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MATHS 1 Marks Question

Integers · Bihar Board (BSEB) STD 7 (BSEB - English Medium). 82 questions.

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Question 11 Mark
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. How many questions were given in the test$?$
Answer
Let $x$ be the correct answers and $y$ be the incorrect answers, given by Sona. It is given that, if she gives $10$ correct answers and her score is $20.$ Since, for every correct answer, $+3$ is given and for every incorrect answer, $-1.$ is given. Hence, Total number od question $=$ Correct anwser + Incorrect answer $= x + y = 10 + 10 = 20$
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Question 21 Mark
$(-157) \times (-19) + 157 =$ _____.
Answer
$(-157) \times (-19) + 157 = (-1) \times (157) \times (-19) + 157.$
Solution:
$(-157) \times (-19) + 157 = (-1) \times (157) \times (-19) + 157 = 157\{-(-19) + 1\}$
$ [$taking $157$ as common$] = 157\{19 + 1\} = 157 \times 20 = 3140$
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Question 31 Mark
$a ÷ (-b) = - (a ÷ b).$
Answer
Division of a negative integer and a positive integer is always a negative integer $\frac{\text{a}}{\text{-b}}=\frac{\text{-b}}{\text{a}}=-\big(\frac{\text{a}}{\text{b}}\big)$ i.e where, $a$ and $b$ are integers.
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Question 41 Mark
What’s the Error? Ramu evaluated the expression $-7 - (-3)$ and came up with the answer $-10.$ What did Ramu do wrong?
Answer
Ramu went wrong in solving $- (-3)$ and took it as $-3$ only. Correct answer $= -7 - (-3) = -7 + 3 = -4$
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Question 51 Mark
If $x, y$ and $z$ are integers then $(x+\_\_\_\_\_ ) + z = \_\_\_\_\_ + (y + \_\_\_\_\_).$
Answer
If $x, y$ and $z$ are integers then $(x + y) + z = x + (y + z).$
Solution:
Addition is associative for integers, $i.e. (a + b) + c = a + (b + c).$
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Question 61 Mark
Product of two negative integers is a negative integer.
Answer
Product of two negative integers is a positive integer, i.e. $(-a) × (-b) = ab$ where, $a$ and $b$ are two positive integers.
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Question 71 Mark
$a ÷ b = b ÷ a.$
Answer
Division is not commutative for integers, i.e. $\text{a}\div\text{b}\neq\text{b}\div\text{a}$ where, $a$ and $b$ are integers.
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Question 81 Mark
$-3 × 3 = -12 - (-3)$
Answer
$\because LHS = (-3) \times 3 = (-9) $
$RHS = (-12) - (-3) = (-12) + 3 = (-9)$
Hence, $LHS = RHS.$
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Question 91 Mark
When $-16$ is divided by _____ the quotient is $4$
Answer
When $-16$ is divided by negative integer the quotient is $4$
Solution:
When $(-16)$ is divided by negative integer, i.e. $-4$ the quotient is $4$ as both signs are cancelled out.
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Question 101 Mark
$(-9) \times 20 =$ _____.
Answer
$(-9) × 20 = -180.$
Solution:
$(-9) × 20 = -180 [\because$ in multiplication of integers, if both the numbers have different signs, then the result is a negative number$].$
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Question 111 Mark
$[(-32) ÷ 8 ] ÷ 2 = -32 ÷ [ 8 ÷ 2]$
Answer
$\because\text{LHS}=[(-32)\div8]\div2$
$= \bigg[\frac{-32}{8}\bigg]+2=-4+2=-2$
$\text{and}\ \text{RHS}=(-32)\div[8\div2]$
$\bigg[\frac{-32}{8}\bigg]+2=-4+2=-2$
$\text{and}\ \text{RHS}=(-32)\div[8\div2]$
$=(-32)\div\bigg[\frac{8}{2}\bigg]$
$=(-32)\div4=\frac{(-32)}{4}=-8$
Hence, $LHS \neq RHS.$
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Question 121 Mark
$(-20) × (5 - 3) = (-20) × (-2)$
Answer
$\because LHS = (-20) \times (5 - 3) = (-20) \times 2 = (-40)$
$RHS = (-20) \times (-2) = 40$
Hence, $LHS = RHS.$
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Question 141 Mark
$65 ÷ (-13) =$ _____.
Answer
$65\div(-13)\ 65\times\frac{1}{(-13)}[\because$ division is inverse of multiplication$] = -5$
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Question 151 Mark
When we divide a negative integer by a positive integer, we divide them as whole numbers and put _____ a sign before quotient.
Answer
When we divide a negative integer by a positive integer, we divide them as whole numbers and put negative a sign before quotient.Solution:
When we divide a negative integer by a positive integer or a positive integer by a negative integer, we divide them as whole numbers and put a negative sign before quotient.
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Question 161 Mark
_____ $\times (-23) = -920$
Answer
$(40) \times (-23) = -920$
$\bigg[\because\bigg(\frac{-920}{-23}\bigg)=40\bigg]$
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Question 181 Mark
Integers are closed under division.
Answer
False. Solution: Because, when we divide two integers, we may or may not get an integer. e.g. $\frac{2}{1}=2(\text{integer})$ and $\frac{2}{3}(\text{not an integer}).$
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Question 201 Mark
$(-225) ÷ 5 =$ _____.
Answer
$(-225) ÷ 5$
$=-225\times\frac{1}{5}[\because$ division is inverse of multiplication$] = -45$
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Question 211 Mark
If we multiply six negative integers and six positive integers, then the resulting integer is _____.
Answer
If we multiply six negative integers and six positive integers, then the resulting integer is positive.

Solution:

If we multiply six negative integers and six positive integers, then the resulting integer is positive, because even numbers of negative integers, in multiplication becomes positive.

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Question 221 Mark
$(a + b) \times c = a \times c + a \times b.$
Answer
Integers show distributive property of multiplication over addition, i.e.
$a \times (b + c)$
$= a \times b + a \times c$
where a, b and c are integers.
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Question 231 Mark
_____ is the multiplicative identity for integers.
Answer
$1$ is the multiplicative identity for integers,
i.e. $a \times 1 = 1 \times a = a$ for any integer a.
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Question 241 Mark
The successor of $0 \times (-25)$ is $1 \times (-25)$
Answer
We know that, successor means adding $1$ to the given number.
Here, given number is $0 × (-25) = 0 [$on multiplying by $0$ to any number the result is zero$]$
Hence, the successor of $0 =0+1$ $=1\ \text{but}\ 1$ $\neq\ 1\times(-25)$
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Question 251 Mark
$4 \times (-5) = (-10) \times (-2)$
Answer
$\because LHS = 4 \times (-5) = -20$
$RHS = (-10) \times (-2) = 20$
Hence, $LHS = RHS.$
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Question 261 Mark
When we multiply two integers their product is always greater than both the integers
Answer
e.g. Let two integers are $(-5)$ and $2$
So, $(-5) \times 2 = -10 \Rightarrow (-10) < (-5)$ and $(-10) < 2$
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Question 271 Mark
$(-23) + 47$ is same as $47 + (-23)$
Answer
Because, addition is commutative,
i.e. $a + b = b + a$
$\Rightarrow (-23) + 47$
$= 47 + (-23)$
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Question 281 Mark
Integers are closed under multiplication.
Answer
True. Solution: If we multiply two integers, we get an integer.
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Question 291 Mark
Difference of two negative integers cannot be a positive integer.
Answer
e.g. Taking two negative integers, i.e. $-4$ and -$5 - 4 -(-5) = -4 + 5 = 1$ [positive integer]
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Question 301 Mark
$[(-8) + $______ $] + $________$ = $________$ + [(-3) + $________ $] = -3$
Answer
$[(-8) + (-3)] + 8 = (-8) + [(-3) + 8]$
$[\because$ addition is associative, i.e. $a + (to + c) = (a + b) + c] = -8 + 5 = -3$
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Question 311 Mark
_____$ \times (-1) = -35$
Answer
$-35 \times (-1) = -35$
$\bigg[\because\bigg(\frac{-35}{-1}\bigg)=35\bigg]$
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Question 321 Mark
When we change the order of integers, their sum remains the same.
Answer
Because, sum of two integers is commutative, i.e. $a + b = b + a$ for two integers $a$ and $b.$
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Question 341 Mark
$(-5) \times (-6) \times (-7) =$ _____.
Answer
Odd negative integers make the resultant integer, negative.
$(-5) \times (-6) \times (-7) = 30 \times (-7) = -210.$
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Question 351 Mark
Multiplication is not commutative for integers.
Answer
Multiplication is commutative for integers, i.e. $a × b = b × a$ for any two integers $a$ and $b.$
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Question 361 Mark
$a - b = b - a.$
Answer
Subtraction is not commutative for integers, i.e. $\text{a}-\text{b}\neq\text{b}-\text{a}$ where, $a$ and $b$ are integers.
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Question 371 Mark
We can write a pair of integers whose sum is not an integer.
Answer
False. Solution: Because, sum of two integers, is always be an integer.
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Question 381 Mark
We get additive inverse of an integer a when we multiply it by _____.
Answer
Additive inverse of an integer is the same integer value, with opposite sign. So, we get additive inverse of integer a, when we multiply it by $(-1).$
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Question 421 Mark
$(-23) \times (42) = (-42) \times $_____.
Answer
$(-23) \times (42) = (-1) \times (23) \times (42) = (-1) \times (42) \times (23)$
$[\because$ multiplication is commutative, i.e. $a \times b = b \times a] = (-42) \times (23)$
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Question 431 Mark
On the following number line, $(-4) \times 3$ is represented by the point _____.
​​​​​​​
Answer
$(-4) \times 3 = (-12)$ On the number line, each division has epual spacing pf $2$ units.
So, $A = -20 + 2 = -18 $
$B = -18 + 2 = -16 $
$C = -16 + 2 = -14 $
$D = -14 + 2 = -12$
Hence, $(-4) \times 3$ is represented by the point $D.$
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Question 441 Mark
$11 × (-5) = -( \_\_\_\_\_ × \_\_\_\_\_ ) = \_\_\_\_\_.$
Answer
We can write the equation as, $11 × (-5) = -(11 × 5) = -55$
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Question 451 Mark
$99 \times 101$ can be written as $(100 - 1) \times (100 + 1).$
Answer
$\because 99 \times 101 = 9999$ and $(100 - 1) \times (100 + 1)$
$= 100 \times (100 + 1) -1 \times (100 + 1)$
$= 100 \times 100 + 1 \times 100 - 1 \times 100 -1 \times 1 [$using distributive property$]$
$= 10000 + 100 - 100 -1 = 9999$
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Question 461 Mark
$3 × (-1 ) × (-15) = $_____.
Answer
Two negative integers and one positive integer make the resultant integer, positive.
$3 \times (-1) \times (-15) = (-3) \times (-15)= 45c.$
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Question 471 Mark
If $a, b, c$ are integers and $b \neq 0$ then, $a \times (b - c) = a \times b - a \times c$
Answer
Multiplication can be distributive over subtraction, i.e. $a \times (b - c) = a \times b - a \times c.$
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Question 501 Mark
$(-a) + b = b\ +$ Additive inverse of _____.
Answer
Additive inverse is the negation of a number. As we know, addition is commutative for integers, i.e. $-a + b = b + (-a)$
Now $‘- a’$ is the additive inverse of $a.$
So, a will be the answer.
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