Questions — OCR C1 (333 questions)

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OCR C1 Specimen Q6
12 marks Moderate -0.5
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
13 marks Moderate -0.3
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
14 marks Moderate -0.3
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
3 marks Easy -1.2
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8$$
OCR C1 Q2
3 marks Easy -1.2
  1. Express
$$\frac { 2 } { 3 \sqrt { 5 } + 7 }$$ in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational.
OCR C1 Q3
4 marks Moderate -0.8
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
5 marks Easy -1.2
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
7 marks Standard +0.3
5. Solve the simultaneous equations $$\begin{aligned} & x + y = 2 \\ & 3 x ^ { 2 } - 2 x + y ^ { 2 } = 2 \end{aligned}$$
OCR C1 Q6
8 marks Moderate -0.3
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
9 marks Moderate -0.8
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
9 marks Moderate -0.3
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
10 marks Moderate -0.3
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
14 marks Standard +0.3
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
3 marks Moderate -0.8
  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
4 marks Moderate -0.8
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
4 marks Easy -1.2
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 marks Moderate -0.5
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$
OCR C1 Q5
7 marks Moderate -0.5
  1. Given that
$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).
OCR C1 Q6
8 marks Moderate -0.8
6. \includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).
  1. Write down the number of solutions that exist for the equation
    1. \(\mathrm { f } ( x ) = 1\),
    2. \(\mathrm { f } ( x ) = - x\).
  2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
    1. \(\quad y = \mathrm { f } ( x - 2 )\),
    2. \(y = \mathrm { f } ( 2 x )\).
OCR C1 Q7
9 marks Moderate -0.8
7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
  3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  4. Determine whether each stationary point is a maximum or a minimum point.
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 Q9
10 marks Moderate -0.3
9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).
  1. Find an equation for \(C\).
  2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
  3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 Q10
13 marks Standard +0.3
10. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
  2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  3. Show that the normal to the curve at \(P\) does not intersect the curve again.
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 \(y = m x + c\).