RBSE Solutions Class 7 Maths Chapter 1 Integers Important Questions

Rajasthan Board RBSE Class 7 Maths Chapter 1 Integers Additional Questions

Multiple Choice Questions

Question 1. Value of (-3) x 5 will be
(a) - 15
(b) 15
(c) 30
(d) - 30
Answer: (a) - 15
In simple words: When you multiply a negative number by a positive number, the answer is always negative. Here, 3 times 5 is 15, so -3 times 5 is -15.

🎯 Exam Tip: Remember the rules for multiplying integers: negative × positive = negative, positive × negative = negative, positive × positive = positive, negative × negative = positive.

Question 2. Additive inverse of 0 is
(a) 1
(b) 0
(c) 4
(d) -1
Answer: (b) 0
In simple words: The additive inverse of a number is what you add to it to get zero. For zero itself, you add zero to get zero, so zero is its own additive inverse.

🎯 Exam Tip: The additive inverse of any number 'x' is '-x', because x + (-x) = 0. For 0, 0 + 0 = 0.

Question 3. Value of (-3) x [2 + (-1)] will be
(a) - 1
(b) - 2
(c) - 3
(d) 0
Answer: (c) - 3
In simple words: First, solve the part inside the square brackets: 2 + (-1) equals 1. Then, multiply -3 by 1, which gives -3.

🎯 Exam Tip: Always follow the order of operations (BODMAS/PEMDAS) – parentheses/brackets first, then multiplication.

Question 4. Predecessor of whole number 0 will be
(a) -1
(b) -2
(c) 1
(d) undefined
Answer: (d) undefined
In simple words: A predecessor is the number that comes just before another number. For whole numbers, the smallest number is 0, so there is no whole number before 0.

🎯 Exam Tip: Understand the difference between whole numbers (0, 1, 2, 3...) and integers (...-2, -1, 0, 1, 2...). While 0 has a predecessor in integers (-1), it does not in whole numbers.

Question 5. The options for this question are:
(a) a
(b) 0
(c) 1
(d) undefined
Answer: (d) undefined
In simple words: If the question was asking about division by zero, the result is undefined. If it was about a number that doesn't exist in a set, it could be undefined.

🎯 Exam Tip: Pay close attention to what number sets are involved (e.g., whole numbers, integers) as this affects what results are "undefined".

Question 6. What will be the product of 5 negative numbers?
(a) Negative
(b) Positive
(c) 0
(d) None of the options
Answer: (a) Negative
In simple words: When you multiply an odd number of negative integers, the final product is always negative. Multiplying an even number of negative integers gives a positive result.

🎯 Exam Tip: Count the number of negative signs. If it's odd, the product is negative. If it's even, the product is positive.

Fill Up The Blank

Question. Fill in the blanks:
(i) \( 5 + (-8) = (-8) + (\text{......}) \)
(ii) \( -53 + \text{......} = -53 \)
(iii) \( 17 + \text{......} = 0 \)
(iv) \( [13 + (-12)] + (\text{......}) = 13 + [(-12) + (-7)] \)
(v) \( (-4) + [15 + (-3)] = [-4 + 15] + \text{........} \)
Answer:
(i) \( 5 + (-8) = (-8) + (-5) \)
(ii) \( -53 + 0 = -53 \)
(iii) \( 17 + (-17) = 0 \)
(iv) \( [13 + (-12)] + (-7) = 13 + [(-12) + (-7)] \)
(v) \( (-4) + [15 + (-3)] = [-4 + 15] + (-3) \)
In simple words: We used different properties of integers to fill in the missing numbers. For (i), it's the commutative property of addition. For (ii), adding zero doesn't change the number. For (iii), we find the additive inverse. For (iv) and (v), we use the associative property of addition.

🎯 Exam Tip: Remember the properties of integers: commutative property \( (a+b = b+a) \), additive identity \( (a+0=a) \), additive inverse \( (a+(-a)=0) \), and associative property \( ((a+b)+c = a+(b+c)) \).

True/False

Question. State whether the following statements are True or False:
(i) If a positive integer is added then move in right side.
(ii) If a negative integer is subtracted then move in left side.
(iii) If a negative integer is multiplied with 0 then product is a.
Answer:
(i) True
(ii) False
(iii) False
In simple words: (i) Adding a positive number always moves you to the right on a number line. (ii) Subtracting a negative number is the same as adding a positive number, which moves you to the right, not left. (iii) Any number multiplied by zero is always zero, not 'a' (the number itself).

🎯 Exam Tip: Visualizing a number line helps with addition and subtraction of integers. Remember that multiplying any integer by zero always results in zero.

Very Short Answer Type Questions

Question 1. Find the value of \( 8 \times (6 - 3) \)
Answer:
\( 8 \times (6 - 3) = 8 \times 3 \)
\( = 24 \)
In simple words: First, subtract 3 from 6 inside the bracket, which gives 3. Then, multiply 8 by this result, which is 24.

🎯 Exam Tip: Always perform operations inside brackets first when evaluating expressions.

Question 2. What is the solution of \( [-8 + 3] \div [(-2) + 1] \)?
Answer:
We need to find the value of \( [-8 + 3] \div [(-2) + 1] \).
First, solve the brackets:
\( [-8 + 3] = -5 \)
\( [(-2) + 1] = -1 \)
Next, divide the results:
\( -5 \div (-1) = 5 \)
So, the solution is 5.
In simple words: First, solve what's inside each bracket separately. Then, divide the first answer by the second answer. Remember that dividing a negative number by another negative number gives a positive answer.

🎯 Exam Tip: Handle each bracket calculation independently before combining them with the division operation. A negative number divided by a negative number results in a positive number.

Question 3. What is the solution of \( (-15) \times 0 \times (-18) \)?
Answer:
\( (-15) \times 0 \times (-18) \)
To solve this, we can group the terms:
\( = [(-15) \times 0] \times (-18) \)
We know that any number multiplied by 0 is 0.
\( = 0 \times (-18) \)
\( = 0 \)
So, the solution is 0.
In simple words: When you multiply any number by zero, the answer is always zero. Since zero is one of the numbers being multiplied, the final result will be zero, no matter what other numbers are in the multiplication.

🎯 Exam Tip: The "zero property of multiplication" is a powerful shortcut: if zero is a factor in any product, the entire product is zero.

Question 4. Find the value of \( 0 \div (-12) \).
Answer:
We need to find the value of \( 0 \div (-12) \).
When 0 is divided by any non-zero integer, the result is always 0.
\( 0 \div (-12) = 0 \)
So, the value is 0.
In simple words: If you have nothing and you divide it among 12 people (even if it's a negative amount of people, which is just a concept here), each person still gets nothing. So, zero divided by any non-zero number is always zero.

🎯 Exam Tip: Remember the rule for division with zero: \( \frac{0}{a} = 0 \) (where \( a \neq 0 \)), but \( \frac{a}{0} \) is undefined.

Short Answer Type Questions

Question 1. Verify the following:
(i) \( 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \)
(ii) \( (-21) \times [(-4) + (6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \)
Answer:
(i) For the left side (LHS):
\( 18 \times [7 + (-3)] \)
\( = 18 \times [7 - 3] \)
\( = 18 \times 4 \)
\( = 72 \)
For the right side (RHS):
\( [18 \times 7] + [18 \times (-3)] \)
\( = 126 + (-54) \)
\( = 126 - 54 \)
\( = 72 \)
Since LHS = RHS, the statement is verified.

(ii) For the left side (LHS):
\( (-21) \times [(-4) + (6)] \)
\( = (-21) \times [6 - 4] \)
\( = (-21) \times 2 \)
\( = -42 \)
For the right side (RHS):
\( [(-21) \times (-4)] + [(-21) \times (-6)] \)
\( = [84] + [126] \)
\( = 84 + 126 \)
\( = 210 \)
The provided solution from the source has a discrepancy in part (ii) where it attempts to show \( (-21) \times (-4) \) and \( (-21) \times (-6) \) as part of the distributive property, but then calculates \( [(-21) \times (-4)] + [(-21) \times (-6)] = 84 + 126 = 210 \).
However, the original statement for (ii) implies: \( (-21) \times [(-4) + (6)] = (-21) \times (-4) + (-21) \times (6) \)
Let's re-evaluate (ii) using the correct distributive property for the given terms:
LHS: \( (-21) \times [(-4) + (6)] = (-21) \times 2 = -42 \)
RHS: \( [(-21) \times (-4)] + [(-21) \times (6)] = 84 + (-126) = 84 - 126 = -42 \)
Since LHS = RHS, the statement is correctly verified to be true. The source calculation for RHS \( [(-21) \times (-6)] \) was used in a way that implies it was solving a slightly different problem or had an error in transcription.
In simple words: For both parts, we check if multiplying a number by a sum is the same as multiplying the number by each part of the sum separately and then adding those results. This is called the distributive property. We calculate both sides of the equals sign and see if they are the same.

🎯 Exam Tip: The distributive property \( a \times (b + c) = (a \times b) + (a \times c) \) is fundamental. Always calculate both sides of the equation separately to verify it and be careful with negative signs during multiplication.

Question 2. Is for any integer a
(i) \( 1 \div a = 1 \)?
(ii) \( a \div (-1) = -a \)?
Check the answer of different values of a.
Answer:
(i) Let's check with an example. Let \( a = 5 \).
Then, \( 1 \div a = 1 \div 5 = \frac{1}{5} \).
Since \( \frac{1}{5} \neq 1 \), the statement \( 1 \div a = 1 \) is not true for all integers \( a \) (except for \( a=1 \)).
So, \( 1 \div a = 1 \) is generally false.

(ii) Let's check with an example. Let \( a = 7 \).
Then, \( a \div (-1) = 7 \div (-1) = -7 \).
Also, \( -a = -7 \).
Since \( 7 \div (-1) = -7 \), the statement \( a \div (-1) = -a \) is true for any integer \( a \).
In simple words: For the first part, we checked if dividing 1 by any number always gives 1. It only works if the number itself is 1. For the second part, we checked if dividing any number by -1 always makes it negative (or positive if it was already negative). This is always true because dividing by -1 just changes the sign of the number.

🎯 Exam Tip: When checking if a statement is true for "any integer," a single counterexample (where it's false) is enough to prove it's false. For "true for any integer," you usually need to demonstrate its truth generally or with multiple examples.

Long Answer Questions

Question 1. The Temperature of Srinagar on Monday was \( -5^\circ C \) and it decreased on Tuesday by \( 2^\circ C \). What will be the temperature on Tuesday? On Wednesday it increased by \( 4^\circ C \). What was the temperature on Wednesday?
Answer:
Temperature of Srinagar on Monday \( = -5^\circ C \).
On Tuesday, the temperature decreased by \( 2^\circ C \).
So, Temperature on Tuesday \( = (-5^\circ C) - 2^\circ C \)
\( = -7^\circ C \).
On Wednesday, the temperature increased by \( 4^\circ C \).
So, Temperature on Wednesday \( = (-7^\circ C) + 4^\circ C \)
\( = -3^\circ C \).
Thus, the temperature on Tuesday was \( -7^\circ C \) and on Wednesday was \( -3^\circ C \). A decrease means subtracting, and an increase means adding, just like movements on a number line.
In simple words: On Monday it was -5 degrees. It got 2 degrees colder, making it -7 degrees on Tuesday. Then it got 4 degrees warmer, bringing the temperature up to -3 degrees on Wednesday.

🎯 Exam Tip: Remember that 'decrease' means subtraction and 'increase' means addition. Pay close attention to negative signs when adding and subtracting temperatures.

Question 3. An uplift moves down words in a mine-well out the rate of 6 metre/minute. If lift starts going down wards from the height 10 m above the ground, then how much time will it take to reach -350 m?
Answer:
The lift starts at a height of 10 m above the ground, which can be represented as \( +10 \) m.
The lift needs to reach a depth of 350 m below the ground, which is \( -350 \) m.
The total vertical distance the lift needs to travel is the difference between the final position and the initial position.
Difference between two heights \( = \text{Final height} - \text{Initial height} \)
\( = (-350 \text{ m}) - (10 \text{ m}) \)
\( = -350 \text{ m} - 10 \text{ m} \)
\( = -360 \text{ m} \)
The absolute distance the lift travels downwards is 360 m.
The rate of going downwards \( = 6 \text{ metre/minute} \).
Time taken \( = \frac{\text{Total distance}}{\text{Rate}} \)
\( = \frac{360 \text{ m}}{6 \text{ m/minute}} \)
\( = 60 \text{ minutes} \)
Since 60 minutes is equal to 1 hour, the time taken will be 1 hour.
In simple words: The lift starts 10 meters above the ground and needs to go to 350 meters below. This means it has to travel a total of 360 meters downwards. Since it moves 6 meters every minute, it will take 60 minutes, or 1 hour, to cover this distance.

🎯 Exam Tip: Clearly define the starting and ending points using positive and negative integers. Calculate the total distance (absolute value of the difference) and then use the formula: Time = Distance / Rate.

Question 4. A cement factory gains at the rate of Rs.8 per bag on white cement and looses at the rate of Rs.5 per bag on gray colour cement
(i) Company sells 3000 bags of white cement and 5000 bags of gray cement What is his toss or profit?
(ii) If numbers of bags of gray colour cement 6400 then how many bags of white cement should company sold so that there is no loss no profit?
Answer:
Profit on one bag of white cement \( = \text{Rs.8} \)
Loss on one bag of gray cement \( = \text{Rs.5} \)

(i) Calculation for total profit or loss:
Profit from white cement \( = 3000 \text{ bags} \times \text{Rs.8/bag} = \text{Rs.24,000} \)
Loss from gray cement \( = 5000 \text{ bags} \times \text{Rs.5/bag} = \text{Rs.25,000} \)
Total loss/profit \( = \text{Profit} - \text{Loss} \)
\( = \text{Rs.24,000} - \text{Rs.25,000} \)
\( = -\text{Rs.1,000} \)
Since the result is negative, it is a total loss of Rs.1,000.

(ii) To achieve no loss no profit:
Number of gray cement bags sold \( = 6400 \)
Total loss from gray cement \( = 6400 \text{ bags} \times \text{Rs.5/bag} = \text{Rs.32,000} \)
For no loss no profit, the profit from white cement must equal the loss from gray cement.
Required profit from white cement \( = \text{Rs.32,000} \)
Number of white cement bags needed \( = \frac{\text{Required profit}}{\text{Profit per bag}} \)
\( = \frac{\text{Rs.32,000}}{\text{Rs.8/bag}} \)
\( = 4000 \text{ bags} \)
So, the company should sell 4000 bags of white cement to have no loss and no profit. This balances the overall finances.
In simple words: (i) The company made Rs.24,000 from white cement and lost Rs.25,000 from gray cement, so they had a total loss of Rs.1,000. (ii) If they sell 6400 bags of gray cement, they lose Rs.32,000. To break even (no profit, no loss), they need to make Rs.32,000 from white cement. Since each white cement bag gives Rs.8 profit, they need to sell 4000 bags of white cement.

🎯 Exam Tip: Clearly distinguish between profit (positive value) and loss (negative value) in your calculations. For break-even analysis, ensure that total profit equals total loss.