NCERT Solutions Maths Class 7 Chapter 1 Exercise 1.2 - Integers

1. Find each of the following products:

a. $3\,\,\times \,\,\left( -1 \right)$

Ans: While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e., 

$3\times \left( -1 \right)=-3$

b. $\left( -1 \right)\,\,\times \,\,225$

Ans: While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e., 

$\left( -1 \right)\times 225=-225$

c. $\left( -21 \right)\,\,\times \,\,\left( -30 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., 

$\left( -21 \right)\times \left( -30 \right)=630$

d. $\left( -316 \right)\,\,\times \,\,\left( -1 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., 

$\left( -316 \right)\times \left( -1 \right)=316$

e. $\left( -15 \right)\,\,\times \,\,0\,\,\times \,\,\left( -30 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., 

$\left( -15 \right)\times \,0\times \left( -18 \right)=0$

f. $\left( -12 \right)\,\,\times \,\,\left( -11 \right)\,\,\times \,\,\left( 10 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., 

$\left[ \left( -12 \right)\times \left( -11 \right) \right]\times \left( 10 \right)=132\times 10=1320$

g. $9\,\,\times \,\,\left( -3 \right)\,\,\times \,\,\left( -6 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., 

$9\times \left[ \left( -3 \right)\times \left( -6 \right) \right]=9\times 18=162$

h. $\left( -18 \right)\,\,\times \,\,\left( -5 \right)\,\,\times \,\,\left( -4 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., $\left[ \left( -18 \right)\times \left( -5 \right) \right]\times \left( -4 \right)=90\times \left( -4 \right)$   ….. (1)

While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e., from (1), 

$\left[ \left( -18 \right)\times \left( -5 \right) \right]\times \left( -4 \right)=-360$

i. $\left( -1 \right)\,\,\times \,\,\left( -2 \right)\,\,\times \,\,\left( -3 \right)\,\,\times \,\,4$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product and while multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e., 

$\left[ \left( -1 \right)\times \left( -2 \right) \right]\times \left[ \left( -3 \right)\times 4 \right]=2\times \left( -12 \right)=-24$

j. $\left( -3 \right)\,\,\times \,\,\left( -6 \right)\,\,\times \,\,\left( 2 \right)\,\,\times \,\,\left( -1 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product and while multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e., 

$\left[ \left( -3 \right)\times \left( -6 \right) \right]\times \left[ \left( 2 \right)\times \left( -1 \right) \right]=\left( 18 \right)\times \left( -2 \right)=-36$

2. Verify the following:

a. $18\,\,\times \,\,\left[ 7+\left( -3 \right) \right]=\left[ 18\times 7 \right]+\left[ 18\times \left( -3 \right) \right]$

Ans: Given expression, $18\times \left[ 7+\left( -3 \right) \right]=\left[ 18\times 7 \right]+\left[ 18\times \left( -3 \right) \right]$.

Simplifying the given expression by first solving the square brackets.

While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product.

$\Rightarrow \,\,18\times \left[ 4 \right]=\left[ 126 \right]+\left[ -54 \right]$ 

$\Rightarrow \,\,72=72$

$\Rightarrow \,\,\text{L}\text{.H}\text{.S}\text{.}=\text{R}\text{.H}\text{.S}\text{.}$

Hence verified.

b. $\left( -21 \right)\times \left[ \left( -4 \right)+\left( -6 \right) \right]=\left[ \left( -21 \right)\times \left( -4 \right) \right]+\left[ \left( -21 \right)\times \left( -6 \right) \right]$ 

Ans: Given expression, $\left( -21 \right)\times \left[ \left( -4 \right)+\left( -6 \right) \right]=\left[ \left( -21 \right)\times \left( -4 \right) \right]+\left[ \left( -21 \right)\times \left( -6 \right) \right]$

Simplifying the given expression by first solving the square brackets.

While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product

$\Rightarrow \,\,\left( -21 \right)\times \left( -10 \right)=84+126$

$\Rightarrow \,\,210=210$

$\Rightarrow \,\,\text{L}\text{.H}\text{.S}\text{.}=\text{R}\text{.H}\text{.S}\text{.}$                                                                     

Hence verified.

3. Solve the following:

i.  For any integer $a$, what is $\left( -1 \right)\,\,\times \,\,a$ equal to?

Ans: $\left( -1 \right)\times a=-a,\,\text{ where }a\text{ is an integer}\text{.}$

ii. Determine the integer whose product with $\left( -1 \right)$ is:

a. $-22$

Ans: The integer whose product with $-1$ is \[-22\] is $22$ i.e., $\left( -1 \right)\times \left( 22 \right)=-22$.

b. $37$ 

Ans: The integer whose product with $-1$ is \[37\] is $-37$ i.e., $\left( -1 \right)\times \left( -37 \right)=37$.

c. $0$ 

Ans: The integer whose product with $-1$ is \[0\] is $0$ i.e., $-1\times 0=0$.

4. Starting from $\left( -1 \right)\,\,\times \,\,5$ write various products showing some patterns to show $\left( -1 \right)\,\,\times \,\,\left( -1 \right)=1$.

Ans: Consider the product, $\left( -1 \right)\times 5=-5$

Also, $\left( -1 \right)\times 4=-4$, $\left( -1 \right)\times 3=-3$, $\left( -1 \right)\times 2=-2$, $\left( -1 \right)\times 1=-1$, etc.

Thus, we can observe that the product of one negative integer and one positive integer is a negative integer.

Similarly, $\left( -1 \right)\times \left( -1 \right)=1$ i.e., the product of two negative integers is a positive integer.

Conclusion

NCERT Solutions for Class 7th Maths Exercise 1.2 offers a thorough explanation of how to add and subtract integers. This practice is essential because it fortifies the basis for increasingly complex mathematical ideas. Students can gain confidence in integer operations—which are crucial for many mathematical applications. Vedantu can enhance comprehension and problem-solving skills. Pay attention to the step-by-step solutions provided, grasp the underlying principles, and ensure clarity on the concepts before moving forward. Students who attentively study through this exercise and comprehend the answer strategies will be able to solve problems more effectively and do better on tests.

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