indices logarithms

7/31/2019 indices logarithms

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Indices and Logarithms

1. Given 1 – log5 x = 2 log5 y, express x in terms of y.

Diberi 1 – log5 x = 2 log5 y, ungkapkan x dalam sebutan y.

[3 marks /3 markah]

Answer/ Jawapan:

2. Given 2 log10 xy2 = 3 + log10 y − log10 x, prove that xy = 10.

Diberi 2 log10 xy2

= 3 + log10 y − log10 x, buktikan bahawa xy = 10.

[3 marks /3 markah]Answer/ Jawapan:

3. Solve the equation: log4 x – log4 ( x + 6) = – 1

Selesaikan persamaan:[3 marks /3 markah]

Answer/ Jawapan:

4. Solve the equation log2 x − log2 (2 x + 3) = −2.

Selesaikan persamaan log2 x − log2 (2 x + 3) = −2. [3 marks /3 markah]

Answer/ Jawapan:

5. Express the recurring decimal 0.232323... as a fraction in its simplest form.

Ungkapkan perpuluhan jadi semula 0.232323... sebagai satu pecahan dalam bentuk

termudah.

[4 marks /4 markah]

Answer/ Jawapan:

6. Find the value of:

Cari nilai bagi:


7.Given express y in terms of x.

Diberi ungkapkan y dalam sebutan x. [4 marks /4 markah]

Answer/ Jawapan:



Coordinate Geometry

8. The points A( – 1, p), B(2, – 1) and C (4, 5) are collinear.

Find the value of p.

Titik-titik A( – 1, p), B(2, – 1) dan C (4, 5) adalah segaris.

Cari nilai p. [2 marks /2 markah]

Answer/ Jawapan:

9. Diagram shows three points, A, B and C , on a straight line.

Rajah menunjukkan tiga titik, A, B dan C, pada satu garis lurus.

Diagram/ Rajah

Given 4 AB = AC , find the coordinates of point C .

Diberi 4 AB = AC , cari koordinat bagi titik C.

[3 marks /3 markah]

Answer/ Jawapan:

10. P(−5, −6), Q(−3, −2) and R(4, k ) are three points on a straight line. Q lies between P and R

and divides PR in the ratio m : n.

P(−5, −6), Q(−3, −2) dan R(4, k ) ialah tiga titik pada suatu garis lurus. Q terletak di antara

P dan R dan membahagi PR dalam nisbah m : n.

Find

Cari

(a) the ratio m : n.

nisbah m : n.

[2 marks/ 2 markah]

(b) the value of k .

nilai k.

[2 marks/ 2 markah]Answer/ Jawapan:

(a)

(b)

11. Given three points, A(0, 2), B(5, 6) and C (2, 10), find the equation of the straight line that

passes through point B and is perpendicular to AC .

Diberi tiga titik A(0, 2), B(5, 6) dan C (2, 10), cari persamaan garis lurus yang melalui titik

B dan berserenjang dengan AC.

[3 marks /3 markah]



Answer/ Jawapan:

12. Find the equation of a straight line that passes through the point (−1, −6) and is parallel to

the line .

Cari persamaan garis lurus yang melalui titik (−1, −6) dan selari dengan garis .[3 marks/ 3 markah]

Answer/ Jawapan:

13. Given the points P(1, 2), Q(t , 0) and R( – 7, – 4) lie on a straight line.

Diberi titik-titik P(1, 2), Q(t , 0) dan R( – 7, – 4) terletak pada satu garis lurus.

(a) Find the value of t .

Cari nilai t.

(b) The point Q divides the line PR internally in the ratio m : n.Find the ratio m : n.

Titik Q membahagi garis PR mengikut nisbah m : n.

Cari nisbah m : n.[4 marks /4 markah]

Answer/ Jawapan:

(a)

(b)

14. Diagram shows a straight line graph 4 x + 6 y = 24 passing through point A and point B.

Rajah menunjukkan graf garis lurus 4 x + 6 y = 24 yang melalui titik A dan titik B.

Diagram/ Rajah

(a) Find the coordinates of points A and B.

Cari koordinat bagi titik A dan titik B.

(b) Find the equation of the perpendicular bisector of the straight line AB.Cari persamaan pembahagi dua sama serenjang bagi garis lurus AB.

[4 marks /4 markah]

Answer/ Jawapan:(a)

(b)



Statistics

15. A set of data consists of six numbers. The sum of the numbers is 72 and the sum of the

squares of the numbers is 944.

Satu set data mempunyai enam nombor. Hasil tambah bagi nombor-nombor itu ialah 72 dan

hasil tambah bagi kuasa dua nombor-nombor itu ialah 944.

Find, for the six numbers,

Cari, bagi enam nombor itu,

(a) the mean.

min. [1 mark/ 1 markah]

(b) the standard deviation.

sisihan piawai.

[2 marks/ 2 markah]Answer/ Jawapan:

(a)

(b)

16. A set of data consists of eight numbers. The sum of the numbers is 32 and the sum of the

squares of the numbers is 146.

Satu set data terdiri daripada lapan nombor. Hasil tambah nombor-nombor itu ialah 32 dan

hasil tambah bagi kuasa dua nombor-nombor itu ialah 146.

(a) Find the mean.

Cari minnya.

(b) Find the standard deviation.

Cari sisihan piawainya.


(a)

(b)

18. The mean of a set of numbers, x + 4, 2 x + 5, 2 x – 1, x + 7 and x – 3, is 8.

Min bagi satu set nombor, x + 4, 2 x + 5, 2 x – 1, x + 7 dan x – 3, ialah 8.

Find

Cari

(a) the value of x.nilai x.

(b) the variance of the set of numbers.

varians bagi set nombor itu. [4 marks /4 markah]

Answer/ Jawapan:

(a)



(b)

19. The mean and variance of 2 x − y, 10, 12 and 3 x + y are 18 and 100 respectively.

Calculate the values of x and y, if x and y are positive numbers.

Min dan varians bagi 2 x − y, 10, 12 dan 3 x + y masing-masing ialah 18 dan 100.

Hitung nilai x dan nilai y, jika x dan y ialah nombor positif. [4 marks/ 4 markah]

Answer/ Jawapan:

20. The mean of a set of numbers, 7, 14, 15, a, 2a, 47 and 52, is 27.

Min bagi satu set nombor 7, 14, 15, a, 2a, 47 dan 52 ialah 27.

(a) Find the value of a and the standard deviation of the set of numbers.

Cari nilai a dan sisihan piawai bagi set nombor itu.

(b) If each of the numbers in the set is multiplied by 4, find the standard deviation of the newset of numbers.

Jika setiap nombor dalam set itu didarab dengan 4, cari sisihan piawai bagi set nombor

yang baru itu.

[4 marks /4 markah]

Answer/ Jawapan:

(a)

(b)

21. Table shows the allowances given to the workers in a factory on a certain day.

Jadual menunjukkan elaun yang diberi kepada pekerja di sebuah kilang pada suatu hari

tertentu.

Table/ Jadual

Find the mean of the workers’ allowances.

Cari min elaun pekerja-pekerja itu.

[3 marks /3 markah]

Answer/ Jawapan:

Jawapan



1.

2. 2 log10 xy2

= 3 + log10 y − log10 x

log10( xy2)2

= log10 103

+ log10 y − log10 x

log10 ( x2 y

4) = log10

x3 y

3 = 1 000

xy = 10

3.

4. x =

5.

6.



7.

8. Given A( – 1, p), B(2, – 1) and C (4, 5) are collinear.

9.

The coordinates of point C are (14, 9).

10. (a) 2 : 7(b) k = 12

11.Gradient of AC = = 4

So, the gradient of a line that is perpendicular to AC is .

The equation of the line that passes through B(5, 6) and is perpendicular to AC is

y – 6 = ( x – 5)4 y – 24 = – x + 5

x + 4 y = 29

12. 5 x

+ 4 y

+ 29 = 013. P(1, 2), Q(t , 0), R( – 7, – 4) lie on a straight line.

(a)



(b)

Thus, the ratio is 1 : 2.

14. (a) Equation of the straight line:

Thus, the coordinates of points A and B are (6, 0) and (0, 4) respectively.

(b)

Gradient of the line perpendicular to AB =

The equation of the perpendicular bisector of AB is

y – 2 = ( x – 3)

2 y – 4 = 3 x – 9

2 y – 3 x + 5 = 0

15. (a) 12

(b) 3.651

16. N = 8, Σ x = 32, Σ x2

= 146



(a)

(b) Standard deviation

17. 7

18.

19. x = 10, y = 5.05

20. (a)

(b) New standard deviation

= 4 × 16.48= 65.92

21. Mean of the allowances

indices logarithms

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