CBSE Class 7 Mathematics Operations with Integers MCQs Set 11

Answer: (b) a negative integer

Explanation:
1. When the two factors have different signs, the product is negative, for example 4 × (−2) = −8.
2. The size of the product is the product of the two sizes, so it is never zero here.
3. So the product is always a negative integer.

Teacher's Note:
1. Different signs give a negative product.
2. Zero appears only when one factor is 0.

Answer: (c) 4 and 8

Explanation:
1. Try pairs that add up to 12 and check the difference.
2. 4 + 8 = 12 and 4 − 8 = −4, so the pair is 4 and 8.
3. 8 and 4 has the right sum but the difference 8 − 4 = +4, so the order matters.

Teacher's Note:
1. A negative difference means the first number is the smaller one.
2. Always check both the sum and the difference.

Answer: (d) 18

Explanation:
1. −(−18) means the additive inverse of −18.
2. The number that adds to −18 to give 0 is 18, since (−18) + 18 = 0.
3. So −(−18) = 18.

Teacher's Note:
1. The additive inverse of a negative number is positive.
2. Two minus signs together give a plus.

Answer: (a) −a

Explanation:
1. Multiplying by −1 keeps the size of the number the same but changes its sign.
2. So (−1) × a is the additive inverse of a, that is −a.
3. For example, (−1) × 5 = −5 and (−1) × (−5) = 5.

Teacher's Note:
1. −a is not always negative; if a is negative, −a is positive.
2. 1 × a = a, but (−1) × a = −a.

Answer: (c) positive quantities

Explanation:
1. Brahmagupta (628 CE) used dhana (fortune) for positive quantities.
2. He used rina (debt) for negative quantities.
3. So dhana stands for positive quantities.

Teacher's Note:
1. Fortune is something you have, so it is positive.
2. Debt is something you owe, so it is negative.

Answer: (d) −6

Explanation:
1. Output = (−8) + (−5) − (−7).
2. (−8) + (−5) = −13, and subtracting −7 is the same as adding 7.
3. −13 + 7 = −6.

Teacher's Note:
1. Subtracting a negative number means adding its inverse.
2. −20 comes from wrongly subtracting 7.

Answer: (a) −4

Explanation:
1. The dividend is negative and the divisor is positive, so the quotient is negative.
2. 36 ÷ 9 = 4, so (−36) ÷ 9 = −4.
3. Check: 9 × (−4) = −36.

Teacher's Note:
1. Different signs give a negative quotient.
2. Check every division by multiplying back.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 5 × 6 = 30 and both factors are negative, so the product is +30. A is true.
2. R is the sign rule for two negative factors, and it is true.
3. R is exactly the reason why the answer in A is positive, so R explains A.

Teacher's Note:
1. First test A and R separately.
2. Then ask whether R gives the reason for A.

Answer: (c) A is true, but R is false.

Explanation:
1. Putting 2 red tokens into the bag 4 times gives 8 reds, that is −8. So A is true.
2. A positive multiplier tells us to put tokens into the bag, not to remove them. So R is false.
3. Hence A is true but R is false.

Teacher's Note:
1. Positive multiplier: put in. Negative multiplier: take out.
2. A true statement can be paired with a false reason.

Answer: (c) Both statements are true

Explanation:
1. Swapping the two factors does not change the product, so Statement I is true.
2. Any integer multiplied by 0 gives 0, so Statement II is true.
3. So both statements are true.

Teacher's Note:
1. Zero times any integer, positive or negative, is zero.
2. Commutative means the order can be swapped.

Answer: (b) (i)-(q), (ii)-(p), (iii)-(s), (iv)-(r)

Explanation:
1. (i) (−3) × 6 = −18, so (q). (ii) (−48) ÷ (−8) = 6, so (p).
2. (iii) 7 × (−2) = −14, so (s). (iv) (−5) × (−5) = 25, so (r).
3. The matching is (i)-(q), (ii)-(p), (iii)-(s), (iv)-(r).

Teacher's Note:
1. Find the sign first, then the size.
2. Same signs give positive; different signs give negative.

Answer: (a) −3

Explanation:
1. The 1st strike moves the coin 6 units to the right: +6.
2. The 2nd strike moves it 9 units to the left, from 6 back to −3: −9.
3. Final position = 6 + (−9) = −3.

Teacher's Note:
1. Right is positive and left is negative.
2. Add the two movements to get the final position.

Answer: (b) remove tokens from the bag

Explanation:
1. The multiplier tells us how many times we act on the bag.
2. A positive multiplier means putting tokens in; a negative multiplier means removing tokens from the bag.
3. If the bag is empty, we first put in zero pairs and then remove the tokens.

Teacher's Note:
1. Zero pairs make removal possible without changing the value.
2. The multiplicand tells which tokens are removed.

Answer: (d) distributive property

Explanation:
1. Here a multiplication is spread over a sum.
2. This is the distributive property of multiplication over addition.

Teacher's Note:
1. Distribute means share out to each term.
2. Commutative swaps; associative regroups.

Answer: (b) a ÷ b

Explanation:
1. The dividend −a and the divisor −b are both negative.
2. Same signs give a positive quotient, so (−a) ÷ (−b) = a ÷ b.
3. Example: (−12) ÷ (−3) = 4 = 12 ÷ 3.

Teacher's Note:
1. The sign rules for division are the same as for multiplication.
2. Test the rule with small numbers.

Answer: (a) −13 °C

Explanation:
1. A fall of 4 °C each hour is a change of −4 per hour.
2. Change in 5 hours = 5 × (−4) = −20.
3. New temperature = 7 + (−20) = −13 °C.

Teacher's Note:
1. A fall is a negative change.
2. Add the total change to the starting value.

Answer: (d) 65

Explanation:
1. Marks for correct answers = 18 × 4 = 72.
2. Wrong answers = 25 − 18 = 7, giving 7 × (−1) = −7 marks.
3. Score = 72 + (−7) = 65.

Teacher's Note:
1. Find the number of wrong answers first.
2. Negative marking reduces the total.

Answer: (d) A is false, but R is true.

Explanation:
1. For a = −5, (−1) × (−5) = 5, which is positive; and (−1) × 0 = 0. So A is false.
2. (−1) × a = −a always, which is the additive inverse of a. So R is true.
3. Hence A is false but R is true.

Teacher's Note:
1. One counterexample is enough to show a statement is false.
2. The inverse of a negative number is positive.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Both sides equal −100, so A is true.
2. Multiplication of integers is associative, so R is also true.
3. But A comes from swapping the two factors, which is the commutative property, so R does not explain A.

Teacher's Note:
1. Swapping is commutative; regrouping is associative.
2. A true reason may still be the wrong reason.

Answer: (c) A is true, but R is false.

Explanation:
1. 25 × (−4) = −100, so (−100) ÷ 25 = −4. A is true.
2. Every division can be turned into a multiplication question: (−100) ÷ 25 asks 'what times 25 gives −100?'. So R is false.
3. Hence A is true but R is false.

Teacher's Note:
1. Division is checked by multiplying back.
2. The word 'never' often makes a statement false.

Answer: (b) Only Statement II is true

Explanation:
1. The quotient of two negative integers is positive, e.g. (−48) ÷ (−8) = 6. Statement I is false.
2. The product of two negative integers is positive, e.g. (−4) × (−9) = 36. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. Products and quotients follow the same sign rules.
2. Two negatives give a positive in both.

Answer: (b) Only Statement I is true

Explanation:
1. (−1) × a is always the additive inverse −a, so Statement I is true.
2. 1 × a = a for every integer, for example 1 × (−7) = −7, so the word 'only' makes Statement II false.
3. So only Statement I is true.

Teacher's Note:
1. 1 leaves every integer unchanged.
2. Note that the options here are in a different order.

Answer: (c) (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q)

Explanation:
1. (i) (−5) × 15 = −75, so (r). (ii) (−7) × 4 × (−1) = 28, so (s).
2. (iii) 84 ÷ (−4) = −21, so (p). (iv) four negative factors: 2 × 1 × 5 × 3 = 30, positive, so (q).
3. The matching is (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q).

Teacher's Note:
1. Solve the bracket first in (i).
2. An even number of negative factors gives a positive product.

Answer: (a) 5 × (−3) = −15

Explanation:
1. Each jump is 3 units to the left, that is −3, and there are 5 jumps.
2. Starting at 0, the coin ends at −15, so the figure shows 5 × (−3) = −15.
3. The others are false: (−5) × 3 = −15, (−5) × (−3) = 15 and 3 × (−5) = −15.

Teacher's Note:
1. Number of jumps × size of one jump = final position.
2. Check the sign of every option.

Answer: (c) −46944

Explanation:
1. 326 × 144 is one group of 326 less than 326 × 145: 47270 − 326 = 46944.
2. One factor is negative, so the product is negative.
3. (−326) × 144 = −46944.

Teacher's Note:
1. Use the given product instead of multiplying again.
2. Fix the sign separately from the size.

Answer: (c) a ÷ b = b ÷ a

Explanation:
1. (a), (b) and (d) are the commutative, associative and distributive properties, true for all integers.
2. Division is not commutative: 8 ÷ (−4) = −2, but (−4) ÷ 8 is not even an integer.
3. So a ÷ b = b ÷ a is not always true.

Teacher's Note:
1. Order matters in division.
2. One example is enough to disprove a general rule.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. By the distributive property, (−13) × 7 + (−13) × 3 = (−13) × (7 + 3) = (−13) × 10 = −130. A is true.
2. R is true for all integers.
3. A is worked out directly by using R, so R is the correct explanation of A.

Teacher's Note:
1. Take out the common factor −13.
2. Check: −91 + (−39) = −130.

Answer: (d) Only Statement I is true

Explanation:
1. A negative product needs one positive and one negative factor, so Statement I is true.
2. Two negative integers also give a positive product, e.g. (−6) × (−3) = 18, so Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. A positive product means the signs are the same, not that both are positive.
2. Watch for 'must' in a statement.

Answer: (d) (i)-(s), (ii)-(q), (iii)-(p), (iv)-(r)

Explanation:
1. (i) (−4) × (−9) = 36, so (s). (ii) 9 × (−7) = −63, so (q).
2. (iii) (−48) ÷ (−6) = 8, so (p). (iv) (−132) ÷ 11 = −12, so (r).
3. The matching is (i)-(s), (ii)-(q), (iii)-(p), (iv)-(r).

Teacher's Note:
1. For a missing dividend, multiply the quotient by the divisor.
2. Check each blank by putting the number back.

Answer: (b) 35

Explanation:
1. Three equal strikes take the coin from 0 to −21, so each strike is (−21) ÷ 3 = −7.
2. A strike of the same size in the opposite direction is +7.
3. Five such strikes: 5 × 7 = 35.

Teacher's Note:
1. Find the size and direction of one strike first.
2. Opposite direction means opposite sign.