1.02f Solve quadratic equations: including in a function of unknown

448 questions

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OCR C1 2007 June Q10
12 marks Standard +0.3
10
  1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
  2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
  3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.
OCR C1 2008 June Q4
5 marks Standard +0.3
4 Solve the equation \(2 x - 7 x ^ { \frac { 1 } { 2 } } + 3 = 0\).
OCR MEI C1 2007 June Q9
4 marks Easy -1.3
9
  1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
  2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.
OCR MEI C1 2007 June Q10
5 marks Moderate -0.8
10 The triangle shown in Fig. 10 has height \(( x + 1 ) \mathrm { cm }\) and base \(( 2 x - 3 ) \mathrm { cm }\). Its area is \(9 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d8caf0f-7594-42cb-bd40-e6c11e2b6832-3_444_1088_351_715} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Show that \(2 x ^ { 2 } - x - 21 = 0\).
  2. By factorising, solve the equation \(2 x ^ { 2 } - x - 21 = 0\). Hence find the height and base of the triangle. Section B (36 marks)
OCR MEI C1 2007 June Q12
12 marks Moderate -0.8
12
  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 2008 June Q3
5 marks Moderate -0.8
3
  1. Solve the equation \(2 x ^ { 2 } + 3 x = 0\).
  2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 x - k = 0\) has no real roots.
OCR MEI C1 2008 June Q9
4 marks Moderate -0.8
9 Solve the equation \(y ^ { 2 } - 7 y + 12 = 0\).
Hence solve the equation \(x ^ { 4 } - 7 x ^ { 2 } + 12 = 0\). Section B (36 marks)
OCR MEI C1 2015 June Q12
12 marks Moderate -0.8
12
  1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
  2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
  3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).
OCR MEI C1 Q11
12 marks Moderate -0.8
11
  1. Multiply out \(( x - p ) ( x - q )\).
  2. You are given that \(p = 2 + \sqrt { 3 }\) and \(q = 2 - \sqrt { 3 }\) are the roots of a quadratic equation. Find \(p + q\) and \(p q\) and hence find the quadratic equation with roots \(x = p\) and \(x = q\).
  3. Solve the quadratic equation \(x ^ { 2 } + 5 x - 7 = 0\) giving the roots exactly.
  4. Show that \(x = 1\) is the only root of the equation \(x ^ { 3 } + 2 x - 3 = 0\).
  5. A quadratic equation \(x ^ { 2 } + r x + s = 0\), where \(r\) and \(s\) are integers, has two roots. One root is \(x = 3 + \sqrt { 5 }\). Without finding \(r\) or \(s\), write down the other root.
OCR MEI C1 Q5
5 marks Moderate -0.8
5 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } - 7 x + 6\).
  1. Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI C1 Q9
5 marks Easy -1.2
9
  1. Show that \(( x - 1 ) ( x - 2 ) ( x - 3 ) - \left( x ^ { 3 } - x ^ { 2 } + 11 x - 12 \right) = 6 - 5 x ^ { 2 }\).
  2. Solve the equation \(6 - 5 x ^ { 2 } = 0\).
OCR MEI C1 Q10
12 marks Moderate -0.8
10
  1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
    Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
  2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
  3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
  4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.
OCR MEI C1 Q3
4 marks Moderate -0.5
3 Solve the inequality \(2 x ^ { 2 } - 7 x \geq 4\).
OCR C1 Q7
9 marks Moderate -0.8
7.
  1. Calculate the discriminant of \(x ^ { 2 } - 6 x + 12\).
  2. State the number of real roots of the equation \(x ^ { 2 } - 6 x + 12 = 0\) and hence, explain why \(x ^ { 2 } - 6 x + 12\) is always positive.
  3. Show that the line \(y = 8 - 2 x\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 12\).
OCR C1 Q8
10 marks Moderate -0.8
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 Q6
6 marks Moderate -0.3
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 Q1
3 marks Moderate -0.5
  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 Q4
6 marks Easy -1.2
4.
  1. Evaluate $$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$
  2. Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$
OCR C1 Q6
7 marks Moderate -0.5
  1. 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 $$2 y ^ { 2 } - 7 y + 3 = 0 .$$
  2. Hence, solve the equation $$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$
OCR C1 Q2
4 marks Moderate -0.3
2. Find in exact form the real solutions of the equation $$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$
OCR C1 Q4
6 marks Moderate -0.3
4.
  1. By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$
  2. Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$
OCR C1 Q9
10 marks Standard +0.3
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 Q3
5 marks Moderate -0.3
3.
  1. Solve the equation $$y ^ { 2 } + 8 = 9 y .$$
  2. Hence solve the equation $$x ^ { 3 } + 8 = 9 x ^ { \frac { 3 } { 2 } } .$$
OCR C1 Q5
7 marks Moderate -0.5
5. Find the pairs of values \(( x , y )\) which satisfy the simultaneous equations $$\begin{aligned} & 3 x ^ { 2 } + y ^ { 2 } = 21 \\ & 5 x + y = 7 \end{aligned}$$
OCR MEI C1 Q2
3 marks Moderate -0.8
2 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).