Core 1 Mathematics Edexcel June 2011 Question 6
6. Given that
(6x + 3x5/2) / √ x can be written in the form 6xp + 3xq
(a) write down the value of p and the value of q.
dy/dx = (6x + 3x5/2) / √ x and that y = 90 when x = 4,
(b) find y in terms of x, simplifying the coefficient of each term.
Core 1 Mathematics Edexcel June 2011 Question 7
7. f(x) = x2 + (k + 3)x + k,
where k is a real constant.
(a) Find the discriminant of f (x) in terms of k.
(b) Show that the discriminant of f (x) can be expressed in the form (k + a)2 + b where
a and b are integers to be found.
(c) Show that, for all values of k, the equation f(x) = 0 has real roots.
Core 1 Mathematics Edexcel June 2011 Question 8
8. Figure 1 shows a sketch of the curve C with equation y = f (x).
The curve C passes through the origin and through (6, 0).
The curve C has a minimum at the point (3, –1).
On separate diagrams, sketch the curve with equation
(a) y = f(2x),
(b) y = −f(x),
(c) y = f (x + p), where p is a constant and 0 < p < 3.
On each diagram show the coordinates of any points where the curve intersects the x-axis
and of any minimum or maximum points.
Core 1 Mathematics Edexcel June 2011 Question 9
9. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,
2 + 4 + 6 + …… + 100
(b) In the arithmetic series
k + 2k + 3k + …… + 100
k is a positive integer and k is a factor of 100.
(i) Find, in terms of k, an expression for the number of terms in this series.
(ii) Show that the sum of this series is
50 + 5000/k
(c) Find, in terms of k, the 50th term of the arithmetic sequence
(2k + 1), (4k + 4), (6k + 7), …… ,
giving your answer in its simplest form.
Core 1 Mathematics Edexcel June 2011 Question 10
10. The curve C has equation
y = (x + 1)(x + 3)2
(a) Sketch C, showing the coordinates of the points at which C meets the axes.
(b) Show that dy/dx = 3x2 + 14x + 15
The point A, with x-coordinate -5, lies on C.
(c) Find the equation of the tangent to C at A, giving your answer in the form y = mx + c,
where m and c are constants.
Another point B also lies on C. The tangents to C at A and B are parallel.
(d) Find the x-coordinate of B.
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