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 Q4
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$
OCR C1 Q5
  1. Given that
$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).
OCR C1 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).
  1. Write down the number of solutions that exist for the equation
    1. \(\mathrm { f } ( x ) = 1\),
    2. \(\mathrm { f } ( x ) = - x\).
  2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
    1. \(\quad y = \mathrm { f } ( x - 2 )\),
    2. \(y = \mathrm { f } ( 2 x )\).
OCR C1 Q7
7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
  3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  4. Determine whether each stationary point is a maximum or a minimum point.
OCR C1 Q8
8. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$
  1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
  2. State the maximum value of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
  4. Sketch the curve \(y = \mathrm { f } ( x )\).
OCR C1 Q9
9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).
  1. Find an equation for \(C\).
  2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
  3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 Q10
10. 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.
OCR C1 Q1
  1. Solve the equation
$$9 ^ { x } = 3 ^ { x + 2 } .$$
OCR C1 Q2
  1. The straight line \(l\) has the equation \(x - 5 y = 7\).
The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).
OCR C1 Q3
3.
\includegraphics[max width=\textwidth, alt={}, center]{4fec0924-d727-4d4f-81e6-918e1ccfedbd-1_330_1230_829_386} The diagram 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.
OCR C1 Q4
4. (i) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(ii) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$
OCR C1 Q5
  1. (i) Solve the inequality
$$x ^ { 2 } + 3 x > 10 .$$ (ii) Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3
& x ^ { 2 } + 3 x > 10 \end{aligned}$$
OCR C1 Q6
6. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9 .$$
  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.
OCR C1 Q7
7. The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).
  1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
  2. Find the \(x\)-coordinates of \(A\) and \(B\).
  3. Show that \(A B = 2 \sqrt { 10 }\).
OCR C1 Q8
8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$
  1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
  2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
  3. Determine whether the turning point is a maximum or minimum point.
OCR C1 Q9
9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.
OCR C1 Q10
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$
  1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
  2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
  3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
  4. Show that \(m\) has the equation \(y = 3 x + 8\).
OCR C1 Q1
  1. Solve the equation
$$x ^ { 2 } - 4 x - 8 = 0$$ giving your answers in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
OCR C1 Q2
2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).
OCR C1 Q3
3. (i) Solve the simultaneous equations $$\begin{aligned} & y = x ^ { 2 } - 6 x + 7
& y = 2 x - 9 \end{aligned}$$ (ii) Hence, describe the geometrical relationship between the curve \(y = x ^ { 2 } - 6 x + 7\) and the straight line \(y = 2 x - 9\).
OCR C1 Q4
4. (i) Evaluate $$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$ (ii) Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$
OCR C1 Q5
  1. (i) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\).
Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$
OCR C1 Q6
  1. (i) Given that \(y = x ^ { \frac { 1 } { 3 } }\), show that the equation
$$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$ can be rewritten as $$2 y ^ { 2 } - 7 y + 3 = 0 .$$ (ii) Hence, solve the equation $$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$
OCR C1 Q7
  1. Given that
$$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
  3. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0 .$$
OCR C1 Q8
\begin{enumerate} \setcounter{enumi}{7} \item \(f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }\).
  1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  2. Determine whether each stationary point is a maximum or minimum point.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions. \item The points \(P\) and \(Q\) have coordinates \(( 7,4 )\) and \(( 9,7 )\) respectively.