Post on 09-Jan-2016
description
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1. The exact value of tan is:
2. The period of tan3xo, x є R , is:
3. This diagram is most likelyto be part of the graph of:
1. The exact value of tan is:
2. The period of tan3xo, x є R , is:
3. This diagram is most likelyto be part of the graph of:
67π
answer
3 D. 3
1 C.
3
1 B. 3 A.
540 D. 180 C. 120 B. 60 A.
12cos D. sin - 1 C.
sin 1 B. cos A.oo
oo
xx
xx1
2y
x90
180
270
360
1. The exact value of tan is:
2. The period of tan3xo, x є R , is:
3. This diagram is most likelyto be part of the graph of:
1. The exact value of tan is:
2. The period of tan3xo, x є R , is:
3. This diagram is most likelyto be part of the graph of:
67π
3 D. 3
1 C.
3
1 B. 3 A.
540 D. 180 C. 120 B. 60 A.
12cos D. sin - 1 C.
sin 1 B. cos A.oo
oo
xx
xx1
2y
x90
180
270
360
answer
1. Which of the following has (have) a negative value:
2. The minimum value of
occurs
when x is:
3. Which of the following could be this graph:
1. Which of the following has (have) a negative value:
2. The minimum value of
occurs
when x is:
3. Which of the following could be this graph:
π D. 34π
C. 3π
B. 0 A.
23π
x 0 , 3π
xcos 1
1
2y
x90
180
270
360
ncombinatioor responseother Some D. IV and III Only C.
III and I Only B. III II, I, Only A.
45
cos IV. 3
5tan III.
65
sin II. 125
sin I.
cos2 1 D. 2cos C.
sin2 2 B. 1 21
cos A.
oo
oo
xx
xx
1. Which of the following has (have) a negative value:
2. The minimum value of
occurs
when x is:
3. Which of the following could be this graph:
1. Which of the following has (have) a negative value:
2. The minimum value of
occurs
when x is:
3. Which of the following could be this graph:
ncombinatioor responseother Some D. IV and III Only C.
III and I Only B. III II, I, Only A.
45
cos IV. 3
5tan III.
65
sin II. 125
sin I.
π D. 34π
C. 3π
B. 0 A.
23π
x 0 , 3π
xcos 1
1
2y
x90
180
270
360
cos2 1 D. 2cos C.
sin2 2 B. 1 21
cos A.
oo
oo
xx
xx
answer
1. Which of the following is/are solution(s) of sin2x = 1, x є R:
2. If has a maximum value when θ is:
3. The line with equation y = -1 intersects the curve y = √2sinx , at :
1. Which of the following is/are solution(s) of sin2x = 1, x є R:
2. If has a maximum value when θ is:
3. The line with equation y = -1 intersects the curve y = √2sinx , at :
IV III, II, I, of None D.
only III & II C. only II B. only I A.
65
IV. 43
III. 4
II. 6
I.
65
D. 3π
C. 6π
B. 0 A.
oo
oo
150 D. 210 C.
60- B. 315 A.
6π
2sin , 2 x 0
√2y
x90 180 270 360
-√2
1. Which of the following is/are solution(s) of sin2x = 1, x є R:
2. If has a maximum value when θ is:
3. The line with equation y = -1 intersects the curve y = √2sinx , at :
1. Which of the following is/are solution(s) of sin2x = 1, x є R:
2. If has a maximum value when θ is:
3. The line with equation y = -1 intersects the curve y = √2sinx , at :
IV III, II, I, of None D.
only III & II C. only II B. only I A.
65
IV. 43
III. 4
II. 6
I.
65
D. 3π
C. 6π
B. 0 A.
oo
oo
150 D. 210 C.
60- B. 315 A.
6π
2sin , 2 x 0
√2y
x90 180 270 360
-√2
answer
1. The exact value of cos is:
2. The maximum value of
occurs when x = t. What is the value of t?
3. This diagram is most likelyto be part of the graph of:
1. The exact value of cos is:
2. The maximum value of
occurs when x = t. What is the value of t?
3. This diagram is most likelyto be part of the graph of:
65π
3 D. 3
1 C.
23
B. 3 A.
1 - cos D. sin - 2 C.
2sin B. 2 - cos A.
oo
oo
xx
xx
2π x 0 , 6π
xsin - 1
65
D. 34π
C. 2π
B. 23π
A.
2y
x180 360 540
-2
1. The exact value of cos is:
2. The maximum value of
occurs when x = t. What is the value of t?
3. This diagram is most likelyto be part of the graph of:
1. The exact value of cos is:
2. The maximum value of
occurs when x = t. What is the value of t?
3. This diagram is most likelyto be part of the graph of:
65π
3 D. 3
1 C.
23
B. 3 A.
1 - cos D. sin - 2 C.
2sin B. 2 - cos A.
oo
oo
xx
xx
2π x 0 , 6π
xsin - 1
65
D. 34π
C. 2π
B. 23π
A.
2y
x180 360 540
-2
answer
1. The exact value of sin (-120o) is:
2. If has a minimum value when θ is:
3. The line with equation y = √3 intersects the curve y = 2cosx , at :
1. The exact value of sin (-120o) is:
2. If has a minimum value when θ is:
3. The line with equation y = √3 intersects the curve y = 2cosx , at :
35
D. 65π
C. 6π
B. 0 A.
oo
oo
420D. 45C.
60- B. 330 A.
6π
2sin , 2 x 0
2y
x180 540360
-2
21
D. 23
- C. 3
1 B. 3 A.
1. The exact value of sin (-120o) is:
2. If has a minimum value when θ is:
3. The line with equation y = √3 intersects the curve y = 2cosx , at :
1. The exact value of sin (-120o) is:
2. If has a minimum value when θ is:
3. The line with equation y = √3 intersects the curve y = 2cosx , at :
35
D. 65π
C. 6π
B. 0 A.
oo
oo
420D. 45C.
60- B. 330 A.
6π
2sin , 2 x 0
2y
x180 540360
-2
21
D. 23
- C. 3
1 B. 3 A.
answer
1. The exact value of cos 135o is:
2. The largest possible domain of, is:
3. This diagram is most likelyto be part of the graph of:
1. The exact value of cos 135o is:
2. The largest possible domain of, is:
3. This diagram is most likelyto be part of the graph of:
3 D. 2
1- C.
2
1 B.
21
A.
2 xD. 2- xC. 2 xB. 2 x 2- A.
x)(2f(x)
oo
oo
45)sin(x- D. x)-sin(45 C.
45)-sin(x B. 45)sin(x A.
1
-1
y
x90
180
270
360
1. The exact value of cos 135o is:
2. The largest possible domain of, is:
3. This diagram is most likelyto be part of the graph of:
1. The exact value of cos 135o is:
2. The largest possible domain of, is:
3. This diagram is most likelyto be part of the graph of:
3 D. 2
1- C.
2
1 B.
21
A.
2 xD. 2- xC. 2 xB. 2 x 2- A.
x)(2f(x)
oo
oo
45)sin(x- D. x)-sin(45 C.
45)-sin(x B. 45)sin(x A.
1
-1
y
x90
180
270
360
1. Which of the following graphs represents y = -f(x + 2):
A B C D
2. The exact value of cos is:
3. Functions f and g , are given by f(x) = 3x2 + 1 and g(x) = x2 - 4. Find an expression for f(g(x)).
1. Which of the following graphs represents y = -f(x + 2):
A B C D
2. The exact value of cos is:
3. Functions f and g , are given by f(x) = 3x2 + 1 and g(x) = x2 - 4. Find an expression for f(g(x)).
answer
35π
21
D. 31
- C. 23
B. 21
A.
4924x3x D. 1 6x9x C.
3 - 3x B. 3 - 4xA.
2424
42
(-1,3)
3
y
x-3(5,-2)
Y = f(x)
(-1,5)
y
x
(-3,2)
5
(3,2)
(1,5)
y
x
(-3,2)
-5
(3,2)
(-1,-1)
y
x
(-3,2)(5,4)
(3,2)
(-3,-3)
y
x-5 1
(3,2)
1. Which of the following graphs represents y = -f(x + 2):
A B C D
2. The exact value of cos is:
3. Functions f and g , are given by f(x) = 3x2 + 1 and g(x) = x2 - 4. Find an expression for f(g(x)).
1. Which of the following graphs represents y = -f(x + 2):
A B C D
2. The exact value of cos is:
3. Functions f and g , are given by f(x) = 3x2 + 1 and g(x) = x2 - 4. Find an expression for f(g(x)).
35π
21
D. 31
- C. 23
B. 21
A.
4924x3x D. 1 6x9x C.
3 - 3x B. 3 - 4xA.
2424
42
(-1,3)
3
y
x-3(5,-2)
Y = f(x)
(-1,5)
y
x
(-3,2)
5
(3,2)
(1,5)
y
x
(-3,2)
-5
(3,2)
(-1,-1)
y
x
(-3,2)(5,4)
(3,2)
(-3,-3)
y
x-5 1
(3,2)
answer
1. For which real values of x is the functiondefined on the set of real numbers?
2. The minimum value of
occurs when x is:
3. The line with equation y = 2 intersects the curve y = 1 - 2sinx , at :
1. For which real values of x is the functiondefined on the set of real numbers?
2. The minimum value of
occurs when x is:
3. The line with equation y = 2 intersects the curve y = 1 - 2sinx , at :
6 D. C.
2 B.
3 A.
)x(1
1x:f
2
only 1 xD. only 1- xand 1 xC.
only 1 x 1- B. 1- xand 1 except x xAll A.
3π
2cos - 1 , 2 0
67
D. 65
C.
47
B. 34
A.
3
y
x180 360 -1
1. For which real values of x is the functiondefined on the set of real numbers?
2. The minimum value of
occurs when x is:
3. The line with equation y = 2 intersects the curve y = 1 - 2sinx , at :
1. For which real values of x is the functiondefined on the set of real numbers?
2. The minimum value of
occurs when x is:
3. The line with equation y = 2 intersects the curve y = 1 - 2sinx , at :
6 D. C.
2 B.
3 A.
)x(1
1x:f
2
only 1 xD. only 1- xand 1 xC.
only 1 x 1- B. 1- xand 1 except x xAll A.
3π
2cos - 1 , 2 0
67
D. 65
C.
47
B. 34
A.
3
y
x180 360 -1
answer
1. Which of the following is/are solution(s) of 2sin2x =
√3:
2. Which of these would be the exact value of ?
3. Functions f and g , are given by f(x) = x2 – 2x and g(x) = -3x. Find an expression for f(g(x)).
1. Which of the following is/are solution(s) of 2sin2x =
√3:
2. Which of these would be the exact value of ?
3. Functions f and g , are given by f(x) = x2 – 2x and g(x) = -3x. Find an expression for f(g(x)).
3π
sin3π
2cos
21
D. 0 C. 23
B. 2
1- A.
5x - xD. 6x 9x C.
2x - 3x- B. 6x 3x- A.
22
22
IV III, II, I, of None D.
only III & II C. only II & I B. only I A.
4 IV.
32
III. 3
II. 6
I.
1. Which of the following is/are solution(s) of 2sin2x =
√3:
2. Which of these would be the exact value of ?
3. Functions f and g , are given by f(x) = x2 – 2x and g(x) = -3x. Find an expression for f(g(x)).
1. Which of the following is/are solution(s) of 2sin2x =
√3:
2. Which of these would be the exact value of ?
3. Functions f and g , are given by f(x) = x2 – 2x and g(x) = -3x. Find an expression for f(g(x)).
3π
sin3π
2cos
21
D. 0 C. 23
B. 2
1- A.
5x - xD. 6x 9x C.
2x - 3x- B. 6x 3x- A.
22
22
IV III, II, I, of None D.
only III & II C. only II & I B. only I A.
4 IV.
32
III. 3
II. 6
I.
answer
1. Which of the following graphs represents y = -2f(x) + 1:
A B C D
2. Given that then g-1(x) equals:
3. Functions f and g, are given by and g(x) =
x2 - 1.Find an expression for f(g(x)).
1. Which of the following graphs represents y = -2f(x) + 1:
A B C D
2. Given that then g-1(x) equals:
3. Functions f and g, are given by and g(x) =
x2 - 1.Find an expression for f(g(x)). 4 2x xD.
2x - x1
C. 3 4x x
1 B.
3 2x - x1
A. 24242424
, R x, 2
1x g(x)
3
3333 2x 1 D. 1)(x2 C. 1)(2x B.
x2
A. 1
(-2,3)
1
y
x-4
Y = f(x)
(-3,6)
y
x-5 0
(3,6)
y
x50
(1,1)
y
x(-2,-5)
(-4,1) (2,7)
y
x
(-1,1) (4,1)
2x1
f(x) 2
1. Which of the following graphs represents y = -2f(x) + 1:
A B C D
2. Given that then g-1(x) equals:
3. Functions f and g, are given by and g(x) =
x2 - 1.Find an expression for f(g(x)).
1. Which of the following graphs represents y = -2f(x) + 1:
A B C D
2. Given that then g-1(x) equals:
3. Functions f and g, are given by and g(x) =
x2 - 1.Find an expression for f(g(x)). 4 2x xD.
2x - x1
C. 3 4x x
1 B.
3 2x - x1
A. 24242424
, R x, 2
1x g(x)
3
3333 2x 1 D. 1)(x2 C. 1)(2x B.
x2
A. 1
(-2,3)
1
y
x-4
Y = f(x)
(-3,6)
y
x-5 0
(3,6)
y
x50
(1,1)
y
x(-2,-5)
(-4,1) (2,7)
y
x
(-1,1) (4,1)
2x1
f(x) 2
answer
1. The largest possible domain of, is:
2. The minimum value of
occurs when x = t. What is the value of t?
3. The line with equation y = 1 intersects the curve y = 4sin2x , at :
1. The largest possible domain of, is:
2. The minimum value of
occurs when x = t. What is the value of t?
3. The line with equation y = 1 intersects the curve y = 4sin2x , at :
π D. 2π
C. 6π
B. 0 A.
oooo 300 D. 45C. 210 B. 150 A.
π x 0 , 6π
x3cos - 1
0 xD. 0 xC. 0 xB. 0 xA.
x2f(x)
1. The largest possible domain of, is:
2. The minimum value of
occurs when x = t. What is the value of t?
3. The line with equation y = 1 intersects the curve y = 4sin2x , at :
1. The largest possible domain of, is:
2. The minimum value of
occurs when x = t. What is the value of t?
3. The line with equation y = 1 intersects the curve y = 4sin2x , at :
π D. 2π
C. 6π
B. 0 A.
oooo 300 D. 45C. 210 B. 150 A.
π x 0 , 6π
x3cos - 1
0 xD. 0 xC. 0 xB. 0 xA.
x2f(x)
answer
1. Which of the following functions represents the black curve:
A. y = g(-x) + 2 B. y = -g(x) - 2
C. y = 2 – g(x) D. y = g(x – 2)
2. Given that then h-1(x) equals:
3. Functions f and g, are given by and g(x) =
1 + x.Find an expression for g(f(x)).
1. Which of the following functions represents the black curve:
A. y = g(-x) + 2 B. y = -g(x) - 2
C. y = 2 – g(x) D. y = g(x – 2)
2. Given that then h-1(x) equals:
3. Functions f and g, are given by and g(x) =
1 + x.Find an expression for g(f(x)). 2 D.
x- 1x
C. x- 12
B. x- 1 x- 2
A. 2
2
22
2
, R x, 2
x5 h(x)
2x - 5 D. 52x
C. 5 2x B. 5 x
2 A.
2 x- 11
f(x)
(-1,5)
(1,-1)
y
x
y = g(x)
(-1,-3)
(1,3)
1. Which of the following functions represents the black curve:
A. y = g(-x) + 2 B. y = -g(x) - 2
C. y = 2 – g(x) D. y = g(x – 2)
2. Given that then h-1(x) equals:
3. Functions f and g, are given by and g(x) =
1 + x.Find an expression for g(f(x)).
1. Which of the following functions represents the black curve:
A. y = g(-x) + 2 B. y = -g(x) - 2
C. y = 2 – g(x) D. y = g(x – 2)
2. Given that then h-1(x) equals:
3. Functions f and g, are given by and g(x) =
1 + x.Find an expression for g(f(x)). 2 D.
x- 1x
C. x- 12
B. x- 1 x- 2
A. 2
2
22
2
, R x, 2
x5 h(x)
2x - 5 D. 52x
C. 5 2x B. 5 x
2 A.
2 x- 11
f(x)
(-1,5)
(1,-1)
y
x
y = g(x)
(-1,-3)
(1,3)
answer
1. For which real values of x is the functiondefined on the set of real numbers?
2. The equation of the straight line through the points (1 , -2) and (-3 , 4) is:
A. 3x + 2y = -1 B. 3x – 2y = 7C. 2x + 3y = -4 D. None of these
3. Which of the following is/are solution(s) of √3tan2x = -
1:
1. For which real values of x is the functiondefined on the set of real numbers?
2. The equation of the straight line through the points (1 , -2) and (-3 , 4) is:
A. 3x + 2y = -1 B. 3x – 2y = 7C. 2x + 3y = -4 D. None of these
3. Which of the following is/are solution(s) of √3tan2x = -
1:
2 x- 91
x:f
only 3 x 3- D. only 3- xand 3 xC.
only 3 xB. 3- xand 3 except x xAll A.
only II D. only III C. only IV & III B. only I A.
1211π
IV. 125π
III. 3
5π II.
65π
I.
1. For which real values of x is the functiondefined on the set of real numbers?
2. The equation of the straight line through the points (1 , -2) and (-3 , 4) is:
A. 3x + 2y = -1 B. 3x – 2y = 7C. 2x + 3y = -4 D. None of these
3. Which of the following is/are solution(s) of √3tan2x = -
1:
1. For which real values of x is the functiondefined on the set of real numbers?
2. The equation of the straight line through the points (1 , -2) and (-3 , 4) is:
A. 3x + 2y = -1 B. 3x – 2y = 7C. 2x + 3y = -4 D. None of these
3. Which of the following is/are solution(s) of √3tan2x = -
1:
2 x- 91
x:f
only 3 x 3- D. only 3- xand 3 xC.
only 3 xB. 3- xand 3 except x xAll A.
only II D. only III C. only IV & III B. only I A.
1211π
IV. 125π
III. 3
5π II.
65π
I.
answer
1. The gradient of a straight line parallel to the line x + 3y + 7 = 0 is:
2. Functions f and g, are given by andFind an expression for f(g(x)).
3. The line with equation y = 4 intersects the curve y = 1 - 6sinx , at :
1. The gradient of a straight line parallel to the line x + 3y + 7 = 0 is:
2. Functions f and g, are given by andFind an expression for f(g(x)).
3. The line with equation y = 4 intersects the curve y = 1 - 6sinx , at :
31
- D. 7 C. 31
B. 3- A.
65
D. 45
C. 34
B. 67
A.
1x1
g(x) 2
x1
f(x)
2
22
2 x 1x
D. 1 xC. 1 x
1 B.
1xx
A.
1. The gradient of a straight line parallel to the line x + 3y + 7 = 0 is:
2. Functions f and g, are given by andFind an expression for f(g(x)).
3. The line with equation y = 4 intersects the curve y = 1 - 6sinx , at :
1. The gradient of a straight line parallel to the line x + 3y + 7 = 0 is:
2. Functions f and g, are given by andFind an expression for f(g(x)).
3. The line with equation y = 4 intersects the curve y = 1 - 6sinx , at :
31
- D. 7 C. 31
B. 3- A.
65
D. 45
C. 34
B. 67
A.
1x1
g(x) 2
x1
f(x)
2
22
2 x 1x
D. 1 xC. 1 x
1 B.
1xx
A.
answer
1. The line joining the points (-2,-3) and (6, k) has
gradient . The value of k is:
2. Which of the following could be this graph:
3. The minimum value of
occurs when x is:
1. The line joining the points (-2,-3) and (6, k) has
gradient . The value of k is:
2. Which of the following could be this graph:
3. The minimum value of
occurs when x is:
9 D. 325
C. 3
17 B.
37
A.
65π
D. 3
5π C.
611π
B. 3π
A.
3π
2sin 1 , 2 0
4y
x180 -2
oo
oo
3sin2 - 1 D. 2sin3 1 C.
sin3 2 B. 1 2cos A.
xx
xx
1. The line joining the points (-2,-3) and (6, k) has
gradient . The value of k is:
2. Which of the following could be this graph:
3. The minimum value of
occurs when x is:
1. The line joining the points (-2,-3) and (6, k) has
gradient . The value of k is:
2. Which of the following could be this graph:
3. The minimum value of
occurs when x is:
9 D. 325
C. 3
17 B.
37
A.
65π
D. 3
5π C.
611π
B. 3π
A.
3π
2sin 1 , 2 0
4y
x180 -2
oo
oo
3sin2 - 1 D. 2sin3 1 C.
sin3 2 B. 1 2cos A.
xx
xx
answer
1. For which real values of x is the functiondefined on the set of real numbers?
2. Which of the following is the inverse of f(x) = x – 2 , where x є R ?
3. If the points (p , q) , (3 , -2) and (-1 , 4) are collinear, then the relationship connecting p and q could be:
A. 2p + 3q = 13 B. 3p – 2q = 5C. 3p + 2q = 5 D. 3p – 2q = 13
1. For which real values of x is the functiondefined on the set of real numbers?
2. Which of the following is the inverse of f(x) = x – 2 , where x є R ?
3. If the points (p , q) , (3 , -2) and (-1 , 4) are collinear, then the relationship connecting p and q could be:
A. 2p + 3q = 13 B. 3p – 2q = 5C. 3p + 2q = 5 D. 3p – 2q = 13
2 x1
D. 1 2x C. 2 xB. 2 - x
1 A.
5x3x1
x:f
only 3 x 5- D. only 0 xC.
xB. 5- xand 3 except x xAll A.
1. For which real values of x is the functiondefined on the set of real numbers?
2. Which of the following is the inverse of f(x) = x – 2 , where x є R ?
3. If the points (p , q) , (3 , -2) and (-1 , 4) are collinear, then the relationship connecting p and q could be:
A. 2p + 3q = 13 B. 3p – 2q = 5C. 3p + 2q = 5 D. 3p – 2q = 13
1. For which real values of x is the functiondefined on the set of real numbers?
2. Which of the following is the inverse of f(x) = x – 2 , where x є R ?
3. If the points (p , q) , (3 , -2) and (-1 , 4) are collinear, then the relationship connecting p and q could be:
A. 2p + 3q = 13 B. 3p – 2q = 5C. 3p + 2q = 5 D. 3p – 2q = 13
2 x1
D. 1 2x C. 2 xB. 2 - x
1 A.
5x3x1
x:f
only 3 x 5- D. only 0 xC.
xB. 5- xand 3 except x xAll A.
answer
1. Which of the following graphs represents y = f(1 - x) :
A B C D
2. Which of the following is the equation of a line perpendicular to the line x - 3y + 4 = 0
A. y = -3x B. y = x C. y = -x D. y
= -x
3. Functions f and g, are given by andFind an expression for f(g(x)).
1. Which of the following graphs represents y = f(1 - x) :
A B C D
2. Which of the following is the equation of a line perpendicular to the line x - 3y + 4 = 0
A. y = -3x B. y = x C. y = -x D. y
= -x
3. Functions f and g, are given by andFind an expression for f(g(x)).
1 x1
g(x)
2x
1 f(x)
2322
22
x x1
D. 1 2x x
1 C.
1 xx
B. 1 2x xA.
3
(2,1)
y
x-2
y = f(x)
(-1,3)
y
x-3
(1,1)
(1,3)
y
x3
(-1,1)
(-1,3)
y
x1
(-3,1)xx
2
y
x
(-2,1)
-2
1. Which of the following graphs represents y = f(1 - x) :
A B C D
2. Which of the following is the equation of a line perpendicular to the line x - 3y + 4 = 0
A. y = -3x B. y = x C. y = -x D. y
= -x
3. Functions f and g, are given by andFind an expression for f(g(x)).
1. Which of the following graphs represents y = f(1 - x) :
A B C D
2. Which of the following is the equation of a line perpendicular to the line x - 3y + 4 = 0
A. y = -3x B. y = x C. y = -x D. y
= -x
3. Functions f and g, are given by andFind an expression for f(g(x)).
1 x1
g(x)
2x
1 f(x)
2322
22
x x1
D. 1 2x x
1 C.
1 xx
B. 1 2x xA.
3
(2,1)
y
x-2
y = f(x)
(-1,3)
y
x-3
(1,1)
(1,3)
y
x3
(-1,1)
(-1,3)
y
x1
(-3,1)xx
2
y
x
(-2,1)
-2
answer
1. The line 2y = 3x + 6 meets the y-axis at C. The
gradient of the line joining C to A (4,-3) is:
2. Which of these would be the exact value of ?
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
1. The line 2y = 3x + 6 meets the y-axis at C. The
gradient of the line joining C to A (4,-3) is:
2. Which of these would be the exact value of ?
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
23
- D. 49
C. 32
- B. 49
A.
4π
D. 65π
C. 67π
B. 3π
A.
4π
sin4π
2cos
21
D. 0 C. 1 B. 2
4 A.
1. The line 2y = 3x + 6 meets the y-axis at C. The
gradient of the line joining C to A (4,-3) is:
2. Which of these would be the exact value of ?
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
1. The line 2y = 3x + 6 meets the y-axis at C. The
gradient of the line joining C to A (4,-3) is:
2. Which of these would be the exact value of ?
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
23
- D. 49
C. 32
- B. 49
A.
4π
D. 65π
C. 67π
B. 3π
A.
4π
sin4π
2cos
21
D. 0 C. 1 B. 2
4 A.
answer
1. The straight lines with equations ay = 3x + 7 and y =
5x + 2 are perpendicular. The value of a is:
2. Which of the following could be this graph:
3. The maximum value of
occurs when x is:
1. The straight lines with equations ay = 3x + 7 and y =
5x + 2 are perpendicular. The value of a is:
2. Which of the following could be this graph:
3. The maximum value of
occurs when x is:
15- D. 53
- C. 35
- B. 51
- A.
43π
D. 4
5π C.
47π
B. 4π
A.
4π
2sin 1 , 2 0
4y
x720
2
oo
oo
21
4cos- 2 D. 2 2sin2 C.
sin221
2 B. 21
2sin - 2 A.
xx
xx
1. The straight lines with equations ay = 3x + 7 and y =
5x + 2 are perpendicular. The value of a is:
2. Which of the following could be this graph:
3. The maximum value of
occurs when x is:
1. The straight lines with equations ay = 3x + 7 and y =
5x + 2 are perpendicular. The value of a is:
2. Which of the following could be this graph:
3. The maximum value of
occurs when x is:
15- D. 53
- C. 35
- B. 51
- A.
43π
D. 4
5π C.
47π
B. 4π
A.
4π
2sin 1 , 2 0
4y
x720
2
oo
oo
21
4cos- 2 D. 2 2sin2 C.
sin221
2 B. 21
2sin - 2 A.
xx
xx
answer
1. R and S have coordinates (5,-7) and (-1,-3) respectively.The perpendicular bisector of RS has a gradient of -.What is the equation of the perpendicular bisector of RS?
A. 3y = 2x + 13 B. 3y = -2x + 19
C. 2y = -3x - 19 D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1 B. m = -√2
C. m = -1 D. m = -
3. What is the solution of the equation 2cosx - √3 = 0 where ?
1. R and S have coordinates (5,-7) and (-1,-3) respectively.The perpendicular bisector of RS has a gradient of -.What is the equation of the perpendicular bisector of RS?
A. 3y = 2x + 13 B. 3y = -2x + 19
C. 2y = -3x - 19 D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1 B. m = -√2
C. m = -1 D. m = -
3. What is the solution of the equation 2cosx - √3 = 0 where ?
y
x45o
A
B2
1
35π
D. 6
11π C.
65π
B. 6π
A.2πx23π
1. R and S have coordinates (5,-7) and (-1,-3) respectively.The perpendicular bisector of RS has a gradient of -.What is the equation of the perpendicular bisector of RS?
A. 3y = 2x + 13 B. 3y = -2x + 19
C. 2y = -3x - 19 D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1 B. m = -√2
C. m = -1 D. m = -
3. What is the solution of the equation 2cosx - √3 = 0 where ?
1. R and S have coordinates (5,-7) and (-1,-3) respectively.The perpendicular bisector of RS has a gradient of -.What is the equation of the perpendicular bisector of RS?
A. 3y = 2x + 13 B. 3y = -2x + 19
C. 2y = -3x - 19 D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1 B. m = -√2
C. m = -1 D. m = -
3. What is the solution of the equation 2cosx - √3 = 0 where ?
y
x45o
A
B2
1
35π
D. 6
11π C.
65π
B. 6π
A.2πx23π
answer
1. The side of a triangle has equation y = -x – 3.
Which of these could be the equation of an altitude passing through this side?
A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0
C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0
2. The vertices of triangle STV are S(-4,10) , T(10,3) and V(0,-
10).Which of the following is the equation of the median TM?
A. 4y = x + 2 B. y = 4x + 2
C. y = -2x + 23 D. y = 2x - 2
3. Functions f and g, are given by andFind an expression for f(g(x)).
1. The side of a triangle has equation y = -x – 3.
Which of these could be the equation of an altitude passing through this side?
A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0
C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0
2. The vertices of triangle STV are S(-4,10) , T(10,3) and V(0,-
10).Which of the following is the equation of the median TM?
A. 4y = x + 2 B. y = 4x + 2
C. y = -2x + 23 D. y = 2x - 2
3. Functions f and g, are given by andFind an expression for f(g(x)).
1 xx
g(x)
x1
f(x)
1 xD. 1 x
1 C.
1 xx
B. x
1 x A.
2
1. The side of a triangle has equation y = -x – 3.
Which of these could be the equation of an altitude passing through this side?
A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0
C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0
2. The vertices of triangle STV are S(-4,10) , T(10,3) and V(0,-
10).Which of the following is the equation of the median TM?
A. 4y = x + 2 B. y = 4x + 2
C. y = -2x + 23 D. y = 2x - 2
3. Functions f and g, are given by andFind an expression for f(g(x)).
1. The side of a triangle has equation y = -x – 3.
Which of these could be the equation of an altitude passing through this side?
A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0
C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0
2. The vertices of triangle STV are S(-4,10) , T(10,3) and V(0,-
10).Which of the following is the equation of the median TM?
A. 4y = x + 2 B. y = 4x + 2
C. y = -2x + 23 D. y = 2x - 2
3. Functions f and g, are given by andFind an expression for f(g(x)).
1 xx
g(x)
x1
f(x)
1 xD. 1 x
1 C.
1 xx
B. x
1 x A.
2
answer
1. If f’(4) equals:
A. B. 2 C. 3 D. 6
2. If the line ax - 2y + 5 = 0 is parallel to the line 3x + y - 4 = 0, a is equal to:
A. -6 B. - C. D.
3. PQ, of length 2, is parallel to OY.
QR, of length 4, is parallel to OX.
Angle PQR = 90o. P is the point (1,2).
The line PR cuts OY at:
A. (0,) B. (0,) C. (0,-) D. (0,-)
1. If f’(4) equals:
A. B. 2 C. 3 D. 6
2. If the line ax - 2y + 5 = 0 is parallel to the line 3x + y - 4 = 0, a is equal to:
A. -6 B. - C. D.
3. PQ, of length 2, is parallel to OY.
QR, of length 4, is parallel to OX.
Angle PQR = 90o. P is the point (1,2).
The line PR cuts OY at:
A. (0,) B. (0,) C. (0,-) D. (0,-)
; 2x f(x) 32
y
x0
Q R4
P (1,2)
2
1. If f’(4) equals:
A. B. 2 C. 3 D. 6
2. If the line ax - 2y + 5 = 0 is parallel to the line 3x + y - 4 = 0, a is equal to:
A. -6 B. - C. D.
3. PQ, of length 2, is parallel to OY.
QR, of length 4, is parallel to OX.
Angle PQR = 90o. P is the point (1,2).
The line PR cuts OY at:
A. (0,) B. (0,) C. (0,-) D. (0,-)
1. If f’(4) equals:
A. B. 2 C. 3 D. 6
2. If the line ax - 2y + 5 = 0 is parallel to the line 3x + y - 4 = 0, a is equal to:
A. -6 B. - C. D.
3. PQ, of length 2, is parallel to OY.
QR, of length 4, is parallel to OX.
Angle PQR = 90o. P is the point (1,2).
The line PR cuts OY at:
A. (0,) B. (0,) C. (0,-) D. (0,-)
; 2x f(x) 32
y
x0
Q R4
P (1,2)
2
answer
1. This diagram is most likely to be part of the graph of:
2. Find the gradient of the line ST:
A. m = -1 B. m = 1
C. m = -√2 D. m = -
3. If and x ≠ 0 then f’(x) equals:
1. This diagram is most likely to be part of the graph of:
2. Find the gradient of the line ST:
A. m = -1 B. m = 1
C. m = -√2 D. m = -
3. If and x ≠ 0 then f’(x) equals:
y
x135o
S
T2
1
33 x1
- D. x1
- C. x2
- B. 2x1
A.
2x1
f(x)
1 - cos241
D. 1 - 2cos4 C.
3 cos4 B. 41
cos - 2 A.
oo
oo
xx
xx 1
y
x90
-3
answer
1. If f(x) = x√x , x > 0 ; f’(x) equals:
2. Which of the following is/are true of the line withequation 3x - 2y + 3 = 0?
I. It passes through the point (-2,-3)II. It is parallel to the line 6x + 4y + 3 = 0III. It is perpendicular to the line 2x + 3y + 3 = 0
A. I only B. I & III only C. III onlyD. Some other combination of responses
3. The line with equation y = √3 intersects the curve y =
2cosx, at:
1. If f(x) = x√x , x > 0 ; f’(x) equals:
2. Which of the following is/are true of the line withequation 3x - 2y + 3 = 0?
I. It passes through the point (-2,-3)II. It is parallel to the line 6x + 4y + 3 = 0III. It is perpendicular to the line 2x + 3y + 3 = 0
A. I only B. I & III only C. III onlyD. Some other combination of responses
3. The line with equation y = √3 intersects the curve y =
2cosx, at:
25
x52
D. x53
C. x 1 B. x2
1 1 A.
oooo 420D. 45C. 60- B. 330 A.
answer
1. The gradient of the curve y = 5x3 - 10x at the point (1,-
5) is: A. -5 B. 5 C. 15 D. None of
these
2. f and g are functions on the set of real numbers such
that f(x) = 2x – 1 and f(g(x)) = 4x + 1, g(x)
equals:
A. 8x + 1 B. 8x - 3 C. 2x + 3 D. 2x
+ 1
3. Functions f and g, are given by andFind an expression for g(f(x)).
1. The gradient of the curve y = 5x3 - 10x at the point (1,-
5) is: A. -5 B. 5 C. 15 D. None of
these
2. f and g are functions on the set of real numbers such
that f(x) = 2x – 1 and f(g(x)) = 4x + 1, g(x)
equals:
A. 8x + 1 B. 8x - 3 C. 2x + 3 D. 2x
+ 1
3. Functions f and g, are given by andFind an expression for g(f(x)).
1 xx
g(x)
x1
f(x)
1 xD. 1 x
1 C.
1 xx
B. x
1 x A.
2
2
answer
1. The x-coordinate of the point at which the curve y = 6 – 3x2 has gradient 12 is:
A. -6 B. -2 C. -√2 D. -1
2. The vertices of triangle ABC are A(1,-7), B(-4,7) & C(-
1,3).Which of the following is the equation of the median CM?
A. y = 6x + 4 B. y = 6x + 9
C. 2y = x + 7 D. 2y = 3x - 9
3. The maximum value of
occurs when x is:
1. The x-coordinate of the point at which the curve y = 6 – 3x2 has gradient 12 is:
A. -6 B. -2 C. -√2 D. -1
2. The vertices of triangle ABC are A(1,-7), B(-4,7) & C(-
1,3).Which of the following is the equation of the median CM?
A. y = 6x + 4 B. y = 6x + 9
C. 2y = x + 7 D. 2y = 3x - 9
3. The maximum value of
occurs when x is:
35π
D. 65π
C. 67π
B. 6
11π A.
3π
2sin 3 , 2 0
Question 27Question 27
How do you show that a curve is always increasing ?
How do you show that a curve is always increasing ?
answer
Answer to Question 27Answer to Question 27(i) Differentiate(ii) show that f’(x) is a perfect square
(i) Differentiate(ii) show that f’(x) is a perfect square
Question 28Question 28
How do you find the equation of a tangent to a curve at the point when x = a ?
How do you find the equation of a tangent to a curve at the point when x = a ?
answer
Answer to Question 28Answer to Question 28(i) Differentiate(ii) fit a into f’(x) to get the gradient (m)
(iii) fit a into f(x) to get the tangent point (a,b)
(iv) use y-b=m(x-a)
(i) Differentiate(ii) fit a into f’(x) to get the gradient (m)
(iii) fit a into f(x) to get the tangent point (a,b)
(iv) use y-b=m(x-a)
Question 29Question 29
For what values of a function is the function said to be undefined ?
For what values of a function is the function said to be undefined ?
answer
Answer to Question 29Answer to Question 29When you fit in a value of x and you cannot get an answer
When you fit in a value of x and you cannot get an answer
Question 30Question 30
How do you draw the graph of f(x-1) given the graph of f(x) ?
How do you draw the graph of f(x-1) given the graph of f(x) ?
answer
Answer to Question 30Answer to Question 30Move the graph 1 unit to the right
Move the graph 1 unit to the right
Question 31Question 31
How do you find f(g(x)) for given functions f(x) and g(x) ?
How do you find f(g(x)) for given functions f(x) and g(x) ?
answer
Answer to Question 31Answer to Question 31Fit g(x) into f(x)i.e. each x in f(x) is replaced by the function g(x)
Fit g(x) into f(x)i.e. each x in f(x) is replaced by the function g(x)
Question 32Question 32
What two things do you require in order to find the equation of a straight line ?
What two things do you require in order to find the equation of a straight line ?
answer
Answer to Question 32Answer to Question 32The gradient of the line and a point on the line
The gradient of the line and a point on the line
x
y
(a,b)m
1
Question 33Question 33
How do you find the midpoint of a line joining two points ?
How do you find the midpoint of a line joining two points ?
answer
Answer to Question 33Answer to Question 33Add the coordinates and divide by two
x1+ x
2 , y1+ y
2
2 2
Add the coordinates and divide by two
x1+ x
2 , y1+ y
2
2 2( )x
y(x2,y2)
(x1,y1)
Question 34Question 34
What is the gradient of a vertical line ?
What is the gradient of a vertical line ?
answer
Answer to Question 34Answer to Question 34undefinedundefined
x
y
Question 35Question 35
How do you find the median AM of triangle ABC ?
How do you find the median AM of triangle ABC ?
answer
Answer to Question 35Answer to Question 35 (i) find the
mid pointof BC (M)
(ii) find thegradient of AM
(iii) use y-b = m(x-a)
(i) find themid pointof BC (M)
(ii) find thegradient of AM
(iii) use y-b = m(x-a)
A
BCM
Question 36Question 36
Which two points does the graphy = ax always pass through ?
Which two points does the graphy = ax always pass through ?
answer
Answer to Question 36Answer to Question 36(0,1) and (1,a)(0,1) and (1,a)
Question 37Question 37
What is the perpendicular bisector of a line ?
What is the perpendicular bisector of a line ?
answer
Answer to Question 37Answer to Question 37A line which cuts the given line in half at 90o
A line which cuts the given line in half at 90o
Question 38Question 38
How do you draw the graph of f(x+1) given the graph of f(x) ?
How do you draw the graph of f(x+1) given the graph of f(x) ?
answer
Answer to Question 38Answer to Question 38Move the graph 1 unit to the left
Move the graph 1 unit to the left
Question 39Question 39
How do you find the equation of a perpendicular bisector of a line ?
How do you find the equation of a perpendicular bisector of a line ?
answer
Answer to Question 39Answer to Question 39 (i) find the midpoint of the
line(ii) find the gradient of the line(iii) find the gradient perpendicular to the given line (iv) Use midpoint and gradient in
y-b = m(x-a)
(i) find the midpoint of the line(ii) find the gradient of the line(iii) find the gradient perpendicular to the given line (iv) Use midpoint and gradient in
y-b = m(x-a)
M(a,b)
Question 40Question 40
For what values is this function undefined ?f(x) = x
(x+2)(x-3)
For what values is this function undefined ?f(x) = x
(x+2)(x-3)answer
Answer to Question 40Answer to Question 40-2 and 3-2 and 3
Question 41Question 41
What are the two formulae used to find the area of a triangle ?
What are the two formulae used to find the area of a triangle ?
answer
Answer to Question 41Answer to Question 41A = ½base x heightA = ½bcsinA
A = ½base x heightA = ½bcsinA
A
BCa
bc
B a s e
height
Question 42Question 42
What three processes do you go through in order to factorise a quadratic ?
What three processes do you go through in order to factorise a quadratic ?
answer
Answer to Question 42Answer to Question 42(i) common factor(ii) difference of two
squares(iii) trinomial
(i) common factor(ii) difference of two
squares(iii) trinomial
Question 43Question 43
What is the equation of a vertical line passing through (a,b) ?
What is the equation of a vertical line passing through (a,b) ?
answer
Answer to Question 43Answer to Question 43x = ax = a
x
y
(a,b)
Question 44Question 44
What is the Theorem of Pythagoras ?
What is the Theorem of Pythagoras ?
answer
Answer to Question 44Answer to Question 44For ΔABC,right-angled at A,a2 = b2 + c2
For ΔABC,right-angled at A,a2 = b2 + c2
AB
C
a
c
b
Question 45Question 45
What do you know about the gradients of two parallel lines?
What do you know about the gradients of two parallel lines?
answer
Answer to Question 45Answer to Question 45They are the same They are the same
Question 46Question 46
How do you draw the graph of f’(x) given the graph of f(x) ?
How do you draw the graph of f’(x) given the graph of f(x) ?
answer
Answer to Question 46Answer to Question 46 (i) plot x coords of st. points on
x-axis (SPs become roots)(ii) look at each part of f(x)
separately: if rising, graph of f’(x) is above x-axis if falling, graph of f’(x) is below x-axis
(i) plot x coords of st. points on x-axis (SPs become roots)
(ii) look at each part of f(x) separately: if rising, graph of f’(x) is above x-axis if falling, graph of f’(x) is below x-axis
Question 47Question 47
How do you get the gradient of a line with an equation like3x + 2y = 5 ?
How do you get the gradient of a line with an equation like3x + 2y = 5 ?
answer
Answer to Question 47Answer to Question 47(i) Rearrange into the
formy = mx + c
(ii) read offgradient = m
(i) Rearrange into the formy = mx + c
(ii) read offgradient = m
Question 48Question 48
What is loga1
equal to ?
What is loga1
equal to ?
answer
Answer to Question 48Answer to Question 4800
Question 49Question 49
How do you find the length of a line joining two points ?
How do you find the length of a line joining two points ?
answer
Answer to Question 49Answer to Question 49√(x2 – x1)2 + (y2 –y1)2 √(x2 – x1)2 + (y2 –y1)2
A(x1,y1)
B(x2,y2)
x
y
Question 50Question 50
What is the Converse of Pythagoras ?
What is the Converse of Pythagoras ?
answer
Answer to Question 50Answer to Question 50If a2 = b2 + c2
then ΔABC isright-angled at A
If a2 = b2 + c2
then ΔABC isright-angled at A
AB
C
a
c
b
Question 51Question 51
How do you find the gradient of a line joining two points ?
How do you find the gradient of a line joining two points ?
answer
Answer to Question 51Answer to Question 51m = y2 – y1
x2 – x1
m = y2 – y1
x2 – x1
A(x1,y1)
B(x2,y2)
x
y
Question 52Question 52
How do you find the altitude AN of ΔABC ?
How do you find the altitude AN of ΔABC ?
answer
Answer to Question 52Answer to Question 52 (i) find the gradient
of BC(ii) find the gradient
of AN,perpendicularto BC
(iii)use y-b=m(x-a)
(i) find the gradient of BC
(ii) find the gradient of AN,
perpendicularto BC
(iii)use y-b=m(x-a)
A
NB
C
Question 53Question 53
For a curve, how do you find the stationary points and their nature ?
For a curve, how do you find the stationary points and their nature ?
answer
Answer to Question 53Answer to Question 53(i) differentiate(ii) let f’(x) = 0(iii) solve to find
stationary points(iv) find y-coordinates(v)draw nature table
(i) differentiate(ii) let f’(x) = 0(iii) solve to find
stationary points(iv) find y-coordinates(v)draw nature table
Question 54Question 54
How do you draw the graph of 3+f(x) given the graph of f(x) ?
How do you draw the graph of 3+f(x) given the graph of f(x) ?
answer
Answer to Question 54Answer to Question 54move graph up 3move graph up 3
Question 55Question 55
How do you find where a curve is increasing ?
How do you find where a curve is increasing ?
answer
Answer to Question 55Answer to Question 55 (i) differentiate(ii) let f’(x) = 0(iii)solve to find stationary
points(iv) draw nature table(v) read values for which
graph is increasing
(i) differentiate(ii) let f’(x) = 0(iii)solve to find stationary
points(iv) draw nature table(v) read values for which
graph is increasing
Question 56Question 56
How do you find where two lines intersect ?
How do you find where two lines intersect ?
answer
Answer to Question 56Answer to Question 56Simultaneous equations
Simultaneous equations
Question 57Question 57
How do you draw the graph of 3-f(x) given the graph of f(x) ?
How do you draw the graph of 3-f(x) given the graph of f(x) ?
answer
Answer to Question 57Answer to Question 57Reflect the graph in the x-axis, then move it up 3
Reflect the graph in the x-axis, then move it up 3
Question 58Question 58
How do you draw the graph of f(-x) given the graph of f(x) ?
How do you draw the graph of f(-x) given the graph of f(x) ?
answer
Answer to Question 58Answer to Question 58Reflect the graph in the y-axis
Reflect the graph in the y-axis
Question 59Question 59
How do you solve equations like
100 = 0 x2
How do you solve equations like
100 = 0 x2
answer
?4 -
Answer to Question 59Answer to Question 59(i) multiply by the
denominator of the fraction (here x2)
(ii) factorise and solve
(i) multiply by the denominator of the
fraction (here x2)(ii) factorise and solve
Question 60Question 60How do you find the exact values ofsin(A+B), cos(A-B) etc.given that
cosA = 3/5 andsinB = 12/13 ?
How do you find the exact values ofsin(A+B), cos(A-B) etc.given that
cosA = 3/5 andsinB = 12/13 ?
answer
Answer to Question 60Answer to Question 60 (i) draw
two Δs (ii) find
missing sides (iii) expand
formula (iv) fit in values
from Δs
(i) drawtwo Δs
(ii) findmissing sides
(iii) expandformula
(iv) fit in valuesfrom Δs
A
3 5
B
12
13
Question 61Question 61How do you solve equations like
Cos2xo - 5cosxo = 2 ?(0 ≤ x ≤ 360)
How do you solve equations like
Cos2xo - 5cosxo = 2 ?(0 ≤ x ≤ 360)
answer
Answer to Question 61Answer to Question 61(i) fit in 2cos2xo-1 for
cos2xo
(ii) factorise(iii) solve the equation
(i) fit in 2cos2xo-1 for cos2xo
(ii) factorise(iii) solve the equation
Question 62Question 62What is
sin xcos xequal to ?
What is sin xcos xequal to ?
answer
Answer to Question 62Answer to Question 62
tan xtan x
Question 63Question 63How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ?
How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ?
answer
Answer to Question 63Answer to Question 63(i) rewrite the function as
f(x)=x3+0x2-3x+2(ii) use synthetic division
with 1 on the outside(iii) show that
remainder = 0
(i) rewrite the function asf(x)=x3+0x2-3x+2
(ii) use synthetic division with 1 on the outside
(iii) show thatremainder = 0
Question 64Question 64What is the sine rule ?
What is the sine rule ?
answer
Answer to Question 64Answer to Question 64
a b c sinA sinb sinC
a b c sinA sinb sinC
= =A
B
Ca
bc
Question 65Question 65Given f’(x) and a point on the curve, how do you findf(x) ?
Given f’(x) and a point on the curve, how do you findf(x) ?
answer
Answer to Question 65Answer to Question 65(i) integrate(ii) fit in given point
to work out value
of C
(i) integrate(ii) fit in given point
to work out value
of C
Question 66Question 66How do you solve quadratic inequations likex2 - 5x + 6 ≤ 0 ?
How do you solve quadratic inequations likex2 - 5x + 6 ≤ 0 ?
answer
Answer to Question 66Answer to Question 66
(i) factorise(ii) draw graph(iii) read values
below x-axis
(i) factorise(ii) draw graph(iii) read values
below x-axis
Question 67Question 67How do you change from radians to degrees ?
How do you change from radians to degrees ?
answer
Answer to Question 67Answer to Question 67
Divide by π and multiply by 180
Divide by π and multiply by 180
Question 68Question 68What is the condition for real roots ?
What is the condition for real roots ?
answer
Answer to Question 68Answer to Question 68
b2 – 4ac ≥ 0 b2 – 4ac ≥ 0
Question 69Question 69How do you find the value of a in the polynomial x3+ax2+4x+3 given a factor of the polynomial or the remainder when the polynomial is divided by a number ?
How do you find the value of a in the polynomial x3+ax2+4x+3 given a factor of the polynomial or the remainder when the polynomial is divided by a number ?
answer
Answer to Question 69Answer to Question 69 (i) do synthetic division(ii) let the expression
= 0 or the remainder
(iii) solve the equation
(i) do synthetic division(ii) let the expression
= 0 or the remainder
(iii) solve the equation
Question 70Question 70How do you find f(x) iff’(x) = 5-3x2 andthe curve passes through the point (1,9) ?
How do you find f(x) iff’(x) = 5-3x2 andthe curve passes through the point (1,9) ?
answer
Answer to Question 70Answer to Question 70 (i) f(x) = ∫f'(x) dx(ii) find C by replacing
point (1,9) into f(x)(iii) write down completed formula for f(x)
(i) f(x) = ∫f'(x) dx(ii) find C by replacing
point (1,9) into f(x)(iii) write down completed formula for f(x)
Question 71Question 71What is
sin2x + cos2xequal to ?
What issin2x + cos2xequal to ?
answer
Answer to Question 71Answer to Question 71 1 1
Question 72Question 72How do you find the equation of the tangent to a circle at a particular point on the circumference ?
How do you find the equation of the tangent to a circle at a particular point on the circumference ?
answer
Answer to Question 72Answer to Question 72 (i) find the
centre(ii) find gradient
from centreto point
(iii) find perpendicular gradient(iv)use y-b=m(x-a)
(i) find thecentre
(ii) find gradientfrom centreto point
(iii) find perpendicular gradient(iv)use y-b=m(x-a)
x
y
(a,b)
C
Question 73Question 73How do you find
x2 + 1√x
How do you find x2 + 1
√x
answer
∫ dx ?
Answer to Question 73Answer to Question 73 (i) change root to
power(ii) split up into fractions(iii)simplify each term(iv) integrate each term(v) REMEMBER +C
(i) change root to power
(ii) split up into fractions(iii)simplify each term(iv) integrate each term(v) REMEMBER +C
Question 74Question 74How do you show that the root of a function lies between two given values ?
How do you show that the root of a function lies between two given values ?
answer
Answer to Question 74Answer to Question 74 fit in two values and
show one is positive and one is negative
fit in two values and show one is positive and one is negative
x
+ve
-ve
Question 75Question 75How do you find exact values of sin2x and cos2x given cosx =3/5 ?
How do you find exact values of sin2x and cos2x given cosx =3/5 ?
answer
Answer to Question 75Answer to Question 75 (i) draw a
right-angledtriangle
(ii) find the missing side
(iii) expand the double angle formula
(iv) fit in values from Δ
(i) draw aright-angledtriangle
(ii) find the missing side
(iii) expand the double angle formula
(iv) fit in values from Δ
A3
5
Question 76Question 76What is the turning point of
y=2(x-a)2+b ?Max or min ?
What is the turning point of
y=2(x-a)2+b ?Max or min ?
answer
Answer to Question 76Answer to Question 76
(i) (a,b)minimum
(i) (a,b)minimum
(a,b)
Question 77Question 77How do you integrate xn ?
How do you integrate xn ?
answer
Answer to Question 77Answer to Question 77
xn+1
n+1
xn+1
n+1+ C+ C
Question 78Question 78How do you solve equations like
cos2xo-5sinxo = 0 ?(0≤x≤360)
How do you solve equations like
cos2xo-5sinxo = 0 ?(0≤x≤360)
answer
Answer to Question 78Answer to Question 78
(i) fit in 1-2sin2xo for cos2xo
(ii) factorise(iii) solve equation
(i) fit in 1-2sin2xo for cos2xo
(ii) factorise(iii) solve equation
Question 79Question 79How do you complete the square for functions like2x2 + 12x + 3 ?
How do you complete the square for functions like2x2 + 12x + 3 ?
answer
Answer to Question 79Answer to Question 79 (i) multiply out
a(x+p)2+q(ii) compare with
given function(iii) find a, p and q
(i) multiply out a(x+p)2+q
(ii) compare with given function
(iii) find a, p and q
Question 80Question 80How do you solve equations of the form
sin2xo = 0.5 ?(0≤x≤360)
How do you solve equations of the form
sin2xo = 0.5 ?(0≤x≤360) answe
r
Answer to Question 80Answer to Question 80 (i) decide on the 2 quadrants (sin is +ve)
(ii) press INV sin to get angle
(iii) work out your 2 angles(iv) divide each by 2
(i) decide on the 2 quadrants (sin is +ve)
(ii) press INV sin to get angle
(iii) work out your 2 angles(iv) divide each by 2
Question 81Question 81How do you solve quadratic inequations like
x2+5x-6 ≥ 0 ?
How do you solve quadratic inequations like
x2+5x-6 ≥ 0 ?
answer
Answer to Question 81Answer to Question 81
(i) factorise(ii) draw graph(iii) read values
above x-axis
(i) factorise(ii) draw graph(iii) read values
above x-axis
Question 82Question 82What is the centre and radius of a circle with equation x2 + y2 = r2 ?
What is the centre and radius of a circle with equation x2 + y2 = r2 ? answe
r
Answer to Question 82Answer to Question 82
(i) centre (0,0)(ii) radius = r
(i) centre (0,0)(ii) radius = r
Question 83Question 83How do you calculate the area under a curve ?
How do you calculate the area under a curve ?
answer
Answer to Question 83Answer to Question 83 (i) integrate(ii) fit in two limits
and subtract to find area
(i) integrate(ii) fit in two limits
and subtract to find area
Question 84Question 84How do you find the root of an equation between two given values to 1 dp ?
How do you find the root of an equation between two given values to 1 dp ? answe
r
Answer to Question 84Answer to Question 84
iterationiteration
Question 85Question 85How do you solve equations of the form
sin2xo = 0.5 ?(0≤x≤360)
How do you solve equations of the form
sin2xo = 0.5 ?(0≤x≤360) answe
r
Answer to Question 85Answer to Question 85(i) rearrange to get
sinxo = ± …(ii) find answers in
all 4 quadrants
(i) rearrange to get sinxo = ± …
(ii) find answers in all 4 quadrants
Question 86Question 86How do you name the angle between a line and a plane ?
How do you name the angle between a line and a plane ?
answer
Answer to Question 86Answer to Question 86 (i) start at end of line (A) (ii) go to where line meets
the plane (B) (iii) go to the point
on the plane directly under the start of the line (C)
ABC
(i) start at end of line (A) (ii) go to where line meets
the plane (B) (iii) go to the point
on the plane directly under the start of the line (C)
ABC
A
B
C
Question 87Question 87What is the condition for equal roots ?
What is the condition for equal roots ?
answer
Answer to Question 87Answer to Question 87
b2 – 4ac = 0b2 – 4ac = 0
Question 88Question 88What is the turning point of y = b-3(x-a)2 ?
max or min ?
What is the turning point of y = b-3(x-a)2 ?
max or min ?
answer
Answer to Question 88Answer to Question 88
(a,b)
Maximum
(a,b)
Maximum
(a,b)
Question 89Question 89What is the quadratic formula and explain when it is used ?
What is the quadratic formula and explain when it is used ?
answer
Answer to Question 89Answer to Question 89x = -b±√(b2-4ac)
2aIt is used to find roots of a quadratic equation when it is difficult to factorise.
x = -b±√(b2-4ac)2a
It is used to find roots of a quadratic equation when it is difficult to factorise.
Question 90Question 90How do you prove that a line is a tangent to a circle ?
How do you prove that a line is a tangent to a circle ?
answer
Answer to Question 90Answer to Question 90Rearrange line to make
y = or x =Fit line into circleProve it has equal roots using b2-4ac = 0 or repeated roots
Rearrange line to makey = or x =
Fit line into circleProve it has equal roots using b2-4ac = 0 or repeated roots
Question 91Question 91How do you find theexact value of
sin (α-β),given that sinα =4/5
and cosβ = 12/13 ?
How do you find theexact value of
sin (α-β),given that sinα =4/5
and cosβ = 12/13 ? answer
Answer to Question 91Answer to Question 91 (i) draw triangles
for α and β (ii) work out
cosα and sinβ
(iii) expand formula for sin(α-β)
(iv) insert exact values
(i) draw triangles for α and β
(ii) work out cosα and sinβ
(iii) expand formula for sin(α-β)
(iv) insert exact values
αα
4
5
12
13
ββ
Question 92Question 92How do you solve equations of the form
cosxo = - 0.8 ?(0≤x≤360)
How do you solve equations of the form
cosxo = - 0.8 ?(0≤x≤360) answe
r
Answer to Question 92Answer to Question 92 (i) decide on the
2 quadrants (cos is -ve)
(ii) ignore the sign and press INV cos to get angle
(iii) work out your 2 angles
(i) decide on the 2 quadrants (cos is -
ve)(ii) ignore the sign and
press INV cos to get angle
(iii) work out your 2 angles
Question 93Question 93How do you change from degrees to radians ?
How do you change from degrees to radians ?
answer
Answer to Question 93Answer to Question 93
Divide by 180 and multiply by π
Divide by 180 and multiply by π
Question 94Question 94How do you find the exact values of sin x or tan x given
cos x = a ? b
How do you find the exact values of sin x or tan x given
cos x = a ? b
answer
Answer to Question 94Answer to Question 94 (i) draw triangle
(ii) use Pythagoras to fill in missing side
(iii) read values off triangle using SOHCAHTOA
(i) draw triangle
(ii) use Pythagoras to fill in missing side
(iii) read values off triangle using SOHCAHTOA
a
b
xx
Question 95Question 95How do you factorise a cubic expression like
x3-2x2-x+2 ?
How do you factorise a cubic expression like
x3-2x2-x+2 ?
answer
Answer to Question 95Answer to Question 95
Synthetic division using factors of last number
Synthetic division using factors of last number
Remainder=0
factor 1 -2 -1 2
Question 96Question 96What is the centre and radius of a circle of the form
x2+y2+2gx+2fy+c=0 ?
What is the centre and radius of a circle of the form
x2+y2+2gx+2fy+c=0 ?
answer
Answer to Question 96Answer to Question 96
Centre (-g,-f)Radius √(g2+f2-c)
Centre (-g,-f)Radius √(g2+f2-c)
Question 97Question 97How do you remember the exact values of 30o, 45o and 60o ?
How do you remember the exact values of 30o, 45o and 60o ?
answer
Answer to Question 97Answer to Question 97sin30o = ½Draw right-angledtriangle
Complete using PythagorasDo similarfor tan 45o =1
sin30o = ½Draw right-angledtriangle
Complete using PythagorasDo similarfor tan 45o =1
30o
60o
1 2
√3
45o
45o
1
1 √2
Question 98Question 98How do you calculate the area between two curves ?
How do you calculate the area between two curves ?
answer
Answer to Question 98Answer to Question 98(i) let equations equal
each other(ii) solve to find limits(iii) integrate
(upper - lower) functions
between limits
(i) let equations equal each other
(ii) solve to find limits(iii) integrate
(upper - lower) functions
between limits
Question 99Question 99How do you solve an equation like
3sinx+1 = 0 ?
How do you solve an equation like
3sinx+1 = 0 ?
answer
Answer to Question 99Answer to Question 99(i) rearrange to sinx =(ii) decide on 2 quadrants(iii) ignore any –ve and press INV sin to get angle
(iv) work out two answers
(i) rearrange to sinx =(ii) decide on 2 quadrants(iii) ignore any –ve and press INV sin to get angle
(iv) work out two answers
Question 100Question 100What is the condition for no real roots ?
What is the condition for no real roots ?
answer
Answer to Question 100Answer to Question 100b2 – 4ac < 0b2 – 4ac < 0
Question 101Question 101How do you find
∫ x3 dx ?
How do you find
∫ x3 dx ?
answer
aa
bb
Answer to Question 101Answer to Question 101
x3+1
3+1
then 1/4[(b4) - (a4)]
x3+1
3+1
then 1/4[(b4) - (a4)]
[[ ]]bb
aa
Question 102Question 102How do you find where a line and a circle intersect ?
How do you find where a line and a circle intersect ?
answer
Answer to Question 102Answer to Question 102Rearrange line to getx = … or y = …
Fit into circle and solve
Rearrange line to getx = … or y = …
Fit into circle and solve
Question 103Question 103State the cosine rule to find an angle
State the cosine rule to find an angle
answer
Answer to Question 103Answer to Question 103cos A = b2 + c2 - a2
2bc
cos A = b2 + c2 - a2
2bcA
B
Ca
bc
Question 104Question 104What is the centre and radius of a circle of the form
(x-a)2+(y-b)2 = r2 ?
What is the centre and radius of a circle of the form
(x-a)2+(y-b)2 = r2 ?
answer
Answer to Question 104Answer to Question 104Centre (a,b)Radius = r
Centre (a,b)Radius = r
x
y
(a,b)C
r
Question 105Question 105State the cosine rule to find a missing side
State the cosine rule to find a missing side
answer
Answer to Question 105Answer to Question 105a2 = b2+c2-2bccosAa2 = b2+c2-2bccosA
A
B
Ca
bc
Question 106Question 106How do you find
∫ (ax + b)n dx ?
How do you find
∫ (ax + b)n dx ?
answer
Answer to Question 106Answer to Question 106 (i) increase power by 1 (ii) divide by new power (iii) divide by the
derivative ofthe bracket
i.e. (ax+b)n+1
a(n+1)
(i) increase power by 1 (ii) divide by new power (iii) divide by the
derivative ofthe bracket
i.e. (ax+b)n+1
a(n+1)+ C+ C
Question 107Question 107How do you findthe coordinates of a point which divides a line in a ratio e.g. 3:2 ?
How do you findthe coordinates of a point which divides a line in a ratio e.g. 3:2 ?
answer
Answer to Question 107Answer to Question 107 (i) write in form AB = 3
BC 2 (ii) cross-multiply (iii)write AB = (b-a) (iv) solve to find missing
vector (v) rewrite as point (*,*)
(i) write in form AB = 3BC 2
(ii) cross-multiply (iii)write AB = (b-a) (iv) solve to find missing
vector (v) rewrite as point (*,*)
A
B
C
3
2
Question 108Question 108What is a position vector ?
What is a position vector ?
answer
Answer to Question 108Answer to Question 108A vector which starts at the origin
A vector which starts at the origin
Question 109Question 109How do you express acosx+bsinx+cin the formkcos(x-α) etc?
How do you express acosx+bsinx+cin the formkcos(x-α) etc?
answer
Answer to Question 109Answer to Question 109 (i) expand brackets and
equate like terms (ii) find k =√(a2+b2) (iii) identify quadrant α is in
(iv) find α , tanα = sinα cosα
(i) expand brackets and equate like terms
(ii) find k =√(a2+b2) (iii) identify quadrant α is in
(iv) find α , tanα = sinα cosα
ATS
C
Question 110Question 110How do you differentiate a bracket without multiplying it out ?
How do you differentiate a bracket without multiplying it out ?
answer
Answer to Question 110Answer to Question 110(i) multiply by old power(ii) decrease power by 1(iii) multiply by
derivative of bracket
(i) multiply by old power(ii) decrease power by 1(iii) multiply by
derivative of bracket
Question 111Question 111What isLogax – logay
equal to ?
What isLogax – logay
equal to ?
answer
Answer to Question 111Answer to Question 111 x
x
loglogaa yy
Question 112Question 112What do you get when you differentiate cosx ?
What do you get when you differentiate cosx ?
answer
Answer to Question 112Answer to Question 112-sinx-sinx
Question 113Question 113How do you show that two vectors are perpendicular ?
How do you show that two vectors are perpendicular ?
answer
Answer to Question 113Answer to Question 113Show that a.b=0Show that a.b=0
a
b
Question 114Question 114How do you integrate sin ax ?
How do you integrate sin ax ?
answer
Answer to Question 114Answer to Question 114-1/a cos ax + C-1/a cos ax + C
Question 115Question 115How do you draw a graph of the form
y = acosxor y = asinx ?
How do you draw a graph of the form
y = acosxor y = asinx ?
answer
Answer to Question 115Answer to Question 115Draw y = cosxor y = sinx graphwith a maximum of a and a minimum of -a
Draw y = cosxor y = sinx graphwith a maximum of a and a minimum of -a
Question 116Question 116How do you find the maximum or minimum values of
acosx + bsinx + c ?
How do you find the maximum or minimum values of
acosx + bsinx + c ?answer
Answer to Question 116Answer to Question 116(i) change acosx+bsinx into Rcos(x-a)
(ii) max is R+c
(i) change acosx+bsinx into Rcos(x-a)
(ii) max is R+c
Question 117Question 117How do you find a unit vector parallel to a given vector ?
How do you find a unit vector parallel to a given vector ?
answer
Answer to Question 117Answer to Question 117(i) find the length of the given vector
(ii) divide all the components by this length
(i) find the length of the given vector
(ii) divide all the components by this length
Question 118Question 118How do you integrate cos ax ?
How do you integrate cos ax ?
answer
Answer to Question 118Answer to Question 1181/a sin ax + C1/a sin ax + C
Question 119Question 119How do you draw a graph of the form
y = cos(x+a) or y = sin(x+a) ?
How do you draw a graph of the form
y = cos(x+a) or y = sin(x+a) ?
answer
Answer to Question 119Answer to Question 119Move the graph of y=cosx or y=sinx
a units to the LEFT
Move the graph of y=cosx or y=sinx
a units to the LEFT
Question 120Question 120What is a unit vector ?
What is a unit vector ?
answer
Answer to Question 120Answer to Question 120A vector of length 1 unitA vector of length 1 unit
Question 121Question 121How do you draw a graph of the form
y = cos bx or y = sin bx ?
How do you draw a graph of the form
y = cos bx or y = sin bx ?
answer
Answer to Question 121Answer to Question 121Draw the normal graph but fit in b waves between 0o and 360o
Draw the normal graph but fit in b waves between 0o and 360o
Question 122Question 122What isloga x + loga y equal to ?
What isloga x + loga y equal to ?
answer
Answer to Question 122Answer to Question 122Loga xyLoga xy
Question 123Question 123What do you get when you differentiate sin x ?
What do you get when you differentiate sin x ?
answer
Answer to Question 123Answer to Question 123cos xcos x
Question 124Question 124How do you find the angle between two vectors ?
How do you find the angle between two vectors ?
answer
Answer to Question 124Answer to Question 124 a.b
a b
a.b
a bcos=
a
b
Question 125Question 125Given an equation like m = moe-3k and an amount by which it has been decayed, how do you find k ?
Given an equation like m = moe-3k and an amount by which it has been decayed, how do you find k ?
answer
Answer to Question 125Answer to Question 125(i) fit in m and mo
(ii) rearrange to get e-3k =(iii) take logs(iv) solve
(i) fit in m and mo
(ii) rearrange to get e-3k =(iii) take logs(iv) solve
Question 126Question 126If u = ai+bj+ckthen what is u in component form ?
If u = ai+bj+ckthen what is u in component form ?
answer
Answer to Question 126Answer to Question 126
U =abc
Question 127Question 127What do you get when you differentiate
cosax ?
What do you get when you differentiate
cosax ?answer
Answer to Question 127Answer to Question 127
-asinax
Question 128Question 128How do you solve an equation of the form acosx + bsinx + c=0 ?
How do you solve an equation of the form acosx + bsinx + c=0 ?
answer
Answer to Question 128Answer to Question 128 Change acosx+bsinx into Rcos(x- )
Rearrange and solve
Change acosx+bsinx into Rcos(x- )
Rearrange and solve
Question 129Question 129
What is loga xn equal to ?
What is loga xn equal to ?
answer
Answer to Question 129Answer to Question 129 nloga x nloga x
Question 130Question 130How would you differentiate a function like
y = sin3 x ?
How would you differentiate a function like
y = sin3 x ?answer
Answer to Question 130Answer to Question 130 (i) write as (sin x)3
(ii) multiply by the power (iii) decrease power by one (iv) multiply by the derivative
of the bracket i.e. 3cosx sin2x
(i) write as (sin x)3
(ii) multiply by the power (iii) decrease power by one (iv) multiply by the derivative
of the bracket i.e. 3cosx sin2x
Question 131Question 131State the three rules of logs ?
State the three rules of logs ?
answer
Answer to Question 131Answer to Question 131 (i) logaxy = logax + logay
(ii) loga = logax – logay
(iii) logaxn = nlogax
(i) logaxy = logax + logay
(ii) loga = logax – logay
(iii) logaxn = nlogax
xy
Question 132Question 132How do you solve equations of the form
3x = 0.155 ?
How do you solve equations of the form
3x = 0.155 ?
answer
Answer to Question 132Answer to Question 132 (i) take logs of both sides(ii) bring x down to front(iii) solve the equation
(i) take logs of both sides(ii) bring x down to front(iii) solve the equation
Question 133Question 133Given experimental data, how do you find an equation in the form y=abx or y=axb ?
Given experimental data, how do you find an equation in the form y=abx or y=axb ?
answer
Answer to Question 133Answer to Question 133 (i) take logs of both sides(ii) rearrange to get a
straight line equation(iii) determine type(iv) find solution
(i) take logs of both sides(ii) rearrange to get a
straight line equation(iii) determine type(iv) find solution
Question 134Question 134How would you differentiate a function like
y = sin ax ?
How would you differentiate a function like
y = sin ax ?answer
Answer to Question 134Answer to Question 134
dy/dx = acos ax dy/dx = acos ax
Question 135Question 135
If u =
then what is u ?
If u =
then what is u ?answer
abc
Answer to Question 135Answer to Question 135 work out length√(a2+b2+c2)
work out length√(a2+b2+c2)
Question 136Question 136How do you add or subtract vectors ?
How do you add or subtract vectors ?
answer
Answer to Question 136Answer to Question 136 add or subtract matching components
add or subtract matching components
Question 137Question 137What doesa.a equal ?
What doesa.a equal ?
answer
Answer to Question 137Answer to Question 137 a2 a2
Question 138Question 138How do you prove that three 3-D points are
collinear ?
How do you prove that three 3-D points are
collinear ?answer
Answer to Question 138Answer to Question 138 Prove they are the same vector multiplied by different or the same numbers
Prove they are the same vector multiplied by different or the same numbers
Question 139Question 139Express the equation y=kxn in the form of the equation of a straight line, Y=nX+c.
Express the equation y=kxn in the form of the equation of a straight line, Y=nX+c.
answer
Answer to Question 139Answer to Question 139 logy = nlogx + logk logy = nlogx + logk
Question 140Question 140Who loves maths ?
Who loves maths ?
answer
Answer to Question 140Answer to Question 140 ME !!!!! ME !!!!!