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**Question 9: Calculate:1 âˆ’ 2 + 3 âˆ’ 4 + 5 âˆ’ 6 +......+ 15 âˆ’ 16**

**ANSWER:**1 âˆ’ 2 + 3 âˆ’ 4 + 5 âˆ’ 6 + 7 âˆ’ 8 + 9 âˆ’ 10 + 11 âˆ’ 12 + 13 âˆ’ 14 + 15 âˆ’ 16

= (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15) âˆ’ (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16)

= 64 âˆ’ 72

= âˆ’8

**Question 10: Calculate the sum:5 + (âˆ’5) + 5 (âˆ’5) + .....(i) if the number of terms is 10.(ii) if the number of terms is 11.**

**ANSWER:**

(i) If the number of terms is 10, then 5 + (âˆ’5) + 5 + (âˆ’5) + 5 + (âˆ’5) + 5 + (âˆ’5) + 5 + (âˆ’5) = 0.

(ii) If the number of terms is 11, then 5 + (âˆ’5) + 5 + (âˆ’5) + 5 + (âˆ’5) + 5 + (âˆ’5) + 5 + (âˆ’5) + 5 = 5.

**Question 11: Replace * by '<' or '>' in each of the following to make the statement true:(i) (âˆ’6) + (âˆ’9) * (âˆ’6) âˆ’ (âˆ’9)(ii) (âˆ’12) âˆ’ (âˆ’12) * (âˆ’12) + (âˆ’12)(iii) (âˆ’20) âˆ’ (âˆ’20) * 20 âˆ’ (65)(iv) 28 âˆ’ (âˆ’10) * (âˆ’16) âˆ’ (âˆ’76)**

**ANSWER:**

(i) (âˆ’6) + (âˆ’9) = âˆ’15 < (âˆ’6) âˆ’ (âˆ’9) = âˆ’6 + 9 = 3

(ii) (âˆ’12) âˆ’ (âˆ’12) = âˆ’12 + 12 = 0 > (âˆ’12) + (âˆ’12) = âˆ’12 âˆ’ 12 = âˆ’24

(iii) (âˆ’20) âˆ’ (âˆ’20) = âˆ’20 + 20 = 0 > 20 âˆ’ 65 = âˆ’45

(iv) 28 âˆ’ (âˆ’10) = 28 + 10 = 38 < âˆ’16 âˆ’ (âˆ’76) = âˆ’16 + 76 = 60

**Question 12: If âˆ† is an operation on integers such that a âˆ† b = âˆ’a + b âˆ’ (âˆ’2) for all integers a, b. Find the value of(i) 4 âˆ† 3(ii) (âˆ’2) âˆ† (âˆ’3)(iii) 6 âˆ† (âˆ’5)(iv) (âˆ’5) âˆ† 6**

**ANSWER:**

(i) âˆ’4 + 3 âˆ’ (âˆ’2)

= âˆ’4 + (3 + 2)

= âˆ’4 + 5

= 1

(ii) âˆ’(âˆ’2) + (âˆ’3) âˆ’ (âˆ’2)

= (2 âˆ’ 3) + 2

= âˆ’1 + 2

= 1

(iii) âˆ’6 + (âˆ’5) âˆ’ (âˆ’2)

= âˆ’6 + (âˆ’5 + 2)

= âˆ’6 âˆ’ 3

= âˆ’9

(iv) âˆ’(âˆ’5) + 6 âˆ’ (âˆ’2)

= 5 + (6 + 2)

= 5 + 8

= 13

**Question 13: If a and b are two integers such that a is predecessor of b. Find the value of a âˆ’ b.**

**ANSWER: **a and b are integers such that a is the predecessor of b, that is, a = b âˆ’ 1.

âˆ´ (a âˆ’ b)

= (b âˆ’ 1) âˆ’ b

= b âˆ’ 1 âˆ’ b

= âˆ’1

**Question 14: If a and b are two integers such that a is the successor of b. Find the value of a âˆ’ b.**

**ANSWER: **a and b are two integers such that a is the successor of b, that is, a = b + 1.

âˆ´ a âˆ’ b

= b + 1 âˆ’ b

= 1

**Question 15:**

**Which of the following statements are true:(i) âˆ’13 > âˆ’8 âˆ’ (âˆ’2)(ii) âˆ’4 + (âˆ’2) < 2.(iii) The negative of a negative integer is positive.(iv) If a and b are two integers such that a > b, then a âˆ’ b is always positive integer,(v) The difference of two integers is an integer.(vi) Additive inverse of a negative integer is negative.(vii) Additive inverse of a positive integer is negative.(viii) Additive inverse of a negative is positive.**

**ANSWER:**

(i) False; It should be âˆ’13 < âˆ’8 + 2 = âˆ’6.

(ii) True; âˆ’4 âˆ’ 2 = âˆ’6 < 2

(iii) True; For example: âˆ’(âˆ’2) = 2

(iv) True; a > b

(v) True; For example: 3 âˆ’ 2 = 1, which is a integer.

(vi) False; For example: âˆ’2 + 2 = 0. Here, 2 is the additive inverse of âˆ’2; it is positive.

(vii) True

(viii) True

**Question 16: Fill in the blanks:(i) âˆ’7 + ...... = 0(ii) 29 + ...... = 0(iii) 132 + (âˆ’132) = ......(iv) âˆ’14 + ....= 22(v) âˆ’1256 + ..... = âˆ’742(vi) ..... âˆ’1234 = âˆ’4539.**

**ANSWER:**

(i) âˆ’7 + 7 = 0 (âˆ’a and a are the negative and additive inverses of each other.)

(ii) 29 + (âˆ’29) = 0 (âˆ’a and a are the negative and additive inverses of each other.)

(iii) 132 + (âˆ’132) = 0 (âˆ’a and a are the negative and additive inverses of each other.)

(iv) âˆ’14 + 36 = 22

(v) âˆ’1256 + 514 = âˆ’742

(vi) âˆ’3305 âˆ’ 1234 = âˆ’4539

**Question 1: Which of the following statement is true?(a) âˆ’ 7 > âˆ’ 5 (b) âˆ’ 7 < âˆ’ 5 (c) (âˆ’ 7) + (âˆ’ 5) > 0 (d) (âˆ’ 7) âˆ’ (âˆ’ 5) > 0**

**ANSWER:**

(a)

On the number line, âˆ’ 7 is to the left of âˆ’ 5, so âˆ’ 7 < âˆ’ 5.

(c)

Here, (âˆ’ 7) + (âˆ’ 5) = âˆ’ ( 7 + 5) = âˆ’ 12.

On the number line, âˆ’ 12 is to the left of 0, so (âˆ’ 7) + (âˆ’ 5) < 0.

(d)

Here, (âˆ’ 7) âˆ’ (âˆ’ 5) = (âˆ’ 7) + (additive inverse of âˆ’ 5) = (âˆ’ 7) + (5) = âˆ’ (7 âˆ’ 5) = âˆ’ 2

On the number line, âˆ’ 2 is to the left of 0, so (âˆ’ 7) âˆ’ (âˆ’ 5) < 0.

Hence, the correct option is (b).

**Question 2: 5 less than âˆ’ 2 is(a) 3 (b) âˆ’ 3 (c) âˆ’ 7 (d) 7**

**ANSWER:**

Here, 5 less than âˆ’ 2 = (âˆ’ 2) âˆ’ (5) = âˆ’ 2 âˆ’ 5 = âˆ’ 7.

Hence, the correct option is (c).

**Question 3: 6 more than âˆ’ 7 is(a) 1 (b) âˆ’ 1 (c) 13 (d) âˆ’ 13**

**ANSWER:**

6 more than âˆ’ 7 = (âˆ’ 7) + 6

= âˆ’ ( 7 âˆ’ 6)

= âˆ’ 1

Hence, the correct option is (b).