Questions — OCR C1 (333 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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OCR C1 Q3
4 marks Easy -1.3
  1. Solve the equation $$x^{\frac{3}{2}} = 27.$$ [2]
  2. Express \((2\frac{1}{4})^{-\frac{3}{2}}\) as an exact fraction in its simplest form. [2]
OCR C1 Q4
5 marks Moderate -0.3
Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]
OCR C1 Q5
6 marks Moderate -0.8
\includegraphics{figure_5} The diagram shows a sketch of the curve with equation \(y = f(x)\). The curve has a maximum at \((-3, 4)\) and a minimum at \((1, -2)\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations
  1. \(y = 2f(x)\), [3]
  2. \(y = -f(x)\). [3]
OCR C1 Q6
8 marks Moderate -0.8
\(f(x) = 2x^2 - 4x + 1\).
  1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$f(x) = a(x + b)^2 + c.$$ [4]
  2. State the equation of the line of symmetry of the curve \(y = f(x)\). [1]
  3. Solve the equation \(f(x) = 3\), giving your answers in exact form. [3]
OCR C1 Q7
9 marks Moderate -0.3
A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),
  1. find the values of \(a\) and \(b\), [6]
  2. find the coordinates of the other stationary point of the curve. [3]
OCR C1 Q8
10 marks Standard +0.3
The circle \(C\) has the equation $$x^2 + y^2 + 10x - 8y + k = 0,$$ where \(k\) is a constant. Given that the point with coordinates \((-6, 5)\) lies on \(C\),
  1. find the value of \(k\), [2]
  2. find the coordinates of the centre and the radius of \(C\). [3]
A straight line which passes through the point \(A(2, 3)\) is a tangent to \(C\) at the point \(B\).
  1. Find the length \(AB\) in the form \(k\sqrt{5}\). [5]
OCR C1 Q9
10 marks Standard +0.3
A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).
  1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [3]
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]
OCR C1 Q10
13 marks Moderate -0.3
The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). [2]
The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).
  1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]
The line \(l_2\) crosses the \(x\)-axis at the point \(C\).
  1. Find the coordinates of \(C\). [1]
The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).
  1. Show that \(D\) has coordinates \((5, 4)\). [5]
  2. Find the area of triangle \(ACD\). [2]
OCR C1 Q1
4 marks Moderate -0.5
Find the value of \(y\) such that $$4^{y+1} = 8^{2y-1}.$$ [4]
OCR C1 Q2
4 marks Easy -1.2
Express \(\sqrt{22.5}\) in the form \(k\sqrt{10}\). [4]
OCR C1 Q3
5 marks Moderate -0.8
A circle has the equation $$x^2 + y^2 + 8x - 4y + k = 0,$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of the circle. [2]
Given that the \(x\)-axis is a tangent to the circle,
  1. Find the value of \(k\). [3]
OCR C1 Q4
6 marks Moderate -0.8
$$\text{f}(x) = 4x - 3x^2 - x^3.$$
  1. Fully factorise \(4x - 3x^2 - x^3\). [3]
  2. Sketch the curve \(y = \text{f}(x)\), showing the coordinates of any points of intersection with the coordinate axes. [3]
OCR C1 Q5
8 marks Moderate -0.3
  1. Find in exact form the coordinates of the points where the curve \(y = x^2 - 4x + 2\) crosses the \(x\)-axis. [4]
  2. Find the value of the constant \(k\) for which the straight line \(y = 2x + k\) is a tangent to the curve \(y = x^2 - 4x + 2\). [4]
OCR C1 Q6
10 marks Moderate -0.8
Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A\) cm\(^2\), after \(t\) seconds is given by $$A = (p + qt)^2,$$ where \(p\) and \(q\) are positive constants. Given that when \(t = 0\), \(A = 4\) and that when \(t = 5\), \(A = 9\),
  1. find the value of \(p\) and show that \(q = \frac{1}{5}\), [5]
  2. find \(\frac{\mathrm{d}A}{\mathrm{d}t}\) in terms of \(t\), [3]
  3. find the rate at which the area of the stain is increasing when \(t = 15\). [2]
OCR C1 Q7
11 marks Moderate -0.8
The curve \(C\) has the equation \(y = x^2 + 2x + 4\).
  1. Express \(x^2 + 2x + 4\) in the form \((x + p)^2 + q\) and hence state the coordinates of the minimum point of \(C\). [4]
The straight line \(l\) has the equation \(x + y = 8\).
  1. Sketch \(l\) and \(C\) on the same set of axes. [3]
  2. Find the coordinates of the points where \(l\) and \(C\) intersect. [4]
OCR C1 Q8
11 marks Moderate -0.3
$$\text{f}(x) \equiv \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
  2. Show that $$\text{f}'(x) = \frac{3x^2 - 8x - 16}{4x^{\frac{3}{2}}}.$$ [5]
  3. Find the coordinates of the stationary point of the curve \(y = \text{f}(x)\). [3]
OCR C1 Q9
13 marks Standard +0.8
\includegraphics{figure_9} The diagram shows the parallelogram \(ABCD\). The points \(A\) and \(B\) have coordinates \((-1, 3)\) and \((3, 4)\) respectively and lie on the straight line \(l_1\).
  1. Find an equation for \(l_1\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The points \(C\) and \(D\) lie on the straight line \(l_2\) which has the equation \(x - 4y - 21 = 0\).
  1. Show that the distance between \(l_1\) and \(l_2\) is \(k\sqrt{17}\), where \(k\) is an integer to be found. [7]
  2. Find the area of parallelogram \(ABCD\). [2]
OCR C1 Q1
3 marks Easy -1.8
Evaluate \(49^{\frac{1}{2}} + 8^{\frac{2}{3}}\). [3]
OCR C1 Q3
5 marks Moderate -0.8
Find the set of values of \(x\) for which
  1. \(6x - 11 > x + 4\), [2]
  2. \(x^2 - 6x - 16 < 0\). [3]
OCR C1 Q4
7 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]
OCR C1 Q5
8 marks Moderate -0.3
$$f(x) = x^2 - 10x + 17.$$
  1. Express \(f(x)\) in the form \(a(x + b)^2 + c\). [3]
  2. State the coordinates of the minimum point of the curve \(y = f(x)\). [1]
  3. Deduce the coordinates of the minimum point of each of the following curves:
    1. \(y = f(x) + 4\), [2]
    2. \(y = f(2x)\). [2]
OCR C1 Q6
10 marks Moderate -0.3
The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.
  1. Show that \(\angle PQR = 90°\). [4]
Given that \(P\), \(Q\) and \(R\) all lie on a circle,
  1. find the coordinates of the centre of the circle, [3]
  2. show that the equation of the circle can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]
OCR C1 Q7
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A (5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
OCR C1 Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.
  1. Find an equation for \(l\). [5]
The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,
  1. find the coordinates of \(B\). [6]
OCR C1 Q9
13 marks Moderate -0.3
The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
  2. Find the exact coordinates of the stationary point of \(C\). [5]
  3. Determine the nature of the stationary point. [2]
  4. Sketch the curve \(C\). [2]