Grade 10 Algebra Assessment

1. A notebook costs $8. If Sarah buys 3 notebooks, how much does she spend?

$16

$20

$24

$30

To find the total amount spent, we multiply the cost per notebook ($8) by the number of notebooks (3). $8 × 3 = $24. Therefore, Sarah spends $24.

Explanation

To find the total amount spent, we multiply the cost per notebook ($8) by the number of notebooks (3). $8 × 3 = $24. Therefore, Sarah spends $24.

2. Which of the following is the solution to the equation 3x + 5 = 17?

X = 3

X = 4

X = 6

X = 9

To solve the equation, we need to isolate the variable x. Subtracting 5 from both sides, we get 3x = 12. Then, dividing both sides by 3, we find x = 4 as the solution.

Explanation

To solve the equation, we need to isolate the variable x. Subtracting 5 from both sides, we get 3x = 12. Then, dividing both sides by 3, we find x = 4 as the solution.

3. Which of the following is the solution to the equation 4(x + 3) = 28?

X = -1

X = 2

X = 4

X = 7

.

Explanation

.

4. The length of a rectangle is 10 cm and its width is 5 cm. What is its area?

25 sq cm

50 sq cm

100 sq cm

150 sq cm

To calculate the area of a rectangle, we multiply its length (10 cm) by its width (5 cm). 10 cm × 5 cm = 50 sq cm. Therefore, the area of the rectangle is 50 sq cm.

Explanation

To calculate the area of a rectangle, we multiply its length (10 cm) by its width (5 cm). 10 cm × 5 cm = 50 sq cm. Therefore, the area of the rectangle is 50 sq cm.

5. Which term is missing in the following arithmetic sequence: 8, __, 20, 26, 32, 38?

11

14

17

23

The common difference between the terms is 6. The missing term is the term between 8 and 20. Adding the common difference, we get: 8 + 6 = 14.

Explanation

The common difference between the terms is 6. The missing term is the term between 8 and 20. Adding the common difference, we get: 8 + 6 = 14.

6. What is the common difference in the arithmetic sequence: 9, 16, 23, 30, ...?

5

6

7

8

The common difference between the terms is 7. Each term is obtained by adding 7 to the previous term.

Explanation

The common difference between the terms is 7. Each term is obtained by adding 7 to the previous term.

7. If the price of a shirt is $40 and it is reduced by 20%, what is the sale price?

$32

$36

$38

$42

To calculate the sale price, we need to find 20% of $40, which is $8. We subtract $8 from $40 to get the sale price of $32.

Explanation

To calculate the sale price, we need to find 20% of $40, which is $8. We subtract $8 from $40 to get the sale price of $32.

8. What is the value of x in the equation (x/2) + 3 = 7?

X = -2

X = 2

X = 8

X = 12

To solve the equation, subtract 3 from both sides: (x/2) = 4. Multiply both sides by 2 to eliminate the fraction: x = 8.

Explanation

To solve the equation, subtract 3 from both sides: (x/2) = 4. Multiply both sides by 2 to eliminate the fraction: x = 8.

9. What is the 10th term in the arithmetic sequence: 3, 7, 11, 15, ...?

35

37

39

41

The common difference between the terms is 4. So, the formula to find the nth term is: An = A1 + (n-1)d. Plugging in the values, we get: A10 = 3 + (10-1)*4 = 39.

Explanation

The common difference between the terms is 4. So, the formula to find the nth term is: An = A1 + (n-1)d. Plugging in the values, we get: A10 = 3 + (10-1)*4 = 39.

10. Factorize the expression x^2 + 6x + 9.

(x - 3)(x - 3)

(x + 3)(x + 3)

(x - 3)(x + 3)

(x + 3)(x - 3)

To factorize the expression x^2 + 6x + 9, we can observe that it is a perfect square trinomial. The square of (x + 3) gives this expression. Therefore, the correct answer is (B( (x+ 3)(x + 3).

Explanation

To factorize the expression x^2 + 6x + 9, we can observe that it is a perfect square trinomial. The square of (x + 3) gives this expression. Therefore, the correct answer is (B( (x+ 3)(x + 3).

11. What is the value of x in the equation 2(x - 3) = 10?

X = -1

X = 0

X = 5

X = 8

To solve the equation, distribute 2 to both terms inside the parentheses: 2x - 6 = 10. Add 6 to both sides, and we have 2x = 16. Finally, divide both sides by 2 to find x = 8.

Explanation

To solve the equation, distribute 2 to both terms inside the parentheses: 2x - 6 = 10. Add 6 to both sides, and we have 2x = 16. Finally, divide both sides by 2 to find x = 8.

12. Sara borrowed $500 from a friend at an interest rate of 8% per year. How much interest will she have to pay after 2 years?

$80

$100

$160

$200

To calculate the interest, we multiply the borrowed amount ($500) by the interest rate (8% = 0.08) and the number of years (2). $500 × 0.08 × 2 = $80. Therefore, Sara will have to pay $80 in interest after 2 years.

Explanation

To calculate the interest, we multiply the borrowed amount ($500) by the interest rate (8% = 0.08) and the number of years (2). $500 × 0.08 × 2 = $80. Therefore, Sara will have to pay $80 in interest after 2 years.

13. Factorize the expression 2x^2 - 5x - 3.

(2x - 6)(x + 1)

(2x + 3)(x - 1)

(2x + 1)(x - 3)

(2x - 1)(x + 3)

Explanation

14. What is the simplified form of (5x^2 - 3x + 2) + (-2x^2 + 4x - 1)?

3x^2 + x + 1

3x^2 + x + 3

3x^2 + x - 1

3x^2 - x + 1

To simplify the expression, we combine like terms:
(5x^2 - 3x + 2) + (-2x^2 + 4x - 1) = 5x^2 - 2x^2 - 3x + 4x + 2 - 1 = 3x^2 + x + 1

Explanation

To simplify the expression, we combine like terms:
(5x^2 - 3x + 2) + (-2x^2 + 4x - 1) = 5x^2 - 2x^2 - 3x + 4x + 2 - 1 = 3x^2 + x + 1

15. Factorise the expression 2x^2 + 5x + 3.

(2x + 3)(x + 1)

(2x - 3)(x + 1)

(2x + 1)(x + 3)

(2x - 1)(x + 3)

To factorise the expression 2x^2 + 5x + 3, we need to find two binomials that multiply to give this expression. We can use the ac-method or trial and error method. After some calculations, we find that (2x + 3)(x + 1) gives the original expression when multiplied. Therefore, the correct answer is (A) (2x + 3)(x + 1).

Explanation

To factorise the expression 2x^2 + 5x + 3, we need to find two binomials that multiply to give this expression. We can use the ac-method or trial and error method. After some calculations, we find that (2x + 3)(x + 1) gives the original expression when multiplied. Therefore, the correct answer is (A) (2x + 3)(x + 1).

16. Factorise the expression 3x^2 - 6x - 9.

(3x + 1)(x + 3)

(3x - 1)(x - 3)

(3x - 1)(x + 3)

(3x + 3)(x - 3)

The expression 3x^2 - 6x - 9 can be factorized as (3x + 3)(x - 3).

To factorize this expression, we look for the greatest common factor (GCF) among the terms. The GCF of 3x^2, -6x, and -9 is 3.

Now, we factor out the common factor of 3 from the expression:

3(x^2 - 2x - 3)

Next, we need to factorize the quadratic expression inside the parentheses. We look for two numbers whose product is -2x and whose sum is -3. In this case, the numbers are -3 and 1.

We rewrite the expression using these numbers:

3(x^2 - 3x + 1x - 3)

Now, we group the terms and factorize further:

3[(x^2 - 3x) + (1x - 3)]

3[x(x - 3) + 1(x - 3)]

We can now factor out a common factor of (x - 3) from both terms:

3(x - 3)(x + 1)

So, the factorized form of the expression 3x^2 - 6x - 9 is (3x + 3)(x - 3).

Explanation

The expression 3x^2 - 6x - 9 can be factorized as (3x + 3)(x - 3).

To factorize this expression, we look for the greatest common factor (GCF) among the terms. The GCF of 3x^2, -6x, and -9 is 3.

Now, we factor out the common factor of 3 from the expression:

3(x^2 - 2x - 3)

Next, we need to factorize the quadratic expression inside the parentheses. We look for two numbers whose product is -2x and whose sum is -3. In this case, the numbers are -3 and 1.

We rewrite the expression using these numbers:

3(x^2 - 3x + 1x - 3)

Now, we group the terms and factorize further:

3[(x^2 - 3x) + (1x - 3)]

3[x(x - 3) + 1(x - 3)]

We can now factor out a common factor of (x - 3) from both terms:

3(x - 3)(x + 1)

So, the factorized form of the expression 3x^2 - 6x - 9 is (3x + 3)(x - 3).

17. Which of the following is the solution to the equation 7 - 2(3x + 2) = 1?

X = 1/3

X = 0

X = 1

X = 2

Explanation

18. The ratio of boys to girls in a class is 3:5. If there are 24 boys in the class, how many girls are there?

30

36

40

48

To find the number of girls, we first need to find the total number of parts in the ratio, which is 3 + 5 = 8 parts. Since there are 24 boys (which represent 3 parts), we divide 24 by 3 to find the value of one part: 24 ÷ 3 = 8. So, 1 part represents 8 boys. Now, we multiply 5 (the number of parts representing girls) by 8 to find the total number of girls: 5 × 8 = 40. Therefore, there are 40 girls in the class.

Explanation

To find the number of girls, we first need to find the total number of parts in the ratio, which is 3 + 5 = 8 parts. Since there are 24 boys (which represent 3 parts), we divide 24 by 3 to find the value of one part: 24 ÷ 3 = 8. So, 1 part represents 8 boys. Now, we multiply 5 (the number of parts representing girls) by 8 to find the total number of girls: 5 × 8 = 40. Therefore, there are 40 girls in the class.

19. In a line graph, what does the slope of the line indicate?

Rate of change

Probability

Frequency

Mean value

The slope of a line in a line graph represents the rate of change. It indicates how one variable changes in relation to the other variable.

Explanation

The slope of a line in a line graph represents the rate of change. It indicates how one variable changes in relation to the other variable.

20. A car is traveling at a speed of 60 km/hr. How far can it travel in 2.5 hours?

120 km

125 km

150 km

135 km

To calculate the distance, we multiply the speed (60 km/hr) by the time (2.5 hours). 60 km/hr × 2.5 hours = 150 km. Therefore, the car can travel 150 km in 2.5 hours.

Explanation

To calculate the distance, we multiply the speed (60 km/hr) by the time (2.5 hours). 60 km/hr × 2.5 hours = 150 km. Therefore, the car can travel 150 km in 2.5 hours.

21. 1. Solve the following system of equations: 2x + 3y = 8 4x - y = 10

X = 19/7, y = 6/7

X = 3, y = -2

X = 1, y = 2

X = -1, y = 4

Explanation

22. Factorise the expression x^2 - 5x + 6.

(x - 3)(x - 2)

(x + 3)(x - 2)

(x - 3)(x + 2)

(x + 3)(x + 2)

To factorise the expression x^2 - 5x + 6, we need to find two binomials that multiply to give this expression. We can use the ac-method or trial and error method. After some calculations, we find that (x - 3)(x - 2) gives the original expression when multiplied. Therefore, the correct answer is (A) (x - 3)(x - 2).

Explanation

To factorise the expression x^2 - 5x + 6, we need to find two binomials that multiply to give this expression. We can use the ac-method or trial and error method. After some calculations, we find that (x - 3)(x - 2) gives the original expression when multiplied. Therefore, the correct answer is (A) (x - 3)(x - 2).

23. What does the x-intercept represent on a graph?

The value of y when x is zero

The value of x when y is zero

The maximum point of the graph

The minimum point of the graph

The x-intercept represents the value of x when y is zero. It is the point at which the graph crosses the x-axis.

Explanation

The x-intercept represents the value of x when y is zero. It is the point at which the graph crosses the x-axis.

24. Which of the following is a solution for the inequality 2x - 3 < 9?

X > 6

X < 6

X > 3

X < 3

To solve the inequality, we first add 3 to both sides: 2x < 12. Then, we divide both sides by 2 to isolate x: x < 6. Therefore, the solution is x < 6.

Explanation

To solve the inequality, we first add 3 to both sides: 2x < 12. Then, we divide both sides by 2 to isolate x: x < 6. Therefore, the solution is x < 6.

25. What is the solution to the equation 2(3x - 1) + 3(2x + 1) = 11?

X = -2

X = -1

X = 5/6

X = 1

Explanation

26. Factorize the expression 4x^2 + 16.

4(x + 4)(x - 4)

4(x - 4)(x - 4)

4(x + 2)(x -2)

4(x - 4)(x + 4)

.

Explanation

.

27. What is the simplified form of 6(3x + 4) - (2x - 1)?

20x + 23

16x + 6

18x + 5

16x+25

To simplify the expression, we distribute the 6 and simplify the parentheses:

6(3x + 4) - (2x - 1) = 18x + 24 - 2x + 1 = 16x + 25

Explanation

To simplify the expression, we distribute the 6 and simplify the parentheses:

6(3x + 4) - (2x - 1) = 18x + 24 - 2x + 1 = 16x + 25

28. What is the simplified form of (3x^2 + 4x - 8) - (x^2 - 2x + 1)?

2x^2 + 6x - 9

2x^2 + 6x - 7

2x^2 + 2x - 7

2x^2 + 2x - 9

To simplify the expression, we distribute the subtraction and combine like terms:
(3x^2 + 4x - 8) - (x^2 - 2x + 1) = 3x^2 + 4x - 8 - x^2 + 2x - 1 = 2x^2 + 6x - 9

Explanation

To simplify the expression, we distribute the subtraction and combine like terms:
(3x^2 + 4x - 8) - (x^2 - 2x + 1) = 3x^2 + 4x - 8 - x^2 + 2x - 1 = 2x^2 + 6x - 9

29. Which of the following is a solution for the inequality 7x + 3 > 24?

X > 3

X < 3

X > 4

X < 4

To solve the inequality 7x + 3 > 24, follow these steps:

Subtract 3 from both sides of the inequality:

7x + 3 - 3 > 24 - 3

7x > 21

Divide both sides of the inequality by 7:

(7x)/7 > 21/7

x > 3

The solution for the inequality is x > 3.

Explanation

To solve the inequality 7x + 3 > 24, follow these steps:

Subtract 3 from both sides of the inequality:

7x + 3 - 3 > 24 - 3

7x > 21

Divide both sides of the inequality by 7:

(7x)/7 > 21/7

x > 3

The solution for the inequality is x > 3.

30. Solve the inequality 2(3x - 1) ≤ 4x + 2.

X ≤ 2

X ≥ -2

X ≤ 2/3

X ≥ 2/3

Explanation

31. What is the simplified form of (2a^2 - 3ab + 4b^2)(a - b)?

2a^3 - 4a^2b + b^3

2a^3 - 5ab^2 - 4b^2

2a^3 - 5a^2b + 7ab^2 - 4b^3

2a^3 - 3a^2b + 7ab^2 - 4b^3

We start with the expression (2a^2 - 3ab + 4b^2)(a - b). This is a product of two expressions: 2a^2 - 3ab + 4b^2 and a - b.

To simplify this, we distribute each term in the first expression by each term in the second expression. This is like using the distributive property of multiplication over addition (or subtraction).

First, we multiply each term in the first expression by a from the second expression. This gives us 2a^3 - 3a^2b + 4ab^2.

Then, we multiply each term in the first expression by -b from the second expression. This gives us -2a^2b + 3ab^2 - 4b^3.

We then subtract this second set of terms from the first set of terms to get 2a^3 - 3a^2b + 4ab^2 - (-2a^2b + 3ab^2 - 4b^3), which simplifies to 2a^3 - 5a^2b + 7ab^2 - 4b^3.

So, the simplified form of the original expression is 2a^3 - 5a^2b + 7ab^2 - 4b^3.

Explanation

We start with the expression (2a^2 - 3ab + 4b^2)(a - b). This is a product of two expressions: 2a^2 - 3ab + 4b^2 and a - b.

To simplify this, we distribute each term in the first expression by each term in the second expression. This is like using the distributive property of multiplication over addition (or subtraction).

First, we multiply each term in the first expression by a from the second expression. This gives us 2a^3 - 3a^2b + 4ab^2.

Then, we multiply each term in the first expression by -b from the second expression. This gives us -2a^2b + 3ab^2 - 4b^3.

We then subtract this second set of terms from the first set of terms to get 2a^3 - 3a^2b + 4ab^2 - (-2a^2b + 3ab^2 - 4b^3), which simplifies to 2a^3 - 5a^2b + 7ab^2 - 4b^3.

So, the simplified form of the original expression is 2a^3 - 5a^2b + 7ab^2 - 4b^3.

32. What is the simplified form of 4x/8 - 2x/6?

X/6

X/16

X/12

X/24

To simplify 4x/8 - 2x/6:

Simplify each fraction: 4x/8 = x/2 2x/6 = x/3

Find a common denominator for x/2 and x/3. The least common denominator (LCD) of 2 and 3 is 6. x/2 = 3x/6 x/3 = 2x/6

Subtract the fractions: 3x/6 - 2x/6 = (3x - 2x)/6 = x/6

So, the simplified form of 4x/8 - 2x/6 is x/6.

Explanation

To simplify 4x/8 - 2x/6:

Simplify each fraction: 4x/8 = x/2 2x/6 = x/3

Find a common denominator for x/2 and x/3. The least common denominator (LCD) of 2 and 3 is 6. x/2 = 3x/6 x/3 = 2x/6

Subtract the fractions: 3x/6 - 2x/6 = (3x - 2x)/6 = x/6

So, the simplified form of 4x/8 - 2x/6 is x/6.

33. What is the simplified form of 9a^2b^3 / 3ab?

3ab^2

3a^2b

6ab^2

3a^2b^2

To simplify the expression, we divide the coefficients and subtract the exponents:

9a^2b^3 / 3ab = (9/3)(a^2/a)(b^3/b) = 3a^(2-1)b^(3-1) = 3a^1b^2 = 3ab^2

Explanation

To simplify the expression, we divide the coefficients and subtract the exponents:

9a^2b^3 / 3ab = (9/3)(a^2/a)(b^3/b) = 3a^(2-1)b^(3-1) = 3a^1b^2 = 3ab^2

34. What is the 7th term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...?

10

12

13

21

In the Fibonacci sequence, each term is the sum of the two preceding ones. So, the formula to find the nth term is: Fn = Fn-1 + Fn-2. Plugging in the values, we get: F7 = 5 + 8 = 13.

Explanation

In the Fibonacci sequence, each term is the sum of the two preceding ones. So, the formula to find the nth term is: Fn = Fn-1 + Fn-2. Plugging in the values, we get: F7 = 5 + 8 = 13.

35. Which of the following is a solution for the inequality -2x + 7 < 9?

X > -1

X < -1

X > -2

X < -2

Explanation

36. What is the slope of a vertical line on a graph?

0

Undefined

1

Infinity

The slope of a vertical line is undefined because the rise (change in y-coordinates) over the run (change in x-coordinates) is infinite.

Explanation

The slope of a vertical line is undefined because the rise (change in y-coordinates) over the run (change in x-coordinates) is infinite.

37. 2. Solve the following system of equations: 3x + 2y = 13 2x - 4y = -6?

X = 5/2 , y = 11/4

X = 2/5, y = -2/11

X = 3/7, y = 2/9

X = -2/5, y = 3/7

To solve this system of equations, we can again use either substitution or elimination method. Let's use substitution here. From the second equation, we can express x in terms of y:

2x = 4y - 6

x = (4y - 6)/2

Substituting this value of x in the first equation:

3((4y - 6)/2) + 2y = 13

(12y - 18)/2 + 2y = 13

(12y - 18 + 4y) = 26

16y - 18 = 26

16y = 44

y = 44/16

y = 11/4

Substituting this value of y back into x = (4y - 6)/2:

x = (4(11/4) - 6)/2

x = (11 - 6)/2

x = 5/2

Hence, the solution is x = 5/2 and y = 11/4.

Explanation

To solve this system of equations, we can again use either substitution or elimination method. Let's use substitution here. From the second equation, we can express x in terms of y:

2x = 4y - 6

x = (4y - 6)/2

Substituting this value of x in the first equation:

3((4y - 6)/2) + 2y = 13

(12y - 18)/2 + 2y = 13

(12y - 18 + 4y) = 26

16y - 18 = 26

16y = 44

y = 44/16

y = 11/4

Substituting this value of y back into x = (4y - 6)/2:

x = (4(11/4) - 6)/2

x = (11 - 6)/2

x = 5/2

Hence, the solution is x = 5/2 and y = 11/4.

38. 4. Solve the following system of equations: 4x - 5y = -13 2x + 3y = 7

X = -2, y = 1

X = 1, y = 2

X = 3, y = 4

X = -1, y = -2

We can solve this system of equations using the elimination method. To eliminate y, we can multiply the first equation by 3 and the second equation by 5, then subtract them:

12x - 15y = -39
10x + 15y = 35

Adding these two equations cancels out the y variable:
22x = -4
x = -4/22
x = -2/11

Substituting this value of x into one of the original equations, we can solve for y:
2(-2/11) + 3y = 7
-4/11 + 3y = 7
3y = 7 + 4/11
3y = 77/11 + 4/11
3y = 81/11
y = (81/11) * (1/3)
y = 27/11
y = 27/11

Therefore, the solution is x = -2/11 and y = 27/11, which can be simplified to x = -0.18 and y = 2.45.

Explanation

We can solve this system of equations using the elimination method. To eliminate y, we can multiply the first equation by 3 and the second equation by 5, then subtract them:

12x - 15y = -39
10x + 15y = 35

Adding these two equations cancels out the y variable:
22x = -4
x = -4/22
x = -2/11

Substituting this value of x into one of the original equations, we can solve for y:
2(-2/11) + 3y = 7
-4/11 + 3y = 7
3y = 7 + 4/11
3y = 77/11 + 4/11
3y = 81/11
y = (81/11) * (1/3)
y = 27/11
y = 27/11

Therefore, the solution is x = -2/11 and y = 27/11, which can be simplified to x = -0.18 and y = 2.45.

39. What is the definition of a graph in mathematics?

A visual representation of data using bars and columns

A collection of points and lines connecting some or all of the points

A branch of mathematics that deals with solving equations

A shape formed by two intersecting curves

In mathematics, a graph is a collection of points (vertices) and lines (edges) connecting some or all of the points.

Explanation

In mathematics, a graph is a collection of points (vertices) and lines (edges) connecting some or all of the points.

40. Solve the inequality -4x + 9 ≤ 5.

X ≥ 1

X ≤ 1

X ≥ 2

X ≤ 2

To solve the inequality -4x + 9 ≤ 5, follow these steps:

Subtract 9 from both sides of the inequality:

-4x + 9 - 9 ≤ 5 - 9

-4x ≤ -4

Divide both sides of the inequality by -4, remembering to reverse the inequality symbol since we are dividing by a negative number:

(-4x)/(-4) ≥ (-4)/(-4)

x ≥ 1

The solution for the inequality is x ≥ 1.

Explanation

To solve the inequality -4x + 9 ≤ 5, follow these steps:

Subtract 9 from both sides of the inequality:

-4x + 9 - 9 ≤ 5 - 9

-4x ≤ -4

Divide both sides of the inequality by -4, remembering to reverse the inequality symbol since we are dividing by a negative number:

(-4x)/(-4) ≥ (-4)/(-4)

x ≥ 1

The solution for the inequality is x ≥ 1.

41. 5. Solve the following system of equations: 3x + 4y = 12 6x + 8y = 18

X = 2, y = 0

X = 1, y = 2

X = 0.5, y = 2

X = -2, y = 3

To solve this system of equations, we can observe that the second equation is simply twice the first equation. This means the system is dependent and has infinitely many solutions. Any value of x can be chosen, and y can be calculated accordingly.

For example, if we take x = 2, the equation becomes:
3(2) + 4y = 12
6 + 4y = 12
4y = 6
y = 6/4
y = 3/2

Hence, the solution is x = 2 and y = 3/2, but there are infinitely many solutions.

Explanation

To solve this system of equations, we can observe that the second equation is simply twice the first equation. This means the system is dependent and has infinitely many solutions. Any value of x can be chosen, and y can be calculated accordingly.

For example, if we take x = 2, the equation becomes:
3(2) + 4y = 12
6 + 4y = 12
4y = 6
y = 6/4
y = 3/2

Hence, the solution is x = 2 and y = 3/2, but there are infinitely many solutions.

42. What is the sum of the first 15 terms in the geometric sequence: 2, 6, 18, 54, ...?

14,348,906

886,450

896,450

901,450

To find the sum of the first 15 terms in the geometric sequence: 2, 6, 18, 54, ...

Use the formula for the sum of the first n terms of a geometric sequence:

S_n = a * (r^n - 1) / (r - 1)

where:

a = 2 (first term)

r = 3 (common ratio)

n = 15 (number of terms)

Plug in the values:

S_15 = 2 * (3^15 - 1) / (3 - 1)

Calculate 3^15:

3^15 = 14348907

Now substitute it back:

S_15 = 2 * (14348907 - 1) / 2 S_15 = 2 * 14348906 / 2 S_15 = 14348906

So, the sum of the first 15 terms in the geometric sequence is 14348906.

Explanation

To find the sum of the first 15 terms in the geometric sequence: 2, 6, 18, 54, ...

Use the formula for the sum of the first n terms of a geometric sequence:

S_n = a * (r^n - 1) / (r - 1)

where:

a = 2 (first term)

r = 3 (common ratio)

n = 15 (number of terms)

Plug in the values:

S_15 = 2 * (3^15 - 1) / (3 - 1)

Calculate 3^15:

3^15 = 14348907

Now substitute it back:

S_15 = 2 * (14348907 - 1) / 2 S_15 = 2 * 14348906 / 2 S_15 = 14348906

So, the sum of the first 15 terms in the geometric sequence is 14348906.

43. 3. Solve the following system of equations: 6x + 3y = 21 2x + y = 8

X = 1, y = 6

X = -2, y = 12

X = 3, y = 2

X = 4, y = 0

To solve this system of equations, let's multiply the second equation by 3 to eliminate y:

6x + 3y = 21
6x + 3y = 24

As we can see, both equations are the same. This means the system is dependent and has infinitely many solutions. Any value of x can be chosen, and y can be calculated accordingly.

For example, if we take x = 1, the equation becomes:
6(1) + 3y = 21
6 + 3y = 21
3y = 15
y = 15/3
y = 5

Hence, the solution is x = 1 and y = 5, but there are infinitely many solutions.

Explanation

To solve this system of equations, let's multiply the second equation by 3 to eliminate y:

6x + 3y = 21
6x + 3y = 24

As we can see, both equations are the same. This means the system is dependent and has infinitely many solutions. Any value of x can be chosen, and y can be calculated accordingly.

For example, if we take x = 1, the equation becomes:
6(1) + 3y = 21
6 + 3y = 21
3y = 15
y = 15/3
y = 5

Hence, the solution is x = 1 and y = 5, but there are infinitely many solutions.

44. What is the next term in the sequence: 2, 5, 10, 17, 26, ?

38

39

40

41

The pattern between the terms is (term number)^2 - 1. So, the next term is (6)^2 - 1 = 35.

Explanation

The pattern between the terms is (term number)^2 - 1. So, the next term is (6)^2 - 1 = 35.

45. 6. Solve the following system of equations: 2x + 5y = 9 6x + 15y = 27

X = 3, y = 0

X = 0, y = 3

X = 1, y = 2

X = 2, y = 1

Similar to the previous question, the second equation is three times the first equation. This indicates that the system is dependent and has infinitely many solutions. Any value of x can be chosen, and y can be calculated accordingly.

For example, if we take x = 3, the equation becomes:
2(3) + 5y = 9
6 + 5y = 9
5y = 3
y = 3/5

Hence, the solution is x = 3 and y = 3/5, but there are infinitely many solutions.

Explanation

Similar to the previous question, the second equation is three times the first equation. This indicates that the system is dependent and has infinitely many solutions. Any value of x can be chosen, and y can be calculated accordingly.

For example, if we take x = 3, the equation becomes:
2(3) + 5y = 9
6 + 5y = 9
5y = 3
y = 3/5

Hence, the solution is x = 3 and y = 3/5, but there are infinitely many solutions.

46. Which type of graph is used to show the relationship between two variables?

Bar graph

Line graph

Pie chart

Scatter plot

A scatter plot is used to display the relationship between two variables. It shows the correlation or pattern between the variables.

Explanation

A scatter plot is used to display the relationship between two variables. It shows the correlation or pattern between the variables.

47. Which of the following is a directed graph?

Bar graph

Line graph

Pie chart

Flow chart

A flow chart is an example of a directed graph where the connections between nodes have a specific direction. It represents the flow of a process or a system.

Explanation

A flow chart is an example of a directed graph where the connections between nodes have a specific direction. It represents the flow of a process or a system.

48. Solve the inequality 3x + 5 > 10.

X < 1

X > 1

X < 5

X > 5

To solve the inequality, we first subtract 5 from both sides: 3x > 5. Then, we divide both sides by 3 to isolate x: x > 5/3. As 5/3 is approximately 1.67, we can write the solution as x > 1.

Explanation

To solve the inequality, we first subtract 5 from both sides: 3x > 5. Then, we divide both sides by 3 to isolate x: x > 5/3. As 5/3 is approximately 1.67, we can write the solution as x > 1.