Download Solving quadratics

1.
(a)
Show that x2 + 6x + 11 can be written as
(x + p)2 + q
where p and q are integers to be found.
(2)
(b)
In the space at the top of page 7, sketch the curve with equation y = x2 + 6x + 11, showing
clearly any intersections with the coordinate axes.
(2)
(c)
Find the value of the discriminant of x2 + 6x + 11
(2)
(Total 6 marks)
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2.
(a)
On the axes below sketch the graphs of
(i)
y = x (4 – x)
(ii)
y = x2 (7 – x)
showing clearly the coordinates of the points where the curves cross the coordinate axes.
(5)
(b)
Show that the x-coordinates of the points of intersection of
y = x (4 – x) and y = x2 (7 – x)
are given by the solutions to the equation x(x2 – 8x + 4) = 0
(3)
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The point A lies on both of the curves and the x and y coordinates of A are both positive.
(c)
Find the exact coordinates of A, leaving your answer in the form (p + q√3, r + s√3),
where p, q, r and s are integers.
(7)
(Total 15 marks)
f(x) = x2 + 4kx + (3 + 11k),
3.
(a)
where k is a constant.
Express f(x) in the form (x + p)2 + q, where p and q are constants to be found in terms of
k.
(3)
Given that the equation f(x) = 0 has no real roots,
(b)
find the set of possible values of k.
(4)
Given that k = 1,
(c)
sketch the graph of y = f(x), showing the coordinates of any point at which the graph
crosses a coordinate axis.
(3)
(Total 10 marks)
4.
The equation x2 + 3px + p = 0, where p is a non-zero constant, has equal roots.
Find the value of p.
(Total 4 marks)
5.
(a)
Factorise completely x3 – 6x2 + 9x
(3)
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(b)
Sketch the curve with equation
y = x3 – 6x2 + 9x
showing the coordinates of the points at which the curve meets the x-axis.
(4)
Using your answer to part (b), or otherwise,
(c)
sketch, on a separate diagram, the curve with equation
y = (x – 2)3 – 6(x – 2)2 + 9(x – 2)
showing the coordinates of the points at which the curve meets the x-axis.
(2)
(Total 9 marks)
6.
The equation kx2  4 x  (5  k )  0 , where k is a constant, has 2 different real solutions for x.
(a)
Show that k satisfies
k 2  5k  4  0.
(3)
(b)
Hence find the set of possible values of k.
(4)
(Total 7 marks)
7.
Given that the equation 2qx2 + qx – 1 = 0, where q is a constant, has no real roots,
(a)
show that q2 + 8q < 0.
(2)
(b)
Hence find the set of possible values of q.
(3)
(Total 5 marks)
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8.
The equation
x2 + kx + 8 = k
has no real solutions for x.
(a)
Show that k satisfies k2 + 4k – 32 < 0.
(3)
(b)
Hence find the set of possible values of k.
(4)
(Total 7 marks)
9.
Find the set of values of x for which
x 2 – 7x – 18 > 0.
(Total 4 marks)
10.
The equation x2 + 2px + (3p + 4) = 0, where p is a positive constant, has equal roots.
(a)
Find the value of p.
(4)
(b)
For this value of p, solve the equation x2 + 2px + (3p + 4) = 0.
(2)
(Total 6 marks)
11.
x2 + 2x + 3  (x + a)2 + b.
(a)
Find the values of the constants a and b.
(2)
(b)
In the space provided below, sketch the graph of y = x2 + 2x + 3, indicating clearly the
coordinates of any intersections with the coordinate axes.
(3)
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(c)
Find the value of the discriminant of x2 + 2x + 3. Explain how the sign of the
discriminant relates to your sketch in part (b).
(2)
The equation x2 + kx + 3 = 0, where k is a constant, has no real roots.
(d)
Find the set of possible values of k, giving your answer in surd form.
(4)
(Total 11 marks)
12.
y
C
A
P
R
O
Q
x
The diagram above shows part of the curve C with equation y = x2 – 6x + 18. The curve meets
the y-axis at the point A and has a minimum at the point P.
(a)
Express x2 – 6x + 18 in the form (x – a)2 + b, where a and b are integers.
(3)
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(b)
Find the coordinates of P.
(2)
(c)
Find an equation of the tangent to C at A.
(4)
The tangent to C at A meets the x-axis at the point Q.
(d)
Verify that PQ is parallel to the y-axis.
(1)
The shaded region R in the diagram is enclosed by C, the tangent at A and the line PQ.
(e)
Use calculus to find the area of R.
(5)
(Total 15 marks)
13.
Given that the equation kx2 + 12x + k = 0, where k is a positive constant, has equal roots, find
the value of k.
(Total 4 marks)
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14.
Given that
f(x) = x2 – 6x + 18,
(a)
x  0,
express f(x) in the form (x – a)2 + b, where a and b are integers.
(3)
The curve C with equation y = f(x), x  0, meets the y-axis at P and has a minimum point at Q.
(b)
Sketch the graph of C, showing the coordinates of P and Q.
(4)
The line y = 41 meets C at the point R.
(c)
Find the x-coordinate of R, giving your answer in the form p + q2, where p and q are
integers.
(5)
(Total 12 marks)
15.
f(x) = x2 – kx + 9, where k is a constant.
(a)
Find the set of values of k for which the equation f(x) = 0 has no real solutions.
(4)
Given that k = 4,
(b)
express f(x) in the form (x – p)2 + q, where p and q are constants to be found,
(3)
(c)
write down the minimum value of f(x) and the value of x for which this occurs.
(2)
(Total 9 marks)
16.
(a)
Solve the equation 4x2 + 12x = 0.
(3)
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f(x) = 4x2 + 12x + c,
where c is a constant.
(b)
Given that f(x) = 0 has equal roots, find the value of c and hence solve f(x) = 0.
(4)
(Total 7 marks)
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