\) to the curve \(y = ( 5 \sqrt { } x ) - 3\).
OCR C1 2005 January Q4
4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$
OCR C1 2005 January Q5
5 On separate diagrams,
  1. sketch the curve \(y = \frac { 1 } { x }\),
  2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
  3. sketch the curve \(y = - \sqrt { } x\).
OCR C1 2005 January Q6
6
  1. Calculate the discriminant of \(- 2 x ^ { 2 } + 7 x + 3\) and hence state the number of real roots of the equation \(- 2 x ^ { 2 } + 7 x + 3 = 0\).
  2. The quadratic equation \(2 x ^ { 2 } + ( p + 1 ) x + 8 = 0\) has equal roots. Find the possible values of \(p\).
OCR C1 2005 January Q7
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \frac { 1 } { 2 } x ^ { 4 } - 3 x\),
  2. \(y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )\),
  3. \(y = \sqrt [ 5 ] { x }\).
OCR C1 2005 January Q8
8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.
  1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
  2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
  3. By solving the inequalities in parts (i) and (ii), determine the set of possible values of \(x\).
OCR C1 2005 January Q9
9
  1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
  2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
  3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
  4. Deduce the value of \(k\).
OCR C1 2005 January Q10
10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.
  1. Calculate the gradient of \(D E\).
  2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
  3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
  4. Calculate the length of \(D F\).
  5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).
OCR C1 2006 January Q1
1 Solve the equations
  1. \(x ^ { \frac { 1 } { 3 } } = 2\),
  2. \(10 ^ { \prime } = 1\),
  3. \(\left( y ^ { - 2 } \right) ^ { 2 } = \frac { 1 } { 81 }\).
OCR C1 2006 January Q2
2
  1. Simplify \(( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$
OCR C1 2006 January Q3
3 Given that \(y = 3 x ^ { 5 } - \sqrt { x } + 15\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR C1 2006 January Q4
4
  1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
  2. Hence sketch the curve \(y = \frac { 1 } { ( x - 3 ) ^ { 2 } }\).
  3. Describe fully a transformation that transforms the curve \(y = \frac { 1 } { x ^ { 2 } }\) to the curve \(y = \frac { 2 } { x ^ { 2 } }\).
OCR C1 2006 January Q5
5
  1. Express \(x ^ { 2 } + 3 x\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Express \(y ^ { 2 } - 4 y - \frac { 11 } { 4 }\) in the form \(( y + p ) ^ { 2 } + q\). A circle has equation \(x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0\).
  3. Write down the coordinates of the centre of the circle.
  4. Find the radius of the circle.
OCR C1 2006 January Q6
6
  1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } + 4\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. For what values of \(x\) does \(x ^ { 3 } - 3 x ^ { 2 } + 4\) increase as \(x\) increases?
OCR C1 2006 January Q7
7
  1. Solve the equation \(x ^ { 2 } - 8 x + 11 = 0\), giving your answers in simplified surd form.
  2. Hence sketch the curve \(y = x ^ { 2 } - 8 x + 11\), labelling the points where the curve crosses the axes.
  3. Solve the equation \(y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0\), giving your answers in the form \(p \pm q \sqrt { 5 }\).
OCR C1 2006 January Q8
8
  1. Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
  2. Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
  3. What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
  4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
  5. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 2006 January Q9
9 The points \(A , B\) and \(C\) have coordinates \(( 5,1 ) , ( p , 7 )\) and \(( 8,2 )\) respectively.
  1. Given that the distance between points \(A\) and \(B\) is twice the distance between points \(A\) and \(C\), calculate the possible values of \(p\).
  2. Given also that the line passing through \(A\) and \(B\) has equation \(y = 3 x - 14\), find the coordinates of the mid-point of \(A B\).
OCR C1 2007 January Q1
1 Express \(\frac { 5 } { 2 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
OCR C1 2007 January Q2
2 Evaluate
  1. \(6 ^ { 0 }\),
  2. \(2 ^ { - 1 } \times 32 ^ { \frac { 4 } { 5 } }\).
OCR C1 2007 January Q3
3 Solve the inequalities
  1. \(3 ( x - 5 ) \leqslant 24\),
  2. \(5 x ^ { 2 } - 2 > 78\).
OCR C1 2007 January Q4
4 Solve the equation \(x ^ { \frac { 2 } { 3 } } + 3 x ^ { \frac { 1 } { 3 } } - 10 = 0\).