Evaluate each of the followings: a) sin 4 cos 4 sin34  3

INTRODUCTION to BASIC MATHEMATICS EXERCISE SET

V. Trigonometric expressions 1. Evaluate each of the followings:

a) sin 4

cos 4

sin³4 _ ³ _ 

b)

cos12 12 sin5

2  

c) 



 



 



 



 log 3 9

log 1

cos 10 _{3 1/9}

d) _



 



 

sin 6 6 log

cos log

log_{1/2 3}  ₃ 

e) ⁵ sin( 7 )

4

tan21 _{ } 

f) 









  



 



 

3 sin4 3

sin2 3 cos2 log

2 







2. Simplify

a) sint(tantcott)cost(tantcott)

b) (sintcost)(1sintcost)(sintcost)(1sintcost) 3. Express

a) sint and cost in terms of tant b) tant in terms of sint

c) tant in terms of cost 4. Give the exact value of

a) cos15^sin15^ b) cos10^cos20^ sin10^ sin20^ c) cos²15^ sin²15^ d) sin30^cos15^ cos30^ sin15^

e) cos²22.5^ f) ^{ } cos110^

2 40 1 sin 70

sin  

VI. Percentage. Ratio and proportion 1. Determine:

a) 15% of 75 b) ¾% of 1600 c) 350% of 0.018

d) 6% of 152 e) 6.4% of 2.5 2. What percent of

a) 15 is 6 b) 480 is 1.6 c) 27.5 is 2.5

d) 192 is 36 e) 400 is 900?

3. Determine the number if

a) 3% of the number is18 b) 0.35% of the number is 10.5 c) 60% of the number is 72 d) 2.5% of the number is 0.012 e) 150% of the number is 390 .

4. A copper compound contains 80 percent copper by weight. How much copper is there in a 1.95 g sample of the compound?

5. The volume of a certain gas is 46 ml. If this is 8% lower than the volume under different conditions, what was this previous volume?

6. A sample of water is decomposed into 0.965 g of oxygen and 0.120 g of hydrogen. Calculate the mass present of oxygen and hydrogen in the sample of water.

7. Ammonia gas contains 82.4 percent nitrogen. Determine the mass of ammonia gas containing 300 kg of nitrogen.

8. The map scale shows 5 cm representing 150 km. What actual distance is represented by a map distance of 3.2 cm?

9. A photographic negative is 10 cm by 5 cm. If the width of an enlarged print is 15 cm, what should its length be?

10. An airplane flying at 450 km/hr covers a distance in 3 hours15 minutes. At what speed would it have to fly to cover the same distance in 2 hours 30 minutes?

11. If 10 men can complete a project in 12 days, how many days will it take 15 men to complete it?

VIII. Mixed problems (equations, system of equations)

2. Solve the following equations ( equations involving absolute value ):

a) x2 3 b) x2 7x4

c) 3x2 x1 d)

4 1 1

1    x

x x

e) 2x²  x1 f) x² x  x1

4. For which values of m has the equation x²mx5m90 equal roots?

5. Determine m so that one root of the equation 3x²10x3m0 is the reciprocal of the other.

6. Determine m so that one root of the equation x² mx180 is twice the other.

7. If the roots of x² bxc0 are 2 and 5, what are the values of b and c ? 8. For what values of m will the equation x² 4x7m(x1) have

a) one root the reciprocal of the other b) one root equal zero

c) roots numerically equal but of opposite sign.

9. True or false? Give reasons for your answer.

a) If ab0 , then a0 and b0. b) If x² 4x0, then x0 or x4.

10. Find the exact value of cosx, tanx, cotx and sin2x, if sinx1/3 and x



/2,



. 11. Find the exact value of sinx, cosx and cotx, if tanx2 and x



^,3 ^/2



.

12. Solve the following trigonometric equations:

a) sin2x1/2 b) cos² x1/2

c) cos2xcosx d) sinxsin4x

e) sinxsinx 1 f) ) 0

2 )(cos 1

1

(sinx x 

g) cos²xsin²x1/2 h) 2cos²x5cosx20 i) 3sinx2cos² x j) 12sin²x2cos²x3cos2x k) 2sin²xsin²2x2 l) sin2xcos2x2cosx1 m) 44cos² x 2sinx n) 2cos²x7sinx5 o) cos2x3sinx10 p) 2cos2xtan2x

q) sinxcosx1 r) tan 1

cos

1  x

x

s) 0

) sin 1 ( 2

cos ) sin 2 1

( ²

 



x x

x t) x

x x

x sin

cos 1

) 2 sin(

5 . 0

sin 





 

u) x

x x

x

sin 2 cos cos

1

sin  

 v) 4

cos 1

sin sin

cos

1 

 



x x x

x

x) x x x x

x

2 2

sin 2 1

) sin (cos 2

cos 2 sin 1



 

 y)

x x

x x

x sin cos

2 cos cos

1 sin

1

 



14. Solve the following logarithmic equations:

a) ^log5



²²^log3⁽x² ¹¹⁾



² b) ^log9



^log2⁽x² ¹⁾⁷



¹ c) log_2x(4x² 4x23)2 d) log_1x(3x² 2x31)2 e) 2log₄(2x1)log₄(3x1) f) log₃(8x)2log₃(x4) g) log₂ x2log₂(x2) h) log(5x)log(x1)1

i) log₃(x1)log₃(x3)1 j) log₃(x4)log₃(x1)1log₃2 k) log₃(x1)log₃(x4)3log₃2 l) log(4x2)log(6x)log(12x) m) log(x3)log(x² 23)1log(x5)

n) _



 



 







 5 5

log ) 1 log(

) 1 2 (log

2 ³ ₃

x x

o) log(2 ) log(3 ) log 1 2

1 x  x  x

p) log ( 3 ) log 3 0

2 1

4 2

4 x  x  x  q) log₂(92^x)3x

r) x^{log x} 100 s) log x  logx

t) log₄ xlog_1/4 x4 u) 1log₂(x1)log_x14

15. If x and y are positive real numbers and log²xlog² y2logxlogy9 , find the value of x/y .

16. Solve the following systems of equations:

a) xy 17, 3xy23 b) x2y 1 , 3x6y3 c) 2x y4, 4x2y 0 d) y² 6x1 , y2x10 e) x² xyy² 19, xy10 f) 4x² y² 11 , 2x y1 g) x²  y² 23, 2x² y² 5 h) xyx² 5, y² xy20 i) 34y10

x , 2 2 2





 y

x j) 2y2 6

x , 14y9 x

k) 3^x 3^y1/2 4, 2xy 3/2 l) 4^xy 128, 5^3x2y3 1 m) 3^x2^y 972, log ( 2) 2

3 x  n) 2^x4^y 32, log(xy)² 2log2 o) log₄xlog₂ y0, x² 5y² 40

p) 1

4 log 2 2log

1

2

2  

 x y

y , log₃xlog₃(2y)3 q) x² 3y³ 13log_x y0, log_y x3

IX. Word problems

1. Three is subtracted from a certain number and then that result is multiplied by 4 to produce 152.

Find the number.

2. The sum of three consecutive integers is 21 larger than twice the smallest integer. Find the integers.

3. Barry is paid double time for each hour worked over 40 hours in a week. Last week he worked 47 hours and earned $378. What is his normal hourly rate?

4. Tina is 4 years older than Sherry. In 5 years the sum of their ages will be 48. Find their present ages.

5. If two opposite sides of a square are each increased by 3 cm and the other two sides are each decreased by 2 cm the area is increasing by 8 square centimeters. Find the length of the side of the square.

6. Karla sold a bicycle for $97.50. This selling price represented a 30% profit for her, based on what she had originally paid for the bike. Find Karla`s original cost for the bicycle.

7. One leg of a right triangle is 7 meters longer than the other leg. If the length of the hypotenuse is 17 meters, find the length of each leg.

8. The sum of two integers is 16. The sum of the squares of the integers is 136. Find the integers.

9. The sum of the lengths of the two legs of a right triangle is 34 meters. If the length of the hypotenuse is 26 meters, find the length of each leg.

10. A room contains 120 chairs. The number of chairs per row is one less than twice the number of rows. Find the number of rows and the number of chairs per row.

11. The length of one side of a triangle is 3 cm less than twice the length of the altitude to that side.

If the area of the triangle is 52 square centimeters, find the length of the side and the length of the altitude to that side.

12. The length of a rectangle is 3 meters more than double its width. Its perimeter is 108 meters. Find the length and width.

13. A pharmacist has two solutions, one is 50% strength and the other is 25% strength. How many milliliters of each should he mix to obtain 50 milliliters of a 40% strength solution?

14. Kurt bought 29¢ stamps for letters and 19¢ stamps for postcards. He bought 140 stamps for

$31.60. How many of each type did he buy?

15. Victor invest $6000 in a bank. Part is invested at 7% interest and part at 9% interest. In one year Victor earns $510 in interest. How much did he invest at each amount?

16. A company plans to employ 910 people with a ratio of two managers for every 11 workers. How many managers should be hired? How many workers?

17. If there are 13 milliliters of acid in 37 milliliters of solution, how much acid will be contained in 296 milliliters of solution?

18. Western University has 168 faculty. The university always maintains a student-to-faculty ratio of 21:2. How many students should they enrol to maintain that ratio?

19. The area of a rectangle is 90 square meters. Its length is 3 meters longer than twice its width. Find its length and width.

X. Inequalities

Give the solution set of the following inequalities:

1. 7

1 2 3

2 

 x

x 2. 4

5 7 2

5

13    

 x x

3. x3 1 4. 3

5 8 

 x

5. x x1 6. x45

7. 3x2 x10 8. x²2x

9. x² 2x15 10. 1

1 3 



 x x

11. 1

1 2



 x

x 12. x1 x27

13. 2x11x 14. 0 2 1 



 x

x

15. 13

3 1



 x

x 16. 5

2 3

1 

x x

17. 4

1

3 5



  x

x x 18. 2 1

5 5

3  



 x

x x

19. 2

2 5 1



 

 x

x 20. 1 2 1



 x x

21. 1

1 3 2

 

 x

x 22. x

x x1

23. 1

3 2

3 





 x

x 24. 1

4

2 

 x

x

25. ,



0,2



2 2 sin  sin x x

x 26. cos2xcosx

27. log(x² 1)1 28. 25^x 45^x 50 29. True or false? Give reasons for your answer.

a) If a ‹ b and c › d , then c-a › d-b.

b) If a ‹ b and c › d , then c+a › d+b.

c) If x and y are real numbers and x ‹ y, then x² ‹ xy.

d) If x and y are real numbers and x ‹ y , then x y

1 1 .

30. Find the values of m for which the equation x²2(m3)xm² 40 has two real roots.

XI. Coordinate geometry

1. Find the equation of the line containing the points A(-1,2) and B(6,5).

2. Find the equation of the line that passes through the point (-1,-2) and parallel to the line 2y3x4 .

3. Find the equation of the line that passes through the point (-4,3) and perpendicular to the line 30

5

3y x .

4. Evaluate the slope of the line determined by the origin and the intersection point of the lines 70

2

3x y and 4xy 130.

5. Find the point on the line 5x4y20 that is equidistant from A(0,3) and B(9,0).

6. Determine whether the points A(-34,-25), B(18,-5), C(25,17) and D(-26,-2) are vertices of a parallelogram or not. If yes, find the intersection point of the diagonals.

7. Show that the points (-3,8), (7,4) and the point of intersection of the lines 2x3y7 and

6

y

x are vertices of a right triangle. Find the area of the triangle.

8. For what values of a and b will the two lines represented by the equations x3y4 and b

ay x  2

a) be parallel b) coincide c) be perpendicular.

9. Given the points A(-1,3), B(-4,7) and C(2,9).

a) Find the equation of the line going through the points A and B.

b) Does the point C lie on the line determined by A and B.

10. Find the vertices of the triangle whose sides are parts of lines with equations 2x y1, 3

2xy and xy 5.

11. Two points P₁ and P₂ are on the lines x2y2 and 2xy 1 respectively. The sum of the x coordinates of the points is -10 and the sum of their y coordinates is 12. Find the coordinates of the points.

12. Find the center and the radius of the circle x² y² 6x4y40. Show that the point (1,1) lies on the circle.

13. Find the equation of the circle that is tangent to the line y x3 at P(-2,5) with radius 3 2. 14. Find the equation of the circle with center (-2,-2) and tangent to the X-axis.

15. Find the equation of the circle with center (2,-1) and tangent to the Y-axis.

16. Find the equation of the circle(s) that is tangent to both axes and passes through the point (-2,-4).

17. Find the equation of the circle(s) going through the point (-1,1), its center lies on the line x y1 and it is tangent to the X-axis.

18. Find the equation of the circle(s) tangent to the X-axis with radius 5 and its center lies on the line 1

2 

 x

y .

19. Determine the equation of the circle(s) going through the point P(2,-1), touches the Y-axis and its center lies on the line xy20.

20. Find the equation of the circle passing through the point (5,8), touches the line y = 2 and its center lies on the line y x.

21. Find the equation of the circle going through the points A(1,3), B(-1,1) and its center lies on the line x2y10.

22. a) Find the equation of the circle centered at (3,2) and that passes through the point (6,-2).

b) Show that the point (6,6) lies on the circle and find the equation of the tangent line at this point.

23. Given the points A(0,0), B(-4,0) and C( 0,-6).

a) Find the equation of the circle that contains the points A, B and C.

b) Give the equation of the tangent line to this circle at the point A.

24. Which point of the circle (x1)² (y1)² 9 is equidistant from A(-4,-3) and B(2,-9)?

25. Find the equations of the tangent lines to the circle (x2)²(y3)² 2 which are parallel to the line joining the points (2,0) and (-2,4).

26. Find the equation of the line tangent to the circle (x2)²(y5)² 25 at P(6,-2).

27. Find the equation of the line passing through the origin and tangent to the circle 9

) 2 ( ) 3

(x ²  y ²  .

28. Tangent lines are drawn to the circle (x3)² (y2)² 29 at the points (-2,0) and (5,7). Find where these tangent lines intersect.

29. Find all points on the line y2x1 that are 2 units from (2,1).

30. Find all points on the curve y² 4x that are 4 units from (4,0).

31.a) Find the equations of the circles with radius 5 units, tangent to the Y-axis and passing through the point (9,9).

b) Find the intersection points of the above circles.

c) Find the equation of the common chord.

d) Find the equations of the common tangents.

32. Find the coordinates of the vertex of the parabola y² 4y2x60. 33. Show that the line y2x1 is a tangent to the parabola y x².

34. For which values of m will the line ymx3 be a tangent to the parabola y(x1)²?

35. Find the coordinates of the intersection points of the line y2x1 and the circle 25

) 2 ( ) 1

(x ²  y ²  .

36. Find the length of the chord cut from the circle x² y²6x2y0 by the line y2x. 37. Give the distance between the points of intersection of the curves y² 4x and y x3.

38. Find the intersection points of the curves x y² and (x1)²  y² 7.

39. Determine the distance between the intersection points of the curves 2x y² and 5

) 1

(x ²  y²  .

40. Given the points A(2,2), B(-2,-1) and C(1,0).

a) Find the equation of the line through the points A and B.

b) The point D is the foot of the perpendicular from C to AB. Find the coordinates of D.

c) Calculate the distance between the points C and D.

d) What is the equation of the circle centered C and its radius is 3 units?

XII. Functions

1. Find the domain of the following functions:

a) 20

2 ) 7

( ₂



  x x x

f b)

2 ) 1

( ₂





  x x x x f

c) f x x x x

2 ) 1

( _{3 2}



  d)

x x x

f 

  12 ) 10 (

e) ( ) 1

  x x x

f f)

8 24 7

2 )

( ²

 





 x x x

x f

g) 1

1 2 ) 1

(  

 

x x x

f h) f(x) 4 x1

i) f(x) 2x 9 j) f(x) 73x x1

k) 9

1 1 ) 3

( ₂

 



 

x x

x x

f l) f(x)log(x²x2)

m) log ( 1)

) 1 ) (

(

2 2



 

x x x x

f n) f(x)log(x2)log(x2)

o) _



 







 

 1

1 1 log )

( ₄

x x x

f p)

5 1

) 10

(  

  x x x

f q) ( ) log 1

10

3 

 x

x x

f r) f(x)log₃(3^x 9)

s) f x x x

2

4 2

2 ) 1

(   t)

) 8 2 ( log 3 ) 1

( ₃

2 

  _x

x f

u) f(x) cos⁴ xcos² x v)

x x

f

cos 1 ) 1

(  

x) x x

x x

f (1 tan )cos 2 ) cos

(   y)

x x

x f

sin 3 2 sin ) 1

(  

z)

x x

x f

cos 2 sin 1 ) 1

(   2 

2. Find the domain and the range of the following functions:

a) f(x)2x² 8x b) f(x)13xx² c) f(x) sin² x1 d) f(x)2^{cos2xsin2x} e) f(x)log_1/2(x² 2x3) f) f(x)2sin4x2 3. Determine which function is odd, even or neither odd nor even.

a) f(x)xsinx b) f(x)3xx³ c) f(x)3^x 3^x d) f(x)2^x2^x

e) f(x)xx⁴ f) f(x) 1xx²  1xx² 4. Determine which function is one-to-one.

a) f(x)7^x b)

4 ) 4

(  

x x

f c) ₄

2

) 6

( x

x x

f  

5. Sketch the graph of the following functions:

a) 1

sin2 )

(  x

x

f b) f(x)cos2x1 c) f(x)2x² 4x 6. Let f(x)x²2x3.

a) Sketch the graph of f .

b) Find the coordinates of the vertex of the parabola.

c) Find the minimum value of f .

7. Sketch the set of all points on the XY-plane whose coordinates satisfy the given condition:

a) x² 2x y² 0 b) x² y² 0

c) x  y1 d) x y 1

e) x  y 1 f) x² 4y² 1

8. For what values of the real parameter m does the equation (m2)x² 2(m2)x20 have no real roots?

9. For what values of the real parameter k will the inequality kx² 2kx50 hold for all real numbers?

10. Find the values of the real parameter a for which the inequality 1 2

2

2

2 







 x x

ax x is true for all real numbers.

11. Find the value of a/b without solving the system 1 2

a b , 1 4

b a .

12. Given that 1 5



 x

x , find the value of ^{2 1}₂ x  x .

13. The sum of two nonnegative numbers is 20. Find the numbers for which the sum of their squares is to be as small as possible.

14. The sum of two numbers is 10. Find the numbers whose product is to be as large as possible.

15. Find the quadratic function f(x)ax² bxc whose graph passes through the points (-1,-3), (2,-8) and (-4,-34).

XIII. Sequences

1. The angles of a right triangle form an arithmetic sequence. Find the angles.

2. The first and the third terms of an arithmetic sequence are 25 and 19 respectively. Find the sequence and the number of terms required to make the sum of them equal to 82.

3. How many multiples of 9 are there between 300 and 500?

4. The sum of the third and seventh terms of an arithmetic sequence is 10. The product of the second and fifth terms is -5. Find the first term and the common difference.

5. For a certain arithmetic sequence the sum of the first three terms is -21 and the product of the same three terms is -315. Find the sum of the first 15 terms.

6. Find the sum of seven terms of an arithmetic sequence if the middle term is 9.

7. The sum of the first six terms of an arithmetic sequence is the third of the sum of the next five terms. The common difference is 13. Find the first term.

8. Find the sum of all whole numbers divisible by 9 between 100 and 300.

9. The sum of n terms of an arithmetic sequence is given as (3n² 2n). Find the first three terms.

10. How many rows are in the corner section of a stadium containing 2040 seats if the first row has 10 seats and each successive row has 4 additional seats?

11. A child building a tower with blocks uses 15 for the first row. Each row has two blocks less than the previous one. If there are 8 rows in the tower, how many blocks are used for the top row? How many blocks are used for the tower?

12. If 9 is the second and 17 the fourth term of an arithmetic sequence and the sum of the first n terms

189

Sn . Find a₁ , d and n.

13. Find the eleventh term of a geometric sequence, whose first term is 1/8 and the second term is -1.

14. Find the first term of a geometric sequence whose common ratio is 2 and the eighth term is 640.

15. The first, third and fifth terms of a geometric sequence are (x+1), (4x+4), (15x+22). Find the value of x, find also the common ratio and the sixth term.

16. In a geometric sequence whose terms are positive, any term is equal to the sum of the next two following terms. Find the common ratio.

17. The sum of three consecutive terms of a geometric sequence is 21 and the their product is 216.

Find the numbers.

18. Suppose that x, y, z are distinct nonzero real numbers such that x

y 1 ,

y 2

1 , z y

1

form an arithmetic sequence. Prove that x, y, z form a geometric sequence.

19. For what values of x do the numbers x, 2x+3, 3x+22 form a geometric sequence? Find the common ratio in each case.

20. Find x and y if the terms (2y+1), (x+2),(x+2y-1) form an arithmetic sequence, and (x-2), 4y, 24 form a geometric sequence. Find the common difference and the common ratio.

21. Find x and y if the terms y, x, 1 form an arithmetic sequence and (x-2), 8, (y+5) form a geometric sequence. Write down the first three terms of the sequences.

22. Which of the following statements are true or false? Justify your answer.

a) If the first two terms of an arithmetic sequence are negative, then so is the third.

b) If the first two terms of a geometric sequence are negative, then so is the third.

23. Find two numbers whose arithmetic mean is 39 and whose geometric mean is 15.

24. In a geometric sequence of 12 terms, the first term is 1, and the last term is 2048. Find the common ratio and the sum of these terms.

25. At the end of each year the value of a car is 20% less then at the beginning of that year. If a new car costs $5000, find its value at the end of 4 years.

26. By adding the same constant to each of 9, 23, 51, a geometric sequence results. Find the common ratio.

27. Determine whether the following terms form an arithmetic or a geometric sequence:

log(3x), log(3x)², log(3x)⁴.

28. Find three numbers a, b and c that are consecutive terms of both an arithmetic sequence and a geometric sequence.

XIV. Area, volume.

1. Find the side of the square the area of which is equal to the area of the rectangle with a=17 cm, b=8 cm.

2. A rhombus has diagonals 15 cm and 10 cm respectively. What is the area?

3. What are the lengths of sides of an equilateral triangle of height 3 cm.

4. In a triangle ABC, AB=AC and AB:BC=3:4. If the perimeter of the triangle is 30 cm, find the length of AB.

5. Calculate the length of a side of a rhombus with diagonals of length 6 cm and 5 cm.

6. If the sides of a square are increased by 3 times, how many times more will be the area? What about the perimeter?

7. In a rectangle all sides are increased by three times. How many times more will be the area and how much greater will be the perimeter?

8. In a rectangle one side is made three times greater and the other side is made 7 times greater. How many times more will be the area of the rectangle? What about the perimeter?

9. The circumference of a circle with radius 5 cm is divided into 4 equal parts by the points A, B, C, D. Give the area of the square ABCD thus obtained.

10. Find the height of a circular cone with base radius 1.5 cm and volume 8 cube cm. Find the surface area.

11. Calculate the surface area and the volume of a cone of base diameter 7 cm and height 4 cm.

12. A circular cone has a base of diameter 15 cm and a slant height of 18 cm. Calculate the surface area and the volume of the cone.

13. If the sides of a cube are increased six times, a) how many times larger will be the surface area?

b) How many times larger will be the volume?

c) What about the diagonal of the cube?

14. A sphere has a surface area of 50 square cm. What is the surface area of another sphere with a radius three times as big?