Math Ncert Exemplar 2019 Solutions for Class 7 Maths Chapter 1 - Integers

Answer:

Arranging the numbers in ascending order, we get
–7, –5, 0, 5, 10
Arranging the numbers in descending order, we get
10, 5, 0, –5, –7
Thus, in both the arrangments, 0 remains in the middle.

Hence, the correct answer is option (a).

Answer:

(a) B is to the right of –10. Thus, B is greater than –10.
(b) A is to the right of 0. Thus, A is greater than 0.
(c) A is to the right of B. Thus, A is greater than B.
(d) B is smaller than 0.

Hence, the correct answer is option (c).

Answer:

From the number line, we get that A is 5 and B is –4.

Hence, the correct answer is option (d).

Answer:

The given pattern is 11, 8, 5, 2, ..., ..., ...
Now,
8 – 11 = –3
5 – 8 = –3
2 – 5 = –3
This means that the next number is 3 less than the previous number.
2 – 3 = –1
–1 – 3 = –4
–4 – 3 = –7
So, the three consecutive numbers of the pattern are –1, –4 and –7.

Hence, the correct answer is option (d).

Answer:

The given pattern is –62, –37, –12, ...,
Now,
–62 – (37) = –62 + 37 = –25
–37 – (–12) = –37 + 12 = –25
So, the next number is 25 more than –12.
Thus, the next number is given as!
–12 – (–25) = –12 + 25 = 13
The next number in the pattern is 13.

Hence, the correct answer is option (d).

Answer:

Consider a positive integer 5 and a negative integer –2
Their sum = 5 + (–2)
= 5 – 2
= 3, which is negative
Now, consider the positive integer as 2 and negative integer as –5.
Their sum = 2 + (–5)
= –3, which is negative
Therefore, the sum of a positive and negative integer may or may not be negative.

Hence, the correct answer is option (c).

Answer:

From the number line, we see that there are 5 divisions from –15 to 10.
Now,
Their difference = –15 – (–10)
= –25
So, a difference of 25 is shown by 5 division on the number line.
Thus, each division on the number line represents 5 units. So,

Therefore, point 2 represents 0.

Hence, the correct answer is option (c).

Answer:

The number on the right is always greater in value.
Therefore, the descending order is as follows:
$\overline{)}, \overline{)✔}, \overline{)\times }, \mathbf{•}$

Hence, the correct answer is option (c).

Answer:

(–3) × 3 = –9
The number on right is always greater in value.
And –9 is greater than –10.
Thus, –9 lies on the right hand side of –10.

Hence, the correct answer is option (a).

Answer:

$\begin{array}{rcl}5÷(-1)& =& \frac{5}{-1}\\ & =& -5\end{array}$
Now, –5 does not lie between 0 and 10.

Hence, the correct answer is option (b).

Answer:

Wall of the well = 1 m 20 cm
= 1.2 m
Height of the pulley = 80 cm
= 0.8 m
Now,
Initial water level in the well = –20m
After rising, the water level becomes = –20 + 5
= –15 m
So, total height above ground = 1.2 m + 0.8 m
= 2 m
And total height below ground = 15 m
Thus,
Minimum length of rope = 15 + 2
= 17 m

Hence, the correct answer is option (a).

Answer:

We are given (–11) × 7.
Now,
(–11) × 7 = –77
Also,
(a) 11 × (–7) = –77
(b) –(11 × 7) = –(77) = –77
(c) (–11) × (–7) = 77
(d) 7 × (–11) = –77

Hence, the correct answer is option (c).

Answer:

(–10) × (–5) + (–7) = (10 × 5) + (–7)
= 50 + (–7)
= 50 – 7
= 43

Hence, the correct answer is option (d).

Answer:

The additive inverse of a is (–a).
Now,
(a) –(–a) = a
(b) a × (–1) = –a
(c) –a
(d) $\begin{array}{rcl}a÷(-1)& =& \frac{a}{-1}\\ & =& -a\end{array}$

Answer:

The multiplicative identify of an integer a is 1.

Hence, the correct answer is option (b).

Answer:

We have,
[(–8) × (–3)] × (–4) = 24 × (–4)
= –96
(a) (–8) × [(–3) × (–4)] = –8 × (12)
= –96
(b) [(–8) × (–4)] × (–3) = 32 × (–3)
= –96
(c) [(–3) × (–8)] × (–4) = 24 × (–4)
= –96
(d) (–8) × (–3) – (–8) × (–4) = 24 – 32
= –8 = $\ne$–96

Answer:

We have,
(–25) × [6 + 4] = –25 × 10
= –250
Now,
(a) (–25) × 10 = –250
(b) (–25) × 6 + (–25) × 4 = –150 + (–100)
= –150 – 100
= –250
(c) –25 × 6 × 4 = (–25) × 24
= –600 ≠ –250
(d) –250

Hence, the correct answer is option (d).

Answer:

We have,
–35 × 107 = –35 × (100 + 7)
= (–35 × 1000 + (–35 × 7)
Also,
–35 × 107 = (–30 – 5) × 107

Hence, the correct answer is option (c).

Answer:

We have,
(–43) × (–99) + 43 = 43 [(–1) × (–99) + 1]
= 43[99 + 1]
= 43 × 100
= 4300

Hence, the correct answer is option (a).

Answer:

We have,
$$\begin{gathered} (-16)÷4=\frac{-16}{4} \\ =-4 \\ \mathrm{But} (-4)÷16=\frac{-4}{16} \\ =\frac{1}{4}\ne -4 \end{gathered}$$

Hence, the correct answer is option (a).

Answer:

(a) 0 ÷ (–7) = 0, which is an integer
(b) $20÷(-4)=\frac{20}{-4}$
= –5, which is an integer
(c) $(–9)÷3=\frac{-9}{3}$
= –3, which is an integer
(d) $(–12)÷5=\frac{-12}{5}$
= –2.4, which is not an integer

Hence, the correct answer is option (d).

Answer:

(a) 20 + (–25) = 20 – 25
= –5
(b) (–37) – (–32) = –37 + 32
= –5
(c) (–5) × (–1) = 5
(d) $\begin{array}{rcl}45÷(-9)& =& \frac{45}{-9}\\ & =& -5\end{array}$

Hence, the correct answer is option (c).

Answer:

(a) Rise in temperature = 32º – 23º
= 9º
(b) Rise in temperature = 1º – (–10)º
= 1º + 10º
= 11º
(c) Rise in temperature = –11º – (–18º)
= –11º + 18º
= 7º
(d) Rise in temperature = 5º – (–5º)
= 5º + 5º
= 10º
Thus, the maximum rise in temperature is from –10º to +1º.

Hence, the correct answer is option (b).

Answer:

Given that, a and b are two integers.
Let a = 2, and b = –5.
(a) a + b = 2 + (–5)
= 2 – 5
= –3, which is an integer.
(b) a – b = 2 – (–5)
= 2 + 5
= 7, which is an integer
(c) a × b = 2 × (–5)
= –10, which is an integer
(d) a $÷$b = 2$÷$ (–5)
$\frac{2}{-5}$
= –4, which is not an integer.

Hence, the correct answer is option (d).

Answer:

We know that division of any number by 0 is not defined.
Therefore, a ÷ 0 is not defined.

Hence, the correct answer is option (a).

Answer:

(a) –3 + 3 = 0
(b) –5 + 5 = 0
(c) –6 + 1 = –5
(d) –8 + 8 = 0

Hence, the correct answer is option (c).

Answer:

(a) –1 + (–2) = –1 – 2
= –3
(b) –5 + (+2) = –5 + 2
= –3
(c) –4 + (+1) = –4 + 1
= –3
(d) –9 + (+7) = –9 + 7
= –2

Hence, the correct answer is option (d).

Answer:

(a) (–9) × 5 × 6 × (–3) = –[–(9 × 5 × 6 × 3]
= 9 × 5 × 6 × 3
= 810
(b) 9 × (–5) × 6 × (–3) = –[–(9 × 5 × 6 × 3)]
= 9 × 5 × 6 × 3
= 810
(c) (–9) × (–5) × (–6) × 3 = –{–[–(9 × 5 × 6 × 3)]}
= –(9 × 5 × 6 × 3)
= –810
(d) 9 × (–5) × (–6) × 3 = [–(9 × 5 × 6 × 3)
= 810

Hence, the correct answer is option (c).

Answer:

(a) (–100) $÷$ 5 = $\frac{-100}{5}$
= –20
(b) (–81) $÷$ 9 = $\frac{-81}{9}$
= –9
(c) (–75) $÷$ 5 = $\frac{-75}{5}$
= –15
(d) (–32) $÷$ 9 = $\frac{-32}{9}$
= –3.555, which is not an integer.

Hence, the correct answer is option (d).

Answer:

(a) (–1) × (–1) = –(–1) = 1
(b) (–1) × (–1) × (–1) = –(–1) × (–1)
= 1 × (–1)
= –1
(c) (–1) × (–1) × (–1) × (–1) = [–(–1)] × [–(–1)]
= 1 × 1
= 1
(d) (–1) × (–1) × (–1) × (–1) × (–1) × (–1) = [–(–1)] × [–(–1)] × [–(–1)]
= 1 × 1 × 1
= 1

Hence, the correct answer is option (b).

Answer:

(–a) + b = b + (–a)
= b + Additive inverse of a.

Answer:

_________ $÷$ (–10) = 0
A number divided by (–10) will result in 0 only if that number is 0 itself so,
___0____ $÷$ (–10) = 0

Answer:

We have,
(–157) × (–19) + 157 = 157 [–(–19) + 1]
= 157 [19 + 1]
= 157 × 20
= 3140

Hence, (–157) × (–19) + 157 = 3140_.

Answer:

[(–8) + _____ ] + ________ = _________ + [(–3) + _________] = –3
Consider [(–8) + ________] + __________ = – 3
Now,
[(–8) + (–3)] + 8 = (–8) + (–3) + 8
= [–8 + 8] + (–3)
= 0 + –3
= –3
Now, consider ______ + [(–3) + _______] = –3 _______ + [(–3) + _______] = (–8) + [(–3) + 8]
= (–8) + (–3) + 8
= (–8 + 8) + (–3)
= 0 + (–3)
= –3

Hence, [(–8) + (–3)] + 8 = (–8) + [(–3) + 8]

Answer:

We have,
(–4) × 3 = –12
From the number line, we see that there are 11 divisions from –20 to 2.
Now,
Their difference = 2 – 2 (–20)
= 2 + 20
= 22
So, a difference of 2 is shown by 11 divisions on the number line.
Thus, each division on the number line represents 2 units. So, (–12) is shown as follows on the number line:

Thus, D represents –12 on the given number line.

Answer:

Given that, x, y and z are integers.
Now,
(x + y) + z = x + (y + z)
This is because integers follows the associative property.

Answer:

(–43) + _______ = –43
The only number added to another number and resulting in the same number is 0. Thus,
(–43) + ___0____ = –43

Answer:

(–8) + (–8) + (–8) = ________ × (–8)
Now,
(–8) + (–8) + (–8) = –8 – 8 – 8
= – 24
= 3 × (–8)
Hence, (–8) + (–8) + (–8) =  3 × (–8).

Answer:

We have,
11 × (–5) = –(11 × 5)
= –55
So, 11 × (–5) = –(11 × 5) =  –55 

Answer:

(– 9) × 20 =   –180  

Answer:

(– 23) × (42) = (– 42) × _____
Using the commulative property of integers,
(–23) × 42 = (–42) ×  23  

Answer:

While multiplying a positive integer and a negative integer we multiplu them as whole numbers and put a negative sign before the product.

Answer:

If we multiply even number of negative integers, then the resulting integer is positive.

Answer:

If we multiply six negative integrs and six positive integers, then the resulting integer is positive.

Answer:

If we multiply five positive integers and one negative integer, then the resulting integer is negative.

Answer:

    1     is the multiplicative identity for integers.

Answer:

We get additive inverse of an integer a when we multiply it by    –1   .

Answer:

(– 25) × ( – 2) =    50   

Answer:

(– 5) × ( – 6) × ( – 7) = [(–5) × (–6)] × (–7)
= 30 × (–7)
= –210

Answer:

3 × ( – 1) × ( – 15) =  3 × [(–1) × (–15)]  
= 3 × 15
= 45

Answer:

[12 × ( – 7)] × 5 =    12    × [(– 7) ×     5     ]
Associative property

Answer:

23 × ( – 99) = ________× ( – 100 +________ ) = 23 ×________ + 23 ×________

23 × ( – 99) = 23 × ( –100 +   1  )
= 23 ×   (–100)   + 23 ×   1  
(Distributive property)

Answer:

35 × (–1) = –35

Answer:

(–47) × (–1) = 47

Answer:

88 × (–1) = – 88

Answer:

 (–1) × (–93) = 93

Answer:

(–40) × (–2) = 80

Answer:

  40  × (–23) = –920

Answer:

When we divide a negative integer by a positive integer, we divide them as whole numbers and put a negative sign before quotient.

Answer:

−4

Let −16 be divided by x and quotient is 4.
$$\begin{gathered} \Rightarrow \frac{-16}{x}=4 \\ \Rightarrow x=-4 \end{gathered}$$

Answer:

Division is the inverse operation of multiplication.

Answer:

$\begin{array}{rcl}65÷(–13)& =& \frac{65}{\left(-13\right)}\\ & =& -5\end{array}$

Answer:

$\begin{array}{rcl}(–100)÷(–10)& =& \frac{-100}{-10}\\ & =& 10\end{array}$

Answer:

$\begin{array}{rcl}\left(-225\right)÷5& =& \frac{-225}{5}\\ & =& -45\end{array}$

Answer:

83 ÷ (–1) = –83

Answer:

(–75) ÷ (–1) = 75

Answer:

51 ÷ (−1) = –51

Answer:

113 ÷ (−113) = – 1

Answer:

(–95) ÷  (–1)  = 95

Answer:

(–69) ÷ (69) =    â€‹–1    

Answer:

(–28) ÷ (–28) =     1    

Answer:

5 – ( – 8) = 5 + 8
Hence, the statement is true.

Answer:

$\begin{array}{rcl}(–9)+(–11)& =& –9–11\\ & =& -20\end{array}$

$\begin{array}{rcl}–9-(–11)& =& –9+11\\ & =& 2\end{array}$

So, – 9 – (–11) > (– 9) + (–11).
Hence, the statement is false.

Answer:

Consider two negative integers, −4 and −2.

$\begin{array}{rcl}\mathrm{Their} \mathrm{sum}& =& -4+\left(-2\right)\\ & =& -4-2\\ & =& -6\end{array}$
And
−6 < −2 and −6 < −4.
Hence, the statement is true.

Answer:

Let the two negative integers be −1 and −3.

$\begin{array}{rcl}\mathrm{Their} \mathrm{difference}& =& -1-\left(-3\right)\\ & =& -1+3\\ & =& 2, \mathrm{which} \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{integer}\end{array}$

Hence, the statement is false.

Answer:

The sum of two integers is always an integer.
Hence, the statement is false.

Answer:

The difference of two integers is always an integer.
Hence, the statement is true.

Answer:

$\begin{array}{rcl}(–23)+47& =& 47–23\\ & =& 24\end{array}$

$\begin{array}{rcl}\mathrm{And} 47+\left(-23\right)& =& 47-23\\ & =& 24\\ & =& \left(-23\right)+47\end{array}$

Hence, the statement is true.

Answer:

The order of integers does not affect their sum.
Hence, the statement is true.

Answer:

The order of integers affects their difference.
Let a = −2 and b = 3.
So,
$\begin{array}{rcl}a-b& =& -2-\left(3\right)\\ & =& -2-3\\ & =& -5\end{array}$
$\begin{array}{rcl}b-a& =& 3-\left(-2\right)\\ & =& 3+2\\ & =& 5\ne -5\end{array}$
Hence, the statement is false.

Answer:

Let the east direction be taken as positive and west direction be taken as negative.
Going 500 m east and 200 m back is given as 500 − 200 = 300 m east
Going 200 m west and 500 m back is given as : − 200 + 500 = 300 m east
Thus, they both are the same.
Hence, the statement is true.

Answer:

(– 5) × 33 = –165
And 
5 × ( – 33) = –165
⇒(– 5) × 33 = 5 × (– 33)
Hence, the statement is true.

Answer:

$\begin{array}{rcl}\mathrm{(–19) \times (–11)}& =& –\left[\mathrm{–(19 \times 11)}\right]\\ & =& 19\times 11\end{array}$
Hence, the statement is true.

Answer:

$\begin{array}{rcl}(–20)\times (5–3)& =& (–20)\times 2\\ & =& -40\end{array}$
$\begin{array}{rcl}\left(-20\right)\times \left(-2\right)& =& -\left[-\left(20\times 2\right)\right]\\ & =& 40\end{array}$
⇒ $\left(-20\right)\times \left(5-3\right)\ne \left(-20\right)\times \left(-2\right)$
Hence, the statement is false.

Answer:

4 × (–5) = (–20)
$\begin{array}{rcl}\left(-10\right)\times \left(-2\right)& =& -\left[-\left(10\times 2\right)\right]\\ & =& 20\end{array}$
$\Rightarrow 4\times \left(-5\right)\ne \left(-10\right)\times \left(-2\right)$
Hence, the statement is fales.

Answer:

$\begin{array}{rcl}(–1)\times (–2)\times (–3)& =& -\left\{-\left[-\left(1\times 2\times 3\right)\right]\right\}\\ & =& -\left(1\times 2\times 3\right)\\ & =& -6\\ & & \end{array}$
$1\times 2\times 3=6$
$\Rightarrow \left(-1\right)\times \left(-2\right)\times \left(-3\right)\ne 1\times 2\times 3$
Hence, the statement is false.

Answer:

(–3) × 3 = –9
$\begin{array}{rcl}-12-\left(-3\right)& =& -12+3\\ & =& -9\end{array}$
$\Rightarrow \left(-3\right)\times 3=-12-\left(-3\right)$
Hence, the statement is true.

Answer:

Let the two negative integers be −2 and −3.

$\begin{array}{rcl}\mathrm{Their} \mathrm{product}& =& \left(-2\right)\times \left(-3\right)\\ & =& 6\end{array}$

Hence, the statement is false.

Answer:

Let the three negative integers be −2, −3 and −4.

$\begin{array}{rcl}\mathrm{Their} \mathrm{product}& =& \left(-2\right)\times \left(-3\right)\times \left(-4\right)\\ & =& -\left\{-\left[-\left(2\times 3\times 4\right)\right]\right\}\\ & =& -24, \mathrm{which} \mathrm{is} \mathrm{negative}\end{array}$

Hence, the statement is true.

Answer:

Consider a negative number −2 and a positive number 4.

$\begin{array}{rcl}\mathrm{Their} \mathrm{product}& =& \left(-2\right)\times 4\\ & =& -8, \mathrm{which} \mathrm{is} \mathrm{negative}\end{array}$

Hence, the statement is false.

Answer:

Consider two integers −3 and 2.

$\begin{array}{rcl}\mathrm{Their} \mathrm{product}& =& \left(-3\right)\times 2\\ & =& -6\end{array}$

Now,
−6 < −3 and −6 < 2
Hence, the statement is false.

Answer:

Multiplication of two integers will always result in an integer. Therefore, integers are closed under multiplication.
Hence, the statement is true.

Answer:

Multiplication of a number with 0 will always result in 0.
So,
$\left(-237\times 0=0\times \left(-39\right)\right)$
Hence, the statement is true.

Answer:

Consider two integers, −2 and 3.
Now,
$\begin{array}{rcl}\left(-2\right)\times 3& =& -6\\ & =& 3\times \left(-2\right)\end{array}$

Thus, multiplication of integers is commutative.
Hence, the statement is false.

Answer:

The multiplication identity for integers is 1 as for any integer a,
$a\times 1=1\times a=a$
Hence, the statement is false.

Answer:

We have,
$99\times 101=\left(100-1\right)\times \left(100+1\right)$
Hence, the statement is true.

Answer:

We have, three integers - a, b and c.
and b ≠ 0.
Now,
$a\times \left(b-c\right)=a\times b-a\times c$ [Distributivity of multiplication over subtraction]
Hence, the statement is true.

Answer:

LHS = (a + b) × c
         = (a × c) + (b × c)
         ≠ a × c + a × b

Thus, LHS ≠ RHS
So, the statement is False.
 

Answer:

LHS = a × b 
         = b × a        (Commutative property)
         = RHS 

Thus, LHS = RHS
So, the statement is True.

Answer:

LHS =  a ÷ b 
         ≠ b ÷ a          (∵ Division is not commutative for integers)
        
Thus, LHS ≠ RHS
So, the statement is False.

Answer:

LHS =  a – b 
         = –(b – a) 
         ≠ b – a          (∵ Subtraction is not commutative for integers)
        
Thus, LHS ≠ RHS
So, the statement is False.

Answer:

$\begin{array}{rcl}\mathbf{LHS}& =& a÷\left(-b\right)\\ & =& \frac{a}{\left(-b\right)}\\ & =& -\frac{a}{b}\\ & =& -a÷\left(b\right)\\ & =& \mathbf{RHS}\end{array}$
        
Thus, LHS = RHS
So, the statement is True.

Answer:

$\begin{array}{rcl}\mathbf{LHS}& =& a÷\left(-1\right)\\ & =& \frac{a}{\left(-1\right)}\\ & =& -\frac{a}{1}\\ & =& \frac{-a}{1}\\ & & -a\\ & =& \mathbf{RHS}\end{array}$
        
Thus, LHS = RHS
So, the statement is True.

Answer:

Multiplication fact (–8) × (–10) = 80
(–8) × (–10) = (–8 × –10)
⇒ (–8) × (–10) = –(–80)
⇒ (–8) × (–10) = 80
⇒ [(–8) × (–10)] ÷ (–8) = 80 ÷ (–8)
$\Rightarrow \frac{\left(-8\right)\times \left(10\right)}{\left(-8\right)}=80÷\left(-8\right)$
​⇒ (–10) = 80 ÷ (–8)

Thus, Multiplication fact (–8) × (–10) = 80 is same as division fact 80 ÷ (– 8) = (–10).
So, the statement is True.
 

Answer:

For example: (−5) ÷ 2


And, −2.5 is not an integer.

Thus, Integers are not closed under division.
So, the statement is False.

Answer:

LHS = [(–32) ÷ 8 ] ÷ 2 
= [–4] ÷ 2 
= –2

RHS = –32 ÷ [ 8 ÷ 2]
= –32 ÷ [4]
= –(32 ÷ 4)
​= –(8)
= –8 
Thus, LHS ≠ RHS
So, the statement is False.

Answer:

Let any integer be a.
Its additive inverse is -a.
a + (-a) = a – a = 0
So, the statement is True.

Answer:

Since, 1 × (–25)  = –25 

And, 0 × (–25) = 0
Now, the successor of 0 × (–25) = [0 × (–25)] + 1 
= 0 + 1
= 1
≠ 1 × (–25) 

So, the statement is False.

Answer:

(a) – 5 × 4 = – 20
     – 5 × 3 = – 15 = –20 – ( – 5)
     – 5 × 2 =  – 10 = – 15 – ( –5)
     – 5 × 1 =  – 5 = 0 – 5
     – 5 × 0 = 0 = 5 – 5
     – 5 × – 1 = 5 = 0 – ( –5) 
     – 5 × – 2 = – 10  = 5 – ( –5)

(b) 7 × 4 = 28
     7 × 3 =   21 = 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 

Answer:

(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.

Answer:

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

Answer:

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.

Answer:

Since,
−129° < −90° < −81° < −67° < −27° < −11° < −9°
Hence, the order of the continents from the lowest to the highest recorded temperature is
Antarctica, Asia, North America, Europe, South America, Africa, and Australia.

Answer:

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.

Answer:

–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

So,
–​5 → 3
6 → –2
–7 → 1
8 → –1

Answer:

Let the integers be 12 and –3
Then, their product = 12 × (–3) = –36,
And their difference = 12 – (–3) = 12 + 3 = 15
∴ A pair of integers whose product is – 36 and whose difference is 15 is (–3, 12).

Answer:

(a) a × 1 = a
∴ (a) → (vi)

(b) 1 is Multiplicative identity.
∴ (b) → (iii)

(c) ( – a) ÷ ( – b) =  a ÷ b
∴ (c) → (v)
 
(d) a × ( – 1) = – a
∴ (d) → (vii)

(e) a × 0 = 0
∴ (e) → (viii) 

(f) ( –a) ÷ b = a ÷ ( – b)
∴ (f) → (iv)

(g) 0 is the Additive identity
∴ (f) → (ii)

(h) a ÷ (–a) = –1
∴ (h) → (ix)  

(i) –a is Additive inverse of a
∴ (i) → (i) 



 

Answer:

Money left in the account after given transactions = ₹(500 + 200 + 150 – 120 – 240)
= ₹(850 – 360)
= ₹490

Thus, after these transactions ₹490 is left in the account.

Answer:

(a)  Positive integer = 4 
Negative integer = –9
Their sum = 4 + (–9) = 4 – 9 = –5
Thus, a positive integer and a negative integer whose sum is a negative integer are 4 and –9.

(b) Positive integer = –4 
Negative integer = 9
Their sum = –4 + (9) = –4 + 9 = 5
Thus, a positive integer and a negative integer whose sum is a positive integer –4 and 9.

(c) Positive integer = 7 
Negative integer = –5
Their sum = –5 – (7) = –5 – 7 = –12
Thus, a positive integer and a negative integer whose difference is a negative integer are –7 and 5.

(d) Positive integer = 7 
Negative integer = –9
Their sum = 7 – (–9) = 7 + 9 = 16
Thus, a positive integer and a negative integer whose difference is a positive integer are 7 and –9.

(e) Two integers which are smaller than – 5 are –10 and – 15 such that, 

Their difference = –15 – (– 10) = –15 + 10 = – 5
Thus, two integers which are smaller than – 5 but their difference is – 5 are –10 and – 15.

(f) Two integers which are greater than –10 are –9 and 1 such that, 
Their sum = –9 – (1) = – 9 – 1 = – 10
Thus, two integers which are greater than – 10 but their sum is smaller than – 10.

(g) Two integers which are greater than – 4 are 3 and –2 such that, 
Their difference = –2 – (3) = –2 – 3 = –5
Thus, two integers which are greater than – 4 but their difference is smaller than – 4 are 3 and –2.

(h) Two integers which are smaller than – 6 are –7 and 5 such that
Their difference = 5 – (–7) = 5 + 7 = 12
 Thus, two integers which are smaller than – 6 but their difference is greater than – 6 are –7 and 5.

(i) Two negative integers are –13 and –20.
Their difference = –13 – (–20) = –13 + 20 = 7
Thus, two negative integers whose difference is 7 are –13 and –20.

(j) Integer smaller than –11 = –12,
Integer greater than –11 = –1
Their difference = –12 –(–1) = –12 + 1 = –11.
Thus, two integers such that one is smaller than –11, and other is greater than –11 but their difference is –11 are –12 and –1.

(k) Two integers are –3 and 5 
Their product = –3 × 5 = –15
Thus, two integers whose product is smaller than both the integers are –3 and 5.

(l) Two integers are 3 and 5
Their product = 3 × 5 = 15 
Thus, two integers whose product is greater than both the integers are 3 and 5. 

Answer:

7 − (−3) = −7 + 3 = −4
But Ramu evaluated the expression –7 – (–3) as 10 i.e., −7 − 3 = −10.
∴ Ramu has done addition in place of subtraction.

Answer:

Given: −4 + d, d = −6
∴  −4 + (−6) = −10
But −4 − (−6) = −4 + 6 = 2
Hence, Reeta has done subtraction in place of addition.

Answer:

(a) C is closest to sea level.
(b) D is farthest from sea level.
(c) Since, –180 < –55 < 1600 < 3200.
∴ The lowest to greatest elevation locations are A < C < B < D.

Answer:

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

Answer:

(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

Answer:

(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

Answer:

Given: a âˆ† 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
                   = 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) Δ 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)

Answer:

(a) u × v = u and u = –4
∴ –4 × v = –4
⇒ v = l (Multiplicative identity)

(b) x × w = w. Given that x ≠ 1
∴ x × w = w is only possible when w = 0

(c) u + x = w, Putting u = – 4 and w = 0
∴ –4 + x = 0
⇒ x = 4

Answer:

Given: Height of place A above sea level = 1800 m
Height of place B below sea level = 700 m
The difference between the levels of places A and B = [1800 − (−700)] m
                                                                                    = [1800 + 700] m
                                                                                    = 2500 m

Thus, the difference between the levels of these two places is 2500 m.

Answer:

The relation between Celsius (°C) and Farahnheight (°F) is $\mathrm{C}=\frac{5}{9}(\mathrm{F}-32)$

(i) For Hydrogen 
Freezing Point at Sea Level (°F) = –435
Freezing Point at Sea Level (°C) $=\frac{5}{9}(-435-32)$
$$\begin{gathered} =\frac{5}{9}(-467) \\ =-259.44 \end{gathered}$$

(ii) For Krypton 
Freezing Point at Sea Level (°F) = –251
Freezing Point at Sea Level (°C) $=\frac{5}{9}(–251-32)$
$$\begin{gathered} =\frac{5}{9}(-283) \\ =-157.22 \end{gathered}$$

(iii) For Oxygen 
Freezing Point at Sea Level (°F) = –369
Freezing Point at Sea Level (°C) $=\frac{5}{9}(–369-32)$
$$\begin{gathered} =\frac{5}{9}(-401) \\ =-222.78 \end{gathered}$$


(iv) For Helium 
Freezing Point at Sea Level (°F) = –458
Freezing Point at Sea Level (°C) $=\frac{5}{9}(–458-32)$
$$\begin{gathered} =\frac{5}{9}(-490) \\ =-272.22 \end{gathered}$$

(v) For Argon 
Freezing Point at Sea Level (°F) = –309
Freezing Point at Sea Level (°C) $=\frac{5}{9}(–309-32)$
$$\begin{gathered} =\frac{5}{9}(-341) \\ =-189.44 \end{gathered}$$

 

Answer:

Considering 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

∴ Sana lost the race by 80 seconds or 1 min. 20 seconds.
Thus, Fatima won the race finally.

Answer:

Profit on Monday = ₹47
Loss on Tuesday = â‚¹12
Loss on Wednesday = â€‹â‚¹8
His net profit or loss in 3 days = â‚¹47 − ₹12 − â€‹â‚¹8
                                                 = â‚¹27

Thus, his net profit or loss in 3 days is â‚¹27.

Answer:

(i) Total marks scored by Sona = 20
Total correct answers = 10
∴ Marks for correct answers = 10 × 3 = 30
Marks scored by her = 20
∴ 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

Answer:

Yash secured = 94 marks
∴ Minimum correct answers = 94 ÷ 2 = 47

There are two possibilities:
(1) He attempted 47 correct answers and 3 unattempted ones.
(47 × 2) + (0 × −2) + (1 × 0)
= 94 + 0 + 0
= 94 

(2) He attempted 48 correct and 1 unattempted and 1 wrong answer.
(48 × 2) + (1 × −2) + (1 × 0)
= 96 + (−2) + 0
= 94 

Answer:

Height of each floor = 5 m
∴ Height below the basement to be covered = 2 × 5 m = 10 m
His total distance to be covered = (50 + 10) m = 60 m
Rate of moving of a lift = 1 m/s
∴ A man reaches at 2nd floor of the basement in 1 × 60 = 60 seconds or 1 minute.

Answer:

The day before yesterday is 17 January.
∴ Today is 19 January.
⇒ Tomorrow will be 20 January.
The date 3 days after tomorrow = 20 January + 3 days = 23 January

Answer:

The highest point (above sea level) = 8,848 m
The lowest point (below sea level) = 10,911 m
∴ The total vertical distance between two points
= [8,848 − (−10,911)] m
= [8,848 + 10,911] m
= 19,759 m