Higher Mathematics Objective Questions

Objective QuestionsObjective QuestionsObjective QuestionsObjective Questions1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19 20 21 22 23 24 25 26 27 28

29 30 31 32 33 34 35 36 37 38 39 40 41 42

43 44 45 46 47 48 49 50 51 52 53 54 55 56

57 58 59 60 61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80 81 82 83 84

85 86 87 88 89 90 91 92 93 94 95 96 97 98

99 100 101 102 103 104 105 106 107 108 109 110 111 112

113 114 115 116 117 118 119 120 121 122 123 124 125 126

127 128 129 130 131 132 133 134 135 136 137 138 139 140

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







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:



occurs

when x is:


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



occurs

when x is:




occurs

when x is:


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 :




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








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?






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









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 :




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.







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 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







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


3. Functions f and g , are given by f(x) = 3x2 + 1 and g(x) = x2 - 4. Find an expression for f(g(x)).


A B C D



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)


A B C D




A B C D



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?


occurs when x is:

3. The line with equation y = 2 intersects the curve y = 1 - 2sinx , at :



occurs when x is:


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



occurs when x is:




occurs when x is:


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)).


√3:



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


only III & II C. only II & I B. only I A.

4 IV.

32

III. 3

II. 6

I.


√3:




√3:



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


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)).


A B C D



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


A B C D



x2 - 1.Find an expression for f(g(x)).


A B C D



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




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)









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:


1 + x.Find an expression for g(f(x)).


A. y = g(-x) + 2 B. y = -g(x) - 2

C. y = 2 – g(x) D. y = g(x – 2)



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)


A. y = g(-x) + 2 B. y = -g(x) - 2

C. y = 2 – g(x) D. y = g(x – 2)



1 + x.Find an expression for g(f(x)).


A. y = g(-x) + 2 B. y = -g(x) - 2

C. y = 2 – g(x) D. y = g(x – 2)



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


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:

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:





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)).





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.







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:



occurs when x is:





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





occurs when x is:





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


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




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.




A. 2p + 3q = 13 B. 3p – 2q = 5C. 3p + 2q = 5 D. 3p – 2q = 13




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



A B C D


A. y = -3x B. y = x C. y = -x D. y

= -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


A B C D


A. y = -3x B. y = x C. y = -x D. y

= -x



A B C D


A. y = -3x B. y = x C. y = -x D. y

= -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:


3. The line with equation y = 1 intersects the curve

y = 3tan2x , at :





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.





y = 3tan2x , at :





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:



occurs when x is:





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





occurs when x is:





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 ?


A. 3y = 2x + 13 B. 3y = -2x + 19

C. 2y = -3x - 19 D. 2y = 3x - 13


A. m = 1 B. m = -√2

C. m = -1 D. m = -


y

x45o

A

B2

1

35π

D. 6

11π C.

65π

B. 6π

A.2πx23π


A. 3y = 2x + 13 B. 3y = -2x + 19

C. 2y = -3x - 19 D. 2y = 3x - 13


A. m = 1 B. m = -√2

C. m = -1 D. m = -



A. 3y = 2x + 13 B. 3y = -2x + 19

C. 2y = -3x - 19 D. 2y = 3x - 13


A. m = 1 B. m = -√2

C. m = -1 D. m = -


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




A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0

C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0



A. 4y = x + 2 B. y = 4x + 2

C. y = -2x + 23 D. y = 2x - 2


1 xx

g(x)

x1

f(x)

1 xD. 1 x

1 C.

1 xx

B. x

1 x A.

2



A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0

C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0



A. 4y = x + 2 B. y = 4x + 2

C. y = -2x + 23 D. y = 2x - 2




A. 2y + x – 3 = 0 B. 2y – 3x + 3 = 0

C. 2y + 3x – 1 = 0 D. 3y – 2x + 1 = 0



A. 4y = x + 2 B. y = 4x + 2

C. y = -2x + 23 D. y = 2x - 2


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,-)


A. B. 2 C. 3 D. 6


A. -6 B. - C. D.





A. (0,) B. (0,) C. (0,-) D. (0,-)

; 2x f(x) 32

y

x0

Q R4

P (1,2)

2


A. B. 2 C. 3 D. 6


A. -6 B. - C. D.





A. (0,) B. (0,) C. (0,-) D. (0,-)


A. B. 2 C. 3 D. 6


A. -6 B. - C. D.





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


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


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


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)


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


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


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)


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


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)


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


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


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)


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


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


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)


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


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


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


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)


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


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


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


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


What is loga1

equal to ?

What is loga1

equal to ?

answer

Answer to Question 48Answer to Question 4800


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


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


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


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


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


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


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


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


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


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


How do you solve equations like

100 = 0 x2


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)


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


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


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


(i) factorise(ii) draw graph(iii) read values

below x-axis


below x-axis

Question 67Question 67How do you change from radians to degrees ?

How do you change from radians to degrees ?

answer


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


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


(i) (a,b)minimum

(i) (a,b)minimum

(a,b)

Question 77Question 77How do you integrate xn ?

How do you integrate xn ?

answer


xn+1

n+1

xn+1

n+1+ C+ C

Question 78Question 78How do you solve equations like

cos2xo-5sinxo = 0 ?(0≤x≤360)


cos2xo-5sinxo = 0 ?(0≤x≤360)

answer


(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



above x-axis


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


(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


iterationiteration


sin2xo = 0.5 ?(0≤x≤360)


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


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


(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


cosxo = - 0.8 ?(0≤x≤360)


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


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


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


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


∫ x3 dx ?

How do you find

∫ x3 dx ?

answer

aa

bb


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


∫ (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


y = cos(x+a) or y = sin(x+a) ?


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


y = cos bx or y = sin bx ?


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


U =abc

Question 127Question 127What do you get when you differentiate

cosax ?

What do you get when you differentiate

cosax ?answer


-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


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


3x = 0.155 ?


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


dy/dx = acos ax dy/dx = acos ax


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 !!!!!

Higher Mathematics Objective Questions

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