Integers

Given, 10, 0, 5, -5, -7

Descending order 10, 5, 0, -5, -7

Ascending order -7, -5, 0, 5, 10

⇒ Option A is correct since, when the given integers are arranged in ascending or descending order 0 lies in middle.

⇒ Option B is incorrect as 5 does not lie in middle

⇒ Option C is incorrect as -7 does not lie in middle

⇒ Option D is incorrect as -5 does not lie in middle

⇒ Option A is true as B is greater than -10

⇒ Option B is true as A is greater than 0

⇒ Option C is False as B is greater than A

⇒ Option D is true as B is smaller than 0

⇒ Option D is true since B is a negative integer and lies between -10 and 0

⇒ Option A is false as b should be negative integer.

⇒ Option B is false as A must be positive integer.

⇒ Option C is false. Though B is negative integer as per the figure value must be between -10 and 0

⇒ Option D is the correct one. As per the given, consecutive number pattern 11, 8, 5, 2, .., ..,.. first term is the result of second number + 3 or we can say the difference between first and second is 3 so after 2, we get -1 as the number and -1 + 3 = -4 second number and -7 the third number.

So, the pattern will be

11, 8, 5, 2, -1, -4, -7

⇒ Option A 0, -3, -6 does not satisfy the given pattern

⇒ Option B -1, -5, -8 does not satisfy the given pattern

⇒ Option C -2,-5,-8 does not satisfy the given pattern

⇒ If we see the pattern to obtain second integer add + 25 to the first number so, -65 + 25 = -37 and -37 + 25 = -12 and -12 + 25 = 13

∴ pattern is -62, -37, -12, 13

⇒ Option A is incorrect as per the given pattern 25 does not come after -12

⇒ Option B is correct i.e., 13 is the next number in the pattern.

⇒ Option C is incorrect as per the given pattern 0 does not come after -12

⇒ Option D is incorrect as per the given pattern -13 does not come after -12

⇒ Option A is true

Ex: 1 + 1 = 1

⇒ Option B is true

Ex: (-1) + (-1) = -1

⇒ Option C is not true since it is not mandatory to get a negative integer when a positive integer and a negative integer is added.

Example: let us consider two integers -8 + 5 = -3 and 8 + (-5) = 3

⇒ Option D is true

⇒ As per the figure X, Y, Z, W lies between -15 and 10. So, X is -10 and Y is -5 and Z is 0 and W is 5.

⇒ Option A is incorrect as x is -10

⇒ Option B is incorrect as Y is -5

⇒ Option C is the answer. ‘zero’ is shown by Z.

⇒ Option D is incorrect as W is 5

Descending order is circle, tick, cross and dot

⇒ Option A is incorrect as it is not in the descending order

⇒ Option B is incorrect as it is not in the descending order

⇒ Option C is the answer. As per the given figure dot and cross mark is considered as negative integer and tick mark and circle as positive integer and circle is the highest integer valued number.

⇒ Option D is incorrect as it is not in the descending order

We know (-3) × (3) is -9

⇒ Option A -10 lies at left hand side

⇒Option B is the answer. As -4 lies at right hand side of -9

⇒ Option C 0 is incorrect as it lies in middle

⇒ Option D 9 lies after 0

⇒ 5 ÷ (-1) = -5

⇒ Option A is incorrect as -5 lie between 0 and -10

⇒ Option B is correct as -5 does not lie between 0 and 10

⇒ Option C is incorrect as -5 lie between -4 and -15

⇒ Option D is incorrect as -5 lie between -6 and 6

⇒ Since, the height of the fixed pulley from the bottom of the well is 2m and in rainy season water level raises and lies below 15m from the ground level

∴ 15 + 2 = 17m

⇒ extra meter rope to pull the water so add one m

∴ 17 + 1 = 18m

⇒ Option A is incorrect as the rope required is 18m not 17m

⇒ Option B is correct as rope required as 18m

⇒ Option C is incorrect as the rope required is 18m not 96m

⇒ Option D is incorrect as the rope required is 18m not 97m

Given, (-11) × 7 = -77

⇒ Option A is not the answer as

⇒ 11 × (-7) = -77 is equal to (-11) × 7

⇒ Option B is not the answer as

⇒ -(11 × 7) = -77 is equal to (-11) × 7

⇒Option C is the answer.

⇒ -11 × -7 = 77 not equal to (-11) × 7

⇒ Option D is not the answer as

⇒ 7 × (-11) = -77 is equal to (-11) × 7

⇒ (-10) × (-5) + (-7) = 50 -7

= 43

⇒ Option A i.e -57 is incorrect as we get 43

⇒ Option B i.e 57 is incorrect as we get 43

⇒ Option C i.e -43 is incorrect as we get 43

⇒Option D is the answer. Consider, first two integers we get -10 × -5 as 50.

So, 50-7 = 43

⇒ Option A is the answer. Since, -(-a) is + a which is not the additive inverse of a.

⇒ Option B is not the answer as

⇒ a× (-1) = -a is the additive inverse of a

⇒ Option C is not the answer as –a is the additive inverse of a

⇒ Option D is not the answer as a÷ (-1) = -a is the additive inverse of a

⇒ Option A is incorrect as a is not multiplicative identity for an integer a

⇒Option B is the answer. Since, one is the multiplicative identity for any integer.

⇒ Option C is incorrect as 0 is not multiplicative identity for an integer a

⇒ Option D is incorrect as -1 is not multiplicative identity for an integer a

[(-8) × (-3)] × (-4) = -96

⇒ Option A is incorrect as

As (-8) × [(-3) × (-4)] = -96 is equal to [(-8) × (-3)] × (-4)

⇒ Option B is incorrect as

As [(-8) × (-4)] × (-3) = -96 is equal to [(-8) × (-3)] × (-4)

⇒ Option C is incorrect as

As [(-3) × (-8)]× (-4) = -96 is equal to [(-8) × (-3)] × (-4)

⇒ Option D is the correct option since, [(-8) × (-3)] × (-4) is not equal to (-8) × (-3)-(-8)× (-4) = 24-32 = -8

Given, -25× [6 + 4] = -250

⇒ Option A i.e (-25) × 10 = -250 is same as 25× [6 + 4]

⇒ Option B i.e (-25) × 6 + (-25) × 4 = -250 is same as 25× [6 + 4]

⇒Option C is not same as -25× [6 + 4] . Since, - 25× 6× 4 = -600

⇒ Option B i.e -250 is same as (-25) × 10

Given, -35 × 107 = -3745

⇒ Option A is same as -35 × 107

⇒ -35 × (100 + 7) = -3745

⇒ Option B is same as -35 × 107

⇒ (-35) × 7 + (-35) × 100 = -3745

⇒ Option C is not same as -35 × 107

⇒ -35 × 7 + 100 = 345

⇒ Option D is same as -35 × 107

⇒ (-30-5) × 107 = -35 × 107 = -3745

⇒ (-43) × (-99) + 43

= 4257 + 43

= 4300

⇒ Option A is 4300 equal to (-43) × (-99) + 43

⇒ Option B i.e -4300 is not equal to (-43) × (-99) + 43

⇒ Option C i.e 4257 is not equal to (-43) × (-99) + 43

⇒ Option D i.e -4214 is not equal to (-43) × (-99) + 43

⇒ Option A is not same as

(-16) ÷ 4 = -4 and (-4) ÷ 16 =

⇒ Option B is same as

((-16) ÷ 4 = -4 and –(16 ÷ 4) = -4

⇒ Option C is same as ((-16) ÷ 4 = -4 and 16 ÷ (-4) = -4

⇒ Option D is same as ((-16) ÷ 4 = -4 and -4

∴ Option A is not same

⇒ Option A represents an integer 0 ÷ (-7) = 0

⇒ Option B represents an integer 20 ÷ (-4) = -5

⇒ Option C represents an integer (-9) ÷ 3 = -3

⇒ Option D does not represent an integer (-12) ÷ 5 = 2.4

⇒ Option A is incorrect since 20 + (-25) = -5

⇒ Op[tion B is incorrect since (-37) –(-32) = -5

⇒ Option C is correct since (-5) × (-1) = 5

⇒ Option D is incorrect since (45) ÷ (-9) = -5

Option C is different from all other options. Since, after solving -5 × -1 we get a positive integer and for all other options after solving we get positive integer.

⇒ Option A shows the maximum raise in temperature.

⇒ Option B -10° to 1 does not show raise in temperature

⇒ Option C -18° to -11 does not show raise in temperature

⇒ Option D -5° to 5 does not show raise in temperature

⇒ Option A is incorrect as a + b gives an integer

Ex: 1 + 1 = 2

⇒ Option B is incorrect as a-b gives an integer

Ex: 2-1 = 1

⇒ Option C is incorrect as a × b gives an integer

Ex: 1 × 1 = 1

⇒ Option D is correct a ÷ b may gave irrational number

Ex: 5 ÷ 4

⇒ Option A is not defined as anything by 0 is not defined

⇒ Option B is defined 0 ÷ a = 0

⇒ Option C is defined a ÷ 1 = 1

⇒ Option D is defined 1 ÷ a = number in decimal value

∴ Option A is defined

⇒ Option A is not odd as in the given pair of number -3 and 3 are additive numbers

⇒ Option B is not odd as in the given pair of numbers -5 and 5 are additive numbers

⇒ Option C is odd as in the given pair of number -6 and 1 are not additive numbers

⇒ Option D is not odd as in the given pair of number -8 and 8 are additive numbers

∴Option C is the odd one as 1 is not the additive inverse of 6

⇒ Option A is the odd one since the given pair of number are negative

⇒ Option B is not odd since in the given pair of number one is negative and other is positive

⇒ Option C is not odd since in the given pair of number one is negative and other is positive

⇒ Option D is not odd since in the given pair of number one is negative and other is positive

∴ Option A is the odd one since for every option first integer is negative and second is positive but for A both the integers are negative.

⇒ Option A is incorrect since, (-9) × 5 × 6 × (-3) = 810

⇒ Option B is incorrect since, 9 × -5 × 6 × (-3) = 810

⇒ Option C is correct since, (-9) × (-5) × (-6) × 3 = -810

⇒ Option D is incorrect since, 9 × -5 × -6 × 3 = 810

∴ Option C is the odd one since we get negative integer after their product where as in option we get positive integer.

⇒ option A is incorrect since -100 ÷ 5 = -20

⇒ Option B is incorrect since -81 ÷ 9 = -9

⇒ Option C is incorrect since -75 ÷ 5 = -15

⇒ Option D is correct since -32 ÷ 9

∴Option D is the odd one since, in every option first integer is exactly divisible by second integer where as in option D first integer is not exactly divisible by second.

⇒ Option B is odd one

⇒ (-1) × (-1) × (-1) = -1

[since, minus × minus is plus and plus × minus is minus]

⇒ Option A is not odd

⇒ (-1) × (-1) = 1

Option C is not odd

⇒ (-1) × (-1) × (-1) × (-1) = 1

Option D is not odd

⇒ (-1) × (-1) × (-1) × (-1) × (-1) = 1

∴ option B is odd one as we get negative integer after their product. Where, in every option we get positive integer after their product.

Additive inverse of a.

Given, a number divide by (-10) should be 0

zero divide by -10 is 0

Given, (-157) × (-19) + 157

Need to find out the resultant

⇒ (-157) × -19 = 2983 [since, minus × minus is plus]

⇒ 2983 + 157 = 3140

∴ (-157) × (-19) + 157 = 3140

⇒ [(-8) + (-3)] + (8) = [-11] + 8

= -3

⇒ (-8) + [(-3) + 8] = -8 + [5]

= -3

∴ [(-8) + (-3)] + (8) = (-8) + [(-3) + 8] = -3

Given, (-4) × 3

⇒ (-4) × (3) = -12

-12 is represented by the point D on number line.

Given, x, y and z are integers

⇒ (x + y) + z = x + (y + z)

(x + y) + z = x + (y + z)

Given, (-43) + another number is equal to -43

Need to find out the number

⇒ (-43) + 0 = -43

∴ another number is 0

Given, (-8) + (-8) + (-8)

Need to find out a number if multiplied by -8 gives the same value

⇒ (-8) + (-8) + (-8) = -24

⇒ 3 × (-8) = -24

∴ (-8) + (-8) + (-8) = 3 × (-8) = -24

Given, 11 × (-5) is equal to –another number × other number

⇒11 × (-5) = -55

⇒ -(11 × 5) = -55

∴ 11 × (-5) = -(11 × 5) = -55

-9 × 20 is -180

we get negative integer since plus × minus is minus integer

Given, (-23) × (42) is equal to (-42) × another number

Need to find out (-42) × another number

⇒ (-23) × (42) = -966

⇒ (-42) × (23) = -966

∴ (-23) × (42) = (-42) × (23)

if we are multiplying two numbers of which one is positive and other negative then first we multiply it as positive number and put minus sign.

The resultant integer will be positive if two negative integers are multiplied.

The resultant integer is positive integer.

Here, we are multiplying six –integers and six + integers.

The resultant integer as negative integer.

Here, we are multiplying five + integers and one –integer.

one is the multiplicative identity for the integers

We get additive inverse of an integer a when we multiply by minus sign

Given, (-25) × (-2)

∵ minus × minus is plus

⇒ (-25) × (-2) = 50

50 is the answer

Given, (-5) × (-6) × (-7)

⇒ (-5) × (-6) = 30

⇒ 30 × (-7) = -210

∵ minus × minus is plus and plus × minus is minus

Given, 3× (-1) × (-15)

⇒ 3 × (-1) = -3

⇒ (-3) × (-15) = 45

∵ minus × minus is plus and plus × minus is minus

[12 × (– 7)] × 5 = _________ × [(-7) × _________ ]

= 12 × [(-7) ×5]

(As in multiplication, associative law is satisfied)

Thus 12 and 5 are the

First case:

23 × (–99) = ______× (–100 + _______)

= 23 × (-100 + 1)

Second case:

23 × (–99) =23 × _________+ 23 ×1

=23 × 100+ 23 ×1

(In first case we adjust to get the value of the question and in the second one distribution law is applied)

Thus the blank spaces are 1 and 100.

_________ × (– 1) = – 35

⇒ _________=

=35

(By the given value)

_________ × (– 1) = 47

⇒ _________=

=-47

(By the given value)

88 × _________ = – 88

⇒ _________=

=-1

(By the given value)

________ × (–93) = 93

⇒ _________=

=-1

(By the given value)

(– 40) × _________ = 80

⇒_________ =

_________ × (–23) = – 920

⇒_________ =

Minus (-)

When we divide a positive integer by a negative integer or a negative integer by a positive integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.

For example:

(-6) ÷ 2 = -3

6 ÷ (-2) = 3

[As, 6 ÷ 2 = 3]

_________ × (–16) = 4

⇒_________ =

Multiplication

Lets take a number 8.

Then, 8 × 1/2 = 4 (i.e. 1/2 multiplied by 8 is equal to 4)

And, if we say, 8 ÷ 2 ( i.e. 8 is divided by 2, is also equal to 4)

Hence, we can say that division is the inverse operation of multiplication.

65 ÷ ( – 13) = _________

⇒_________ =

(-100) ÷ ( – 10) = _________

⇒_________ =

(-225) ÷ (5) = _________

⇒_________ =

_____÷ (– 1) = – 83

⇒_________ =

_____÷ (– 1) = 75

⇒_________ =

51 ÷ _____ = – 51

⇒_________ =

113 ÷ _____ = – 1

⇒_________ =

(-95) ÷ _____ = 95

⇒_________ =

(– 69) ÷ (69) = _____

⇒_________ =

(– 28) ÷ (-28) = _____

⇒_________ =

True

Explanation:

5-(-8) =5+8

(As two minus or – signs make a plus or + sign)

False

Explanation:

(-9)+ (-11) =-20 ----(a)

And

(-9)- (-11) =-9+11=2 ----(b)

(a)<(b)

Thus False

True.

Explanation:

(-2)+ (-3)=(-5) where -5 is less than both -2 and -3)

False.

Explanation:

(-9)- (-11) = 2 which is positive

False.

Explanation:

The sum of a pair of integers is always an integer.

For e.g. 2 + (-3) = -1 where -1 is an integer.

True.

Explanation:

The result of subtraction of integers is an integer.

For e.g. 2 + (-3) = -1 where -1 is an integer.

True.

Explanation:

As the values are same in case of addition even if the orders of integers are changed, both are equal.

(-23)+47=24 and 47+ (-23) =47-23=24

True.

Explanation:

As the values are same in case of addition even if the orders of integers are changed, both are equal.

(-23) + 47 = 24 and 47+ (-23) = 47 – 23 = 24

False.

Explanation:

For e.g.:

-9-(-11) = 2 while -11-(-9) = -2 where the values are different.

False.

Explanation:

For direction:

500-(-200) = 700 while -200-(500) = -700 where the signs are different. We have considered east as positive and west as negative.

True

Explanation:

As commutative property is satisfied in multiplication (– 5) × (33) = -165 and 5 × (– 33) = -165 which are same.

True

Explanation:

As the product of numbers with same signs are equal to the absolute value and hence (– 19) × (– 11) = 19 × 11 are equal.

False

Explanation:

5-3=2; not -2

Thus, (– 20) × (5 – 3) = (– 20) × (2)

False

Explanation:

As 4× (-5) = -20 while (-10) × (-2) = 20

The signs are different and not equal.

False

Explanation:

Product of two negative integers makes a positive so in case of a third negative the final product is negative.

(– 1) × (– 2) × (– 3) = 1 × 2 × (-3)

True

Explanation:

-3×3=-9 while -12-(-3) =-12+3=-9 and both are equal.

False

Explanation:

Product of two negative integers makes a positive.

For e.g.: (-100)×(-5)=500

True

Explanation:

Product of two negative integers makes a positive so in case of a third negative the final product is negative.

(– 1) × (– 2) × (– 3) = 1 × 2 × (-3)

False

Explanation:

Product of a negative integer and a positive integer is a negative integer.

(– 1) × (2) = -2

False

Explanation:

For e.g.:

(-2)×3= (-6) where -6 is less than -2 and 3

True

Explanation:

The result of multiplication of integers is an integer.

For e.g.: 2×(-4) = -8 which is also an integer.

True

Explanation:

Any number multiplied by 0 is 0.

False

Explanation:

Commutative property is applicable for all integers in multiplication.

For e.g.: 23 × (-2) = (-2) × 23 = -46

False

Explanation:

-1 is the multiplicative inverse.

True

Explanation:

Distributive property is applicable in multiplication.

99 × 101= (100 – 1) × (100 + 1)

Because 100-1=99 and 100+1=101

True

Explanation:

Distributive property is applicable in multiplication.

23 × (100–99) =23 × 100- 23 ×99

True

Explanation:

Distributive property is applicable in multiplication.

23 × (–99) =23 × 100+ 23 ×1

True

Explanation:

Commutative property is applicable for all integers in multiplication.

For e.g.: 23× (-2) = (-2) ×23 = -46

False

Explanation:

Commutative property is not satisfied in division.

For e.g.:

And are not equal.

Take Left Hand Side of the equation.

Left Hand Side can be represented by LHS.

LHS = a – b …(i)

And then consider Right Hand Side of the same equation.

Right Hand Side can be represented by RHS.

RHS = b – a

Take minus common from RHS, we get

RHS = -(a – b) …(ii)

Now, compare equations (i) and (ii),

Notice that,

LHS = a – b and RHS = -(a – b)

This means, (a – b) ≠ -(a – b)

⇒ LHS ≠ RHS

Thus, this is a false statement.

Take Left Hand Side of the equation.

Left Hand Side can be written as LHS.

LHS = a ÷ (-b)

We can re-write LHS, by taking the negative sign outside of the bracket. Since,

⇒ LHS = -(a ÷ b) …(i)

Take Right Hand Side of the same equation.

Right Hand Side can be written as RHS.

RHS = -(a ÷ b) …(ii)

Now, compare equations (i) and (ii),

Notice that,

LHS = -(a ÷ b) and RHS = -(a ÷ b)

This clearly means, LHS = RHS

Thus, this is a true statement.

Take Left Hand Side of the equation.

Left Hand side can be written as LHS.

LHS = a ÷ (-1)

We can further solve it as,

⇒ LHS = -(a ÷ 1) [∵ minus can be taken out as it won’t affect the equation]

⇒ LHS = -a [∵ a ÷ 1 = a/1 = a] …(i)

Take Right Hand Side of the equation.

We can write Right Hand Side as RHS.

RHS = -a …(ii)

Now, comparing equations (i) and (ii), we get

LHS = -a & RHS = -a

This clearly shows that, LHS = RHS

Thus, this is a true statement.

Let us solve it simultaneously starting from multiplication fact.

Given multiplication fact, (-8) × (-10) = 80

We shall first simplify left hand side (LHS) of this multiplication fact.

LHS = (-8) × (-10)

⇒ LHS = -8 × -10

⇒ LHS = -1 × -1 × 8 × 10

⇒ LHS = (-1 × -1) × (8 × 10)

⇒ LHS = 1 × 80 [∵ -1 × -1 = 1 & 8 × 10 = 80]

⇒ LHS = 80

And RHS = 80 [∵ right hand side, RHS of multiplication fact = 80]

⇒ Multiplication fact is true.

Now, given division fact, 80 ÷ (-8) = (-10)

We shall first simplify left hand side (LHS) of this division fact.

LHS = 80 ÷ (-8)

⇒ LHS = -(80 ÷ 8)

⇒

⇒ LHS = -10

And RHS = -10 [∵ right hand side, RHS of division fact = -10]

⇒ Division fact is also true.

So, Multiplication fact is same as division fact.

Thus, this is a true statement.

We have been given that, integers are closed under division.

Let it be a true statement.

Then, assume two integers, say 5 and 9.

Now, divide the integers.

⇒

According to our assumption, 5/9 is an integer.

But according to the definition of integers, integers are whole numbers that can be positive, negative or zero.

While, 5/9 is a positive but not whole number.

⇒ 5/9 is not integer.

So, our assumption is wrong.

And we can say that integers are not closed under division as integer divided integer does not always give an integer.

Thus, this is a false statement.

We will be basically following BODMAS

Take left hand side (LHS) of this equation.

That is, LHS = [(-32) ÷ 8] ÷ 2

⇒ LHS = [-(32 ÷ 8)] ÷ 2 [∵ minus sign can be taken outside the brackets]

⇒ [The following steps are general division]

⇒ LHS = -4 ÷ 2

⇒ LHS = -2 …(i)

Now, take right hand side (RHS) of this equation.

That is, RHS = -32 ÷ [8 ÷ 2]

⇒

⇒ RHS = -32 ÷ 4

⇒ RHS = -8 …(ii)

Comparing equations (i) & (ii), we can say

LHS ≠ RHS

[∵ LHS = -2 & RHS = -8]

Thus, this is a false statement.

We can easily depict the statement using an easy example.

First, we need to know what integers are - Integers are whole numbers that can be positive, negative or zero.

Let us take an integer, 4.

Now, its additive inverse = -4 [∵ Additive numbers are basically negative or opposite number]

Sum of this integer and its additive inverse = 4 + (-4)

⇒ Sum = 4 – 4 [∵ a + (-a) = a – a; since plus minus is equal to minus]

⇒ Sum = 0

This result is true for any integer a, where ‘a’ is any whole number, that is positive, negative or zero.

So, we can say that the sum of an integer and its additive inverse is zero.

Thus, this is a true statement.

Successors of numbers are usually found out by adding 1 to it.

For example: Successor of 1 = 1 + 1 = 2

Successor of 15 = 15 + 1 = 16

Successor of -20 = -20 + 1 = -19

Similarly, we will try to solve 0 × (-25) and then add 1 to it.

0 × (-25) = - 0 × 25 = 0 [∵ multiplying anything by 0 gives 0]

Now, add 1 to it. That is,

0 + 1 = 1

⇒ 1 is the successor of 0 × (-25)

Now, let us solve 1 × (-25).

1 × (-25) = -25

And -25 ≠ 1

So, 1 × (-25) is not a successor of 0 × (-25).

Thus, this is a false statement.

Given is,

-5 × 4 = -20

Now, note the equation in second line:

-5 × 3 = -15 = -20 – (-5) ...(i)

Now, note the equation in third line:

-5 × 2 = _____________ = -15 – (-5)

Compare the equation of second and third line,

In second line ⇒ -5 × 3 = -15

So, in third line ⇒ -5 × 2 = -10

[This was not a pattern but just simple multiplication of -5 with 2 as was of -5 with 3]

So, we can re-write the equation as:

-5 × 2 = = -15 – (-5) ...(ii)

Now, note the equation in fourth line:

-5 × 1 = _____________ = ____________

Note the pattern in the bolded equation of (i) and (ii),

(-5) remains same, but 20 and 15 are basically factors of 5 in descending order.

⇒ …(iii)

Now, note the equation in fifth line:

-5 × 0 = 0 = _______

Note the pattern in the bolded equation of (i), (ii) and (iii),

(-5) remains same, but 20, 15 and 10 are factors of 5 in descending order.

⇒ -5 × 0 = 0 = …(iv)

Now, note the equation in sixth line:

-5 × -1 = 5 = ________

Note the pattern in the bolded equation of (i), (ii), (iii) and (iv),

(-5) remains same, but 20, 15, 10 and 5 are factors of 5 in descending order.

⇒ …(v)

Now, note the equation in seventh line:

-5 × -2 = ____ = ____

The first blank has got to do with multiplication of -5 and -2.

(-5 × -2 = 10)

For the second blank, (-5) remains same, but 20, 15, 10, 5, 0 are factors of 5 in descending order in bolded equation of (i), (ii), (iii), (iv) and (v). ⇒ ...(vi)

Thus, we have

-5 × 4 = -20

-5 × 3 = -15 = -20 – (-5)

-5 × 2 = = -15 – (-5)

-5 × 0 = 0 =

In order to fill these blanks, you need to identify the pattern.

Given is,

7 × 4 = 28

This can also be written as:

7 × 4 = 28 = 35 – 7

⇒ 7 × 4 = 28 = (7 × 5) – 7 …(i)

Now, note the equation in second line:

7 × 3 = ___ = 28 - 7

Here, the blank can be filled by simple multiplication of 7 by 3.

⇒

⇒ 7 × 3 = 21 = (7 × 4) – 7 ...(ii)

Now, note the equation in third line:

7 × 2 = ___ = _____ - 7

Here, the blank can be filled by simple multiplication of 7 by 2.

Note the pattern in the bolded equation of (i) and (ii),

(7) remains same, but (7 × 5) and (7 × 4) are in series.

⇒

⇒ ...(iii)

Now, note the equation in fourth line:

7 × 1 = 7 = ____ - 7

Note the pattern in the bolded equations of (i), (ii) and (iii),

(7) remains same, but (7 × 5), (7 × 4) and (7 × 3) are in series.

⇒

⇒ …(iv)

Now, note the equation in fifth line:

7 × 0 = ___ = ___ - ___

Here, the first blank can be filled by simple multiplication of 7 by 0.

Note the pattern in the bolded equations of (i), (ii), (iii) and (iv),

(7) remains same in the third blank, but (7 × 5), (7 × 4), (7 × 3) and (7 × 2) are in series.

⇒

⇒ …(v)

Now, note the equation in sixth line:

7 × -1 = -7 = ___ - ___

Note the pattern in the bolded equations of (i), (ii), (iii), (iv) and (v),

(7) remains same in the third blank, but (7 × 5), (7 × 4), (7 × 3), (7 × 2) and (7 × 1) are in series.

⇒

⇒ ...(vi)

Now, note the equation in seventh line:

7 × -2 = ___ = ___ - ___

Here, the first blank can be filled by simple multiplication of 7 by -2.

Note the pattern in the bolded equations of (i), (ii), (iii), (iv), (v) and (vi),

(7) remains same in the third blank, but (7 × 5), (7 × 4), (7 × 3), (7 × 2) and (7 × 0) are in series.

⇒

⇒ …(vii)

Now, note the equation in eighth line:

7 × -3 = ___ = ___ - ___

Here, the first blank can be filled by simple multiplication of 7 by -3.

Note the pattern in the bolded equations of (i), (ii), (iii), (iv), (v), (vi) and (vii),

(7) remains same in the third blank, but (7 × 5), (7 × 4), (7 × 3), (7 × 2), (7 × 0) and (7 × -1) are in series.

⇒

⇒ …(viii)

Thus, we have

7 × 4 = 28

(a) Given that: An atom consists of,

Number of electrons = Number of protons

Let there be ‘a’ number of protons and ‘a’ number of electrons.

And, charge on 1 proton = +1

Then, charge on ‘a’ protons = 1 × a = a

Charge on electron = -1

Then, charge on ‘a’ electrons = -1 × a = -a

⇒ Total charge = Charge on ‘a’ protons + Charge on ‘a’ electrons

⇒ Total charge = a + (-a)

⇒ Total charge = 0

Thus, charge of an atom when there are equal number of electrons and protons is 0.

(b) Let the atom have ‘a’ number of electrons and ‘a’ number of protons.

If an atom loses an electron, it is left with (a – 1) electrons.

Number of protons = a

Now, charge on 1 electron = -1

⇒ Charge on ‘(a – 1)’ electrons = -1 × (a – 1)

⇒ Charge on ‘(a – 1)’ electrons = -(a – 1)

⇒ Charge on ‘(a – 1)’ electrons = (1 – a)

Charge on 1 proton = 1

⇒ Charge on ‘a’ protons = 1 × a

⇒ Charge on ‘a’ protons = a

Total charge on atom = (1 – a) + a

⇒ Total charge on atom = 1

(c) Let the atom have ‘a’ number of electrons and ‘a’ number of protons.

If an atom gains an electron, it becomes (a + 1) electrons.

Number of protons = a

Now, charge on 1 electron = -1

⇒ Charge on ‘(a + 1)’ electrons = -1 × (a + 1)

⇒ Charge on ‘(a + 1)’ electrons = -(a + 1)

Charge on 1 proton = 1

⇒ Charge on ‘a’ protons = 1 × a

⇒ Charge on ‘a’ protons = a

Total charge on atom = -(1 + a) + a

⇒ Total charge on atom = -1 – a + a

⇒ Total charge on atom = -1

We know that, Charge on an ion is basically charge on electron plus charge on proton.

Symbolically, Charge on an ion = Charge on electron + Charge on proton

For Hydroxide:

Proton charge = +9

Electron charge = p (let)

Ion charge = -1

So, Ion charge = electron charge + proton charge

⇒ -1 = p + 9

⇒ p = -1 – 9

⇒ p = -10

For Sodium ion:

Proton charge = +11

Electron charge = q (let)

Ion charge = +1

So, Ion charge = electron charge + proton charge

⇒ 1 = q + 11

⇒ q = 1 – 11

⇒ q = -10

For Aluminium ion:

Proton charge = +13

Electron charge = -10

Ion charge = r (let)

So, Ion charge = electron charge + proton charge

⇒ r = -10 + 13

⇒ r = 3

For Oxide ion:

Proton charge = +8

Electron charge = -10

Ion charge = s (let)

So, Ion charge = electron charge + proton charge

⇒ s = -10 + 8

⇒ s = -2

We have been given that, 1 AD came immediately after 1 BC.

1 BC = -1 and 1 AD = +1

A. Greeco-Roman era started in the year = 330 BC

Greeco-Roman era ended in the year = 395 AD

Count from 1 BC to 330 BC = 330

Then, count from 1 AD to 390 AD = 390

Total count of years = 330 + 390

⇒ Total count of years = 720

Thus, 720 is the span of years the Greeco-Roman era has lasted.

B. AD, that is, Anno Domini is considered to be counted in ascending order.

This means, 1185 AD comes after 1114 AD.

We are given the information that,

Bhaskaracharya was born in 1114 AD.

And he died in 1185 AD.

His age can be calculated by taking difference between the year of his birth and death.

Bhaskaracharya’s age = 1185 – 1114

⇒ Bhaskaracharya’s age = 71

Thus, he dies at the age of 71 years.

C. Since, 1 AD came immediately after 1 BC.

Given that, Queen Nefertis ruled Egypt about 2900 years before Turks ruled in the year 1517 AD.

So, we need to subtract 2900 from 1517. We get

1517 – 2900 = -1383

Since, -1 = 1 BC

∴ -1383 = 1383 BC

Thus, Queen Nefertis ruled in 1383 BC.

D. Remember that AD came immediately after BC.

Also, that AD is counted in ascending order and BC is counted in descending order.

Given that, Archimedes lived between 287 BC and 212 BC; Aristotle lived between 380 BC and 322 BC.

Now, since BC is counted in descending order ⇒ 287-212 BC > 380-322 BC

Thus, Aristotle lived during an early period.

Kindly note that low temperatures are usually lower numbers.

Example: 2° is higher temperature than -2° or in other words, -2° is lower temperature than 2°

(It’s colder in -2° than in 2°)

And similarly, it’s colder in -10° than in -2°.

The arrangement of numbers is such that, …-3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < …

Similarly, arrangement of numbers in the table is -129° < -90° < -81° < -67° < -27° < -11° < -9°

Arranging it in tabular form from lowest temperature to highest temperature:

There are number of integer pairs whose product equals -12.

These are: (1, -12), (2, -6), (3, -4), (-3, 4), (-2, 6) and (-1, 12)

But (2, -6) and (-6, 2) are ideal pairs whose products equals to -12 and there lies seven integers between them.

Observe the number line:

For (-6, 2):

For (-2, 6):

While other pairs do not satisfy the criteria.

Thus, the solutions are (-6, 2) and (-2, 6).

It’s better to check every combination of numbers.

For -5:

(-5) × 1 = -5 (not between -19 and -6)

(-5) × (-1) = 5 (not between -19 and -6)

(-5) × 3 = -15 (-19 < -15 < -6)

(-5) × (-2) = 10 (not between -19 and -6)

For 6:

6 × 1 = 6 (not between -19 and -6)

6 × (-1) = -6 (not between -19 and -6)

6 × (-2) = -12 (-19 < -12 < -6)

For -7:

(-7) × 1 = -7 (-19 < -7 < -6)

For 8:

8 × (-1) = -8 (-19 < -8 < -6)

Arranging the solution:

Let the integers be x and y, where x > y.

According to the question,

Product of integers = -36

⇒ x × y = -36

⇒ xy = -36 …(i)

Also, difference of integers = 15

⇒ x – y = 15 …(ii)

From equation (i),

xy = -36

⇒ …(iii)

Substituting equation (iii) into equation (ii), we get

⇒

⇒ x² + 36 = 15x

⇒ x² – 15x + 36 = 0

⇒ x² – 12x – 3x + 36 = 0

⇒ x(x – 12) – 3(x – 12) = 0

⇒ (x – 3)(x – 12) = 0

⇒ x = 3 or x = 12

If x = 3,

Put x = 3 in equation (ii),

x – y = 15

⇒ 3 – y = 15

⇒ y = 3 – 15

⇒ y = -12

One of the pair is (3, -12).

If x = 12,

Put x = 12 in equation (i),

x – y = 15

⇒ 12 – y = 15

⇒ y = 12 – 15

⇒ y = -3

One of the pair is (12, -3).

Thus, two solutions are (3, -12) and (12, -3).

For (a) → a × 1 = a

So, we can match (a) with (vi).

For (b) → 1 = Multiplicative identity

1 is multiplicative identity because any integer multiplied with 1 gives the same integer.

So, we can match (b) with (iii).

For (c) → (-a) ÷ (-b) = a ÷ b

As

⇒ …(i)

And …(ii)

Comparing equations (i) and (ii), we get

(-a) ÷ (-b) = a ÷ b

So, we can match (c) with (v).

For (d) → a × (-1) = -a

As a × (-1) = -a × 1 = -a

So, we can match (d) with (vii).

For (e) → a × 0 = 0

0 multiplied with any integer equals to 0.

So, we can match (e) with (viii).

For (f) → (-a) ÷ b = a ÷ (-b)

As …(i)

And

⇒ …(ii)

Comparing equations (i) and (ii), we get

(-a) ÷ b = a ÷ (-b)

So, we can match (f) with (iv).

For (g) → 0 = additive identity

0 is additive identity because any integer added to 0 gives the same integer.

So, we can match (g) with (ii).

For (h) → a ÷ (-a) = -1

As

⇒

⇒ a ÷ (-a) = -1

So, we can match (h) with (ix).

For (I) → -a = Additive inverse of a

(-a) is additive inverse because integer a added with –a gives 0.

So, we can match (I) with (i).

Arranging it into table:

According to the question,

Available amount at the beginning of the month = Rs. 500

On 4/9, with the cheque number 384102, Rs 120 amount was withdrawn.

On 12/9, with the cheque number 275146, Rs 200 amount was deposited.

On 22/9, with the cheque number 384103, Rs 240 amount was withdrawn.

On 22/9, with the cheque number 801351, Rs 150 amount was deposited.

Money available in the account can be calculated by amount already available in the account and the money deposited in the account, minus the amount withdrawn from the bank.

So, net available amount in bank account = Already available amount + Deposited amount – Withdrawn amount

⇒ Net available amount in bank account = 500 + (200 + 150) – (120 + 240)

⇒ Net available amount in bank account = 500 + 350 – 360

⇒ Net available amount in bank account = 500 – 10

⇒ Net available amount in bank account = 490

Thus, net money in bank account after these transactions is Rs. 490.

There are number of solutions possible for this question.

Take any positive integer = 6 (say)

Now, take any negative integer, but to make sure it is numerically greater than the positive integer.

⇒ Negative integer = -9 (say)

Add these numbers,

Sum = 6 + (-9)

⇒ Sum = 6 – 9 [plus & minus is equal to minus in addition, multiplication and division]

⇒ Sum = -3

-3 is a negative integer due to the negative sign before the number 3.

Thus, 6 and -9 are such integers.

There are number of solutions possible.

Take any positive integer = 10 (say)

Now, take any negative integer, but to make sure to choose a number numerically smaller than the positive integer.

⇒ Negative integer = -5 (say)

Add these numbers,

Sum = 10 + (-5)

⇒ Sum = 10 – 5 [plus and minus is equal to minus in addition, multiplication and division]

15 is a positive integer.

Thus, 10 and -5 are such numbers.

There are a number of solutions possible in this case.

But we need to keep in mind to take a negative integer first and then subtract a positive integer from that.

What if we do the opposite? Let us take a positive integer first = 11 (say)

Now, let us take a negative integer at second = -14 (say)

Subtract second integer from first,

Difference = 11 – (-14)

⇒ Difference = 11 + 14 [minus & minus = plus, in addition as well as in multiplication and division]

⇒ Difference = 25, which is a positive integer not a negative integer.

That’s why we need to do the exact opposite.

First, take negative integer = -14

Then, take positive integer = 11

Now, subtract first integer from second,

Difference = -14 – 11 [minus & minus = plus, in addition, multiplication and division; also, the greatest number’s sign is taken in the final result]

⇒ Difference = -25, which is a negative integer.

Thus, answer is -14 and 11.

Though there are number of possible answers, take a positive integer first and then a negative integer.

Positive integer = 8 (say)

Negative integer = -12 (say)

Now, keep in mind to subtract negative integer from positive integer in order to get a positive integer.

Difference = 8 – (-12)

⇒ Difference = 8 + 12 [minus & minus = plus, in addition, subtraction and division]

⇒ Difference = 20, which is positive integer.

Thus, 8 and -12 are such integers.

There are number of integers that are smaller than -5, namely

-6, -7, -8, -9, -10, -11, -12, …

We need to do this type of question by hit and trial basis. So, take a look at the integers smaller than -5. We know that the smallest of positive integers that when subtracted give us 5 are 11 and 6. [∵ 11 – 6 = 5]

So, try subtracting the same numbers but with negative sign.

Difference = -11 – (-6)

⇒ Difference = -11 + 6 [minus & minus = plus]

⇒ Difference = -5 [∵ 11 is the greatest number of 11 and 6, so the sign before 11 would be taken in the result]

Similarly, there are other pair of integers satisfying the condition.

Thus, -6 and -11 are one of such integers.

There are number of integers that are greater than -10, namely

-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, …

We need to do this type of question by hit and trial basis. So, take a look at the integers greater than -10.

If we add any positive integer to a positive integer, then we would get a positive integer in result, which would not be smaller than -10.

Any positive integer is greater than -10.

So, try subtracting some negative integers together. Adding -5 to -6 or -6 to -7 or such pairs would work. So,

Difference = -6 + (-7)

⇒ Difference = -6 - 7 [minus & minus = plus]

⇒ Difference = -13, which is smaller than -10 [∵ 7 is the greatest number of 6 and 7, so the sign before 7 would be taken in the result]

Similarly, there are other pair of integers satisfying the condition.

Thus, -6 and -7 are one of such integers.

There are number of integers greater than -4, namely

-3, -2, -1, 0, 1, 2, 3, 4, 5, …

By hit and trial method, take -1 and 4. We get,

Difference = -1 – 4

⇒ Difference = -5, which is smaller than -4 [minus & minus = plus; and since, 4 is greater amongst 1 and 4 so the result would have a minus sign]

Similarly, take -2 and 6.

Difference = -2 – 6

⇒ Difference = -8, which is smaller than -4 [minus & minus = plus; and since, 6 is greater amongst 2 and 6 so the result would have a minus sign]

Thus, -1 and 4 are one of such pair of integers.

There are number of integers smaller than -6, namely

-7, -8, -9, -10, -11, …

By observation, subtract -8 and -9.

Difference = -8 – (-9)

⇒ Difference = -8 + 9 [minus & minus = plus]

⇒ Difference = 1, which is greater than -6 [Since, 9 is greater amongst 8 and 9 so the result would have a positive sign]

Thus, -8 and -9 are one of such integers.

There are number of possible solutions.

To find one of them, find the easiest combination of positive integers which when subtracted gives 7.

The easiest pair which comes to our minds are 10 and 3.

∵ 10 – 3 = 7

So, just subtract the negative form of these integers again.

⇒ Difference = -3 – (-10)

⇒ Difference = -3 + 10 [minus & minus = plus]

⇒ Difference = 7 [Since, 10 is greatest amongst 10 and 3, then + sign will appear in the resulting answer]

We have very carefully subtracted -3 from -10 rather than otherwise to get 7.

Thus, -3 and -10 are one of such pair of integers.

Integers greater than -11 are: -10, -9, -8, -7, …

Integers smaller than -11 are: -12, -13, -14, -15, -16, …

To find the solution, find the easiest combination of positive integers which when subtracted gives 11.

The easiest pair which comes to our minds are 21 and 10.

∵ 21 – 10 = 11

So, just subtract the negative form of these integers again, as these satisfies the question (-21 < -11 & -10 > -11)

⇒ Difference = -21 – (-10)

⇒ Difference = -21 + 10 [Since, minus & minus = plus]

⇒ Difference = -11 [Since, 21 is the greatest integer amongst 21 and 10, then – sign will appear in the resulting answer]

Thus, -21 and -10 are such integers.

There can be number of possible solutions for this.

Let us take any positive integer, say, 6.

Now, it’s better to take a negative integer along with it.

So, negative integer = -2

Product = 6 × -2

⇒ Product = -12

Check: -12 < 6 and -12 < -2

Similarly,

Let positive integer = 9

And negative integer = -5

Product = 9 × -5

⇒ Product = -45

Check: -45 < 6 and -45 < -5

Thus, 6 and -2 are one of such integers.

There can be number of possible solutions for this.

But either only positive pair of integers or negative integers will show the desired result.

Take 6 and 9.

Product = 6 × 9

⇒ Product = 54

Now, take -5 and -3.

Product = -5 × -3

⇒ Product = 15 [minus × minus = plus]

Similarly, there are other pair of integers.

Thus, 6 and 9 are one of pair of integers.

Let us solve the expression for Ramu.

We have, -7 – (-3) = -7 + 3 [∵ minus & minuns = plus]

⇒ -7 – (-3) = -4 [∵ 7 is the greatest amongst 7 and 3, so the sign before 7 will be considered in the result]

The answer is coming out to be -4 while what Ramu did was add -7 and -3 and neglected a minus sign in-between -7 and -3.

Let us solve the expression for Reeta.

We have, -4 + d for d = -6

Put d = -6 in -4 + d.

-4 + d = -4 – 6

⇒ -4 + d = -10

The answer is coming out to be -10, while Reeta gave an answer of 2.

She must have subtracted 4 from 6, replacing -4 from 4.

Let us represent the table into a diagram:

The negative elevation is shown below the sea level and positive elevation is shown above the sea level.

A. The closest location from sea level seems to be C from the diagram.

C is just 55 m away from sea level, while A is 180 m, B is 1600 m and D is 3200 m.

Thus, location C is closest to sea level.

B. The farthest location from sea level seems to be D from the diagram.

D is 3200 m away from sea level, while A is 180 m, B = 1600 m and C = 55 m.

Thus, location D is farthest to sea level.

D. The least elevated according to the diagram is C, then A, after that B and then D.

Notice that, C = -55 m elevated, which is much closer to the sea than other locations.

Then, comes A = -180 m elevated, which is closer to the sea than B and D.

Then, B = 1600 m is closer.

D = 3200 m is farthest.

The arrangement is: C<A<B<D.

Given that, starting point during the motor ride = 380 m

The changes in the elevation during the ride are given to be 540 m, -268 m, 116 m, -152 m, 490 m, -844 m and 94 m.

The changes in the elevation can be calculated as :

Changes = 540 + (-268) + 116 + (-152) + (490) + (-844) + 94

⇒ Changes = 540 – 268 + 116 – 152 + 490 – 844 + 94

⇒ Changes = -24 m

To find elevation at the end of the ride is found by,

Elevation = (Elevation at the starting point) + (Changes in elevation)

⇒ Elevation = 380 + (-24)

⇒ Elevation = 356 m

Thus, elevation at the end of the ride is 356 m.

In distributive property, numbers are split in such a way that it becomes easy to evaluate them after splitting.

(i). -39 × 99

We can write it as,

99 × -39 = (100 – 1) × (-39) [∵ 100 – 1 = 99]

⇒ 99 × -39 = [100 × (-39)] – [1 × (-39)] [∵ (a – b) × c = ac – bc]

⇒ 99 × -39 = -3900 – (-39)

⇒ 99 × -39 = -3900 + 39

⇒ 99 × -39 = -3861

Thus, the answer is -3861.

(ii). (-85) × 43 + 43 × (-15)

We can write it as,

(-85) × 43 + 43 × (-15) = 43 × (-85 + (-15)) [Taking 43 common]

⇒ (-85) × 43 + 43 × (-15) = 43 × (-85 – 15) [plus & minus = minus]

⇒ (-85) × 43 + 43 × (-15) = 43 × -100

⇒ (-85) × 43 + 43 × (-15) = -4300

Thus, the answer is -4300.

(iii). 53 × (-9) – (-109) × 53

We can write it as,

53 × (-9) – (-109) × 53 = 53 × (-9 – (-109)) [Taking 53 common]

⇒ 53 × (-9) – (-109) × 53 = 53 × (-9 + 109) [minus & minus = plus]

⇒ 53 × (-9) – (-109) × 53 = 53 × 100

⇒ 53 × (-9) – (-109) × 53 = 5300

Thus, the answer is 5300.

(iv). 68 × (-17) + (-68) × 3

We can write it as,

68 × (-17) + (-68) × 3 = 68 × (-17) – 68 × 3 [minus & plus = minus]

⇒ 68 × (-17) + (-68) × 3 = 68 × (-17 – 3) [Taking 68 as common]

⇒ 68 × (-17) + (-68) × 3 = 68 × (-20)

⇒ 68 × (-17) + (-68) × 3 = -1360

Thus, the answer is -1360.

We have been given that,

a * b = a × b + (a × a + b × b) …(i)

Apply the same formula in the questions that follows:

(i) We have (-3) * (-5)

Put a = -3 and b = -5 in equation (i), we get

(-3) * (-5) = [ (-3) × (-5) ]+ (-3 × -3 + -5 × -5)

⇒ (-3) * (-5) = 15 + (9 + 25)

⇒ (-3) * (-5) = 15 + 9 + 25

⇒ (-3) * (-5) = 49

Thus, the answer is 49.

(ii). We have (-6) * 2.

Put a = -6 and b = 2 in equation (i), we get

(-6) * 2 = [ (-6) × 2 ] + (-6 × -6 + 2 × 2)

⇒ (-6) * 2 = -12 + (36 + 4)

⇒ (-6) * 2 = -12 + 40

⇒ (-6) * 2 = 28

Thus, the answer is 28.

We have been given that,

a ∆ b = a × b – 2 × a × b + b × b (-a) × b + b × b

Simplifying it, we get

a ∆ b = a × b – 2 × a × b – a × b³ + b² …(A)

Apply the same formula in the questions that follows:

(i). We have 4 ∆ (-3).

Put a = 4 and b = -3 in equation (A), we get

4 ∆ (-3) = 4 × (-3) – 2 × 4 × (-3) – 4 × (-3)³ + (-3)²

⇒ 4 ∆ (-3) = -12 + 24 + 108 + 9

⇒ 4 ∆ (-3) = 129

Thus, answer is 129.

(ii). We have (-7) ∆ (-1).

Put a = -7 and b = -1 in equation (A), we get

(-7) ∆ (-1) = (-7) × (-1) – 2 × (-7) × (-1) – (-7) × (-1)³ + (-1)²

⇒ (-7) ∆ (-1) = 7 – 14 – 7 + 1

⇒ (-7) ∆ (-1) = -13

Thus, the answer is -13.

To show: 4 ∆ (-3) ≠ (-3) ∆ 4

LHS: 4 ∆ (-3) = 129 [from answer of part (i)]

RHS: (-3) ∆ 4 = (-3) × 4 – 2 × (-3) × 4 – (-3) × (4)³ + (4)² [by putting a = -3 and b = 4 in equation A]

⇒ (-3) ∆ 4 = -12 + 24 + 192 + 16

⇒ (-3) ∆ 4 = 220

Comparing LHS and RHS, we see that

LHS ≠ RHS [∵ 129 ≠ 220]

To show: (-7) ∆ (-1) ≠ (-1) ∆ (-7)

LHS: (-7) ∆ (-1) = -13 [from answer of part (ii)]

RHS: (-1) ∆ (-7) = (-1) × (-7) – 2 × (-1) × (-7) – (-1) × (-7)³ + (-7)²

⇒ (-1) ∆ (-7) = 7 – 14 – 343 + 49

⇒ (-1) ∆ (-7) = -301

Clearly, LHS ≠ RHS [∵ -13 ≠ -301]

Given are four variables u, v, w and x representing different integers.

Also, given that there are three equations:

u × v = u …(i)

x × w = w …(ii)

u + x = w …(iii)

Given that, u = -4 and x ≠ 1.

(a). To find: v

Use equation (i), we have

u × v = u

⇒

⇒ v = 1 [u in numerator cancels out u in denominator]

Thus, v = 1.

(b). To find: w

Use equation (ii) to proceed to find w.

We have

x × w = w

⇒

⇒ x = 1

But, x ≠ 1 according to the question.

So, we don’t have any choice but to put w = 0 so that x ≠ 1.

Thus, w = 0

(c). To find: x

Use equation (iii) to find x, we have

u + x = w

⇒ (-4) + x = 0 [u = -4 is given in the question]

⇒ x = 0 + 4

⇒ x = 4

Thus, x = 4.

We have the diagram:

The diagram makes the question clearer that the difference between these two places is the total distance between point A to the sea level and the distance between point B and the sea level.

Notice, the distance between point A and the sea level = 1800 m

The distance between point B and the sea level = 700 m

The distance between the points A and B, AB = AO + OB

⇒ AB = 1800 + 700

⇒ AB = 2500

Thus, Difference between the levels of A and B = 2500 m

Let us solve this table in tabular form for ease of calculation:

Using the relation given, we can easily calculate the freezing points in °C.

We are given:

Sana and Fatima are two participants in an apple race.

The race is conducted in 6 parts.

To calculate the end result, let difference in time be denoted positive, when Sana wins and negative, Sana loses (that is, Fatima wins).

Let’s find difference in time taken from Sana’s context:

Total difference in time taken by Sana in all parts = (Sana wins by 10 s) + (Sana loses by 1 min(=60 s)) + (Sana wins by 20 s) + (Sana loses by 25 s) + (Sana loses by 37 s) + (Sana wins by 12 s)

⇒ Total difference in time taken by Sana in all parts = 10 + (-60) + 20 + (-25) + (-37) + 12

⇒ Total difference in time taken by Sana in all parts = 10 – 60 + 20 – 25 – 37 + 12

⇒ Total difference in time taken by Sana in all parts = -80

The minus indicates that Sana has lost in the end by 80 seconds.

Thus, Fatima wins the race finally.

Given that:

Profit on Monday = Rs. 47

Loss on Tuesday = Rs. 12

Loss on Wednesday = Rs. 8

To find net profit or net loss, we need to find total profit and total loss.

So, total profit = Rs 47

Total loss = Rs 12 + Rs 8 = Rs 20 [The grocer incurred loss on two days in a row, Tuesday and Wednesday]

He has incurred profit in the end as total profit is more than total loss.

Net profit = Total profit – Total Loss

⇒ Net profit = 47 – 20

⇒ Net profit = 27

Thus, net profit is Rs. 27.

Given that, score of Sona = 20

Correct answers by Sona = 10

Sona attempted all question.

The score on 1 correct answer = 3

The score on 1 incorrect answer = -1

(i). Let number of incorrect answers attempted = x

So, we can say that

Sona’s total score = 3 × (Correct answer) + (-1) × (Incorrect answer)

⇒ 20 = 3 × 10 + (-1) × y

⇒ 20 = 30 – y

⇒ y = 30 – 20

⇒ y = 10

Thus, incorrect answers she attempted were 10.

(ii). Sona attempted each and every question in the test.

Correct answers = 10

Incorrect answers = 10

Total questions in the test = Total questions attempted by Sona

⇒ Total questions in the test = Correct answers + Incorrect answers

⇒ Total questions in the test = 10 + 10

⇒ Total questions in the test = 20

Thus, questions in the test = 20

Given is that, there are total 50 questions in a test.

Marks awarded for correct answer = 2

Marks awarded for incorrect answer = -2

Marks awarded for unattempting an answer = 0

Score of Yash in the test = 94

We can find minimum correct answers in Yash’s test paper.

So, minimum correct answer is given by

⇒

⇒ Minimum answers = 47

So possibilities are:

(i). Yash attempted 47 answers correct, while (50 – 47 =) 3 unattempted.

Check: Marks on 47 correct answers = 47 × 2 = 94

Marks on remaining 3 unattempted answers = 3 × 0 = 0

Total marks of Yash should be 94.

Total marks = Marks on 47 correct answers + Marks on 3 unattempted answers

⇒ Total marks = 94 + 0

⇒ Total marks = 94

(ii). Yash attempted 48 answers correct, while 1 answer wrong and 1 unattempted.

Check: Marks on 48 correct answers = 48 × 2 = 96

Marks on 1 wrong answer = 1 × -2 = -2

Marks on 1 unattempted answer = 1 × 0

Total marks should come out to be 94.

Total marks = Marks on 48 correct answers + Marks on 1 wrong answer + Marks on 1 unattempted

⇒ Total marks = 96 – 2 + 0

⇒ Total marks = 94

We have

Since, it is given that the man starts from 50 m from the ground.

⇒ Distance of the man from the ground = 50 m

There are about 3 floors in the basement.

The man has to reach the 2^nd floor of the basement and it is given that, distance between floors is 5 m.

So, distance between ground and 1^st floor of basement = 5 m

And distance between 1^st floor and 2^nd floor of basement = 5 m

⇒ Distance between ground and 2^nd floor of basement = 5 + 5

⇒ Distance between ground and 2^nd floor of basement = 10 m

Total distance between man and 2^nd floor of the basement is given by,

Total distance = Distance between man and ground + Distance between ground and 2^nd floor basement

⇒ Total distance = 50 + 10

⇒ Total distance = 60 m

But, the speed of lift = 1 m/s

We need to find the time taken by the lift to reach 60 m at speed, 1 m/s.

Since,

⇒

⇒

⇒ Time = 60 s

Thus, it takes 60 seconds or 1 minute to reach at 2^nd floor of the basement.

We have

We have drawn a number line so that we can understand the question clearly and estimate answer easily.

If Today = 0

And date day before yesterday = 17 January

So, 3 days after tomorrow is the 4^th day after today.

And 6^th day after day before yesterday, that is, 17 January.

So, we just need to add 6 to 17 to get the desired date.

Date = 17 + 6 = 23

Thus, the date 3 days after tomorrow is 23^rd January.