Questions — Edexcel (9685 questions)

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Edexcel C1 Q7
8 marks Moderate -0.3
7. Given that the equation $$4 x ^ { 2 } - k x + k - 3 = 0$$ where \(k\) is a constant, has real roots,
  1. show that $$k ^ { 2 } - 16 k + 48 \geq 0 ,$$
  2. find the set of possible values of \(k\),
  3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value.
Edexcel C1 Q8
10 marks Moderate -0.8
8. (a) The first and third terms of an arithmetic series are 3 and 27 respectively.
  1. Find the common difference of the series.
  2. Find the sum of the first 11 terms of the series.
    (b) Find the sum of the integers between 50 and 150 which are divisible by 8 .
Edexcel C1 Q9
13 marks Standard +0.3
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 \(28 \frac { 1 } { 8 }\).
Edexcel C1 Q10
13 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e8f154b-c232-49ee-a798-f61ff08ca0b9-4_663_1113_950_402} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the parallelogram \(A B C D\).
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 \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The points \(C\) and \(D\) lie on the straight line \(l _ { 2 }\) which has the equation \(x - 4 y - 21 = 0\).
  2. Show that the distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(k \sqrt { 17 }\), where \(k\) is an integer to be found.
  3. Find the area of parallelogram \(A B C D\).
Edexcel C1 Q1
4 marks Easy -1.2
  1. 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.
Edexcel C1 Q2
4 marks Easy -1.8
2. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).
Edexcel C1 Q3
6 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01488c70-db95-43cb-9216-23d7dbaaf9fe-2_549_944_708_347} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). 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. \(y = 2 \mathrm { f } ( x )\),
  2. \(y = - \mathrm { f } ( x )\).
Edexcel C1 Q4
6 marks Easy -1.8
4. (a) Solve the inequality $$4 ( x - 2 ) < 2 x + 5$$ (b) Find the value of \(y\) such that $$4 ^ { y + 1 } = 8 ^ { 2 y - 1 } .$$
Edexcel C1 Q5
6 marks Moderate -0.3
  1. A sequence of terms \(\left\{ t _ { n } \right\}\) is defined for \(n \geq 1\) by the recurrence relation
$$t _ { n + 1 } = k t _ { n } - 7 , \quad t _ { 1 } = 3$$ where \(k\) is a constant.
  1. Find expressions for \(t _ { 2 }\) and \(t _ { 3 }\) in terms of \(k\). Given that \(t _ { 3 } = 13\),
  2. find the possible values of \(k\).
Edexcel C1 Q6
7 marks Easy -1.2
6. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2. Find an equation for the tangent to the curve at \(A\).
Edexcel C1 Q7
8 marks Easy -1.2
7. As part of a new training programme, Habib decides to do sit-ups every day. He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
Edexcel C1 Q8
11 marks Moderate -0.8
8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ 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 \(q = \frac { 1 } { 5 }\),
  2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
  3. find the rate at which the area of the stain is increasing when \(t = 15\).
Edexcel C1 Q9
11 marks Moderate -0.8
9. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).
  1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
  2. Sketch \(l\) and \(C\) on the same set of axes.
  3. Find the coordinates of the points where \(I\) and \(C\) intersect.
Edexcel C1 Q10
12 marks Moderate -0.3
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0 ,$$ and that the point \(A\) on \(C\) has coordinates (2, 6),
  1. find an equation for \(C\),
  2. find an equation for the tangent to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers,
  3. show that the line \(y = x + 3\) is also a tangent to \(C\).
Edexcel C1 Q1
3 marks Easy -1.2
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8 .$$
Edexcel C1 Q2
4 marks Easy -1.2
  1. Find
$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$
Edexcel C1 Q3
6 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c4ae1bec-12f4-492d-8027-bba4840ff545-2_337_1235_781_383} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 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.
Edexcel C1 Q4
6 marks Standard +0.3
4. (a) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(b) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$
Edexcel C1 Q5
6 marks Moderate -0.8
  1. (a) By completing the square, find in terms of the constant \(k\) the roots of the equation
$$x ^ { 2 } + 2 k x + 4 = 0 .$$ (b) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$
Edexcel C1 Q6
7 marks Moderate -0.5
  1. (a) Evaluate
$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (b) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.
Edexcel C1 Q8
9 marks Moderate -0.3
  1. Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 3 } - 4 } { x ^ { 3 } } , \quad x \neq 0$$
  1. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Given also that \(y = 0\) when \(x = - 1\),
  2. find the value of \(y\) when \(x = 2\).
Edexcel C1 Q9
13 marks Standard +0.3
9. 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.
Edexcel 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 \(A B C\).
Edexcel C1 Q1
3 marks Easy -1.3
  1. Evaluate \(49 ^ { \frac { 1 } { 2 } } + 8 ^ { \frac { 2 } { 3 } }\).
  2. A sequence is defined by the recurrence relation
$$u _ { n + 1 } = \frac { u _ { n } + 1 } { 3 } , \quad n = 1,2,3 , \ldots$$ Given that \(u _ { 3 } = 5\),
  1. find the value of \(u _ { 4 }\),
  2. find the value of \(u _ { 1 }\).
Edexcel C1 Q3
6 marks Moderate -0.3
3. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9$$
  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.