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 Q3
3. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
OCR C1 Q4
4. Express each of the following in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are rational.
  1. \(( 4 - 3 \sqrt { 2 } ) ^ { 2 }\)
  2. \(\frac { 1 } { 2 + \sqrt { 2 } }\)
OCR C1 Q5
5. Given that the equation $$x ^ { 2 } + 4 k x - k = 0$$ has no real roots,
  1. show that $$4 k ^ { 2 } + k < 0 ,$$
  2. find the set of possible values of \(k\).
OCR C1 Q6
6. The curve with equation \(y = x ^ { 2 } + 2 x\) passes through the origin, \(O\).
  1. Find an equation for the normal to the curve at \(O\).
  2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again.
OCR C1 Q7
7. A circle has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).
  1. Find the length of the diameter of the circle.
  2. Find an equation for the circle.
  3. Show that the line \(y = 2 x - 3\) is a tangent to the circle and find the coordinates of the point of contact.
OCR C1 Q8
8. (i) Sketch the graphs of \(y = 2 x ^ { 4 }\) and \(y = 2 \sqrt { x } , x \geq 0\) on the same diagram and write down the coordinates of the point where they intersect.
(ii) Describe fully the transformation that maps the graph of \(y = 2 \sqrt { x }\) onto the graph of \(y = 2 \sqrt { x - 3 }\).
(iii) Find and simplify the equation of the graph obtained when the graph of \(y = 2 x ^ { 4 }\) is stretched by a factor of 2 in the \(x\)-direction, about the \(y\)-axis.
OCR C1 Q9
9. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
  2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
  3. show that triangle \(A B C\) is isosceles.
OCR C1 Q10
10.
\includegraphics[max width=\textwidth, alt={}, center]{6ef55dbd-f18d-4264-b80c-d181473ca7b3-3_531_786_246_523} The diagram shows an open-topped cylindrical container made from cardboard. The cylinder is of height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). Given that the area of card used to make the container is \(192 \pi \mathrm {~cm} ^ { 2 }\),
  1. show that the capacity of the container, \(\mathrm { V } \mathrm { cm } ^ { 3 }\), is given by $$V = 96 \pi r - \frac { 1 } { 2 } \pi r ^ { 3 } .$$
  2. Find the value of \(r\) for which \(V\) is stationary.
  3. Find the corresponding value of \(V\) in terms of \(\pi\).
  4. Determine whether this is a maximum or a minimum value of \(V\).
OCR C1 Q1
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
OCR C1 Q2
  1. Differentiate with respect to \(x\)
$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$
OCR C1 Q3
  1. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\).
Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).
OCR C1 Q4
4. (i) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$ (ii) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$
OCR C1 Q5
  1. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2 .
Find an equation for the tangent to the curve at \(A\).
OCR C1 Q6
6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } }$$
  1. Evaluate \(\mathrm { f } ( 3 )\), giving your answer in its simplest form with a rational denominator.
  2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).
OCR C1 Q7
7. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 7 = 0
& x ^ { 2 } + 2 x y - y ^ { 2 } = 7 \end{aligned}$$
OCR C1 Q8
8.
\includegraphics[max width=\textwidth, alt={}]{4fa65854-801c-4a93-866e-796c000a649f-2_675_689_251_495}
The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).
  1. Find the coordinates of the centre and the radius of the circle.
  2. Find the coordinates of the point on the line which is closest to the circle.
OCR C1 Q9
9. \(f ( x ) = 2 x ^ { 2 } + 3 x - 2\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
  4. Find the values of \(a , b\) and \(c\).
OCR C1 Q10
10. The curve with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).
  1. Sketch the curve, showing the coordinates of \(A\) and \(B\).
  2. Show that the tangent to the curve at \(A\) has the equation $$x + y = 2$$ Given that the curve is stationary at the points \(B\) and \(C\),
  3. find the exact coordinates of \(C\).
OCR C1 Q1
  1. Evaluate \(49 ^ { \frac { 1 } { 2 } } + 8 ^ { \frac { 2 } { 3 } }\).
  2. Solve the equation
$$3 x - \frac { 5 } { x } = 2 .$$
OCR C1 Q3
  1. Find the set of values of \(x\) for which
    1. \(6 x - 11 > x + 4\),
    2. \(x ^ { 2 } - 6 x - 16 < 0\).
    3. (i) Sketch on the same diagram the graphs of \(y = ( x - 1 ) ^ { 2 } ( x - 5 )\) and \(y = 8 - 2 x\).
    Label on your diagram the coordinates of any points where each graph meets the coordinate axes.
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$( x - 1 ) ^ { 2 } ( x - 5 ) = 8 - 2 x$$
  3. State the integer, \(n\), such that $$n < \alpha < n + 1 .$$
OCR C1 Q5
5. $$f ( x ) = x ^ { 2 } - 10 x + 17$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point of the curve \(y = \mathrm { f } ( x )\).
  3. Deduce the coordinates of the minimum point of each of the following curves:
    1. \(y = \mathrm { f } ( x ) + 4\),
    2. \(y = \mathrm { f } ( 2 x )\).
OCR C1 Q6
6. The points \(P , Q\) and \(R\) have coordinates (-5, 2), (-3, 8) and (9, 4) respectively.
  1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on a circle,
  2. find the coordinates of the centre of the circle,
  3. show that the equation of the circle can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.
OCR C1 Q7
7. The straight line \(l _ { 1 }\) has gradient \(\frac { 3 } { 2 }\) and passes through the point \(A ( 5,3 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) has the equation \(3 x - 4 y + 3 = 0\) and intersects \(l _ { 1 }\) at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find the coordinates of the mid-point of \(A B\).
  4. Show that the straight line parallel to \(l _ { 2 }\) which passes through the mid-point of \(A B\) also passes through the origin.
OCR C1 Q8
8.
\includegraphics[max width=\textwidth, alt={}, center]{b7078372-d0d3-4563-818d-637260be5efc-3_592_727_251_493} The diagram 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 \(B\).
OCR C1 Q9
9. The curve \(C\) has the equation $$y = 3 - x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } } , \quad x > 0 .$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis.
  2. Find the exact coordinates of the stationary point of \(C\).
  3. Determine the nature of the stationary point.
  4. Sketch the curve \(C\).