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
Edexcel C1 Q10
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0 ,$$ and that the point \(A\) on \(C\) has coordinates (2, 6),
  1. find an equation for \(C\),
  2. find an equation for the tangent to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers,
  3. show that the line \(y = x + 3\) is also a tangent to \(C\).
Edexcel C1 Q1
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8 .$$
Edexcel C1 Q2
  1. Find
$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$
Edexcel C1 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c4ae1bec-12f4-492d-8027-bba4840ff545-2_337_1235_781_383} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the rectangles \(A B C D\) and \(E F G H\) which are similar.
Given that \(A B = ( 3 - \sqrt { 5 } ) \mathrm { cm } , A D = \sqrt { 5 } \mathrm {~cm}\) and \(E F = ( 1 + \sqrt { 5 } ) \mathrm { cm }\), find the length \(E H\) in cm, giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are integers.
Edexcel C1 Q4
4. (a) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(b) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$
Edexcel C1 Q5
  1. (a) By completing the square, find in terms of the constant \(k\) the roots of the equation
$$x ^ { 2 } + 2 k x + 4 = 0 .$$ (b) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$
Edexcel C1 Q6
  1. (a) Evaluate
$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (b) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.
Edexcel C1 Q8
  1. Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 3 } - 4 } { x ^ { 3 } } , \quad x \neq 0$$
  1. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Given also that \(y = 0\) when \(x = - 1\),
  2. find the value of \(y\) when \(x = 2\).
Edexcel C1 Q9
9. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
  2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  3. Show that the normal to the curve at \(P\) does not intersect the curve again.
Edexcel C1 Q10
10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).
  1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
    Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
  2. find the coordinates of \(B\) and \(C\),
  3. show that \(\angle B A C = 90 ^ { \circ }\),
  4. find the area of triangle \(A B C\).
Edexcel C1 Q1
  1. Evaluate \(49 ^ { \frac { 1 } { 2 } } + 8 ^ { \frac { 2 } { 3 } }\).
  2. A sequence is defined by the recurrence relation
$$u _ { n + 1 } = \frac { u _ { n } + 1 } { 3 } , \quad n = 1,2,3 , \ldots$$ Given that \(u _ { 3 } = 5\),
  1. find the value of \(u _ { 4 }\),
  2. find the value of \(u _ { 1 }\).
Edexcel C1 Q3
3. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9$$
  1. Determine the number of real roots that exist for the equation \(\mathrm { f } ( x ) = 0\).
  2. Solve the equation \(\mathrm { f } ( x ) = 8\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
Edexcel C1 Q4
4. Find the set of values of \(x\) for which
  1. \(6 x - 11 > x + 4\),
  2. \(x ^ { 2 } - 6 x - 16 < 0\),
  3. both \(6 x - 11 > x + 4\) and \(x ^ { 2 } - 6 x - 16 < 0\).
Edexcel C1 Q5
5. $$f ( x ) = ( 2 - \sqrt { x } ) ^ { 2 } , \quad x > 0$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find \(\mathrm { f } ( 3 )\), giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
  3. Find $$\int \mathrm { f } ( x ) \mathrm { d } x$$
Edexcel C1 Q6
  1. The straight line \(l\) passes through the point \(P ( - 3,6 )\) and the point \(Q ( 1 , - 4 )\).
    1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    The straight line \(m\) has the equation \(2 x + k y + 7 = 0\), where \(k\) is a constant.
    Given that \(l\) and \(m\) are perpendicular,
  2. find the value of \(k\).
Edexcel C1 Q7
7. Given that $$\mathrm { f } ^ { \prime } ( x ) = 5 + \frac { 4 } { x ^ { 2 } } , \quad x \neq 0$$
  1. find an expression for \(\mathrm { f } ( x )\). Given also that
    f(2) = 2f(1),
  2. find \(\mathrm { f } ( 4 )\).
Edexcel C1 Q8
8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
    1. \(\quad y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( - x )\).
Edexcel C1 Q9
9. The first two terms of an arithmetic series are \(( t - 1 )\) and \(\left( t ^ { 2 } - 5 \right)\) respectively, where \(t\) is a positive constant.
  1. Find and simplify expressions in terms of \(t\) for
    1. the common difference of the series,
    2. the third term of the series. Given also that the third term of the series is 19 ,
  2. find the value of \(t\),
  3. show that the 10th term of the series is 75,
  4. find the sum of the first 40 terms of the series.
Edexcel C1 Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0ee09b1-25a2-4244-aa69-63e8f5b3543a-4_595_727_1119_493} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 2 + 3 x - x ^ { 2 }\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.
  1. Find an equation for \(l\). The line \(m\) is the normal to the curve at the point \(B\).
    Given that \(l\) and \(m\) are parallel,
  2. find the coordinates of \(B\).
Edexcel C1 2006 January Q6
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = 2 \mathrm { f } ( x )\),
  3. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). On each diagram show clearly the coordinates of all the points where the curve meets the axes.
Edexcel C1 Specimen Q2
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( 2 x )\). On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes.
AQA C1 2014 June Q7
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    1. Write down the coordinates of \(C\).
    2. Show that the circle has radius \(n \sqrt { 5 }\), where \(n\) is an integer.
  3. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(x + p y = q\), where \(p\) and \(q\) are integers.
  4. The point \(B\) lies on the tangent to the circle at \(A\) and the length of \(B C\) is 6. Find the length of \(A B\).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}
AQA C1 2015 June Q4
  1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
  3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
  4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
    [0pt] [2 marks]
Edexcel C1 Q6
  1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
  2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
  3. find the value of \(k\),
  4. find the coordinates of \(D\).
OCR C1 2007 January Q9
  1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.