Questions — OCR (4907 questions)

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OCR C1 Q1
3 marks Easy -1.2
  1. Solve the equation
$$9 ^ { x } = 3 ^ { x + 2 } .$$
OCR C1 Q2
4 marks Moderate -0.3
  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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
6 marks Moderate -0.3
  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 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 Q7
9 marks Moderate -0.3
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
9 marks Moderate -0.3
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
10 marks Standard +0.3
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
14 marks Moderate -0.3
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
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 Q2
4 marks Moderate -0.5
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
5 marks Moderate -0.8
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
6 marks Easy -1.2
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
7 marks Moderate -0.8
  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
7 marks Moderate -0.5
  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
8 marks Moderate -0.8
  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
10 marks Moderate -0.3
  1. \(f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }\).
    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.
    5. The points \(P\) and \(Q\) have coordinates \(( 7,4 )\) and \(( 9,7 )\) respectively.
OCR C1 Q10
12 marks Moderate -0.5
10. \includegraphics[max width=\textwidth, alt={}, center]{af6fdbed-fcab-4db8-9cdf-fd049ce720fd-3_668_787_918_431} The diagram shows the circle \(C\) and the straight line \(l\).
The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).
  1. Find the gradient of 1 .
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the centre of \(C\).
  4. Show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0$$
OCR C1 Q1
3 marks Moderate -0.8
  1. \(\quad \mathrm { f } ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }\).
Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.
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 Q3
5 marks Moderate -0.3
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
5 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Standard +0.3
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.