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 C1 2012 January Q8
Standard +0.3
8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).
OCR C1 2012 January Q9
Moderate -0.3
9
  1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
  2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
  3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).
OCR C1 2012 January Q10
Moderate -0.3
10 A circle has centre \(C ( - 2,4 )\) and radius 5 .
  1. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  2. Show that the tangent to the circle at the point \(P ( - 5,8 )\) has equation \(3 x - 4 y + 47 = 0\).
  3. Verify that the point \(T ( 3,14 )\) lies on this tangent.
  4. Find the area of the triangle \(C P T\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR C1 2013 January Q1
Moderate -0.8
1
  1. Solve the equation \(x ^ { 2 } - 6 x - 2 = 0\), giving your answers in simplified surd form.
  2. Find the gradient of the curve \(y = x ^ { 2 } - 6 x - 2\) at the point where \(x = - 5\).
OCR C1 2013 January Q2
Easy -1.2
2 Solve the equations
  1. \(3 ^ { n } = 1\),
  2. \(t ^ { - 3 } = 64\),
  3. \(\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8\).
OCR C1 2013 January Q3
Moderate -0.3
3
  1. Sketch the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\), giving the coordinates of all points of intersection with the axes.
  2. Describe the transformation that transforms the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\) to the curve \(y = ( 1 - x ) ( 2 + x ) ( 3 - x )\).
OCR C1 2013 January Q4
Moderate -0.8
4
  1. Solve the simultaneous equations $$y = 2 x ^ { 2 } - 3 x - 5 , \quad 10 x + 2 y + 11 = 0$$
  2. What can you deduce from the answer to part (i) about the curve \(y = 2 x ^ { 2 } - 3 x - 5\) and the line \(10 x + 2 y + 11 = 0\) ?
  3. Simplify \(( x + 4 ) ( 5 x - 3 ) - 3 ( x - 2 ) ^ { 2 }\).
  4. The coefficient of \(x ^ { 2 }\) in the expansion of $$( x + 3 ) ( x + k ) ( 2 x - 5 )$$ is - 3 . Find the value of the constant \(k\).
OCR C1 2013 January Q6
Moderate -0.8
6
  1. The line joining the points ( \(- 2,7\) ) and ( \(- 4 , p\) ) has gradient 4 . Find the value of \(p\).
  2. The line segment joining the points \(( - 2,7 )\) and \(( 6 , q )\) has mid-point \(( m , 5 )\). Find \(m\) and \(q\).
  3. The line segment joining the points \(( - 2,7 )\) and \(( d , 3 )\) has length \(2 \sqrt { 13 }\). Find the two possible values of \(d\).
OCR C1 2013 January Q7
Easy -1.2
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \frac { ( 3 x ) ^ { 2 } \times x ^ { 4 } } { x }\),
  2. \(y = \sqrt [ 3 ] { x }\),
  3. \(y = \frac { 1 } { 2 x ^ { 3 } }\).
OCR C1 2013 January Q8
Standard +0.3
8 The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
OCR C1 2013 January Q9
Moderate -0.3
9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 10 y - 19 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle.
  2. Verify that the point \(( 7 , - 2 )\) lies on the circumference of the circle.
  3. Find the equation of the tangent to the circle at the point \(( 7 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2013 January Q10
Moderate -0.3
10 Find the coordinates of the points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 9 } { x }\) at which the tangent is parallel to the line \(y = 8 x + 3\).
OCR C1 2009 June Q1
Easy -1.2
1 Given that \(y = x ^ { 5 } + \frac { 1 } { x ^ { 2 } }\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR C1 2009 June Q2
Easy -1.2
2 Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.
OCR C1 2009 June Q3
Easy -1.8
3 Express each of the following in the form \(3 ^ { n }\) :
  1. \(\frac { 1 } { 9 }\),
  2. \(\sqrt [ 3 ] { 3 }\),
  3. \(3 ^ { 10 } \times 9 ^ { 15 }\).
OCR C1 2009 June Q4
Moderate -0.5
4 Solve the simultaneous equations $$4 x ^ { 2 } + y ^ { 2 } = 10 , \quad 2 x - y = 4$$
OCR C1 2009 June Q5
Moderate -0.8
5
  1. Expand and simplify \(( 2 x + 1 ) ( x - 3 ) ( x + 4 )\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of $$x \left( x ^ { 2 } + 2 x + 3 \right) \left( x ^ { 2 } + 7 x - 2 \right) .$$
OCR C1 2009 June Q6
Moderate -0.8
6
  1. Sketch the curve \(y = - \sqrt { x }\).
  2. Describe fully a transformation that transforms the curve \(y = - \sqrt { x }\) to the curve \(y = 5 - \sqrt { x }\).
  3. The curve \(y = - \sqrt { x }\) is stretched by a scale factor of 2 parallel to the \(x\)-axis. State the equation of the curve after it has been stretched.
OCR C1 2009 June Q7
Moderate -0.8
7
  1. Express \(x ^ { 2 } - 5 x + \frac { 1 } { 4 }\) in the form \(( x - a ) ^ { 2 } - b\).
  2. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 5 x + \frac { 1 } { 4 } = 0\).
OCR C1 2009 June Q8
Easy -1.2
8 Solve the inequalities
  1. \(- 35 < 6 x + 7 < 1\),
  2. \(3 x ^ { 2 } > 48\).
    \(9 \quad A\) is the point \(( 4 , - 3 )\) and \(B\) is the point \(( - 1,9 )\).
OCR C1 2009 June Q10
Moderate -0.8
10
  1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
  2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
  3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
  4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?
OCR C1 2009 June Q11
Standard +0.3
11 The point \(P\) on the curve \(y = k \sqrt { x }\) has \(x\)-coordinate 4 . The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\).
  1. Find the value of \(k\).
  2. This normal meets the \(x\)-axis at the point \(Q\). Calculate the area of the triangle \(O P Q\), where \(O\) is the point \(( 0,0 )\). RECOGNISING ACHIEVEMENT
OCR C1 2010 June Q2
Moderate -0.8
2
  1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
  2. Sketch the curve \(y = 3 - \frac { 1 } { x ^ { 2 } }\).
  3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 2 . State the equation of the transformed curve.
OCR C1 2010 June Q3
Easy -1.2
3
  1. Express \(\frac { 12 } { 3 + \sqrt { 5 } }\) in the form \(a - b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
  2. Express \(\sqrt { 18 } - \sqrt { 2 }\) in simplified surd form.
OCR C1 2010 June Q4
Moderate -0.8
4
  1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
  2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.