Questions — OCR MEI (4301 questions)

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OCR MEI C1 Q5
5 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - , \frac { 1 } { 2 }\) and 5 .
  1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
  4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis.
OCR MEI C1 Q1
1 Express \(5 x ^ { 2 } + 15 x + 12\) in the form \(a ( x + b ) ^ { 2 } + c\).
Hence state the minimum value of \(y\) on the curve \(y = 5 x ^ { 2 } + 15 x + 12\).
OCR MEI C1 Q2
2 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).
  1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
  2. Sketch the curve \(y = \mathrm { f } ( x )\).
  3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).
OCR MEI C1 Q3
9 marks
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6be6c0b0-76b7-49c0-bf1b-dc6f8f79981b-2_836_906_361_675} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).
  1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
    [0pt] [3]
  2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
    [0pt] [6]
  3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
  4. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  5. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  6. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 Q1
1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).
  1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$
OCR MEI C1 Q2
2 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).
OCR MEI C1 Q3
3
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
    Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 Q4
4 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).
  1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).
OCR MEI C1 Q5
5 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).
  1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
  3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).
OCR MEI C1 Q1
1 Sketch the graph of \(y = 9 - x ^ { 2 }\).
OCR MEI C1 Q2
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{618d118a-2557-42f3-9b55-4a55dda93a97-1_449_376_631_889} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).
  1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
  2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
  3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).
OCR MEI C1 Q3
3 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).
  1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
  3. Express \(\mathrm { f } ( x )\) in fully factorised form.
  4. Sketch the graph of \(y = \mathrm { f } ( x )\).
  5. Solve the equation \(\mathrm { f } ( x ) = 12\).
OCR MEI C1 Q4
4
  1. Sketch the graph of \(y = x ( x - 3 ) ^ { 2 }\).
  2. Show that the equation \(x ( x - 3 ) ^ { 2 } = 2\) can be expressed as \(x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0\).
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).
OCR MEI C1 Q5
5
  1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
  2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
  3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
  4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.
OCR MEI C1 Q1
9 marks
1
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes. [4] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{973ad9eb-33f2-432e-9449-e54c1728008b-1_1292_1401_887_359} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
  4. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
  5. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
  6. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation. [5]
  7. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
  8. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
  9. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
    (A) write down the equation of the line of symmetry,
    (B) write down the minimum \(y\)-value on the curve.
OCR MEI C1 Q4
4
  1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
  2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
  3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).
OCR MEI C1 Q1
1
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
  2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.
OCR MEI C1 Q2
2 Answer part (i) of this question on the insert provided.
The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q3
3 Factorise and hence simplify \(\frac { 3 x ^ { 2 } - 7 x + 4 } { x ^ { 2 } - 1 }\).
  1. Prove that 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all even positive integers \(n\).
  2. Determine whether 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all positive integers \(n\).
  3. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
  4. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
  5. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
  6. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.
OCR MEI C1 Q6
6
  1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
  3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
  4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
OCR MEI C1 Q1
1 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
\(2 \quad\) (i) Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
(ii) Factorise \(x ^ { 2 } - 4 \quad x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.
OCR MEI C1 Q3
3 Simplify \(\left( m ^ { 2 } + 1 \right) ^ { 2 } - \left( m ^ { 2 } - 1 \right) ^ { 2 }\), showing your method.
Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer.
\includegraphics[max width=\textwidth, alt={}, center]{6978a576-e86a-48d4-93a4-268845ea6699-1_414_590_1328_772}
OCR MEI C1 Q5
5
  1. Write \(x ^ { 2 } - 7 x + 6\) in the form \(( x - a ) ^ { 2 } + b\).
  2. State the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 7 x + 6\).
  3. Find the coordinates of the points where the graph of \(y = x ^ { 2 } - 7 x + 6\) crosses the axes and sketch the graph.
  4. Show that the graphs of \(y = x ^ { 2 } - 7 x + 6\) and \(y = x ^ { 2 } - 3 x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6978a576-e86a-48d4-93a4-268845ea6699-3_643_864_390_622} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
  5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
  6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
  7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
  8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point.
OCR MEI C1 Q7
7
  1. Describe fully the transformation which maps the curve \(y = x ^ { 2 }\) onto the curve \(y = ( x + 4 ) ^ { 2 }\).
  2. Sketch the graph of \(y = x ^ { 2 } - 4\).
OCR MEI C1 Q3
4 marks
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e93e3c51-ae2b-420b-abb8-bf0c483caff8-3_1270_1219_326_463} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).
  1. Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]