Integers

The integers marked on the number line are: −18, −9, −4, 0, 3, 8, 12.

(i) The integers in ascending order are:

−18, −9, −4, 0, 3, 8, 12

(ii) The integers in descending order are:

12, 8, 3, 0, −4, −9, −18

(iii) The dots marked on the number line represent the following integers:

-17, −14, −11, -7, -5, −3, 1, 4, 7, 9, 11

Given negative integers are −5, −4.
In ascending order they are −5 < −4.

Given positive integers are 7, 4.
In ascending order they are 4 < 7.

Also, 0 lies between negative integers and positive integers.

Hence, the given integers in ascending order are:
−5 < −4 < 0 < 4 < 7.

i.e., −5, −4, 0, 4, 7.

On the number line, marking these integers:

−5 lies to the left of −4, which lies to the left of 0, which lies to the left of 4, which lies to the left of 7. This confirms our ascending order.

Rohit's scores in five successive rounds = 15, −3, −7, 12 and 8

Total score = 15 + (−3) + (−7) + 12 + 8

Grouping positives and negatives:

= (15 + 12 + 8) + ((−3) + (−7))

= 35 + (−10)

= 35 − 10

= 25

∴ Rohit's total score at the end = 25

Amount deposited on Monday = ₹ 4370
Amount withdrawn on Tuesday = ₹ 2875 (denoted by −2875)
Amount deposited on Wednesday = ₹ 1550

Balance on Thursday = ₹ 4370 + ₹ (−2875) + ₹ 1550

= ₹ 4370 − ₹ 2875 + ₹ 1550

= ₹ 1495 + ₹ 1550

= ₹ 3045

∴ Ruchi's balance on Thursday = ₹ 3045

(i) |−13| − |9|

= 13 − 9 $\hspace{2cm}$ [∵ |−13| = 13 and |9| = 9]

= 4

(ii) |13 − 5| − |−9|

= |8| − |−9|

= 8 − 9 $\hspace{2cm}$ [∵ |8| = 8 and |−9| = 9]

= −1

(iii) |35 − 21| − |8 − 3|

= |14| − |5|

= 14 − 5 $\hspace{2cm}$ [∵ |14| = 14 and |5| = 5]

= 9

Given negative integers are −39, −102, −51, −5, −6.
In ascending order they are −102 < −51 < −39 < −6 < −5.

Given positive integers are 35, 7.
In ascending order they are 7 < 35.

Also, 0 lies between negative integers and positive integers.

Hence, all the given integers in ascending order are:
−102 < −51 < −39 < −6 < −5 < 0 < 7 < 35.

i.e., −102, −51, −39, −6, −5, 0, 7, 35.

Given positive integers are 139, 4208.
In descending order they are 4208 > 139.

Given negative integers are −31, −203, −97.
In descending order they are −31 > −97 > −203.

Also, 0 is less than positive integers and greater than negative integers.

Hence, all the given integers in descending order are:
4208 > 139 > 0 > −31 > −97 > −203.

i.e., 4208, 139, 0, −31, −97, −203.

(i) True
Reason — In the set of integers, the successor is found by adding 1. Since −1 + 1 = 0, 0 is indeed the successor of -1.

(ii) False
Reason — Unlike the set of whole numbers or natural numbers, every integer has a predecessor. The predecessor of 0 is -1 (found by calculating 0 - 1).

(iii) True
Reason — The predecessor of a number is the integer immediately to its left on the number line, or the value found by subtracting 1.

Since -1 - 1 = -2, -2 is the predecessor of -1.

(iv) True
Reason — On a standard horizontal number line, 0 is located to the right of all negative integers, making it greater than any negative value.

(i) LHS = (−11) + (−7) = −(11 + 7) = −18
RHS = (−11) − (−7) = −11 + 7 = −4

Since −18 < −4,

∴ (-11) + (-7) $\boxed{ \lt }$ (-11) - (-7)

(ii) LHS = 23 − 41 + 11 = 34 − 41 = −7
RHS = 23 − 41 − 11 = 23 − 52 = −29

Since −7 > −29,

∴ 23 - 41 + 11 $\boxed{ \gt }$ 23 - 41 - 11

(iii) LHS = 40 − (−39) + (−5) = 40 + 39 − 5 = 74
RHS = 40 + (−39) − (−5) = 40 − 39 + 5 = 6

Since 74 > 6,

∴ 40 - (-39) + (-5) $\boxed{ \gt }$ 40 + (-39) - (-5)

(iv) LHS = (−3) + 13 − 15 = 10 − 15 = −5
RHS = 25 − (−2) + (−33) = 25 + 2 − 33 = −6

Since −5 > −6,

∴ (-3) + 13 - (15) $\boxed{ \gt }$ 25 - (-2) + (-33)

(i) Sum is -3

One such pair is -5 and 2

∵ -5 + (2) = −3

(ii) Difference is -5

One such pair is -2 and 3

∵ (−2) - (3) = −2 - 3 = −5

(iii) Difference is 4

One such pair is -7 and -11

∵ (−7) − (−11) = −7 + 11 = 4

Total score of Team A = (−30) + 20 + 0

= −10 + 0

= −10

Total score of Team B = 20 + 0 + (−30)

= 20 − 30

= −10

Since both teams scored −10,

∴ Both teams scored equal.

Yes, we can add integers in any order. This is because addition of integers is commutative as well as associative.

Sum = (−72) + 237 + 84 + 72 + (−184) + (−37)

Grouping suitable integers:

= [(−72) + 72] + [237 + 84] + [(−184) + (−37)]

= 0 + 321 + (−221)

= 321 − 221

= 100

∴ The required sum = 100.

Let the two integers be −5 and −4.

Both −5 and −4 are smaller than −3.

Now, their difference:

(−5) − (−4) = −5 + 4 = -1

Since -1 > −3,

∴ The two integers are −5 and −4.

(i) 7 × (-35)

= −(7 × 35)

= −245 $\hspace{1cm}$ [∵ Positive × Negative = Negative]

(ii) (−13) × (−15)

= +(13 × 15)

= 195 $\hspace{1cm}$ [∵ Negative × Negative = Positive]

(iii) (−12) × (−11) × (−10)

= [(−12) × (−11)] × (−10)

= 132 × (−10)

= −1320 $\hspace{1cm}$ [∵ Positive × Negative = Negative]

(iv) (−13) × 0 × (−24)

= 0 $\hspace{1cm}$ [∵ Anything multiplied with 0 becomes 0]

(v) (−1) × (−2) × (−3) × 4

= [(−1) × (−2)] × [(−3) × 4]

= 2 × (−12)

= −24 $\hspace{1cm}$ [∵ Positive × Negative = Negative]

(vi) (−3) × (−6) × (−2) × (−1)

= [(−3) × (−6)] × [(−2) × (−1)]

= 18 × 2 $\hspace{1cm}$ [∵ Negative × Negative = Positive]

= 36

(i) 37 × [6 + (−3)] = 37 × 6 + 37 × (−3)

LHS = 37 × [6 + (−3)]
= 37 × 3
= 111

RHS = 37 × 6 + 37 × (−3)
= 222 + (−111)
= 222 − 111
= 111

Since LHS = RHS,

∴ 37 × [6 + (−3)] = 37 × 6 + 37 × (−3) $\hspace{1cm}$ [Distributive Property]

(ii) (−21) × [(−6) + (−4)] = (−21) × (−6) + (−21) × (−4)

LHS = (−21) × [(−6) + (−4)]
= (−21) × (−10)
= 210

RHS = (−21) × (−6) + (−21) × (−4)
= 126 + 84
= 210

Since LHS = RHS,

∴ (−21) × [(−6) + (−4)] = (−21) × (−6) + (−21) × (−4) $\hspace{1cm}$ [Distributive Property]

(i) 8 × 53 × (−125)

= 53 × [8 × (−125)] $\hspace{1cm}$ [Using commutative and associative property]

= 53 × (−1000)

= −53000

(ii) (−8) × (−2) × 3 × (−5)

= [(−8) × (3)] × [(-2) × (−5)] $\hspace{1cm}$ [Using associative property]

= -24 × (10)

= −240

(iii) (−6) × 2 × (−8) × 5

= [(−6) × (−8)] × [2 × 5] $\hspace{1cm}$ [Using commutative and associative property]

= 48 × 10

= 480

(iv) 15 × (−25) × (−4) × (−10)

= 15 × [(−25) × (−4)] × (−10) $\hspace{1cm}$ [Using associative property]

= 15 × 100 × (−10)

= 1500 × (−10)

= −15000

(v) 26 × (−48) + (−48) × (−36)

= (−48) × 26 + (−48) × (−36) $\hspace{1cm}$ [Using commutative property]

= (−48) × [26 + (−36)] $\hspace{1cm}$ [Using distributive property]

= (−48) × (−10)

= 480

(vi) 724 × (−56) + (−724) × 44

= 724 × (−56) + 724 × (−44) $\hspace{1cm}$ [∵ (−724) × 44 = 724 × (−44)]

= 724 × [(−56) + (−44)] $\hspace{1cm}$ [Using distributive property]

= 724 × (−100)

= −72400

(vii) (−47) × 102

= (−47) × (100 + 2)

= (−47) × 100 + (−47) × 2 $\hspace{1cm}$ [Using distributive property]

= −4700 + (−94)

= −4700 − 94

= −4794

(viii) (−39) × (−97)

= (−39) × [(−100) + 3]

= (−39) × (−100) + (−39) × 3 $\hspace{1cm}$ [Using distributive property]

= 3900 + (−117)

= 3900 − 117

= 3783

(i) (−4) × (−11) = 44

∵ 44 ÷ (-4) = -11

(ii) 7 × (−6) = −42

∵ -42 ÷ 7 = -6

(iii) (−11) × (−13) = 143

∵ 143 ÷ (-13) = -11

(iv) (−5) × 0 = 0

∵ Anything multiplied with 0 becomes 0.

Initial room temperature = 32°C

Rate of decrease in temperature = 5°C per hour

Total decrease in temperature in 8 hours = 8 × 5°C = 40°C

Since the temperature is decreasing, change in temperature = −40°C

Room temperature after 8 hours = 32°C + (−40°C)

= 32° C − 40° C

= −8° C

∴ The room temperature 8 hours after the freezing process begins = −8° C

Marks given for each correct answer = 5

Marks deducted for each incorrect answer = −2

(i) Rohit's score:

Number of correct answers = 4

∴ Marks for correct answers: 4 x 5 = 20

Number of incorrect answers = 6

∴ Marks for incorrect answers: 6 x (-2) = -12

Rohit's score = Marks for correct answers + Marks for incorrect answers

= 20 + (−12)

= 20 − 12

= 8

∴ Rohit's score = 8

(ii) Seema's score:

Number of correct answers = 5

∴ Marks for correct answers: 5 x 5 = 25

Number of incorrect answers = 5

∴ Marks for incorrect answers: 5 x (-2) = -10

Seema's score = Marks for correct answers + Marks for incorrect answers

= 25 + (−10)

= 25 − 10

= 15

∴ Seema's score = 15

(iii) Ritu's score:

Ritu attempted 7 questions out of which 2 were correct.

∴ Marks for correct answers: 2 x 5 = 10

Number of incorrect answers = 7 − 2 = 5

∴ Marks for incorrect answers: 5 x (-2) = -10

Ritu's score = Marks for correct answers + Marks for incorrect answers

= 10 + (−10)

= 10 − 10

= 0

∴ Ritu's score = 0

(i) (−36) ÷ (−9)

= $\frac{-36}{-9}$

= 4

(ii) 150 ÷ (−25)

= $\frac{150}{-25}$

= −6

(iii) (−270) ÷ 27

= $\frac{-270}{27}$

= −10

(iv) (−59) ÷ 59

= $\frac{-59}{59}$

= −1

(v) 0 ÷ (−17)

= $\frac{0}{-17}$

= 0 $\hspace{1cm}$ [∵ 0 divided by any non-zero integer is 0]

(vi) (−784) ÷ (−56)

= $\frac{-784}{-56}$

= 14

(i) 13 ÷ [(−2) + 1]

= 13 ÷ (−1)

= −13

(ii) (−47) ÷ [(−45) + (−2)]

= (−47) ÷ (−47)

= 1

(iii) [(−6) + 5] ÷ [(−2) + 1]

= (−1) ÷ (−1)

= 1

(iv) [(−48) ÷ (−6)] ÷ (−2)

= 8 ÷ (−2)

= −4

Given: a = −225, b = 15 and c = −3

LHS = (a ÷ b) ÷ c

= [(−225) ÷ 15] ÷ (−3)

= (−15) ÷ (−3)

= 5

RHS = a ÷ (b ÷ c)

= (−225) ÷ [15 ÷ (−3)]

= (−225) ÷ (−5)

= 45

Since 5 ≠ 45,

∴ (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) for a = −225, b = 15 and c = −3

This shows that division is not associative for integers.

(i) Given: a = −10, b = 1 and c = 1

LHS = a ÷ (b + c)

= (−10) ÷ (1 + 1)

= (−10) ÷ 2

= −5

RHS = (a ÷ b) + (a ÷ c)

= [(−10) ÷ 1] + [(−10) ÷ 1]

= (−10) + (−10)

= −20

Since −5 ≠ −20,

∴ a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for a = −10, b = 1 and c = 1

(ii) Given: a = 12, b = 1 and c = −2

LHS = a ÷ (b + c)

= 12 ÷ [1 + (−2)]

= 12 ÷ (−1)

= −12

RHS = (a ÷ b) + (a ÷ c)

= (12 ÷ 1) + [12 ÷ (−2)]

= 12 + (−6)

= 12 − 6

= 6

Since −12 ≠ 6,

∴ a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for a = 12, b = 1 and c = −2

(i) 239 ÷ 239 = 1

(ii) (−85) ÷ 85 = −1

(iii) (−213) ÷ (−213) = 1

(iv) (−43) ÷ (−1) = 43

(v) (−84) ÷ (−21) = 4

(vi) (−66) ÷ 22 = −3

Explanation

(i) Any non-zero number divided by itself equals 1.

(ii) To result in a negative quotient, a negative number must be divided by a positive number.

(iii) A negative number divided by the same negative number equals 1.

(iv) Dividing a negative number by -1 changes its sign to positive.

(v) Multiply the quotient and divisor: 4 x (-21) = -84.

(vi) Divide the dividend by the quotient: (-66) ÷ (-3) = 22

Marks given for each correct answer = 3

Marks given for each incorrect answer = −2

(i) Sachin's total score = 24

Sachin's correct answers = 14

Marks for correct answers = 14 x 3 = 42

Marks obtained for incorrect answers = Total score − Marks for correct answers

= 24 − 42

= −18

Number of incorrect answers = Marks obtained for incorrect answers ÷ Marks given for each incorrect answer

= (−18) ÷ (−2)

= 18 ÷ 2

= 9

∴ Sachin attempted 9 questions incorrectly.

(ii) Nalini's total score = −7

Marks scored for 9 correct answers = 9 × 3 = 27

Marks obtained for incorrect answers = Total score − Marks for correct answers

= −7 − 27

= −34

Number of incorrect answers = Marks obtained for incorrect answers ÷ Marks given for each incorrect answer

= (−34) ÷ (−2)

= 34 ÷ 2

= 17

∴ Nalini attempted 17 questions incorrectly.

Starting position of elevator = 10 m above ground level = +10 m

Final position of elevator = 350 m below ground level = −350 m

Total distance to be descended = 10 − (−350)

= 10 + 350

= 360 m

Rate of descent = 6 m/min

Time taken = $\frac{\text{Total distance}}{\text{Rate of descent}}$

= $\frac{360 \text{ m}}{6 \text{ m/min}}$

= 60 min

= 1 hour

∴ The elevator will take 1 hour (or 60 minutes) to reach the shaft.

Given expression:

7 − 8 ÷ (−2) + 3 × (−4)

= 7 − (−4) + (−12) $\hspace{2cm}$ [Performing ÷ and ×]

= 7 + 4 − 12 $\hspace{3cm}$ [Removing brackets]

= 11 − 12

= −1

Hence, 7 - 8 ÷ (-2) + 3 × (-4) = -1.

Given expression:

9 − {7 − 24 ÷ (8 + 6 × 2 − 16)}

= 9 − {7 − 24 ÷ (8 + 12 − 16)} $\hspace{2cm}$ [Simplifying × inside ( )]

= 9 − {7 − 24 ÷ 4} $\hspace{3.7cm}$ [Simplifying ( )]

= 9 − {7 − 6} $\hspace{4.8cm}$ [Simplifying ÷]

= 9 − 1 $\hspace{6cm}$ [Simplifying { }]

= 8

Hence, 9 - {7 - 24 ÷ (8 + 6 × 2 - 16)} = 8.

Given expression:

−11 − [−6 − {3 − 5(8 ÷ 4 − 1)}]

= −11 − [−6 − {3 − 5(2 − 1)}] $\hspace{2cm}$ [Simplifying ÷ inside ( )]

= −11 − [−6 − {3 − 5(1)}] $\hspace{2.7cm}$ [Simplifying ( )]

= −11 − [−6 − {3 − 5}] $\hspace{3.5cm}$ [Simplifying ×]

= −11 − [−6 − (−2)] $\hspace{3.7cm}$ [Simplifying { }]

= −11 − [−6 + 2] $\hspace{4.5cm}$ [Removing ( )]

= −11 − [−4] $\hspace{5.4cm}$ [Simplifying [ ]]

= −11 + 4 $\hspace{6cm}$ [Removing [ ]]

= −7

Hence, -11 - [-6 - {3 - 5(8 ÷ 4 - 1)}] = −7

Given expression:

(−3) × (−12) ÷ (−4) + 3 × 6

= 36 ÷ (−4) + 3 × 6 $\hspace{2cm}$ [Simplifying × from left to right]

= −9 + 18 $\hspace{3.5cm}$ [Simplifying ÷ and ×]

= 9

Hence, (-3) × (-12) ÷ (-4) + 3 × 6 = 9

Given expression:

14 ÷ (3 of 2 − 3 + 4) − 9(5 − 3)

= 14 ÷ (6 − 3 + 4) − 9(2) $\hspace{2cm}$ [Simplifying 'of' inside ( ) and simplifying ( )]

= 14 ÷ 7 − 9 × 2 $\hspace{3.5cm}$ [Simplifying ( )]

= 2 − 18 $\hspace{5cm}$ [Simplifying ÷ and ×]

= −16

Hence, 14 ÷ (3 of 2 - 3 + 4) - 9(5 - 3) = −16

(i) -1 is the greatest negative integer.

(ii) ((-10) + 3) + (-12) = (-10) + (3 + -12)

(iii) The product of three negative integers and the product of two positive integers is a negative integer.

(iv) The division of any integer by zero is not defined

(v) The integer whose product with (-1) is 22 is -22

(vi) (-15) × -8 = 120

(vii) 72 ÷ (-6) = -12

(viii) (-10) × ((-15) + 33) = -10 × (-15) + (-10) × 33

(ix) 0 ÷ (-25) = 0

(x) ((-8) × (-13)) × 27 = (-8) × ((-13) × 27)

(xi) 13 × (-6) = - (13 × 6) = -78

(xii) (-a) + b = b + additive inverse of a

(xiii) When -25 is divided by -5 the quotient is 5

(xiv) There are infinitely many pairs of integers satisfying a + b = -1

(xv) The value of the expression ((-60) ÷ 12) ÷ (-5) is 1

(i) True
Reason — The absolute value of an integer is always non-negative, i.e., either positive or zero.

(ii) False
Reason — For example, (−2) − (−5) = −2 + 5 = 3, which is a positive integer.

(iii) False
Reason — Sum of two integers is always an integer (closure property of addition).

(iv) True
Reason — For any integer a, a ÷ (−1) = −a, which is the additive inverse of a.

(v) False
Reason — 0 is the additive identity of integers, not 1. (1 is the multiplicative identity.)

(vi) False
Reason — (−17) × 6 = −102, which is a negative integer, not a whole number.

(vii) False
Reason — (−5) × (−8) × 0 = 40 × 0 = 0, which is neither positive nor negative.

(viii) True
Reason — (−237) × 0 = 0 and 0 × (−89) = 0. Both are equal to 0.

(ix) True
Reason — The difference of any two integers is always an integer.

(x) True
Reason — For integers a and b, a − b ≠ b − a in general. For example, 5 − 3 = 2 but 3 − 5 = −2.

(xi) False
Reason — Subtraction is not associative for integers. For example, (5 − 3) − 2 = 0 but 5 − (3 − 2) = 4.

(xii) False
Reason — Division of two integers may not always yield an integer. For example, 1 ÷ 2 is not an integer.

(xiii) True
Reason — Division is not commutative for integers. For example, 12 ÷ (−3) = −4 but (−3) ÷ 12 is not an integer.

(xiv) True
Reason — From (−8) × (−12) = 96, we get 96 ÷ (−12) = −8. They are corresponding multiplication and division facts.

(xv) False
Reason — LHS = [(−32) ÷ 8] ÷ 2 = (−4) ÷ 2 = −2
RHS = (−32) ÷ (8 ÷ 2) = (−32) ÷ 4 = −8
Since −2 ≠ −8, the statement is false. (Division is not associative.)

(xvi) False
Reason — When a = 0, 0 ÷ 0 is not defined. The statement holds only for non-zero integers.

(xvii) False
Reason — 0 × (−10) = 0, and successor of 0 is 1.
But 1 × (−10) = −10.
Since 1 ≠ −10, the statement is false.

(i) False
Reason — The sign of the sum depends on the absolute values of the integers. For example, 3 + (−5) = −2, which is negative, not positive.

(ii) False
Reason — For example, take a = 5 and b = −3.
Sum = 5 + (−3) = 2
Difference = 5 − (−3) = 5 + 3 = 8
Here the sum (2) is less than the difference (8).

(iii) False
Reason — For example, take a = −5 and b = −10.
Then −a = 5 and b = −10.
Here −a > b (i.e., 5 > −10), so the inequality −a < b does not hold.

(iv) False
Reason — For example, take a = 1 and b = 2.
Sum = 1 + 2 = 3
Product = 1 × 2 = 2
Here the product (2) is less than the sum (3).

The integer which is the smallest will lie on the extreme left of the number line.

Among 10, −7, 5, 3, −4 and 0, the smallest integer is −7.

∴ The integer that lies on the extreme left is −7.

Hence, option 3 is the correct option.

(−3) × 3 = −9

On the number line, −9 lies on the right of −10.

(−9 lies to the left of −6, 0 and 9.)

Hence, option 1 is the correct option.

5 ÷ (−1) = −5

Now we check each option:

1. 0 and −10: −5 lies between 0 and −10.
2. 0 and 10: −5 does not lie between 0 and 10 (since −5 is negative)
3. −3 and −10: −5 lies between −3 and −10.
4. −7 and 7: −5 lies between −7 and 7.

So, −5 does not lie between 0 and 10.

Hence, option 2 is the correct option.

Observing the pattern:

−62 + 25 = −37
−37 + 25 = −12
−12 + 25 = 13

So, the next number is 13.

Hence, option 3 is the correct option.

Multiplication of integers satisfies all three properties:

• Closure: a × b is an integer
• Commutativity: a × b = b × a
• Associativity: (a × b) × c = a × (b × c)

Hence, option 4 is the correct option.

Integers between −20 and −10 are:

−19, −18, −17, −16, −15, −14, −13, −12, −11

So, there are 9 integers between −20 and −10.

Hence, option 2 is the correct option.

Let the other integer be x.

Given: 2 + x = −10

x = −10 − 2

x = −12

∴ The other integer is −12.

Hence, option 4 is the correct option.

Let the required integer be x.

Given:

−5 − x = −12

x = −5 − (−12)

x = −5 + 12

x = 7

∴ The required integer is 7.

Hence, option 1 is the correct option.

The additive inverse of a is −a.

1. −(−a) = a (not the additive inverse of a)
2. −a (this is the additive inverse of a)
3. a ÷ (−1) = −a (this is the additive inverse of a)
4. a × (−1) = −a (this is the additive inverse of a)

So, −(−a) is not the additive inverse of a.

Hence, option 1 is the correct option.

0 ÷ (−10) = $\frac{0}{-10}$ = 0

[∵ 0 divided by any non-zero integer is 0]

Hence, option 1 is the correct option.

(−33) × 102 + (−33) × (−2)

= (−33) × [102 + (−2)] $\hspace{1cm}$ [Using distributive property]

= (-33) x [102 - 2]

= (−33) × 100

= −3300

Hence, option 2 is the correct option.

101 × (−1) + 0 ÷ (−1)

= −101 + 0

= −101

Hence, option 1 is the correct option.

(−3) × 5 = −15

1. 3 × (−5) = −15 (equal)

2. −(3 × 5) = −15 (equal)

3. (−3) × (−5) = 15 (not equal)

4. 5 × (−3) = −15 (equal)

So, (−3) × 5 is not equal to (−3) × (−5).

Hence, option 3 is the correct option.

Integers are closed under addition, subtraction and multiplication, but not under division.

For example, 1 ÷ 2 = $\frac{1}{2}$, which is not an integer.

So, a ÷ b may not be an integer.

Hence, option 4 is the correct option.

For a non-zero integer a:

1. a ÷ 0 is not defined (division by zero is undefined)

2. 0 ÷ a = 0 (defined)

3. a ÷ 1 = a (defined)

4. a ÷ a = 1 (defined)

Hence, option 1 is the correct option.

Let us first consider Statement I.

The multiplicative identity for integers is 1 (since a × 1 = a for any integer a), not 0.

So, Statement I is false.

Now let us consider Statement II.

The number 0 is indeed an integer (it is neither positive nor negative).

So, Statement II is true.

Thus, Statement I is false but Statement II is true.

Hence, option 2 is the correct option.

Let us first consider Statement I.

The absolute value of any non-zero integer is a positive integer. So, the product of two positive integers is also a positive integer.

So, Statement I is true.

Now let us consider Statement II.

By the definition of absolute value, |−a| = |a| for all integers a.

For example, |−5| = 5 = |5|.

So, Statement II is true.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Let us first consider Statement I.

Multiplying two integers with different signs results in a negative integer, not a positive integer.

For example, 3 × (−5) = −15, which is negative.

So, Statement I is false.

Now let us consider Statement II.

The associative property of multiplication is a × (b × c) = (a × b) × c, not (a × b) × (a × c).

So, Statement II is false.

Thus, both Statement I and Statement II are false.

Hence, option 4 is the correct option.

Let us first consider Statement I.

For any non-zero integer a:

• 0 ÷ a = 0 (since a × 0 = 0)
• a ÷ a = 1 (since a × 1 = a)

So, Statement I is true.

Now let us consider Statement II.

Integers are closed under addition, subtraction and multiplication. However, integers are not closed under division.

For example, 1 ÷ 2 = $\frac{1}{2}$, which is not an integer.

So, Statement II is false.

Thus, Statement I is true but Statement II is false.

Hence, option 1 is the correct option.

(i) (−7) × (−9) × (−11)

= [(−7) × (−9)] × (−11)

= 63 × (−11)

= −693

(ii) (−5) × 7 × (−6) × (−8)

= [(−5) × 7] × [(−6) × (−8)]

= (−35) × 48

= −1680

(iii) (−1024) ÷ 32

= $\frac{-1024}{32}$

= −(1024 ÷ 32)

= −32

(iv) (−216) ÷ (−12)

= $\frac{-216}{-12}$

= +(216 ÷ 12)

= 18

Product of 39 negative integers:

Since 39 is an odd number, the product of 39 negative integers will be a negative integer.

Product of 98 positive integers:

The product of 98 positive integers will be a positive integer.

Now, product = (negative integer) × (positive integer) = negative integer

∴ The sign of the resulting product will be negative.

(i) LHS = (−15) + 38 = 23

RHS = 27 + (−50) = −23

Since 23 > −23,

∴ (−15) + 38 $\boxed{\gt}$ 27 + (−50)

(ii) LHS = (−13) × 0 × (−5) = 0

RHS = (−7) × (−6) × 14 = 42 × 14 = 588

Since 0 < 588,

∴ (−13) × 0 × (−5) $\boxed{\lt}$ (−7) × (−6) × 14

(iii) LHS = (−18) ÷ (−3) = 6

RHS = (−10) + (−15) + 31 = −25 + 31 = 6

Since 6 = 6,

∴ (−18) ÷ (−3) $\boxed{=}$ (−10) + (−15) + 31

(iv) LHS = (−5) × (−7) × (−10) = 35 × (−10) = −350

RHS = (−1400) ÷ (−4) = 350

Since −350 < 350,

∴ (−5) × (−7) × (−10) $\boxed{\lt}$ (−1400) ÷ (−4)

LHS = ((−45) ÷ (−15)) ÷ (−3)

= 3 ÷ (−3)

= −1

RHS = (−45) ÷ ((−15) ÷ (−3))

= (−45) ÷ 5

= −9

Since −1 ≠ −9,

∴ ((−45) ÷ (−15)) ÷ (−3) ≠ (−45) ÷ ((−15) ÷ (−3))

This shows that division is not associative for integers.

(i) Given expression:

(−7) + (−6) ÷ 2 − {(−5) × (−4) − (3 − 5)}

= (−7) + (−3) − {20 − (−2)} $\hspace{1cm}$ [Simplifying ÷, ×, and ( )]

= (−7) + (−3) − {20 + 2} $\hspace{2cm}$ [Removing ( )]

= (−7) + (−3) − 22 $\hspace{3.4cm}$ [Simplifying { }]

= −7 − 3 − 22

= −32

The answer is −32

(ii) Given expression:

11 - [7 - {5 - 3 (9 - $\overline{3 - 6}$)}]

= 11 - [7 - {5 - 3(9 - (-3))}] $\quad$[Removing 'bar']

= 11 - [7 - {5 - 3(9 + 3)}] $\quad$[Removing ()]

= 11 - [7 - {5 - 3(12)}] $\quad$[Simplifying ()]

= 11 - [7 - {5 - 36}] $\quad$[Simplifying x]

= 11 - [7 - (-31)] $\quad$[Simplifying {}]

= 11 - [7 + 31] $\quad$[Removing ()]

= 11 - 38 $\quad$[Removing []]

= -27

The answer is −27

Let us first evaluate each expression:

−5 + 8 = 3

3 × (−4) = −12

−15 ÷ 5 = −3

|−5 − 7| = |−12| = 12

−7 − 3 × (−2) = −7 − (−6) = −7 + 6 = −1

−7 − 4 ÷ (−2) = −7 − (−2) = −7 + 2 = −5

So, the numbers are: 3, −12, −3, 12, −1, −5

Arranging them in descending order:

12 > 3 > −1 > −3 > −5 > −12

∴ |−5 − 7| > −5 + 8 > −7 − 3 × (−2) > −15 ÷ 5 > −7 − 4 ÷ (−2) > 3 × (−4)

|−5 − 7|, −5 + 8, −7 − 3 × (−2), −15 ÷ 5, −7 − 4 ÷ (−2), 3 × (−4)

We need a pair of integers whose product is −12 and there are seven integers between them.

If there are seven integers between two integers a and b, then their absolute difference is 8.

The factor pairs of −12 are: (1, −12), (−1, 12), (2, −6), (−2, 6), (3, −4), (−3, 4)

Checking the absolute difference of each pair:

• |1 − (−12)| = 13
• |(−1) − 12| = 13
• |2 − (−6)| = 8
• |(−2) − 6| = 8
• |3 − (−4)| = 7
• |(−3) − 4| = 7

Here only two factors have the absolute difference 8.

So, the required pair is 2 and −6 or −2 and 6.

Verification: 2 × (−6) = −12

The seven integers lying between 2 and −6 are: −5, −4, −3, −2, −1, 0, 1.

Profit per pen = ₹2 = 200 paise

Loss per pencil = 50 paise (denoted by −50 paise)

Loss per eraser = 15 paise (denoted by −15 paise)

Number of pens sold = 10

Profit from pens = Number of pens sold x Profit per pen

= 10 × 200

= 2000 paise

Total profit on that day = ₹10 = 1000 paise

Given,

Number of pencils and erasers sold are in the ratio 7 : 10.

Let the number of pencils sold = 7x

Let the number of erasers sold = 10x

Loss from pencils = 7x × 50 = 350x paise

Loss from erasers = 10x × 15 = 150x paise

Total loss = 350x + 150x = 500x paise

Net profit = Profit from pens − Total loss

Substituting the values:

1000 = 2000 − 500x

500x = 2000 − 1000

500x = 1000

x = $\frac{1000}{500}$

x = 2

∴ Number of pencils sold = 7x = 7 × 2 = 14

∴ Number of erasers sold = 10x = 10 × 2 = 20

∴ The shopkeeper sold 14 pencils and 20 erasers on that day.