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UseKey DGCode Key Code Key Code ITEMITEM Use ITEMUsequ. qu. qu.
A 1C145 921Web2
• x2 + 10x − 24 = 0 (x + 12)(x − 2) = 0• x =−12, x = 2
A 2C146 996Web2
• log6 2x = 1⇒ x = 3
• log6 2x = 0⇒ 2x = 1
(•) x = 12
A 3B147 919Web2
• (b,3)→ (b,−3)• so B
A 4B148 55Web2
• f (2x)
• (2x)2 + 1
(•) 4x2 + 1
A 5B149 925Web2
• 2p =−8⇒ p =−4
• p2 + q = 7⇒ q =−9
A 6D150 1235Web2
• sin x −π6
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟= 1⇒ x =
2π3
so (1)√
• ymax = 3×1 + 5 = 8 so (2)√
A 7D151 9Web2
• (0,0) ⇒ not C• (1,−2)⇒C or D(•) D
A 10B152 2001Web2
u2 = 0.4×50 + 5 = 25
• u3 = 0.4×25 + 5 = 15
u4 = 0.4×15 + 5 = 11
• u5 = 0.4×11 + 5 = 9.4
A 11D153 1074Web2
• uk +1
= 3uk− 2 = 13
• uk +2
= 3uk +1− 2 = 37
A 12C154 1109Web2
• −1 < p−1 < 1• 0 < p < 2
A 13A155 929Web2
• L =−0.7L + 21
• L =211.7
(•) L =21017
A 15C156 1001Web2
• y = k(x + 2)(x − 4)• −1 = k(6 + 2)(6− 4)
(•) k =−116
A 16D157 930Web2
• (x + 2)(2x − 3) > 0• x = 0 is false
(•) x <−2, x >32
A 17A158 1163Web2
• 0−3
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟ on f ⇒ cuts x -axis twice
• b2 − 4ac > 0
A 18D159 1003Web2
• b2 − 4ac = 0• 16−(−24k) = 0
(•) k =−23
A 19A160 1076Web2
• y = k(x + 3)(x −1)(x − 2)• 12 = 6k(•) k = 2
A 21B161 91Web2
• (−5)3 + 4×(−5)2 −5k −10 = 0• −125 + 100−10 = 5k(•) k =−7
A 23C162 1902Web2
• x =− 23y − 7
3
• y = 3 − 23y − 7
3( )2+ 4 − 2
3y − 7
3( )−7
A 28C163 24Web2
• log99
32
• 32
log99
A 31D164 339Web2
• 64 = y23
• 8 = y13
(•) y = 512
A 32A165 1251Web2
• log4 8q = 1
• 8q = 4
(•) q =12
A 33A166 2034Web2
y = 100x 3
• log10
y = log10(100x 3)
• log10
y = log10100 + log
10x 3
(•)log10
y = 2 + 3log10x
C 1C167 307Web2
• ′f (x) = 12x 2
• ′f (2) = 12×4(•) ′f (2) = 48
C 2A168 1008Web2
• •
dydx
=14
x−
3
4 +12x−
3
2
C 3D169 30Web2
• f (x) = x−2
• ′f (x) =−2x−3
(•) ′f (x) =−2x 3
C 4A170 968Web2
• dydx
= 3x2 + 4x
• dydx x=1
= 3×1 + 4×1 = 7
C 6A171 1011Web2
• dvdt
= 2t + 2
• t = 3⇒dvdt
= 8
C 7B172 366Web2
• ′f (x) =−4x−3
• −4x 3
> 0
• x 3 is neg., x is neg.
page 1 Web Issue August 20th 2008 m/c Data
UseKey 89,40Code Key Code Key Code ITEMITEM Use ITEMUsequ. qu. qu.
C 8D173 330Web2
• stationary pts at ′f (x) = 0• x(x + 2) = 0(•) x = 0, x =−2
C 12B174 942Web2
• • f (x) = 2x3 − x−1 +c
C 13C175 248Web2
• y =37
x73
⎡
⎣⎢⎢
⎤
⎦⎥⎥−1
1
• y =37×1
73
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟−
37×(−1)
73
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
• y =37− −
37
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟=
67
C 14A176 115Web2
• ′f (x) = x−4
• f (x) =−13x−3
(•) f (x) =−1
3x3+c
C 15D177 2039Web2
• I = 12x + 14x 4⎡
⎣⎢⎤⎦⎥0
2
= (24 + 4)− 0• = 28
C 17A178 208Web2
• sinx = cosx
• x =π4
• sinx dx0
π4⌠
⌡⎮ + cosx dx
π4
π2⌠
⌡⎮
C 18A179 1071Web2
• know to integrate
• y =13x 3 − 2x 2 +c
C 20A180 85Web2
• dydx
=−sinx
• gradient =− sinπ2
(•) gradient =−1
C 21C181 1094Web2
• ′f (x) = 3(x 4 + 2x)2×……
• ′f (x) = ……×(4x 3 + 2)
(•) ′f (x) = 3(4x 3 + 2)(x 4 + 2x)2
C 22C182 320Web2
• y =14(2x + 1)4 ×……
• y = ……×12
(•) y =18(2x + 1)4 +c
C 23B183 98Web2
• I =−cos3x×……• I = ……× 1
3
(•) I =−13
cos3x +c
G 1C184 1045Web2
• PQ = (−2− 3)2 + (3−5)2 + (1− 2)2
• PQ = 30
G 2B185 86Web2
• mAB
=8− 0
0−(−4)= 2
• mAC
=8−(−4)0− p
=−12p
• −2p = 12……p =−6
G 3A186 129Web2
• m =−2− 41−(−3)
=−32
• y − 4 =−32(x + 3)
2y − 8 =−3x − 9• 3x + 2y + 1 = 0
G 5D187 976Web2
• y =−35x +
85 AND m =−
35
• m⊥ =53
G 7A188 1048Web2
• m⊥ to ST
=12
• y − 8 =12(x − 6)
(•) y =12x + 5
G 9D189 144Web2
• x2 + y2 − 4x − 3y +13
= 0
• C = (2,1 12)
G 10C190 970Web2
• centre(P) = (2,−3)• radius(Q) = 4
•( ) (x − 2)2 + (y + 3)2 = 16
G 11C191 1061Web2
• mOC
=43
• mtgt
=−34
(•) y =−34
x
G 12A192 2083Web2
(2y + 5)2 + y2
• −6(2y + 5)− 3y −5 = 0
• 5y2 + 5y −10 = 0
G 16D193 52Web2
• PQ
=−6−44
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
• PQ
= 36 + 16 + 16 = 68
G 17A194 1185Web2
• PQ
=311
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟ • RS
=933
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
(•) R = (1, 4,2) so S = (10,7,5)
G 18C195 165Web2
• 67
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟
2
+ −37
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟
2
+ z 2 = 1
• z 2 =449
(•) z = ±27
G 19D196 956Web2
• 25
=−4p
• 2p =−20, p =−10
G 20D197 35Web2
• x =12(2v − 3u)
• x =12
4−80
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟−
303
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
(•) x =
12−4
−32
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
page 2 Web Issue August 20th 2008 m/c Data
UseKey PMCode Key Code Key Code ITEMITEM Use ITEMUsequ. qu. qu.
G 21B198 1033Web2
• NP
=14
NL
• NL
=−u + v
(•) NP
=14−u + v( )
G 22D199 1137Web2
G 24A200 1138Web2
• PX
=−15−25−20
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟= 5−3−5−4
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
• XQ
=−6−10−8
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟= 2−3−5−4
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟ (•) ratio = 5 : 2
G 25B201 1035Web2
• KL
=100−10
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟ • KP
=40−4
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
(•) K = (−1,0, 4) so P = (3,0,0)
G 26C202 1036Web2
• r.s = r s cos 45°
• r.s = 1× 2×1
2(•) r.s = 1
G 27A203 1059Web2
• g32
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟.
5−4−g
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟= 0
• 5g −12− 2g = 0 (•) g = 4
G 28A204 983Web2
• cost° =3
2× 2=
32
• t = 30
G 29B205 1039Web2
• i.i + 2i.j − i.k
• i2
+ 0 + 0(•) 1
T 1B206 68Web2
• 2t −π4
=π2
• 2t =3π4
(•) t =3π8
T 3A207 1143Web2
• −12
……
• ……−12
=−1
T 4B208 974Web2
• max/ min = ±3 so A or B
• x = π, y = 3sinπ2
= 3 so B
T 5B209 7Web2
• min 1− cos t −π3
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
when cos t −π3
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟ is max
• cos t −π3
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟= 1 (•) t =
π3
T 7A210 1150Web2
• sinx =1
2 or −
1
2
• x =π4
or no sol. so A
T 8B211 34Web2
• cos 45cos30 + sin 45sin 30
• 1
2.
32
+1
2.12
3 + 1
2 2
T 9B212 216Web2
• sin 2x = 2sinx cosx
• cosx = 1−k 2
(•) sin 2x = 2k 1−k 2
T 12D213 1196Web2
• k 2 = 3 + 1, k = 2
• tana° =1
3, a = 30
T 13B214 42Web2
• k sina = 2, k cosa =−3
• tana =−23
page 3 Web Issue August 20th 2008 m/c Data
y =1
x 2 + 10x − 24.
For what values of x is y undefined?
A x =−6 and x = 4B x =−2 and x = 12C x = 2 and x =−12D x = 6 and x =−4
145
Which of the following sketches shows part of the graph of y = log62x ?
x
y
(6, 1)
(1, 0)2
A
x
y
(3, 1)
(1, 0)
D
x
y
(3, 1)
(1, 0)2
C
x
y
(6, 1)
(1, 0)
B
146
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 1
The diagram shows the graph of y = g(x).
O x
y (b, 3)
(a, –2)
(0, 1)
y = g(x)
A
B
D
C
(a, 2)
(0, –1)
(b, –3)
Ox
y
(a, 4)
(0, 1)
(b, –1)
O x
y
O
(a, –3)
(b, 2)
x
y
(–b, 3)
(0, 1)
(–a, –2)
Ox
y
Which diagram below shows the graph of y =−g(x)?
147
f and g are functions defined by f (x) = x 2 + 1 and g(x) = 2x, where x isa real number.
Find an expression for f g x( )( ).
A f g x( )( ) = 2x 2 + 2
B f g x( )( ) = 4x 2 + 1
C f g x( )( ) = 2x 3 + 1
D f g x( )( ) = 2x 3 + 2x
148
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 2
When x 2 − 8x + 7 is written in the form
(x + p)2 + q, what is the value of q?
A −57B −9C 7D 23
149
Here are two statements about the graph of y = 3sin x −π6
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+ 5 for 0≤ x ≤ 2π :
(1) the maximum occurs when x =2π3
(2) the maximum value of the function is 8.
A only statement (1) is correctB only statement (2) is correctC neither statement is correctD both statements are correct
Which of the following is true?
150
The diagram shows the sketch of a cubic function f with turning points at (−1,2) and (1,−2).Which of the following is most likely to be f (x)?
(–1,2)
(1,–2)
y
xO
A x 3 − x
B −x 3 + 3x
C −x 3 − 3x 2 − x + 3
D x 3 − 3x
151
A sequence is defined by the recurrence relationun+1 = 0.4un + 5, u1 = 50.
What is the smallest value of n for which un < 11 ?
A n = 3B n = 5C n = 7D n = 9
152
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 3
The terms of a sequence satisfy the recurrence relation un+1 = 3un − 2.
If uk = 5, what is the value of uk+2 ?
A73
B 7C 13D 37
153
A sequence is defined by the recurrence relationu
n+1= (p−1)u
n+ 3 with u
0= 12.
For what values of p does this sequence have a limit?
A 0≤ p ≤ 2 onlyB −1≤ p ≤ 1 onlyC 0 < p < 2 onlyD −1 < p < 1 only
154
A sequence is defined by the recurrence relation un+1 =−0.7un + 21 with u0 = 10.
What is the limit of this sequence?
A21017
B21013
C 30D 70
155
The diagram shows part of the graph of a quadratic function. The equation of of the graph is of the form y = k(x − l)(x −m).
–2x
y
O4
(6, –1)
A −9B −2
C −116
D −18
What is the value of k ?
156
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 4
For what range of values of x is 2x 2 + x − 6 > 0?
A x <−32
or x > 2
B −32
< x < 2
C −2 < x <32
D x <−2 or x >32
157
The diagram shows part of the graph of a parabola y = f (x) which touches the x -axis as shown.The function g is given by g(x) = f (x)− 3.
x
y
O
y = f(x)
A discriminant > 0B discriminant = 0C discriminant < 0D discriminant = any real number
Which of the following describes the value of the discriminant of g ?
158
The roots of the equation kx 2 + 4x − 6 = 0 are equal.What is the value of k ?
A −9
B −16
C −13
D −23
159
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 5
The diagram shows part of the graph of a cubic function f .
y
xO 1 2
12
–3
A y = 2(x −1)(x − 2)(x + 3)B y =−2(x + 1)(x + 2)(x − 3)C y = 12(x −1)(x − 2)(x + 3)D y = 12(x + 1)(x + 2)(x − 3)
What is the equation of the graph?
160
If x + 5 is a factor of the polynomial x 3 + 4x 2+kx −10 what is the value of k ?
A −3B −7C −35D −43
161
The line with equation 3x + 2y + 7 = 0 intersects the parabola
with equation y = 3x 2 + 4x −7 at two points, P and Q.Which of the following equations can be solved to find the coordinates of P and Q ?
A y = 3 2y + 7( )2+ 4 2y + 7( )−7
B y = 3 −2y −7( )2+ 4 −2y −7( )−7
C y = 3 − 23y − 7
3( )2+ 4 − 2
3y − 7
3( )−7
D y = 3 23y + 7
3( )2+ 4 2
3y + 7
3( )−7
162
What is the exact value of log927 ?
A13
B23
C32
D 3
163
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 6
Given that logy64 =
23, find the value of y .
A 24B 96C 256D 512
164
Given that log4 8 + log4 q = 1, what is the value of q ?
A12
B132
C18
D 2
165
A graph is drawn of log10 y against log10x where y = 100x3.
What is the equation of the graph ?
A log10 y = 3log10x + 2
B log10 y = 100log10x
C log10 y = 300(log10x)2
D log10 y = 300log10x
166
Given that f (x) = 4x3+5, find the value of ′f (2) .
A 21B 26C 48D 53
167
What is the derivative, with respect to x, of x14 − x
−12 ?
A14
x−
34 +
12x−
32
B14
x−
34 −
12x−
32
C14
x−
14 +
12x−
32
D14
x14 +
12x−
12
168
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 7
If f (x) =
1x 2
and x ≠ 0, find ′f (x).
A12x
B2x
C −1x 3
D −2x 3
169
A curve has equation y = x 3 + 2x 2 + 5.What is the gradient of the curve at the point where x = 1?
A 7B 8C 10D 12
170
The speed, v m/s, at time t seconds is given by v(t) = t 2 + 2t.What is the rate of change of speed when t = 3?
A 8 m / s2
B 12 m / s2
C 15 m / s2
D 18 m / s2
171
Find the values of x for which the graph of the
function f (x) =2
x2,x ≠ 0, has a positive gradient.
A all possible valuesB x < 0 onlyC x > 0 only
D x3<4 only
172
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 8
Which of the following graphs could be part of the graph of a function fsuch that ′f (x) = x(x + 2)?
A
B
C
D
x
f(x)
O2
x
f(x)
O–2
x
f(x)
O2
x
f(x)
O–2
173
If f (x) = 6x 2 + x−2( ) dx⌠
⌡, then find f (x) .
A 18x 3 − x−1 +c
B 2x 3 − x−1 +c
C 12x − 2x−3 +c
D 2x 3 −13x−3 +c
174
Find x4
3( )−1
1⌠
⌡
⎮⎮⎮
dx .
A 0
B37
C67
D103
175
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 9
Find
1x 4
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟
⌠⌡⎮ dx .
A −1
3x 3+c
B −3x 3
+c
C −1
5x 5+c
D5x 5
+c
176
What is the value of (12 + x3)dx
0
2
⌠
⌡⎮ ?
A 8B 12C 16D 28
177
The diagram shows the curves with equations
y = sinx and y = cosx for 0≤ x ≤π2
.
O
y
xπ/2
y = sinx
y = cosx
A sinx dx0
π4⌠⌡⎮
+ cosx dxπ4
π2⌠⌡⎮
B cosx − sinx( ) dx0
π2⌠⌡⎮
C sinx − cosx( ) dx0
π2⌠⌡⎮
D sinx + cosx( ) dx0
π2⌠⌡⎮
Find an expression for the shaded area.
178
If
dydx
= x 2 − 4x, express y in terms of x .
A y = 13x 3 − 2x 2 +c
B y = 2x − 4 +c
C y = x 2 − 4x +c
D y = x 3 − 4x 2 +c
179
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 10
A curve has equation y = cosx.
What is the gradient of the curve at the point where x =π2
?
A −1
B −π4
C 0D 1
180
What is the derivative, with respect to x, of x 4 + 2x( )3 ?
A 3 4x 3 + 2( )2
B 14
15x 5 + x 2( )4
C 3 x 4 + 2x( )24x 3 + 2( )
D 14
x 4 + 2x( )415x 5 + x 2( )
181
Find 2x + 1( )3⌠
⌡⎮dx .
A14
2x + 1( )4+c
B12
2x + 1( )4+c
C18
2x + 1( )4+c
D x2 + x( )3 +c
182
Find (sin 3x )⌠
⌡⎮dx .
A −cos3x +c
B −13
cos3x +c
C13
cos3x +c
D 3cos3x +c
183
R and T are the points (−2,3,1) and (3,5,2) respectively.What is the length of the line RT ?
A 6
B 24
C 30
D 74
184
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 11
The straight line joining the points A(0,8) and B(−4,0) passes through the point C(p,−4).What is the value of p ?
A −8B −6C −2D 6
185
Find the equation of the line passing through the points with coordinates (1,−2) and (−3,4).
A 3x + 2y + 1 = 0B 3x − 2y −7 = 0C 2x + 3y + 4 = 0D 2x − 3y − 8 = 0
186
A line L has equation 3x + 5y − 8 = 0.What is the gradient of a line perpendicular to line L?
A − 35
B − 13
C 35
D 53
187
In triangle RST, R has coordinates (6, 8) and the gradient of ST is –2.
S
x
y
O
T
R(6, 8)
A y = 12x + 5
B y =−2x + 20C y = 1
2x + 2
D y =−2x + 22
What is the equation of the altitude through R ?
188
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 12
What are the coordinates of the centre of the circle with
equation 3x 2 + 3y 2 −12x − 9y + 1 = 0 ?
A (12,9)B (6,4 1
2)
C (4,3)D (2,1 1
2)
189
Circle P has equation x 2 + y2 − 4x + 6y −1 = 0.
Circle Q has equation (x + 1)2 + (y −1)2 = 16.Circle R has the same centre as circle P and the same radius as circle Q.What is the equation of circle R?
A (x + 2)2 + (y − 3)2 = 4
B (x − 2)2 + (y + 3)2 = 4
C (x − 2)2 + (y + 3)2 = 16
D (x + 2)2 + (y − 3)2 = 16
190
A circle, centre C(−3,−4), passes through the origin.
x
y
O
C (–3, –4)
A y =43
B y =34
x
C y =−34
x
D y =−43
x
What is the equation of the tangent to the circle at the origin ?
191
The line with equation x = 2y + 5 and the circle with equation
x2 + y2 − 6x − 3y −5 = 0 intersect at the points P and Q.Which of the following equations will give the y -coordinates of P and Q ?
A 5y2 + 5y −10 = 0
B y2 −5y −10 = 0
C 2y2 − 9y −5 = 0
D y2 −13y − 30 = 0
192
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 13
P is the point (4,3,−1) and Q is (−2,−1,3).What is the length of PQ ?
A 6
B 12
C 24
D 68
193
P, Q and R have coordinates (1,2,−1), (4,3,0) and (1, 4, 2) respectively.
If RS
= 3PQ
, what are the coordinates of S ?
A (10,7,5)B (19,19,−1)C (−10,−7,−5)D (−8,−5,−1)
194
a has components
67
− 37
z
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
. If a is a unit vector find the values of z .
A ±47
B ±37
C ±27
D ±17
195
Vectors u = 2i− 4j −8k and v = 5i + pj −20k are parallel.What is the value of p?
A 10B −1C −7D −10
196
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 14
u and v have components 101
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
and2−40
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
respectively.
If 3(x + u) = 2v + x, find the components of x .
A1−8−3
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
B
32
−4− 1
2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
C
12
−4− 1
2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
D
12
−4− 3
2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
197
KLMN is a parallelogram as shown in the diagram. Q is the midpoint of KN and R is the midpoint of MN.
u
v
N
Q
L
M
K
R
P
A NP
=− 14(u + v)
B NP
= 14(v −u)
C NP
= 14(u −v)
D NP
= 14(u + v)
If LM
= u and KL
= v, find an expression for NP
.
198
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 15
The diagram shows a cuboid OPQRSTUV.The coordinates of U are (6, 4,2).M and N are the midpoints of UQ and SV .
S
U(6, 4, 2)
Q
O
M
V
R
P
TN
x
yz
A62−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
B−61−2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
C1−6−2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
D−6−2−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
What are the components of MN
?
199
P,X and Q are collinear points with coordinates as shown in the diagram.
P(–3, 14, 17)
Q(–24, –21, –11)
X(–18, –11, –3)
A 5 : 2B 5 : 7C 2 : 7D 2 : 5
In what ratio does X divide PQ ?
200
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 16
K and L have coordinates (−1,0, 4) and (9,0,−6) as shown in the diagram.P divides KL in the ratio 2:3.
K (–1, 0, 4)
L (9, 0, –6)
P
A (5,0,−2)B (3,0,0)
C165
,0,−45
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
D (4,0,−4)
What are the coordinates of P?
201
EFGH is a square of side 1 unit as shown in the diagram.
HG
= r and HF
= s.
s
rH
E F
G
A1
2
B23
C 1D 2
What is the value of r.s ?
202
Vectors u and v are defined by u =g32
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟ and v =
5−4−g
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟.
If u and v are perpendicular, what is the value of g ?
A 4
B133
C −4
D127
203
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 17
The diagram shows two vectors, a and b, inclined at angle of t°.
a = b = 2 units and a.b = 3
t°
a
b
A 30B 45C 60D 90
What is the value of t ?
204
The diagram shows three unit vectors, i, j and k.
x
z
O
y
i
jk
A 0B 1C 2D 4
What is the value of i.(i + 2j −k) ?
205
For 0≤ x ≤ π, the maximum value of sin 2x −π4
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟ occurs when x = t.
What is the value of t ?
Aπ8
B3π8
Cπ2
D3π4
206
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 18
What is the exact value of sin
7π6
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟− sin
5π6
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟ ?
A −1
B − 3
C32
D 0
207
Which of the sketches below shows the graph with
equation y = 3sin x − π2( )?
A
D
C
B 3
–3
2πx
y
O
3
–3
2πx
y
O
3+π/2
–3+π/22π x
y
O
3–π/2
–3–π/2
2π
y
xO
208
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 19
A minimum value of 1− cos x −π3
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟ occurs when x = t, where 0≤ t ≤
3π2
.
What is the value of t ?
A 0
Bπ3
C π
D4π3
209
What is the solution to the equation 2sin2 x = 1
in the interval 0≤ x <π2 ?
A π4
B π6
C π6
D π3
210
What is the value of cos(45°− 30°) ?
A3 −1
2 2
B3 + 1
2 2
C3 −12
D2 − 3
2
211
Given that sinx = k, where 0≤ x <
π2
, find an expression for sin 2x .
A sin2x = 2k
B sin2x = 2k 1−k2
C sin 2x = 2k 1 + k2
D sin 2x = 2k2 −1
212
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 20
k and a are given by k cosa° = 3 and k sina° = 1 where k > 0 and 0≤ a < 90.What are the values of k and a ?
k aA 4 60B 4 30C 2 60D 2 30
213
2sinx − 3cosx is expressed in the form k cosx cosa + k sinx sinawhere k and a are constants and 0≤ a ≤ 2π.What is the value of tana ?
A −32
B −23
C23
D32
214
Item bank for teachers to access on the SQA Web-site : August 20th 2008 page 21
