How do you simplify 4x − 3y − (3x − 7y)?

First expand the bracket: 4x − 3y − 3x + 7y. Then combine like terms: (4x−3x) + (-3y+7y) = x + 4y.

How do you simplify (x² − y²)/(x + y)?

Use the identity x² − y² = (x−y)(x+y). Cancel (x+y) to get x−y.

How do you make x the subject of y = (x+3)/x?

Multiply both sides by x: yx = x + 3. Rearrange: yx − x = 3. Factor: x(y−1)=3. Divide: x = 3/(y−1).

How do you solve 3x + 2 = 4x?

Subtract 3x from both sides: 2 = x. So x = 2.

How do you solve a cost problem with pens and books?

Let pen = x and book = x+50. Substitute into equation 2x + 3(x+50)=650. Solve: 5x+150=650 → x=100 → book=150.

How do you use factor theorem to find K?

Substitute the root into the polynomial and equate to zero, then solve for K.

How do you interpret a number line inequality?

Closed circle means included (≤ or ≥). Open circle means not included ( ).

How do you evaluate (1/27)^(-1/3)?

Rewrite 27 as 3³. Then (1/27)^(-1/3) = (3^-3)^(-1/3) = 3¹ = 3.

How do you factorize x² − 5x + 6?

Find two numbers that multiply to 6 and add to -5: -2 and -3. So (x−2)(x−3).

How do you simplify algebraic fractions?

Divide coefficients and subtract powers: (2x³y²)/(4xy) = (1/2)x²y.

The principal square root is 7.

How do you calculate mean?

Add all values and divide by number of values.

How do you find median?

Arrange data in order and take the middle value.

How do you find mode?

The value that appears most frequently.

What is probability of head in coin toss?

A fair coin has two outcomes, so probability = 1/2.

What is probability of rolling 4 on a die?

One favorable outcome out of 6 → 1/6.

How do you find midpoint of two points?

Midpoint = ((x1+x2)/2 , (y1+y2)/2).

How do you find distance between two points?

Distance = √((x2−x1)² + (y2−y1)²).

What is equation of straight line?

y = mx + c where m is gradient and c is intercept.

How do you find missing angle in triangle?

Sum of angles = 180°, subtract known angles.

How do you calculate probability from balls?

Probability = favorable outcomes / total outcomes.

How do you find mean of numbers?

Sum values then divide by count.

What is arithmetic progression?

A sequence with constant difference.

What is inverse variation?

When one variable increases, the other decreases proportionally.

How do you find composite function?

Substitute one function into another.

How do you find gradient?

Gradient = change in y / change in x.

How do you solve simultaneous equations?

Use substitution or elimination.

How are bearings measured?

Clockwise from north in degrees.

How do you find median from graph?

Locate midpoint on cumulative frequency.

How do you calculate average time?

Sum values ÷ number of values.

0570 Mathematics 2013 Solution Paper 1

Q: How do you solve 3x − 7 = 11?

Add 7 to both sides: 3x=18. Divide by 3: x=6.

Question 1

Simplify: 4x − 3y − (3x − 7y)

A. x + 4y

B. x − 10y

C. 6x − 23y

D. 6x + 5y

Show Answer

Answer: x + 4y

Explanation: Expand: 4x − 3y − 3x + 7y = (4x−3x) + (-3y+7y) = x + 4y.

Question 2

Simplify (x² − y²)/(x + y)

A. x − y

B. x + y

C. 2x

D. 2y

Show Answer

Answer: x − y

Explanation: Use identity: x² − y² = (x−y)(x+y). Cancel (x+y).

Question 3

Given y = (x+3)/x, find x in terms of y

A. y−3/y

B. 3/(y−1)

C. 3/(y+1)

D. (y+3)/y

Show Answer

Answer: 3/(y−1)

Explanation: yx = x + 3 ⇒ yx − x = 3 ⇒ x(y−1)=3 ⇒ x = 3/(y−1).

Question 4

Two more than three times a number equals four times the number. Find number

A. 2

B. 1

C. 0

D. −2

Show Answer

Answer: 2

Explanation: 3x + 2 = 4x ⇒ x = 2.

Question 5

Two pens and three books cost 650 francs. If one book costs 50 francs more than a pen, find cost of a book

A. 550

B. 250

C. 150

D. 100

Show Answer

Answer: 150

Explanation: Let pen = x, book = x+50. Equation: 2x + 3(x+50)=650 ⇒ 5x+150=650 ⇒ x=100 ⇒ book=150.

Question 6

If (x − i) is a factor of polynomial, find K

A. 12

B. 8

C. −8

D. −12

Show Answer

Answer: 8

Explanation: Substitute x=i into polynomial and equate to zero.

Question 7

Solution of inequality from number line

A. −4 ≤ x ≤ 3

B. −4 ≤ x < 3

C. −4 < x ≤ 3

D. −4 < x < 3

Show Answer

Answer: −4 < x ≤ 3

Explanation: Open circle at −4, closed at 3.

Question 8

Evaluate (1/27)^(-1/3)

A. 1/3

B. 3

C. 9

D. −3

Show Answer

Answer: 3

Explanation: 27 = 3³ ⇒ (1/27)^(-1/3) = 3.

Question 9

Factorize: x² − 5x + 6

A. (x − 2)(x − 3)

B. (x + 2)(x − 3)

C. (x − 1)(x − 6)

D. (x + 1)(x − 6)

Show Answer

Answer: (x − 2)(x − 3)

Explanation: We need two numbers that multiply to +6 and add to −5. These are −2 and −3. So x² − 5x + 6 = (x − 2)(x − 3).

Question 10

Solve: 3x − 7 = 11

A. 6

B. 5

C. 4

D. 3

Show Answer

Answer: 6

Explanation: Add 7 to both sides: 3x = 18. Divide by 3: x = 6.

Question 11

Simplify: (2x³y²)/(4xy)

A. (1/2)x²y

B. (1/2)x²y²

C. 2x²y

D. x³y

Show Answer

Answer: (1/2)x²y

Explanation: Divide coefficients: 2/4 = 1/2. Subtract powers: x³/x = x², y²/y = y.

Question 12

Find the value of x if x² = 36

A. 6

B. −6

C. ±6

D. 12

Show Answer

Answer: ±6

Explanation: Taking square roots gives x = ±√36 = ±6.

Question 13

Evaluate: 2³ × 2²

A. 2⁵

B. 4⁵

C. 2⁶

D. 4⁴

Show Answer

Answer: 2⁵

Explanation: When multiplying same base, add powers: 3 + 2 = 5.

Question 14

Simplify: √49

A. 7

B. −7

C. ±7

D. 14

Show Answer

Answer: 7

Explanation: √49 means the principal (positive) square root, which is 7.

Question 15

Find the mean of 2, 4, 6, 8

A. 4

B. 5

C. 6

D. 8

Show Answer

Answer: 5

Explanation: Mean = (2+4+6+8) ÷ 4 = 20 ÷ 4 = 5.

Question 16

The median of 1, 3, 5, 7, 9 is

A. 3

B. 5

C. 7

D. 9

Show Answer

Answer: 5

Explanation: The median is the middle value when arranged in order → 5.

Question 17

The mode of 2, 3, 3, 4, 5 is

A. 2

B. 3

C. 4

D. 5

Show Answer

Answer: 3

Explanation: Mode is the most frequent number → 3 appears twice.

Question 18

Find the probability of getting a head when tossing a fair coin

A. 1

B. 0

C. 1/2

D. 2

Show Answer

Answer: 1/2

Explanation: A fair coin has 2 outcomes (Head, Tail). Probability = 1/2.

Question 19

The probability of rolling a 4 on a fair die is

A. 1/4

B. 1/6

C. 1/2

D. 1/3

Show Answer

Answer: 1/6

Explanation: A die has 6 outcomes; only one favorable outcome → 1/6.

Question 20

If P(A) = 2/5, find P(not A)

A. 2/5

B. 3/5

C. 1/5

D. 4/5

Show Answer

Answer: 3/5

Explanation: Complement rule: P(not A) = 1 − 2/5 = 3/5.

Question 21

Sum of first and tenth term of AP = 15, d=1. Find first term

A. 12

B. 6

C. 5

D. 3

Show Answer

Answer: 3

Explanation: T10 = a+9. So a+(a+9)=15 ⇒ 2a=6 ⇒ a=3.

Question 22

If p ∝ 1/q² and q=1 gives p=100, find p when q=2

A. 5

B. 25

C. 50

D. 200

Show Answer

Answer: 25

Explanation: p=k/q² ⇒ 100=k ⇒ k=100. When q=2 ⇒ p=100/4=25.

Question 23

Diagram representing a function

Show Answer

Answer: B

Explanation: Each input maps to only one output.

Question 24

If f(x)=x²+1 and g(x)=3x−2, find f(g(x))

A. x²+3x−1

B. 9x²−12x+5

C. 3x²+1

D. 3x²−1

Show Answer

Answer: 9x²−12x+5

Explanation: Substitute: f(g(x))=(3x−2)²+1=9x²−12x+4+1.

Question 25

Gradient between points P(-2,2) and Q(3,-2)

A. −4/5

B. −5/4

C. 4/5

D. 5/4

Show Answer

Answer: −4/5

Explanation: (−2−2)/(3−(−2))=−4/5.

Question 26

Find the midpoint of the line joining A(2,4) and B(6,8)

A. (4,6)

B. (3,5)

C. (2,6)

D. (6,4)

Show Answer

Answer: (4,6)

Explanation: Midpoint = ((2+6)/2 , (4+8)/2) = (8/2 , 12/2) = (4,6).

Question 27

Find the distance between points (1,2) and (4,6)

A. 5

B. 6

C. 7

D. 4

Show Answer

Answer: 5

Explanation: Distance = √((4−1)² + (6−2)²) = √(3² + 4²) = √(9+16) = √25 = 5.

Question 28

The equation of a straight line with gradient 2 and y-intercept 3 is

A. y = 2x + 3

B. y = 3x + 2

C. y = 2x − 3

D. y = 3x − 2

Show Answer

Answer: y = 2x + 3

Explanation: General form is y = mx + c. Here m = 2 and c = 3.

Question 29

Find the area of a rectangle of length 8 cm and width 5 cm

A. 40 cm²

B. 13 cm²

C. 26 cm²

D. 80 cm²

Show Answer

Answer: 40 cm²

Explanation: Area = length × width = 8 × 5 = 40 cm².

Question 30

The circumference of a circle of radius 7 cm (π = 22/7) is

A. 44 cm

B. 154 cm

C. 22 cm

D. 88 cm

Show Answer

Answer: 44 cm

Explanation: Circumference = 2πr = 2 × (22/7) × 7 = 44 cm.

Question 31

The area of a triangle with base 10 cm and height 6 cm is

A. 60 cm²

B. 30 cm²

C. 16 cm²

D. 120 cm²

Show Answer

Answer: 30 cm²

Explanation: Area = 1/2 × base × height = 1/2 × 10 × 6 = 30 cm².

Question 32

The volume of a cube of side 4 cm is

A. 64 cm³

B. 16 cm³

C. 32 cm³

D. 48 cm³

Show Answer

Answer: 64 cm³

Explanation: Volume of cube = side³ = 4³ = 64 cm³.

Question 33

The number of faces of a cube is

A. 4

B. 6

C. 8

D. 12

Show Answer

Answer: 6

Explanation: A cube has 6 square faces.

Question 34

The sum of interior angles of a quadrilateral is

A. 180°

B. 360°

C. 270°

D. 90°

Show Answer

Answer: 360°

Explanation: Sum of interior angles = (n−2)×180 = (4−2)×180 = 360°.

Question 35

A triangle has angles 50° and 60°. The third angle is

A. 70°

B. 80°

C. 90°

D. 100°

Show Answer

Answer: 70°

Explanation: Total = 180°. So third angle = 180 − (50 + 60) = 70°.

Question 36

The probability of selecting a red ball from 3 red and 2 blue balls is

A. 3/5

B. 2/5

C. 1/2

D. 4/5

Show Answer

Answer: 3/5

Explanation: Total balls = 5, red balls = 3 → probability = 3/5.

Question 37

The mean of 5, 7, 9, 11 is

A. 8

B. 7

C. 9

D. 10

Show Answer

Answer: 8

Explanation: Mean = (5+7+9+11)/4 = 32/4 = 8.

Question 38

The median of 2, 4, 6, 8, 10 is

A. 4

B. 6

C. 8

D. 10

Show Answer

Answer: 6

Explanation: Middle value = 6.

Question 39

The mode of 1, 2, 2, 3, 4 is

A. 1

B. 2

C. 3

D. 4

Show Answer

Answer: 2

Explanation: 2 appears most frequently.

Question 40

The probability of not getting a head when tossing a coin is

A. 1

B. 0

C. 1/2

D. 2

Show Answer

Answer: 1/2

Explanation: Not getting a head = getting a tail → probability = 1/2.

Question 41

Which diagram represents p + q = 0

Show Answer

Answer: B

Explanation: Vectors equal magnitude opposite direction.

Question 42

Find vector QP

A. 1/2 p + 1/2 q

B. −1/2 p + 1/2 q

C. −1/2 p − 1/2 q

D. 1/2 p − 1/2 q

Show Answer

Answer: 1/2 p − 1/2 q

Explanation: Use midpoint relation.

Question 43

Find coordinates of Q

A. (4,0)

B. (3,2)

C. (3,6)

D. (2,3)

Show Answer

Answer: (3,6)

Explanation: Add vector components.

Question 44

Food expenditure from pie chart

A. 110000

B. 100000

C. 130000

D. 70000

Show Answer

Answer: 100000

Explanation: Use proportion of sector angle.

Question 45

Average pocket allowance

A. 10000

B. 12500

C. 15000

D. 25000

Show Answer

Answer: 12500

Explanation: Sum values ÷ number of students.

Question 46

Median class

A. 12–16

B. 22–26

C. 17–21

D. 7–11

Show Answer

Answer: 17–21

Explanation: Locate middle cumulative frequency.

Question 47

Average score

A. 1.5

B. 2.5

C. 3

D. 4

Show Answer

Answer: 2.5

Explanation: Weighted mean calculation.

Question 48

Probability of green ball

A. 2/5

B. 3/5

C. 4/10

D. 6/10

Show Answer

Answer: 3/5

Explanation: 6 green out of 10 total.

Question 49

Probability from tree diagram

A. 1/5

B. 1/3

C. 1/2

D. 1/6

Show Answer

Answer: 1/3

Explanation: Multiply probabilities along branches.

Question 50

Average time of race

A. 2.9

B. 4.3

C. 13

D. 16

Show Answer

Answer: 13

Explanation: Add all times and divide by 5.

0570 Mathematics 2013 Solution Paper 1

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

Question 22

Question 23

Question 24

Question 25

Question 26

Question 27

Question 28

Question 29

Question 30

Question 31

Question 32

Question 33

Question 34

Question 35

Question 36

Question 37

Question 38

Question 39

Question 40

Question 41

Question 42

Question 43

Question 44

Question 45

Question 46

Question 47

Question 48

Question 49

Question 50

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