Q6. What will be the sign of the product if we multiply together
(i) 8 negative integers and 1 positive integer?
(ii) 21 negative integers and 3 positive integers?
(iii) 199 negative integers and 10 positive integers?
Sol. (i) Product of 8 negative integers and 1 positive integer is positive.
[Because, when the number of negative integers in a product are even, the product is positive.
(ii) Product of 21 negative integers and 3 positive integers is negative.
[Because when the number of negative integers in a product are odd, the product is negative.
(iii) Product of 199 negative integers and 10 positive integers is negative.
[Because when the number of negative integers in a product are odd, the product is negative.
Q7. State which is greater:
$$ (i) \quad (8+9) \times 10 \text { and } 8+9 \times 10 $$ $$ (ii) \quad (8-9) \times 10 \text { and } 8-9 \times 10 $$ $$ (iii) \quad {(-2)-5} \times(-6) \text { and } (-2)-5 \times(-6) $$$$ \text {Sol. (i) Here } \quad (8+9) \times 10 = 17 \times 10 = 170 $$ $$ \text {And } 8+9 \times 10 = 8+90 = 98 $$ $$ \therefore (8+9) \times 10 \text { is greater.} $$$$ \text {(ii) Here } \quad (8-9) \times 10 = (-1) \times 10 = -10 $$ $$ \text {And } 8-9 \times 10 = 8 – 90 = -82 $$ $$ \therefore (8-9) \times 10 \text { is greater.} $$$$ \text {(iii) Here } \quad {(-2)-5} \times(-6) = -7 \times (-6) = 42 $$ $$ \text {And } (-2)-5 \times(-6) = (-2) + 30 = 24 $$ $$ \therefore {(-2)-5} \times(-6) \text { is greater.} $$$$ \text {Q8. (i) If }a \times(-1)=-30 \text {, is the integer a positive or negative?} $$ $$ \text {(ii) If }a \times(-1)=30,\text { is the integer a positive or negative? } $$Sol. (i) Integer is postive because the product of two integers with opposite signs is equal to the additive inverse of the product of their absolute values.
(ii) Integer is negative because the product of two integers with like signs is equal to the product of their absolute values.
Q9. Verify the following:
$$ \text {(i) } \quad 19 \times \{7+(-3) \}=19 \times 7 + 19 \times(-3) $$ $$ \text {(ii) } \quad (-23) \{(-5)+(+19) \}=(-23) \times(-5)+(-23) \times(+19) $$$$ \text {Sol. (i) Given } \quad 19 \times \{7+(-3) \} =19 \times 7+19 \times (-3) $$ $$ \text {LHS } = 19 \times \{7+(-3)\} = 19 \times 4 = 76 $$ $$ \text {RHS } = 19 \times 7 + 19 \times(-3) $$ $$ = 19 \times \{7 + (-3) \} $$ $$ = 19 \times 4 = 76 $$ Hence LHS = RHS
$$ \text {(ii) Given } \quad (-23) \{(-5)+(+19) \} =(-23) \times (-5) + (-23) \times (+19) $$ $$ \text {LHS } = (-23) \{(-5)+(+19) \} $$ $$ = (-23) \times 14 = -322 $$$$ \text {RHS } = (-23) \times(-5)+(-23) \times(+19) $$ $$ = (-23) \times \{(-5) + (+19) \} $$ $$ = (-23) \times 14 = -322 $$ Hence LHS = RHS
Q10. Which of the following statements are true?
(i) The product of a positive and a negative integer is negative.
(ii) The product of three negative integers is a negative integer.
(iii) Of the two integers, if one is negative, then their product must be positive. $$ \text {(iv) For all non-zero integers a and b, } a \times b \text { is always greater than either a or b.} $$ (v) The product of a negative and a positive integer may be zero. $$ \text {(vi) There does not exist an integer b such that for a > 1, } a \times b = b \times a = b $$Sol. (i) True
(ii) True
(iii) False
(iv) False
(v) False
(vi) True
(i) 8 negative integers and 1 positive integer?
(ii) 21 negative integers and 3 positive integers?
(iii) 199 negative integers and 10 positive integers?
Sol. (i) Product of 8 negative integers and 1 positive integer is positive.
[Because, when the number of negative integers in a product are even, the product is positive.
(ii) Product of 21 negative integers and 3 positive integers is negative.
[Because when the number of negative integers in a product are odd, the product is negative.
(iii) Product of 199 negative integers and 10 positive integers is negative.
[Because when the number of negative integers in a product are odd, the product is negative.
Q7. State which is greater:
$$ (i) \quad (8+9) \times 10 \text { and } 8+9 \times 10 $$ $$ (ii) \quad (8-9) \times 10 \text { and } 8-9 \times 10 $$ $$ (iii) \quad {(-2)-5} \times(-6) \text { and } (-2)-5 \times(-6) $$$$ \text {Sol. (i) Here } \quad (8+9) \times 10 = 17 \times 10 = 170 $$ $$ \text {And } 8+9 \times 10 = 8+90 = 98 $$ $$ \therefore (8+9) \times 10 \text { is greater.} $$$$ \text {(ii) Here } \quad (8-9) \times 10 = (-1) \times 10 = -10 $$ $$ \text {And } 8-9 \times 10 = 8 – 90 = -82 $$ $$ \therefore (8-9) \times 10 \text { is greater.} $$$$ \text {(iii) Here } \quad {(-2)-5} \times(-6) = -7 \times (-6) = 42 $$ $$ \text {And } (-2)-5 \times(-6) = (-2) + 30 = 24 $$ $$ \therefore {(-2)-5} \times(-6) \text { is greater.} $$$$ \text {Q8. (i) If }a \times(-1)=-30 \text {, is the integer a positive or negative?} $$ $$ \text {(ii) If }a \times(-1)=30,\text { is the integer a positive or negative? } $$Sol. (i) Integer is postive because the product of two integers with opposite signs is equal to the additive inverse of the product of their absolute values.
(ii) Integer is negative because the product of two integers with like signs is equal to the product of their absolute values.
Q9. Verify the following:
$$ \text {(i) } \quad 19 \times \{7+(-3) \}=19 \times 7 + 19 \times(-3) $$ $$ \text {(ii) } \quad (-23) \{(-5)+(+19) \}=(-23) \times(-5)+(-23) \times(+19) $$$$ \text {Sol. (i) Given } \quad 19 \times \{7+(-3) \} =19 \times 7+19 \times (-3) $$ $$ \text {LHS } = 19 \times \{7+(-3)\} = 19 \times 4 = 76 $$ $$ \text {RHS } = 19 \times 7 + 19 \times(-3) $$ $$ = 19 \times \{7 + (-3) \} $$ $$ = 19 \times 4 = 76 $$ Hence LHS = RHS
$$ \text {(ii) Given } \quad (-23) \{(-5)+(+19) \} =(-23) \times (-5) + (-23) \times (+19) $$ $$ \text {LHS } = (-23) \{(-5)+(+19) \} $$ $$ = (-23) \times 14 = -322 $$$$ \text {RHS } = (-23) \times(-5)+(-23) \times(+19) $$ $$ = (-23) \times \{(-5) + (+19) \} $$ $$ = (-23) \times 14 = -322 $$ Hence LHS = RHS
Q10. Which of the following statements are true?
(i) The product of a positive and a negative integer is negative.
(ii) The product of three negative integers is a negative integer.
(iii) Of the two integers, if one is negative, then their product must be positive. $$ \text {(iv) For all non-zero integers a and b, } a \times b \text { is always greater than either a or b.} $$ (v) The product of a negative and a positive integer may be zero. $$ \text {(vi) There does not exist an integer b such that for a > 1, } a \times b = b \times a = b $$Sol. (i) True
(ii) True
(iii) False
(iv) False
(v) False
(vi) True