Indices and Surds - Core 2 Revision
1.
(a)
Express each of the following as a power of 3:
(i)
3;
(1)
(ii)
3x
3
.
(1)
(b)
Hence, or otherwise solve the equation
3x
3
=
1
.
3
(3)
(Total 5 marks)
2.
In this question, no credit will be given for using approximate decimal values.
3
1
It is given that p = 8 2 and q = 4 4 .
(a)
3
2
Show that p = 2 .
(1)
(b)
Similarly, express q as a power of 2.
(1)
(c)
Hence express pq as a power of 2.
(2)
(Total 4 marks)
3.
Show that the substitution y = 2x transforms the equation
2(2x) + 2–x = 3
into the quadratic equation
2y2 – 3y + 1 = 0
(1)
Hence find the values of x which satisfy the equation
2(2x) + 2–x = 3.
(4)
(Total 5 marks)
4.
Solve the equation 2x = 31–x, giving your answer correct to three significant figures
(3)
(Total 3 marks)
South Wolds Comprehensive School
1
5.
(a)
Write each of the following as a power of 2:
(i)
2;
(1)
(ii)
8x.
(1)
(b)
Hence solve the equation 8x × 2x+1 = 2 .
(3)
(Total 5 marks)
6.
(a)
Express each of the following as a power of 3:
(i)
3;
(1)
(ii)
3x  9.
(1)
(b)
Hence, or otherwise, solve the equation 3x  9 =
3.
(2)
(Total 4 marks)
7.
(a)
(i)
Write
2 as a power of 2.
(1)
(ii)
Hence, express 4 2 as a power of 2.
(1)
(iii)
Hence, solve the equation 23x+4 = 4 2
(2)
(b)
Show that the equation 23x+4 = 20 has root between 0.10 and 0.11
(2)
(Total 6 marks)
8.
(a)
Write each of the following as a power of 3 :
(i)
1
;
27
(1)
(ii)
9x.
(1)
(b)
Hence solve the equation 9x × 31–x =
1
.
27
(3)
(Total 5 marks)
South Wolds Comprehensive School
2
9.
(a)
Simplify:
3
2
1
x2
x 
(i)
;
(1)
3
x 2  x;
(ii)
(1)
 3
x2 
 
 
(iii)
2
(1)
(b)
(i)
Find

1
3x 2 dx.
(3)
(ii)
Hence find the value of

9
1
1
3x 2 dx .
(2)
(Total 8 marks)
10.
(a)
Write
x in the form xk , where k is a fraction.
(1)
(b)
Hence express
x ( x  1) in the form xp – xq.
(2)
(c)
Find
 x x – 1 dx.
(3)
(d)
Hence show that

2
1
x x  1 dx 
4
15


2 1 .
(3)
(Total 9 marks)
South Wolds Comprehensive School
3