Questions — OCR C1 (324 questions)

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OCR C1 Q1
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(k \sqrt { 2 }\).
  2. Find the coordinates of the stationary point of the curve with equation
$$y = x + \frac { 4 } { x ^ { 2 } } .$$
OCR C1 Q3
3.
\includegraphics[max width=\textwidth, alt={}]{4a5e8809-b4f6-4d24-b3f9-741eea5cc450-1_522_919_705_411}
The diagram shows the curve with equation \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). Show that \(a = - 5\) and find the values of \(b\) and \(c\).
OCR C1 Q4
4. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant. Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$
  1. find the value of \(a\),
  2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).
OCR C1 Q5
5. The straight line \(l _ { 1 }\) has the equation \(3 x - y = 0\). The straight line \(l _ { 2 }\) has the equation \(x + 2 y - 4 = 0\).
  1. Sketch \(l _ { 1 }\) and \(l _ { 2 }\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes.
  2. Find, as exact fractions, the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
OCR C1 Q6
6. (a) Given that \(y = 2 ^ { x }\), find expressions in terms of \(y\) for
  1. \(2 ^ { x + 2 }\),
  2. \(2 ^ { 3 - x }\).
    (b) Show that using the substitution \(y = 2 ^ { x }\), the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$ can be rewritten as $$4 y ^ { 2 } - 33 y + 8 = 0$$ (c) Hence solve the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$
OCR C1 Q7
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 Q8
8. (i) Express \(3 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
(ii) Sketch the curve with equation \(y = 3 x ^ { 2 } - 12 x + 11\), showing the coordinates of the minimum point of the curve. Given that the curve \(y = 3 x ^ { 2 } - 12 x + 11\) crosses the \(x\)-axis at the points \(A\) and \(B\),
(iii) find the length \(A B\) in the form \(k \sqrt { 3 }\).
OCR C1 Q9
9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).
OCR C1 Q1
  1. Solve the inequality
$$4 ( x - 2 ) < 2 x + 5$$
OCR C1 Q2
2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
OCR C1 Q3
3. (i) Solve the equation $$y ^ { 2 } + 8 = 9 y .$$ (ii) Hence solve the equation $$x ^ { 3 } + 8 = 9 x ^ { \frac { 3 } { 2 } } .$$
OCR C1 Q4
  1. Given that
$$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } } ,$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).
OCR C1 Q5
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 C1 Q6
  1. (i) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
    (ii) Find the value of \(x\) such that
$$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
OCR C1 Q7
7. The straight line \(l\) passes through the point \(P ( - 3,6 )\) and the point \(Q ( 1 , - 4 )\).
  1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The straight line \(m\) has the equation \(2 x + k y + 7 = 0\), where \(k\) is a constant.
    Given that \(l\) and \(m\) are perpendicular,
  2. find the value of \(k\).
OCR C1 Q8
8. (i) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(ii) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(iii) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).
OCR C1 Q9
9. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
  3. Sketch \(C\). Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
  4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).
OCR C1 Q10
10.
\includegraphics[max width=\textwidth, alt={}, center]{76efaf91-a6f3-4493-88d4-3654b023441d-3_646_773_986_477} The diagram shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).
  1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
  2. Find an equation for the tangent to the curve at \(P\).
  3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
  4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).
OCR C1 2009 January Q1
1 Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
OCR C1 2009 January Q2
2 Simplify
  1. \(( \sqrt [ 3 ] { x } ) ^ { 6 }\),
  2. \(\frac { 3 y ^ { 4 } \times ( 10 y ) ^ { 3 } } { 2 y ^ { 5 } }\).
OCR C1 2009 January Q3
3 Solve the equation \(3 x ^ { \frac { 2 } { 3 } } + x ^ { \frac { 1 } { 3 } } - 2 = 0\).
OCR C1 2009 January Q4
4
  1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
  2. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is translated by 3 units in the negative \(x\)-direction. State the equation of the curve after it has been translated.
  3. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 4 and, as a result, the point \(P ( 1,1 )\) is transformed to the point \(Q\). State the coordinates of \(Q\).
OCR C1 2009 January Q5
5 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = 10 x ^ { - 5 }\),
  2. \(y = \sqrt [ 4 ] { x }\),
  3. \(y = x ( x + 3 ) ( 1 - 5 x )\).
OCR C1 2009 January Q6
6
  1. Express \(5 x ^ { 2 } + 20 x - 8\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the equation of the line of symmetry of the curve \(y = 5 x ^ { 2 } + 20 x - 8\).
  3. Calculate the discriminant of \(5 x ^ { 2 } + 20 x - 8\).
  4. State the number of real roots of the equation \(5 x ^ { 2 } + 20 x - 8 = 0\).
OCR C1 2009 January Q7
7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).
  1. Find the value of \(k\).
  2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  3. Write down the coordinates of the centre and the radius of the circle.
  4. Verify that \(A B\) is a diameter of the circle.