Questions — OCR C1 (324 questions)

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OCR C1 2008 June Q8
8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
  3. Determine whether this stationary point is a minimum or maximum point.
  4. Find the \(x\)-coordinate of the other stationary point.
OCR C1 2008 June Q9
9
  1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
  3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
  4. Find an equation of the tangent to the circle at the point ( 8,9 ).
OCR C1 2008 June Q10
10
  1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
  3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
  4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
  5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).
OCR C1 Specimen Q1
1 Write down the exact values of
  1. \(4 ^ { - 2 }\),
  2. \(( 2 \sqrt { } 2 ) ^ { 2 }\),
  3. \(\left( 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\).
OCR C1 Specimen Q2
2
  1. Express \(x ^ { 2 } - 8 x + 3\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Hence write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 8 x + 3\).
OCR C1 Specimen Q3
3 The quadratic equation \(x ^ { 2 } + k x + k = 0\) has no real roots for \(x\).
  1. Write down the discriminant of \(x ^ { 2 } + k x + k\) in terms of \(k\).
  2. Hence find the set of values that \(k\) can take.
OCR C1 Specimen Q4
4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = 4 x ^ { 3 } - 1\),
  2. \(y = x ^ { 2 } \left( x ^ { 2 } + 2 \right)\),
  3. \(y = \sqrt { } x\)
OCR C1 Specimen Q5
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 Specimen Q6
6
  1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
  2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
  3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
  4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.
OCR C1 Specimen Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{5fa27228-37b2-45d9-a8dc-355b2f7f6fa4-3_757_810_1050_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.
  1. Calculate the radius of the circle and the coordinates of the centre.
  2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
  3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).
OCR C1 Specimen Q8
8
  1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
  4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.
OCR C1 Q1
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8$$
OCR C1 Q2
  1. Express
$$\frac { 2 } { 3 \sqrt { 5 } + 7 }$$ in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational.
OCR C1 Q3
3. A circle has the equation $$x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0$$
  1. Find the coordinates of the centre of the circle.
  2. Find the radius of the circle.
OCR C1 Q4
4. (i) Express \(x ^ { 2 } + 6 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
(ii) State the coordinates of the vertex of the curve \(y = x ^ { 2 } + 6 x + 7\).
OCR C1 Q5
5. Solve the simultaneous equations $$\begin{aligned} & x + y = 2
& 3 x ^ { 2 } - 2 x + y ^ { 2 } = 2 \end{aligned}$$
OCR C1 Q6
6.
\includegraphics[max width=\textwidth, alt={}]{e90356f2-7485-4a25-80c5-84e48ceddd62-2_472_753_248_456}
The diagram shows the curve with equation \(y = 3 x - x ^ { \frac { 3 } { 2 } } , x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) and has a maximum at the point \(B\).
  1. Find the \(x\)-coordinate of \(A\).
  2. Find the coordinates of \(B\).
OCR C1 Q7
7. (i) Calculate the discriminant of \(x ^ { 2 } - 6 x + 12\).
(ii) 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.
(iii) Show that the line \(y = 8 - 2 x\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 12\).
OCR C1 Q8
8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
    1. \(\quad y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( - x )\).
OCR C1 Q9
9. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2 .
  1. Find the gradient of the curve at \(A\).
  2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
  3. Find the coordinates of \(B\).
OCR C1 Q10
10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).
  1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
    Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
  2. find the coordinates of \(B\) and \(C\),
  3. show that \(\angle B A C = 90 ^ { \circ }\),
  4. find the area of triangle \(A B C\).
OCR C1 Q1
  1. Find the set of values of the constant \(k\) such that the equation
$$x ^ { 2 } - 6 x + k = 0$$ has real and distinct roots.
OCR C1 Q2
2. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively. Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 Q3
3. (i) Express \(\frac { 18 } { \sqrt { 3 } }\) in the form \(k \sqrt { 3 }\).
(ii) Express \(( 1 - \sqrt { 3 } ) ( 4 - 2 \sqrt { 3 } )\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
OCR C1 Q4
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$