Mathematics
Higher
Block 3 Practice Assessment A
Read carefully
1. Calculators may be used.
2. Full credit will be given only where the solution contains appropriate working.
3. Answers obtained from reading from scale drawings will not receive any credit.
Questions
1
2
Assessment
Standard
RC1.3
RC1.4
3
RC1.4
4
RC1.4
5
EF1.4
6
7
EF1.4
EF1.4
8
EF1.4
9
EF1.1
10
EF1.1
11
EF1.2
12
RC1.2
Subskill details
Differentiating k sin x, k cos x
Integrating functions of the form f ( x)  ( x  q)n , n  1
Integrating functions of the form f ( x)  p cos x and
f ( x)  p sin x
Calculating definite integrals of polynomial functions with
integer limits
Determining the resultant of vector pathways in three
dimensions
Working with collinearity
Determining the coordinates of an internal division point of a
line
Evaluate a scalar product given suitable information and
determine the angle between two vectors
Simplifying a numerical expression, using the laws of
logarithms and exponents
Solving logarithmic and exponential equations, using the laws
of logarithms and exponents
Converting a cos x  b sin x to k cos( x   ) or k sin( x   ) , α
in 1st quad k  0
Solve trigonometric equations in degrees, including those
involving trigonometric formulae or identities, in a given
interval
1
Differentiate the function f ( x)  7 sin x with respect to x.
2
g( x)  ( x  7)4 , find g ( x), x  7.
3
Find  8 cos  d .
(1)
RC1.3
(#R2.1, 2)
RC1.4
(1)
RC1.4
1
4
Find
 ( x  5) dx.
3
(4)
3
RC1.4
5
ABCDE is a pyramid with rectangular base BCDE.
A
C
B
D
E
The vectors EB, ED, EA are given by:
 10 
 10 
7


 
 
EB   16  ; ED   4  ; EA   13 
 10 
 2 
 1


 
 
Express BA in component form.
(3)
EF1.4
6
An air traffic controller is landing airplanes, he needs to ensure that:
 the next 3 airplanes approaching the runway are in a straight line
 the distance between airplane 2 and airplane 3 is three times the distance between
airplane 1 and airplane 2
Relative to suitable axes, the position of each airplane can be represented by the
points A (-2, 5, 7), B (2, 3, 10), and C (14,–3, 19) respectively.
B (2, 3, 10)
A (-2, 5, 7)
airplane 1
C (14, -3, 18)
airplane 3
airplane 2
Has the air traffic controller aligned the planes correctly?
You must justify your answer.
7
(#E2.1, 4, #E2.2)
EF1.4
The points P, Q and R lie in a straight line, as shown. Q divides PR in the ratio 2:3.
Find the coordinates of Q.
R(12, −11, -12)
P(2, −1, 3)
Q
(3)
EF1.4
8
The diagram shows vectors DE and DF .
D, E and F have coordinates D(-3, –4, 5), E(2, –6, -3) and F(2, 0, 3).
F
D
E
Find the size of the acute angle EDF.
(5)
EF1.4
(a) Simplify log5 2 p  log5 6q .
9
(b)
Express log a x 7  log a x 4 in the form k log a x.
10
Solve log 2 ( x  1)  4
11
Express 5sin x  4 cos x in the form k sin( x  a) where k  0 and 0  a  360 .
(1)
EF1.1
(2)
EF1.1
(2)
EF1.1
(5)
EF1.2
12
Given 5sin x  4 cos x  41 cos( x  51.3) ,
solve 5 sin 𝑥° + 4 cos 𝑥° = 3.2, 0 < 𝑥 < 360
(3)
RC1.2
Question
Points of expected response
Illustrative scheme
1
1 differentiates correctly
2
#2.1 recognises as differential
equation and hence knows to integrate form g ( x)  x  7 4 dx
 
1 starts integration correctly
2 completes integration correctly
d
(7 sin x)  7 cos x
dx
#2.1 evidence of setting up integral
1
1
( x  7) 3
1
...  c **
3
** To achieve block 2 question 10 4 the constant of integration must appear at least once
associated with a correct integral in either question 10 (block2), 2 (block 3) or 3 (block 3).
3
1 integrates correctly
1 8sin   c **
4
1 starts integration
1 ( x  5) 4
5
1
4
2
... 
3 substitutes limits
3
1
1
(1  5) 4  (3  5) 4
4
4
4
60
3 and 4 are only available as a result of an attempt at integration.
Simply substituting these into the integrand gains no marks.
Candidates who differentiate the original expression cannot gain 3 and 4.
1 BA  EB  EA
1 recognise a pathway for PT
 10 
2


2
 identify  EB vector

BE  EB    16 
 10 


 complete calculation for BA
3
Note:
..  
2 completes integration
4 evaluate definite integral
Notes:
2
Do not award 3 for (17, -3, 9).
3
 10   7   17 

    
BA  16  13  3

    
 10   1  9 

    
6
#2.1 select appropriate strategy to
show collinearity
#2.1 show they are collinear by
attempting to demonstrate that for two
appropriate vectors, one is the scalar
multiple of the other
1 interpret vector
1 eg
4
 
AB   2 
3
 
2 interpret multiple of vector
2eg
 12 
 
BC   6   3AB
9
 
Note:
7
3 complete proof
3 BC  3AB hence vectors are
parallel, but B is a common point so
A, B and C are collinear.
4 interpret ratio
4 interpret ratio BC : AB  3:1
#2.2 Explain a solution in context
#2.2 Yes, the air traffic controller has
aligned the airplanes correctly (with
statement related to previous
working).
Any appropriate combination of vectors is acceptable.
1 find vector components
1
 10 


PR   10 
 15 


2 uses correct ratio
2
 10 
2

PQ   10 
5

 15 
3 processes vectors and finds
coordinates of S
3 Q  (6,  5,  3)
NB If candidates use the Section
formula:
1 begins substitution
2 completes substitution
3 as 3 above
Note:
8
3 is only available if expressed as a coordinate.
1 find vector components
1
5 
5 
 
 
DE   2  , DF   4 
 8 
 2 
 
 
2 use scalar product
2
DE.DF
cos EDF 
DE . DF
3 process scalar product
4 process PQ and PR
3
DE.DF  (25  8  16)  33
4 DE  93 and PR  45
ie calculate magnitudes
5 find angle
5 angle = 1.04 radians or 59.3°
Note:
If the evidence for 2 does not appear explicitly, then 2 is only awarded if
working for 5 is attempted.
9(a)
1 use log a x  log a y  log a xy
1 log5 (2 p  6q)  log5 12 pq
(b)
2 log a x m  m log a x
2 7 log a x  4 log a x OR log a x 3
m
OR log a x m  log a x n  log a xxn
3 simplify to k log a x
Note:
10
Note:
3 3log a x
For 1 the final answer must be simplified to log 4 15 pq .
1 starts to solve
1 ( x  1)  24
2 solves correctly
2 x  15
If a candidate simply obtains the solution by inspection, award both points.
11
1 expand k sin( x  a)
1 k sin x cos a  k cos x sin a
stated explicitly
2 compare coefficients
2 k cos a  5 and k sin a  4
stated explicitly
3 process k
3 k  41
 process a
4
 state final form
5
Note:
12
4 a  38  7
5
41sin  x  38.7 
For 1 k (sin x cos a  cos x sin a) is acceptable.
3.2
1 sets up cos (…..) correctly
1 cos( x  51 3) 
41
2 obtains first angles
3 obtains final angles
x  51 3  60.0,300.0
2
3
x  111.3, 351.3
