NCERT Exemplar Class 7 Maths Chapter 1 Integers are part of NCERT Exemplar Class 7 Maths. Here we have given NCERT Exemplar Class 7 Maths Solutions Chapter 1 Integers.
NCERT Exemplar Class 7 Maths Solutions Chapter 1 Integers
Question 1.
When the integers 10, 0, 5, -5,-7 are arranged in descending or ascending order, then find out which of the following integers always remains in the middle of the arrangement.
(a) 0
(b) 5
(c) – 7
(d) -5
Solution:
(a): The integers are 10, 0, 5, -5, -7
Descending order: 10, 5, 0, – 5, – 7
Ascending order: -7, – 5,0, 5, 10
Hence, 0 is the integer which always remains in the middle of the arrangement.
Question 2.
By observing the number line, state which of the following statements is not true?
(a) B is greater than -10
(b) A is greater than 0
(c) B is greater than A
(d) 6 is smaller than 0
Solution:
(c): Since, B is smaller than A
Question 3.
By observing the above number line, state which of the following statements is true?
(a) B is 2
(b) A is – 4
(c) S is -13
(d) B is – 4
Solution:
(d): Here, A is 7 and B is -4.
Question 4.
Next three consecutive numbers in the pattern 11, 8, 5, 2,______ ,__ ,__ are
(a) 0, – 3, – 6
(b)-1,-5,-8
(c) – 2, — 5, – 8
(d) -1,-4,-7
Solution:
(d): Here, the pattern is
11 – 3 = 8, 8 – 3 = 5, 5 – 3 = 2
So, the next three consecutive numbers will be 2 – 3 = -1, -1 – 3 = – 4, -4 – 3 = – 7
i.e., -1, -4, -7
Question 5.
The next number in the pattern – 62,- 37,- 12 is_________________ .
(a) 25
(b) 13
(c) 0
(d) -13
Solution:
(b): Here, the pattern is -62 + 25 = -37, -37 + 25 = -12
∴ Next number in the pattern will be -12 + 25 = 13
Question 6.
Which of the following statements is not true? ;
(a) When two positive integers are added, we always get a positive integer.
(b) When two negative integers are added, we always get a negative integer.
(c) When a positive integer and a negative integer are added, we always get a negative integer.
(d) Additive inverse of an integer 2 is (-2) and additive inverse of (-2) is 2.
Solution:
(c): Statement (c) is false because when a positive integer and a negative integer is added we may also get a positive integer or zero.
Question 7.
On the following number line value, ‘zero’ is shown by the point
(a) X
(b) Y
(c) Z
(d) W
Solution:
(c):
Question 8.
If 
and • represent some integers on number line, then descending order of these numbers is
Solution:
(c) Descending order of given numbers is 
Question 9:
On the number line, the value of (-3) x 3 lies on right hand side of
(a) -10
(b) – 4
(c) 0
(d) 9
Solution:
(a):
Since, (-3) × 3 = – 9
It lies on right hand side of -10.
Question 10.
The value of 5 + (- 1) does not lie between
(a) 0 and -10
(b) 0 and 10
(c) – 4 and -15
(d) – 6 and 6
Solution:
(b): The value of \(5 \div(-1)=\frac{5}{(-1)}=-5\)
∴ -5 does not lie between 0 and 10.
Question 11.
Water level in a well was 20 m below ground level. During rainy season, rainwater collected in different water tanks was drained into the well and the water level rises 5 m above the previous level. The wall of the well is lm 20cm high and a pulley afixed at a height of 80 cm. Raghu wants to draw water from the well. The minimum length of the rope, that he can use is
Solution:
(a): Height of the wall of the well = 1 m 20 cm = 1.20 m
Height of the pulley = 80 cm = 0.80 m
∴ Required minimum length of the rope
= (20-5+ 1.20 +0.80) m
= (15 + 2) m = 17 m
Question 12.
(-11) × 7 is not equal to
(a) 11 × (-7)
(b) -(11 × 7)
(c) (- 11) × (- 7 )
(d) 7 × (-11)
Solution:
(c):(-11) × 7 = – (11 × 7)
= 11 × (-7) = 7 × (-11) = -77
∴ (-11) × 7 ≠ (-11) × (-7) = 77
Question 13.
(- 10) × (- 5) + (- 7) is equal to
(a) -57
(b) 57
(c) -43
(d) 43
Solution:
(d): (-10) × (- 5) + (- 7) = 50 + (-7) = 50 – 7 = 43
Question 14.
Which of the following is not the additive inverse of a?
(a) – (-a)
(b) a × (-1)
(c) – a
(d) a + (-1)
Solution:
(a): Additive inverse of a is (- a).
So, it cannot be additive inverse of a.
Question 15.
Which of the following is the multiplicative identity for an integer o?
(a) a
(b) 1
(c) 0
(d) -1
Solution:
(b): Multiplicative identity for an integer a is 1. [ ∵ a × 1 = a = 1 × a]
Question 16.
[(- 8) × (- 3)] × (- 4)] is not equal to
(a) (- 8) × [(- 3) × (- 4)]
(b) [(- 8) × (- 4)] × (- 3)
(c) [(- 3) × (- 8)] × (- 4)
(d) (- 8) × (- 3) – (- 8) × (- 4)
Solution:
(d): [(-8) × (-3)] × (-4)
= (-8) × [( 3) × (-1)]
= [(-8) × (-4)] × (-3)
= [(-3) – (-8)] × (-4)
But
[(-8) × (-3)] × (-4) ≠ (-8) ×(-3) – (-8) × (-4)
Question 17.
(- 25) × [6 + 4] is not same as
(a) (-25) × 10
(b) (-25) × 6 + (- 25) × 4
(c) -25 × 6 × 4
(d) – 250
Solution:
(c): (- 25) × [6 + 4] ≠ (- 25) × 6 × 4
Question 18.
– 35 × 107 is not same as
(a) – 35 × (100 + 7)
(b) (- 35) × 7 + (- 35) × 100
(c) – 35 × 7 + 100
(d) (-30 -5) × 107
Solution:
(c): – 35 × 107 ≠ -35 × 7 + 100
Question 19.
(- 43) × (- 99) + 43 is equal to
(a) 4300
(b) – 4300
(c) 425
(d) -4214
Solution:
(a): (- 43) × (- 99) + 43 = 43× -99)+ 43
= 4257 + 43 = 4300
Question 20.
(- 16) ÷ 4 is not same as
(a) (-4) ÷ 16
(b) -(16 ÷ 4)
(c) 16 ÷ (-4)
(d) – 4
Solution:
(a):\((-16) \div 4=\frac{(-16)}{4}=-4\)
But \((-4) \div 16=\frac{-4}{16}=-\frac{1}{4}\)
(-16) ÷ 4 is not same as (-4) ÷ 16
Question 21.
Which of the following does not represent an integer?
(a) 0 ÷ (- 7)
(b) 20 ÷ (- 4)
(c) (-9) ÷ 3
(d) (-12) ÷ 5
Solution:
(d):
(a) 0 ÷ (-7) = 0
(b) 20 ÷ (- 4) = -5
(c) (-9) ÷ 3 = -3
(d) \((-12) \div 5=\frac{-12}{5}\)
Hence, (-12) ÷ 5 does not represent an integer.
Question 22.
Which of the following is different from the others?
(a) 20 + (-25)
(b) (-37) – (-32)
(c) (-5) × (— 1)
(d) 45 ÷ (- 9)
Solution:
(c):
(a) 20 + (- 25) = 20 – 25 = – 5
(b) (- 37) – (- 32)= – 37 + 32 = – 5
(c) (- 5) × (- 1)= 5
(d) \((45) \div(-9)=-\frac{45}{9}=-5\)
Hence, (-5) × (-1) is different
Question 23.
Which of the following shows the maximum rise in temperature?
(a) 23° to 32°
(b)-10°to 1°
(c) -18° to-11°
(d) -5° to 5°
Solution:
(b): Rise in temperature,
(a) 32° – 23° = 9°
(b) 1°- (-10)° = 1° + 10° = 11° (maximum)
(c) -11°- (-18)°=-11°+18° = 7°
(d) 5° – (-5°) = 5° + 5° = 10°
Hence, the maximum rise in temperature -10° to +1°.
Question 24.
If a and b are two integers, then which of the following may not be an integer?
(a) a + b
(b) a – b
(c) a × b
(d) a ÷ b
Solution:
(d): If a and b are two integers, then
a + b is always an integer.
a – bis always an integer.
a × b is also an integer,
but a ÷ b may or may not be an integer.
Question 25.
For a non-zero integer a, which of the following is not defined?
(a) a ÷ 0
(b) 0 ÷ a
(c) a ÷1
(d) 1 ÷ a
Solution:
(a):\(a \div 0=\frac{a}{0}\) is not defined
Directions: Encircle the odd one of the following: (Questions 26 to 30)
Question 26.
(a) (-3, 3)
(b) (-5, 5)
(c) (-6, 1)
(d) (-8, 8)
Solution:
(c): (a) -3 + 3 = 0
(b) -5 + 5 = 0
(c) -6 + 1 = -5
(d) -8 + 8 = 0
∴ (-6,1) is different.
Question 27.
(a) (-1, -2)
(b) (-5, 2)
(c) (- 4, 1)
(d) (- 9, 7)
Solution:
(d): (a) -1 – 2 = -3
(b) -5 + 2 = -3
(c) -4 + 1 = -3
(d) -9 + 7 = -2
∴ (-9, +7) is different.
Question 28.
(a) (- 9) × 5 × 6 × (- 3)
(b) 9 × (-5) × 6 × (-3)
(c) (- 9) × (- 5) × (- 6) × 3
(d) 9 × (- 5) × (- 6) × 3
Solution:
(c): (a) (- 9) × 5 × 6 × (- 3) = -810
(b) 9 × (-5) × 6 × (-3) = 810
(c) (- 9) × (- 5) × (- 6) × 3 = -810
(d) 9 × (- 5) × (- 6) × 3 = 810
∴ (- 9) × (- 5) × (- 6) × 3 is different
Question 29.
(a) (-100) ÷ 5
(b) (-81) ÷ 9
(c) (- 75) ÷ 5
(d) (-32) ÷ 9
Solution:
(d):
∴ (-32) ÷ 9 is different
Question 30.
(a) (- 1) × (- l)
(b) (- 1) × (- l) × (- 1)
(c) (-1) × (-1) × (-1) × (-1)
(d) (-1) × (-1) × (-1) × (-1) × (-1) × (-1)
Solution:
(b):
(a) (-1) × (-1) = 1
(b) (- 1) × (-1) × (-1) = — 1
(c) (- 1) × (- 1) × (- 1) × (- 1) = 1
(d) (-1) × (-1) × (-1) × (- 1) × (-1) × (-1) = 1
∴ (- 1) × (-1) × (-1) is different
Directions: In questions 31 to 71, fill in the blanks to make the statements true.
Question 31 .
(- a) + b = b + additive inverse of ___ .
Solution:
a: (-a) + b = b + (-a)
= b + Additive inverse of a.
Question 32.
__________ ÷ (- 10) = 0
Solution:
\(0 : 0 \div(-10)=-\frac{0}{10}=0\)
Question 33.
(- 157) × (— 19) + 157 =___________ .
Solution:
3140: (-157) × (-19) + 157
= 157 × 19 + 157 × 1
= 157 × [19 + 1] = 157 × 20= 3140
Question 34.
[(- 8)+ ___ ]+ ___ = ___ + [(-3) + ___ ] = – 3
Solution:
-3, 8, (-8), 8 : [(-8) + (-3)] + 8 = -8+[(-3) + 8] = -3
Question 35.
On the following number line, (- 4) × 3 is represented by the point_____ .
Solution:
D: (-4) × 3 = – 12
∴ -12 is represented by point D.
Question 36.
If x, y and z are integers, then (x +____ ) + z =____ + (y +____ )
Solution:
y, x, z : (x + y) + z = x + (y + z) [Associative property of integers]
Question 37.
(-43)+ __________ = (-43)
Solution:
0: (-43) + 0 = – 43
Question 38.
(- 8) + (- 8) + (- 8) =_______ × (- 8)
Solution:
3: (-8) + (-8) + (-8)
= (-8) × [1 + 1 + 1] = (-8) × 3 = 3 × (-8)
Question 39.
11 × (- 5) = – (_____ x____ ) =_____
Solution:
11, 5, -55: 11 × (-5) = -(11 × 5) = -55
Question 40.
(- 9) × 20 =_______
Solution:
-180: (-9) × 20 = -180
Question 41.
(- 23) × (42) = (- 42) ×______
Solution:
23: (-23) × (42) = (-42) × 923)
Question 42.
While multiplying a positive integer and a negative integer, we multiply them as ___ numbers and put a ___ sign before the product.
Solution:
Whole, negative
Question 43.
If we multiply___ number of negative integers, then the resulting integer is positive.
Solution:
Even
Question 44.
If we multiply six negative integers and six positive integers, then the resulting integer is___ .
Solution:
Positive: When even number of negative integers are multiplied they give positive integer and when positive integers are multiplied they always give positive integer.
Question 45.
If we multiply five positive integers and one negative integer, then the resulting integer is ___ .
Solution:
Negative: When odd number of negative integers multiplied, they give negative integer. Also, when a negative and a positive integer are multiplied they give a negative integer.
Question 46.
________ is the multiplicative identity for integers.
Solution:
1: a × 1 = 1 × a = a
Question 47.
We get additive inverse of an integer a, when we multiply it by ___ .
Solution:
(-1): a × (-1) = -a
Question 48.
(- 25) × (- 2) =______.
Solution:
50: (-25) × (-2) = 25 × 2 = 50
Question 49.
(- 5) × (- 6) × (- 7) =______.
Solution:
210: (-5) × (-6) × (-7)
= 5 × 6 × (-7) = 30 × (-7) = -210
Question 50.
3 × (- 1) × (- 15) =_______.
Solution:
45: 3 × (-1) × (-15) = (-3) × ( – 15)= 45
Question 51.
[12 × (- 7)] × 5 =_____ × [(- 7) × ____ ]
Solution:
12, 5: [12 × (- 7)] × 5 = 12 × [(- 7) × 5]
Question 52.
23 × (- 99) = ____ × (- 100 + ____ ) = 23 × ____ + 23 × ____
Solution:
23, 1, -100, 1: 23 × (- 99) = 23 × (- 100 + 1)= 23 × (- 100) + 23 × 1
Question 53.
______ × (- 1) = – 35
Solution:
35: – 35 × (-1) = – 35
Question 54.
____ × (- 1) = 47
Solution:
-47: (- 47) × (- 1) = 47
Question 55.
88 × ____ = – 88
Solution:
-1: 88 × (- 1) = – 88
Question 56.
____ × (- 93) = 93
Solution:
-1: (- 1) × (- 93) = 93
Question 57.
(- 40) × ___ = 80
Solution:
-2: (- 40) × (-2) = 80
Question 58.
____ × (-23) = – 920
Solution:
40: Let a number to be multiplied be x.
x × (-23) = -920
\(\Rightarrow x=-920 \div(-23)=\frac{920}{23}=40\)
Question 59.
When we divide a negative integer by a positive integer, we divide them as whole numbers and put a ____ sign before quotient.
Solution:
Minus
Question 60.
When (-16) is divided by ____ the quotient is 4.
Solution:
-4: Let -16 be divided by x gives the quotient 4.
\(\therefore \frac{-16}{x}=4\)
\(\Rightarrow \quad x=\frac{-16}{4}=-4\)
Question 61.
Division is the inverse operation of ____ .
Solution:
Multiplication.
Question 62.
65 ÷ (- 13) =_____.
Solution:
\(-5 : 65 \div(-13)=-\frac{65}{13}=-5\)
Question 63.
(-100) ÷ (-10) =_____.
Solution:
\(10 :(-100) \div(-10)=\frac{100}{10}=10\)
Question 64.
(-225) ÷ 5 = _____.
Solution:
\(-45 :(-225) \div 5=-\frac{225}{5}=-45\)
Question 65.
_____ ÷ (-1) = (- 83)
Solution:
83: 83 ÷ (-1) = – 83
Question 66.
____ ÷ (-1) = 75
Solution:
\(-75 :(-75) \div(-1)=\frac{75}{1}=75\)
Question 67.
51 ÷ ____ =(-51)
Solution:
\(-1 : 51 \div(-1)=-\frac{51}{1}=-51\)
Question 68.
113 ÷ ____ = (- 1)
Solution:
\(-113 : 113 \div(-113)=-\frac{113}{113}=-1\)
Question 69.
-95 ÷ (-1) = 95
Solution:
\(-1 :(-95) \div(-1)=\frac{95}{1}=95\)
Question 70.
(-69) ÷ 69 =_____.
Solution:
\(-1 :(-69) \div(69)=-\frac{69}{69}=-1\)
Question 71.
(-28) ÷ (-28) = _____
Solution:
\(1 :(-28) \div(-28)=\frac{28}{28}=1\)
Directions : In questions 72 to 108, state whether the statements are true or false.
Question 72.
5 – (-8) is same as 5 + 8.
Solution:
True
5 – (-8) = 5 + 8
Question 73.
(-9) + (-11) is greater than (-9) – (- 11).
Solution:
False
(-9) + (-11) = – 9 – 11 = -20
and (-9) – (-11) = -9 + 11 = 2
Since, -20 < 2⇒ -9 + (-11) < (-9) – (-11)
Question 74.
Sum of two negative integers always gives a number smaller than both the integers.
Solution:
True
Question 75.
Difference of two negative integers cannot be a positive integer.
Solution:
False
As -3 – (-5) = -3 + 5 = 2
Question 76.
We can write a pair of integers, whose sum is not an integer.
Solution:
False
Since, sum of two integers is always an integer.
Question 77.
Integers are closed under subtraction.
Solution:
True
As subtraction of any two integers is always an integer.
∴ Integers are closed under subtraction.
Question 78.
(- 23) + 47 is same as 47 + (- 23).
Solution:
True
(-23) + 47 = 24
and 47 + (-23) = 47 – 23 = 24
Question 79.
When we change the order of integers their sum remains the same.
Solution:
True
Question 80.
When we change the order of integers, their difference remains the same.
Solution:
False
As 2 – 3 – 5 = 2 – 8 = -6
but 3 – 2 – 5 = 3 – 7 = -4
Question 81.
Going 500 m towards East first and then 200 m back, is same as going 200 m towards West first and then going 500 m back.
Solution:
True
In first case, he is at a distance of (500 – 200) m = 300 m towards east.
In second case, he is at a distance of (200 – 500) m = -300 m towards west i.e., 300 m towards east.
Question 82.
(-5) × (33) = 5 × (- 33)
Solution:
True
(-5) × (33) = – 165 = 5 × (-33)
Question 83.
(-19) × (-11) = 19 × 11
Solution:
True
(-19) × (-11) = 19 × 11 = 209
Question 84.
(-20) × (5 – 3) = (-20) × (-2)
Solution:
False
(-20) × (5 – 3) = (-20) × 2 = (-40)
but (-20) × (-2) = 20 × 2 = 40
Question 85.
4 × (-5) = (-10) × (-2)
Solution:
False
4 × (-5) = – 20
but (-10) × (-2) = 10 × 2 = 20
Question 86.
(-1) × (-2) × (-3) = 1 × 2 × 3
Solution:
False
(-1) × (-2) × (-3) = 1 × 2 × (-3) = 2 × (-3) = (-6)
but 1 × 2 × 3 = 6
Question 87.
(-3) × 3 = (-12) – (-3)
Solution:
True
Since, (-3) × 3 = (- 9)
and (-12) – (-3) = (-12) + 3 = (-9)
Question 88.
Product of two negative integers is a negative integer.
Solution:
False
As product of two negative integers is always a positive integer.
Question 89.
Product of three negative integers is a negative integer.
Solution:
True
Since, product of odd number of negative integers is always a negative integer.
Question 90.
90 Product of a negative integer and a positive integer is a positive integer.
Solution:
False
As product of a negative integer and a positive integer is a negative integer.
Question 91.
When we multiply two integers their product is always greater than both the integers.
Solution:
False
When we multiply two integers then their product may or may not be greater than both the integers.
Question 92.
Integers are closed under multiplication.
Solution:
True
Since, multiplication of two integers is always an integer.
Integers are closed under multiplication.
Question 93.
(-237) × 0 is same as 0 × (-39).
Solution:
True
(-237) × 0 = 0
and 0 × (-39) = 0
Question 94.
Multiplication is not commutative for integers.
Solution:
False
As Multiplication is commutative for integers.
Question 95.
(-1) is not a multiplicative identity of integers.
Solution:
True
As 1 is multiplicative identity for integers.
Question 96.
99 × 101 can be written as (100 – 1) × (100 + 1).
Solution:
True
99 = 100 – 1 and 101 = 100 + 1
So, 99 × 101 = (100 – 1) × (100 + 1)
Question 97.
If a, b and c are integers and b ≠ 0, then a × (b – c) = a × b – a × c
Solution:
True
a × (b – c) = (a × b) – (a × c)
(Distributive property of multiplication over subtraction)
Question 98.
(a + b) × c = a × c + a × b
Solution:
False
a × (b + c) = a × b + a × c
Question 99.
a × b = b × a
Solution:
True
Question 100.
a ÷ b = b ÷ a
Solution:
False
As division is not commutative for integers,
∴ a ÷ b ≠ b ÷ a
Question 101.
a – b = b – a
Solution:
False
As subtraction is not commutative for integers.
∴ a – b ≠ b – a
Question 102.
a ÷ (- b) = – (a ÷ b)
Solution:
True
\(a \div(-b)=\frac{a}{-b}=-\left(\frac{a}{b}\right)=-(a \div b)\)
Question 103.
a ÷ (-l) = – a
Solution:
True
\(a \div(-1)=\frac{a}{-1}=-a\)
Question 104.
Multiplication fact (-8) × (-10) = 80 is same as division fact 80 ÷ (-8) = (-10).
Solution:
True
(-8) × (-10) = 8 × 10 = 80
and \(80 \div(-8)=-\frac{80}{8}=-10\)
Question 105.
Integers are closed under division.
Solution:
False
As integers are not closed under division.
Question 106.
[(-32) ÷ 8] ÷ 2 = – 32 ÷ [8 ÷ 2]
Solution:
False
Question 107.
The sum of an integer and its additive inverse is zero (0).
Solution:
True
Let any integer be a.
Its additive inverse is -a.
a + (-a) = a – a = 0
Question 108.
The successor of 0 × (-25) is 1 × (-25).
Solution:
False
0 × (-25) = 0
and 1 × (-25) = -25
But successor of 0 is 1.
Question 109.
Observe the following patterns and fill in the blanks to make the statements true:
(a) – 5 × 4 = – 20
-5 × 3 = -15 = -20 – (-5)
-5 × 2 =_____ = -15 – (-5)
-5 × 1 =_____ = ______
-5 × 0 = 0 =_______
-5 × -1 = 5 = _____
– 5 × – 2 =__ =______
(b) 7 × 4 = 28
7 × 3 =______ = 28 – 7
7 × 2 =______ =____ – 7
7 × 1 = 7 =____ -7
7 × 0 =______ =____ -______
7 × – 1 = -7 =__ -______
7 × – 2 =___ =______ -_____
7 × – 3_____ =____ -______
Solution:
(a) -10, -5, 10, -10 – (-5), -5 – (-5), 0 – (-5), 10, 5 – (-5):
-5 × 2 = -15 – (-5)
-5 × 1 = -5 = – 10 – (-5)
-5 × 0 = 0 =-5 – (-5)
-5 × -1 = 5 = 0 – (-5)
– 5 × – 2 =10 = 5 – (-5)
(b) 21, 14, 21, 14, 0, 7, 7, 0, 7, -14, -7, 7, -21, -14, 7:
7 × 3 = 28 – 7
7 × 2 = 14 = 21 – 7
7 × 1 = 7 = 14 – 7
7 × 0 = 0 = 7 – 7
7 × (-1) = -7 = 0 – 7
7 × (- 2) = – 14 = -7 – 7
7 × (- 3) = -21 = – 14 – 7
Question 110.
Science Application An atom consists of charged particles called electrons and protons. Each proton has a charge of +1 and each electron has a charge of -1. Remember number of electrons is equal to number of protons, while answering these questions:
(a) What is the charge on an atom?
(b) What will be the charge on an atom, if it loses an electron?
(c) hat will be the charge on an atom, if it gains an electron?
Solution:
(a) Total charge = +1-1=0
(b) If an atom loses an electron, then the charge on the atom is +1.
(c) The charge on an atom if it gains an electron is -1.
Question 111:
An atom changes to a charged particle called ion, if it loses or gains electrons. The charge on an ion is the charge on electrons plus charge on protons. Now, write the missing information in the table given below:
| Name of Ion | Proton Charge | Electron Charge | Ion Charge |
| Hydroxide ion | +9 | — | -1 |
| Sodium ion | +11 | — | +1 |
| Aluminium ion | +13 | -10 | — |
| Oxide ion | +8 | -10 | — |
Solution:
Hydroxide ion charge = Proton charge + Electron charge
⇒ -1 = +9 + Electron charge
⇒ Electron charge = -1 – 9 = – 10
Sodium ion charge = Proton charge + Electron charge
⇒ +1 = +11 + Electron charge
⇒ Electron charge = +1 – 11 = -10
Aluminium ion charge = Proton charge + Electron charge
⇒ +13 + (-10) = +13 – 10 = +3
Oxide ion charge = Proton charge + Electron charge
⇒ +8 + (-10) = +8 – 10 = -2
Question 112.
Social Studies Application remembering that 1AD came immediately after 1 BC, while solving following problems take 1BC as -1 and 1AD as + 1.
(a) The Greeco-Roman era, when Greece and Rome ruled Egypt, started in the year 330 BC and ended in the year 395 AD. How long did this era last?
(b) haskaracharya was born in the year 1114 AD and died in the year 1185 AD. What was his age when he died?
(c) Turks ruled Egypt in the year 1517 AD and Queen Nefertis ruled . Egypt about 2900 years, before the Turks ruled. In what year did she rule?
(d) Greek Mathematician Archimedes lived between 287 BC and 212 BC and Aristotle lived between 380 BC and 322 BC. Who lived during an earlier period?
Solution:
Taking 1 BC as -1 and 1 AD as +1
(a) Starting year = 330 BC = (-330) AD
Ending year = 395 AD
The era lasted for = 395 – (-330)
= 395 + 330 = 725 years
(b) Born year = 1114 AD
Death year = 1185 AD
∴ Total age = 1185 – 1114 = 71 years
(c) Turks ruled Egypt in 1517 AD.
Queen Nefertis ruled Egypt in
(1517-2900) AD= -1383 AD or 1383 BC.
(d) Archimedes lived between 287 BC and 212 BC.
Aristotle lived between 380 BC and 322 BC.
∴ Aristotle lived during an earlier period.
Question 113.
The table shows the lowest recorded temperatures for each continent. Write the continents in order from the lowest recorded temperature to the highest recorded temperature.
| The Lowest Recorded Temperatures | |
| Continent | Temperature (in Fahrenheit) |
| Africa | -11° |
| Antarctica | -129° |
| Asia | -90° |
| Australia | -9° |
| Europe | -67° |
| North America | -81° |
| South America | -27° |
Solution:
Since,
-129° < -90° < -81° < -67° < -27° < -11° < -9°
Hence, order of the continents from the lowest to the highest recorded temperature is
Antarctica, Asia, North America, Europe, South America, Africa, Australia.
Question 114.
Write a pair of integers whose product is -12 and there lies seven integers between them (excluding the given integers).
Solution:
Let the integers be -2 and 6 such that (-2) × 6 = -12
∴ A pair of integers is (-2, 6).
And there are seven integers, i.e.,
-1, 0, 1, 2, 3, 4, 5 which lie between -2 and 6.
Question 115.
From given integers in Column I, match an integer of Column II, so that their product lies between -19 and -6.
Solution:
-5 → 3; 6 → -2; -7 → 1; 8 → -1;
-5 × 3 = -15 and -19 < -15 < -6
6 × (-2) = -12 and -19 < -12 < -6
-7 × 1 = -7 and -19<-7<-6
8 × (-1) = – 8 and -19 < -8 < – 6
Question 116.
Write a pair of integers, whose product is -36 and whose difference is 15.
Solution:
Let the integers be 12 and -3 such that 12 × (-3) = -36
and their difference = 12 – (-3) = 12 + 3 = 15
∴ A pair of integers is (-3,12).
Question 117.
Match the following:
Solution:
Question 118.
You have ₹ 500 in your saving account at the beginning of the month. The record below, shows all of your transactions during the month. How much money is in your account after these transactions?
| Cheque No. | Date | Transaction Description | Payment | Deposit |
|
384102 275146 |
4/9
12/9 |
Jal Board Deposit | ₹ 120 | ₹ 200 |
|
384103 801351 |
22/9
29/9 |
LIC India Deposit | ₹ 240 | ₹ 150 |
Solution:
Money left in the account after given transactions
= ₹ (500 + 200 + 150 – 120 – 240)
= ₹ (850 – 360) = ₹ 490
Question 119.
(a) Write a positive integer and a negative integer whose sum is a negative integer.
(b) Write a positive integer and a negative integer whose sum is a positive integer.
(c) Write a positive integer and a negative integer whose difference is a negative integer.
(d) Write a positive integer and a negative integer whose difference is a positive integer.
(e) Write two integers which are smaller than – 5 but their difference is – 5.
(f) Write two integers which are greater than -10 but their sum is smaller than -10.
(g) Write two integers which are greater than – 4 but their difference is smaller than – 4.
(h) Write two integers which are smaller than – 6 but their difference is greater than – 6.
(i) Write two negative integers whose difference is 7.
(j) Write two integers such that one is smaller than -11, and other is greater than -11 but their difference is -11.
(k) Write two integers whose product is smaller than both the integers.
(l) Write two integers whose product is greater than both the integers.
Solution:
(a) 2 + (-3)=-l
(b) 3 + (-2) = 1
(c) -1 – (4) = -5
(d) 4 – (-1) = 5
(e) -7 < -5, -12 < -5 and -12 – (-7) = -5
(f) -5 > -10, -6 >-10 and-5 +(-6) = -11 <-10
(g) 2 > -4, -3 > -4 and -3 – 2 = -5 < -4
(h) -7< -6, -8 <-6 and-7-(-8) = -7 + 8 = 1 > -6
(i) -2-(-9) = -2 + 9 = 7
(j) -18 < -11; -7 >-11 and -18 – (-7) = -18 + 7 = -11
(k) (-1) × (2) = -2. Also, -2 < -1 and -2 < 2
(l) 2 × 3 = 6 . Also, 2 < 6 and 3 < 6
Question 120.
What’s the error? Ramu evaluated the expression – 7 – (-3) and came up with the answer – 10 . What did Ramu do wrong?
Solution:
7 – (-3) = -7 + 3 = -4
But -7 – 3 = -10.
∴ Ramu have done addition in place of subtraction.
Question 121.
What’s the error? Reeta evaluated -4 + d for d = – 6 and gave an answer of 2. What might Reeta have done wrong?
Solution:
-4 + d, d = -6
∴ -4 + (-6) = -10
But -4 – (-6) = -4 + 6 = 2
Hence, Reeta have done subtraction in place of addition.
Question 122.
The table given below, shows the elevations relative to sea level of four locations. Taking sea level as zero (0), answer the following questions.
| Location | Elevation (in m) |
| A | -180 |
| B | 1600 |
| C | -55 |
| D | 3200 |
(a) Which location is closest to sea level?
(b) Which location is farthest from sea level?
(c) Arrange the locations from the least to the greatest elevation.
Solution:
(a) C is closest to sea level.
(b) D is farthest from sea level.
(c) Since, -180 < -55 < 1600 < 3200.
∴ The locations from the least to the greatest elevation is A < C < B < D.
Question 123.
You are at an elevation 380 m above sea level as you start a motor ride. During the ride, your elevation changes by the following metres 540 m, -268 m, 116 m, -152 m, 490 m, -844 m, 94 m. What is your elevation relative to the sea level at the end of the ride?
Solution:
Elevation relative to the sea level at the end of the ride
= [380 + 540 – 268 + 116 -152 + 490 – 844 + 94]m
= [380 + 540 + 116 + 490 + 94 – 268 – 152 – 844]m
= [1620 – 1264] m = 356 m
Question 124.
Evaluate the following, using distributive property.
(i) -39 × 99
(ii) (-85) × 43 +43 × (-15)
(iii) 53 × (-9) – (-109) × 53
(iv) 68 × (-17) + (-68) × 3
Solution:
(i) -39 × 99 = -39 × [100 -1]
= -39 × 100 + (-39) × (-1)
= -3900 + 39 = -3861
(ii) (-85) × 43 + 43 × (-15)
= 43 × (-85) + 43 × (-15)
= 43 × [-85 – 15]
= 43 × [-100] = -4300
(iii) 53 × (-9) – (-109) × 53
= 53 × (-9) – 53 × (-109)
= 53 × [(-9) – (-109)]
= 53 × [-9 + 109] = 53 × 100 = 5300
(iv) 68 × (-17) + (-68) × 3 = 68 × (-17) + 68 × (-3)
= 68 × [(-17) + (-3)]
= 68 × (-20) = -1360
Question 125.
If ‘*’ is an operation have, such that for integers a and b. We have a * b = a × b+(a × a + b × b), then find
(i) (-3)*(-5)
(ii) (-6)*2
Solution:
(i) We have, a * b = a × b +(a × a + b × b)
Now, put a = (-3) and b = (-5)
(-3)* (-5) = (- 3) × (- 5)+ [(- 3) × (- 3)+ (- 5) × (- 5)]
= 15 + (9 + 25)= 15 + 34 = 49
(ii) Now, put a = – 6 and b = 2
(-6)*2 = (-6) × 2 +[(-6) × (-6) + 2 × 2
= -6 × 2 + (36 + 4)= -12 + 40= 28
Question 126.
If Δ is an operation such that for integers a and b we have a Δb = a × b – 2 × a × b + b × b (-a) × b + b × b then find
(i) 4 Δ (- 3)
(ii) (- 7)Δ (- 1)
Also show that 4 Δ (- 3) ≠ (- 3) Δ 4 and (-7) Δ (-1) ≠ (-1) Δ (- 7)
Solution:
a Δ b = a × b – 2 × a × b + b × b (-a) × b + b × b
(i) 4 Δ (-3) = 4 × (-3) – 2 × 4 × (-3)
+ (-3) × (-3)(-4) × (-3) + (-3) × (-3) = -12 + 24 + 108 + 9 = -12 + 141 = 129
(ii) (-7) Δ (-1) = (-7) × (-1) – 2 × (-7) × (-1)
+ (-1) × (-1) (7) × (-1) + (-1) × (-1) = 7-14- 7 + 1 = 8-21= -13
Now, (-3) Δ 4 = (-3) × 4 – 2 × (-3) ×(4) + 4 × 4(3) × 4 + 4 × 4
= -12 + 24 + 192 + 16 = -12 + 232 = 220
But 4 Δ (-3) = 129
∴ 4 Δ (-3) ≠ (-3) A 4
And, (-1) Δ (-7) = (-1) × (-7) – 2 × (-1) × (-7) + (-7) × (-7)(1) × (-7) + (-7) × (-7)
= 7 -14 – 343 + 49 = 56 – 357 = -301
But (-7) Δ (-1) = -13
∴ (-7) Δ (-1) ≠ (-1) Δ (-7)
Question 127.
Below u, v, w and x represent different integers, where u = (-4) and x ≠ 1. By using following equations, find each of the values
u x v=u
x × w =w
u + x = w
(a) v
(b) w
(c) x
Explain your reason, using the properties of integers
Solution:
(a) As u × v = u and u = -4 ∴ -4 × v = -4
⇒ v = l (Multiplicative identity)
(b) As x × w = w. Given that x ≠ 1
∴ x × w = w is only possible when w = 0
(c) As u + x = w, Put u = – 4 and w = 0
∴ -4 + x = 0
⇒ x = 4 (Transposing -4 to R.H.S.)
Question 128.
Height of a place A is 1800 m above sea level. Another place B is 700 m below sea level. What is the difference between the levels of these two places?
Solution:
Difference between the levels of places A and B is [1800 – (-700)] m = [1800 + 700] m = 2500 m
Question 129.
The given table shows the freezing points in °F of different gases at sea level. Convert each of these into °C to the nearest integral value using the relation and complete the table,
| Gas | Freezing Point at Sea Level (°F) | Freezing Point at Sea Level (°C) |
| Hydrogen | -435 | |
| Krypton | -251 | |
| Oxygen | -369 | |
| Helium | -458 | |
| Argon | -309 |
\(c=\frac{5}{9}(F-32)\)
Solution:
Question 130.
Sana and Fatima participated in an apple race. The race was conducted in 6 parts. In the first part, Sana won by 10 seconds. In the second part, she lost by 1 min, then won by 20 seconds in the third part and lost by 25 seconds in the fourth part, she lost by 37 seconds in the fifth part and won by 12 seconds in the last part. Who won the race finally?
Solution:
Taking winning by time be a positive integer and losing by time be a negative integer.
∴ Sana’s total time (winning/losing)
= (10 – 60 + 20 – 25 – 37 + 12) seconds
= (42 – 122) seconds = -80 seconds
Hence, Sana lost the race by 80 seconds or 1 min. 20 seconds.
i.e., Fatima won the race finally.
Question 131.
A green grocer had a profit of ₹ 47 on Monday, a loss of ₹ 12 on Tuesday and loss of ₹ 8 on Wednesday. Find his net profit or loss in 3 days.
Solution:
Taking profit as a positive integer and loss as negative integer, we have
grocer’s net profit or loss in 3 days
= ₹(47-12-8) = ₹27
Hence, the grocer has a profit of ₹ 27.
Question 132.
In a test, +3 marks are given for every correct answer and -1 mark are given for every incorrect answer. Sona attempted all the questions and scored +20 marks, though she got 10 correct answers.
(i) How many incorrect answers has she attempted?
(ii) How many questions were given in the test?
Solution:
(i) Total marks scored by Sona = 20
Total correct answers = 10
∴ Marks for correct answers = 10 × 3 = 30
but she got 20 marks.
∴ Marks for incorrect answers = 20 – 30 = – 10
-1 mark is given for every incorrect answer.
∴ Total incorrect answers \(=\frac{-10}{-1}=10\)
(ii) Total correct answers = 10
Total incorrect answers = 10 (From (i) part)
∴ Total questions given in the test = 10 + 10 = 20
Question 133.
In a true-false test containing 50 questions, a student is to be awarded 2 marks for every correct answer and -2 for every incorrect answer and 0 for not supplying any answer. If Yash scored 94 marks in a test, what are the possibilities of his marking correct or wrong answer?
Solution:
Yash secured = 94 marks So, minimum correct answers = 94 ÷ 2 = 47
Now, there are two possibilities :
(1) He attempted 47 correct answers and 3 unattempted.
(2) He attempted 48 correct and 1 unattempted and 1 wrong answer.
Question 134.
A multistory building has 25 floors above the ground level each of height 5 m. It also has 3 floors in the basement each of height 5m. A lift in building moves at a rate of lm/s. If a man starts from 50m above the ground, how long will it take him to reach at 2nd floor of basement?
Solution:
Height of each floor = 5 m
∴ Height below the basement to be
covered = 2 × 5m = 10m
If a man starts from 50 m above ground
level and reach at 2nd floor of basement.
∴ His total distance to be covered
= (50 + 10) m = 60 m
Rate of moving of a lift = 1 m/s
∴ A man reach at 2nd floor of basement in 1 × 60 = 60 seconds or 1 minute.
Question 135.
Taking today as zero on the number line, if the day before yesterday is 17 January, what is the date 3 days after tomorrow?
Solution:
A day before yesterday is 17 January.
∴ Today is 19 January.
The date 3 days after tomorrow will be 20 January + 3 days = 23 January
Question 136.
The highest point measured above sea level is the summit of Mt. Everest, which is 8,848 m above sea level and the lowest point is challenger deep at the bottom of Mariana Trench which is 10,911 m below sea level. What is the vertical distance between these two points?
Solution:
The highest point (above sea level) = 8,848 m
The lowest point (below sea level) = 10,911 m
∴ Total vertical distance between two points
= [8,848 – (-10,911)] m
= [8,848 + 10,911] m = 19,759 m
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