Download Exercise 2 - Elgin Academy

Elgin Academy
Mathematics Department
Higher
Unit 3 Revision
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Outcome 1
Use vectors in three dimensions.
PC(a): Determine whether three points with given coordinates are collinear.
Example
S, T and V have co-ordinates (1, 3, 4), (5, 5, 6) and (17, 11, 12)
(i)
Write down the components of SV .
17   1  16 
     
SV  v  s   11    3    8 
12   4   8 
     
Hence show that the points S, T and V are collinear.
 4
 
ST  t  s   2  since SV  4 ST then SV is parallel to ST
 2
 
(ii)
But S is a common point, so S , T and V are collinear.
Exercise 1
1. A, B and C have co-ordinates (2,-3,4), (10,5,0) and (12,7,-1)
(i)
(ii)
Write down the components of AC .
Hence show that the points A, B and C are collinear.
2. D, E and F have co-ordinates (1,3,-4), (9,-1,0) and (3,2,-3)
(i)
Write down the components of DF .
(ii) Hence show that the points D, E and F are collinear.
3. P, Q and R have co-ordinates (3,-2,0), (2,1,5) and (0,7,15)
(i)
Write down the components of PR .
(ii) Hence show that the points P, Q and R are collinear.
PC(b): Determine the coordinates of the point which divides the join of two given
points internally in a given numerical ratio.
Example
The point D divides XY in the ratio 3:2
as shown in the diagram.
Find the coordinates of D.
Y(6, -4, 11)
D
X(1, 1, 1)
XD
DY

3
2
, so XD 
3
DY , i.e. 2 XD  3DY
2
 4 
 
1
so 2(d  x )  3( y  d )  d  2 x  3 y     2   D  (4,  2, 7)
5
 7 
 
Exercise 2
T(6, 7, -3)
1. The point P divides the line ST in the ratio 3:1
as shown in the diagram.
Find the co-ordinates of P.
P
S(-2, 3, 1)
F(8, -4, 6)
2. The point E divides the line DF in the ratio 2:1
as shown in the diagram.
E
Find the co-ordinates of E.
C(-3, 8, 0)
D(2, -1, 3)
3. The point B divides the line AC in the ratio 3:1
as shown in the diagram.
Find the co-ordinates of B.
B
A(1, 0, -4)
PC(c): Use the scalar product.
Example
C
The diagram shows triangle ABC where
 1 
  5
 
 
AB =   3  and AC =   5 
  1
  4
 
 
B
A
a) Find the value of AB . AC .
AB. AC  1.(5)  (3).(5)  (1).(4)  5  15  4  14
b) Use the result of (a) to find the size of angle BAC.
AB  11 and AC  66
cos BAC 
AB. AC
AB . AC

14
11. 66

14
11 6
BAC  58.7
Exercise 3
C
1. The diagram shows triangle ABC where
 1 
 2
 
 
AB =   2  and AC =  0 
 1 
  1
 
 
A
a) Find the value of AB. AC .
b) Use the result of (a) to find the size of angle BAC.
B
2. The diagram shows triangle ABC where
C
 2 
 1 
 
 
AB =   1  and AC =  0 
  2
  2
 
 
a) Find the value of AB. AC .
b) Use the result of (a) to find the size of angle BAC.
B
A
3. The diagram shows triangle ABC where
C
  3
 1
 
 
AB =   2  and AC =  0 
 2 
 4 
 
 
a) Find the value of AB. AC .
b) Use the result of (a) to find the size of angle BAC.
Outcome 2
Use further differentiation and integration.
PC(a): Differentiate ksin x, kcos x.
Example
i) Differentiate 4cos x with respect to x.
d
4 cos x   4 sin x
dx
dy
1
ii) Given y   sin x , find
.
8
dx
dy
1
  cos x
dx
8
Exercise 4
1. i) Differentiate –2sin x with respect to x.
dy
1
ii) Given y  cos x , find
.
3
dx
2. i) Differentiate –3cos x with respect to x.
dy
1
ii) Given y  sin x , find
.
dx
4
3. i) Differentiate –5sin x with respect to x.
dy
1
ii) Given y  cos x , find
.
dx
2
A
B
PC(b): Differentiate using the function of a function rule.
Example
5
Find f  x  when f  x   3x  2 
u  3x  2
y  u5
dy
4
 5u 4  53 x  2 
du
dy dy du
4
f  x  


 153 x  2 
dx du dx
du
3
dx
Exercise 5
3
1. Find f  x  when f  x   x  4 .
4
2. Find f  x  when f  x    x  2 .
6
3. Find f  x  when f  x    x  5 .
n
PC(c): Integrate functions of the form f ( x)   x  q  , n rational except for –1, and
f ( x)  p cos x and f ( x)  p sin x .
Example
i) Find  7 sin x dx
 7 cos x  C
4
ii) Integrate  cos x with respect to x.
7
4
 sin x  C
7
iii) Evaluate
1 3x  3
2
2
dx .
2
1 3 1 3
1
3


3
x

3

.9  6  57
 9

9
9
1
Exercise 6
1. a) Find  3 cos x dx .
2
b) Integrate  sin x with respect to x.
3
5
c) Evaluate 2 ( x  1) 3 dx
2. a) Find  2 sin x dx .
3
b) Integrate  cos x with respect to x.
5
4
c) Evaluate 2 ( x  1) 4 dx .
3. a) Find  5 cos x dx .
2
sin x with respect to x.
5
4
c) Evaluate 3 ( x  2) 2 dx .
b) Integrate
Outcome 3
Use properties of logarithmic and exponential functions.
PC(a): Simplify a numerical expression using the laws of logarithms..
Example
i) Simplify log a 30  log a 6
30
log a
 log a 5
6
ii) Simplify 7 log 16 4  log 16 64
16384
log 16 4 7  log 16 64  log 16
 log 16 256  log 16 16 2  2
64
Exercise 7
1. a) Simplify log a 10  log a 3 .
b) Simplify 5 log 9 3  log 9 27 .
2. a) Simplify log a 16  log a 2 .
b) Simplify 5 log 27 3  log 27 9 .
3. a) Simplify log a 4  log a 15 .
b) Simplify 7 log 4 2  log 4 8 .
PC(b): Solve simple logarithmic and exponential equations.
log e 7
, find an approximation for x.
log e 4
x = 1.404, rounded from the calculator.
Example i) If x 
ii) Given that log 10 y  1.6 write down an expression for the exact value of y.
y  101.6
iii) If y  10 3.4 , find an approximation for y.
y = 2511.89, rounded from the calculator.
Exercise 8
log e 5
, find an approximation for x.
log e 3
b) Given that log 10 y = 2.4, write down an expression for the exact value of y.
c) If y = 10 3.1 , find an approximation for y.
1. a) If x =
log e 8
, find an approximation for x.
log e 5
b) Given that log 10 y = 4.2, write down an expression for the exact value of y.
c) If y = 10 1.8 , find an approximation for y.
2. a) If x =
log e 9
, find an approximation for x.
log e 7
b) Given that log 10 y = 3.8, write down an expression for the exact value of y.
c) If y = 10 2.5 , find an approximation for y.
3. a) If x =
Outcome 4
Apply further trigonometric relationships.
PC(a): Express a cos  b sin  in the form r cos     or
Example
r sin    .
Express 3 cos x  sin x in the form k cosx    where
k  0, 0    360 .
k cos x     k cos x cos   k sin x sin  
 k cos  cos x  k sin   sin x
k cos   3 ; k sin    1
k cos2    k 2 sin 2    4  k  2
k sin    1

k cos 
3
1
tan   
   330
3
2
3 cos x  sin x  2 cos x  330 
Exercise 9
1. Express 3cos x + 4sin x in the form kcos(x - ) where k  0 and 0    360.
2. Express cos x + 2sin x in the form kcos(x - ) where k  0 and 0    360.
3. Express 7cos x + 24sin x in the form kcos(x - ) where k  0 and 0    360.
Answers
Exercise 1
 10 
1. i)  10  ii) Proof
  5
 
 2
2. i)   1 ii) Proof
 1
 
  3
3. i)  9  ii) Proof
 15 
 
2. E(6, -3, 5)
3. B(-2, 6, -3)
2. a) 6
3. a) 11
Exercise 2
1. P(4, 6, -2)
Exercise 3
1. a) 1
b) 79.5
b) 26.6
b) 42.8
Exercise 4
1
3
1. a) -2cos x
b) - sin x 2. a) 3sin x
b)
1
cos x
4
3. a) -5cos x
Exercise 5
1.  3( x  4) 4
2.  4( x  2) 5
3. 6( x  5) 5
Exercise 6
1. a)3sin x + C
2
cos x + C
3
3
b) - sin x + C
5
2
b) - cos x + C
5
b)
2. a)-2cos x + C
3. a)5sin x + C
3
4
c) 63
c) 576.4
c) 2
1
3
Exercise 7
1. a) log a 30
b) 1
2. a) log a 8
b) 1
3. a) log a 60 b) 2
Exercise 8
1. a) 1.46
3. a) 1.13
b) 10 2.4
b) 10 3.8
c) 1259
c) 316
2.
a) 1.29
b) 10 4.2
c) 63.1
Exercise 9
1. 5cos (x - 53.1)
2. 5 cos (x - 63.4)
3. 25cos (x - 73.7)
1
2
b) - sin x