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Explanation:

For three numbers to be as per the sides of a right-angled triangle, they must satisfy Pythagoras Theorem.

From Pythagoras Theorem, in a right-angled triangle, the Longest side must be the hypotenuse, and the square of the hypotenuse must be equal to the sum of squares of the other two sides.

Now after checking the options,

We get 9, 5, 6 is not satisfying this theorem.

Since, 9² ≠ 5² + 6²

81 ≠ 61

Hence 9, 5, 6 cannot represent the sides of a right-angled triangle.

1. Cognitive empathy 

2. Emotional empathy 

3. Compassionate empathy

Correct option: (c) Cognitive and emotional empathy

Empathy means stepping into another person’s shoes. It is a process through which we become aware of other people’s thoughts and feelings, it is awareness of the feelings and emotions of others 

1. Create opportunities where children exchange feelings, thoughts and perspectives. Listen to them with a genuine intent to know more. 

2. Bring diversified content and facilitate discussion around it. Your questions can help children reflect and understand perspective 

3. Plan activities where children can reflect on their emotions and connect more. Emotional feelings and subsequent reactions are running wild in children these days, and they need an appropriate and regular outlet for coming to terms with and effectively dealing with them.

Types of empathy

1. Cognitive empathy – To understand what others are feeling but logically, not experiencing it. 

2. Emotional empathy – To understand what others are experiencing, it means how well an individual can perceive and understand the emotions of another. It is like experiencing the emotion the other person is going through.

(A) 1/2(log x)² + c

1. Openness gives trainees a sense of freedom and encourages participation 

2. More space for role-playing and other physical activities

Concept:

Ratio proportion

Calculation:

Let the two numbers be 3x and 4x.

⇒ (3x + y)/(4x + y) = 5/6

⇒ 18x + 6y = 20x + 5y

⇒ y = 2x

∴ 50% of larger number 4x must be added.

Concept used:

Check the by divisibility rule of 7.

To check if a number is evenly divisible by 7: Take the last digit of the number, double it Then subtract the result from the rest of the number If the resulting number is evenly divisible by 7

Calculation:

1) 519 

If we multiply last digit by 2 we get, 18

Then subtract 18 from 51, we get 33

Which is not divisible by 7

∴  519 is not divisible by 7.

2) 983

If we multiply last digit by 2 we get, 6

Then subtract 6 from 98, we get 92

Which is not divisible by 7

∴  983 is not divisible by 7.

3) 568

If we multiply last digit by 2 we get, 16

Then subtract 16 from 56, we get 40

Which is not divisible by 7

∴  568 is not divisible by 7.

4) 364

If we multiply last digit by 2 we get, 8

Then subtract 8 from 36, we get 28

Which is divisible by 7

∴  364 is divisible by 7.

Divisibility Test for 7

To Find out if a number is divisible by 7 or not, follow these steps:
1. Separate the last digit from the rest of the number. Let us call the rest of the number the truncated number. The truncated number has one less digit than the original number or the previous truncated number.
2. Double the last digit and subtract it from the truncated number.
3. Check if this result is sufficiently small so that you can immediately say if this is divisible by 7. If it is divisible by 7, then so was the original number. If it is not divisible by 7, then neither was the original number.
4. If the number is still too large to visually check if it is divisible, apply this rule over and over again as necessary.

E.g.Check 6132.

The last digit is 2 and the truncated number is 613.

Twice of 2 is 4. So subtract 4 from the truncated number 613 i.e. 613 – 4 = 609.
Again, the last digit is now 9, and the truncated number is 60.

Twice of 9 is 18. Subtract it from the truncated number 60, i.e. 60 – 18 = 42.

Now 42 is small enough to check visually.

We know that 42 is divisible by 7, so we can tell that 6132 is divisible by 7 also.

NOTE:

Explaining this step is long. But actually using it is a very short and time-saving method. This method is especially useful in Geometry and Mensuration problems where the value of π plays an important role.

Given :

Length and breadth of the room are respectively 100 m and 80 m

 Concept used :

If A and B are the length and breadth of a floor then the length of the square tile required minimum to cover this floor without any wastage will be HCF of A and B

Calculations:

Length of the tile = HCF of 100 and 80 

⇒ 20 

Area of the tile = 20 × 20 = 400 

Now, 

Number of tiles × Area of each tile = 100 × 80 

Number of tiles = 8000/400 

⇒ 20 

∴ A minimum of 20 tiles required without any wastage 

Given:

The number = 2356987

Concept:

Place value = Position of the digit in a number

Face value  = value of the digit itself

Calculation:

Place value of 5 in this number  = 5 × 10000 = 50000

Face value of 5 = 5

Required difference = (50000 ─ 5) = 49995

∴ The required difference is 49995

Given :

Product of LCM and HCF is 540 

One number is 60 

Formula used :

Product of two numbers = Product of their LCM and HCF 

Calculations :

According to the question 

60 × second number = 540 

⇒ Second number = 9 

∴ The second number will be 9 

Calculation:

Greatest four digit square will be square of 99

⇒ 99 × 99 = 9801

∴ The greatest four digit square is 9801

Given:

Numbers =  36, 54, and 60

Calculations:

Prime factors of 36 = 2 × 2 × 3 × 3

⇒ 2² × 3²

Prime factors of 54 = 2 × 3 × 3 × 3

⇒ 2 × 3³

Prime factors of 60 = 2 × 2 × 3 ×  5

⇒ 2² × 3 × 5

The LCM of 36, 54, and 60 = 2² × 3³ × 5

⇒ 4 × 27 × 5 = 540

The smallest common multiple of (36, 54 and 60) ÷ 90 = 540/90

⇒ 6

∴ The smallest common multiple of (36, 54, and 60) ÷ 90 is equal to 6

Calculation 

Doing cross multiplication , we get

⇒ 8(1 - 9y) = 5(19 - 3y)

⇒ 8 - 72y = 95 - 15y 

Add both sides by 15y, we get 

⇒ 8 - 72y + 15y = 95 - 15y + 15y

⇒  8 - 57y = 95

Subtract bothsides by 8 , we get 

⇒ 8 - 57y - 8 = 95 - 8 

⇒ - 57y = 87 

⇒ y = - 87/ 57

∴ the required  value of y  = - 87/57

Given:

x + 2 is a factor of x2 + ax + 8

Concept Used:

If p(x) be a function and (x - a) be a factor of p(x) then, p(a) = 0

Calculation

Let, p(x) = x2 + ax + 8

x + 2 is factor of p(x)

⇒ x + 2 = 0 

⇒ x  = - 2

Then by factor theorem p(- 2) = 0 

⇒ p(x) = x2 + ax + 8 

⇒ p(- 2) = (- 2)2 - 2a + 8

⇒ 4 - 2a + 8 = 0 

⇒ 12 - 2a = 0 

⇒ 2a = 12 

⇒ a = 6

 ∴ The None of these will be the right choice.

Given:

It is given that K is three digit number where k, 600 < K < 750

Calculation:

Let suppose the number is 9

∴ 9 × 9 × 9 = 729, perfect cube

Now, consider the expression again

∴ 9 × 9 × 9 = (9 × 3) × (3 × 9) = 27 × 27 = 729, perfect square

Now, sum of digits

∴ 7 + 2 + 9 = 18

Hence, option (1) is correct

Calculation:

(10)³ = 1000

(20)³ = 8000

(25)³ = 15625

⇒ We can say that largest 4 digit number which is perfect cube lies between (20)³ and (25)³

( 21)³ will give the unit digit as 1

Also from option there is only one possible case

⇒ (21)³ = 9261

∴ The largest four digit number which is a perfect cube is 9261.

Given:

Number = 7364x

Number divisible by 11

Concept used:

Divisibility rule of 11:  When the difference between the sum of alternate digits of a number is either 0 or divisible by 11, then the number is divisible by 11.

Calculation:

Let’s add the alternate digits of the given number.

7 + 6 + x = y

3 + 4 = 7

Difference between the two = y – 7

(7 + 6 + x) – 7 = multiple of 11 = 0 (least multiple of 11)

⇒ 6 + x = 0

⇒ x = -6

As we can not have a negative value for x in the number, we subtract 6 from 11 to get the positive value of x.

x = 11 – 6 = 5 

∴ For the number to be divisible by 11 the value of x must be 5.

Given:

The digit present in place of the unit of the cube root of a number is equal to the digit present in place of the unit of its number.

Calculation:

We have unit digits in any number from 0 to 9

Let us cube the unit digit and see how many of the cubes so obtained will have the same unit digit.

From the above table we get that in 6 numbers, the unit digit of the cubes is the number itself.

∴ The required answer is 6. 

Given: 

1001, 1301, 1601 and 1901

Concept used: 

A prime number is a positive integer having exactly two factors. If p is a prime, then it’s only factors are necessarily 1 and p itself. Any number which does not follow this is termed as composite numbers, which means that they can be factored into other positive integers.

Calculation: 

1001 = 1 × 7 × 11 × 13 

1301 = 1 × 1301 

1601 = 1 × 1601 

1901 = 1 × 1901 

1001 is the number with more than two factors 

∴ The correct answer is 1001

Given:

We have to find HCF of 52, 78 and 156

Concept used:

Factorization concept

Calculation:

The factors of 52 = 1, 2, 4, 13, 26, 52

The factors of 78 = 1, 2, 3, 6, 13, 26, 39, 78

The factors of 156 = 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156

Now we have to take highest common factor from all the factors

HCF = 26

∴ HCF of 52, 78, 156 is 26

Given:

We have to find LCM of 52, 78 and 156

Concept used:

Prime factorization concept

Calculation:

52 = 2² × 13¹

78 = 2¹ × 3¹ × 13¹

156 = 2² × 3¹ × 13¹

Now we have to take factors which occur most often

LCM = 2² × 3¹ × 13¹

⇒ LCM = 4 × 3 × 13

⇒ LCM = 156

∴ Required LCM is 156

Given:

The present age of Rani = y

Rani is x year older than Hamid

Fifteen years ago, Hamid's age was 1/4 of the age of Rani

Calculation:

Age of Hamid + x years = age of Rani

Age of Hamid = age of Rani - x = y - x

According to the question:

Fifteen years ago, the age of Hamid = y - x - 15

Fifteen years ago, the age of Rani = y - 15

Hence, y - x - 15 = 1/4 × (y - 15)

⇒ 4y - 4x - 60 = y - 15

⇒ 3y - 4x = 45

Hence option 1 is correct i.e. 3y - 4x = 45

subtract 6   4/5      fro.    5      7/8

Given:

Raja + Ram = 30 days

Raja worked = 16 days

Ram finishes rest = 44 days

Formula used:

Total work done (1/T) = (1/T1) + (1/T2) + (1/T3)

Where, T, T1, T2, T3 is time taken to complete the work in days

Calculation:

Assuming raja’s one day of work to be X and ram’s as Y

⇒ X + Y = (1/30)      ….(1)

⇒ 16X + 44Y = 1      ….(2)

Solving equation 1 and 2

⇒ X = 1/60

⇒ Y = 1/60

A rhombus is formed by joining two equilateral triangles. Using the right triangle of 30°, 60°, 90°, find the diagonals as 10,10√3 units. 

Hence the coordinates of the vertices of the rhombus are(5, 0) (-5, 0) (0,5√3), (0,-5√3).

Given:

A and B can complete a work in 1.8 days.

A completes the half work and B completes the other half work and it takes 3.75 days to complete the work.

B is more efficient than A.

Calculation:

Let A works for ‘a’ days to complete the half of the job.

Hence, B takes (3.75 – a) days to complete the remaining half of the work

∴ A’s one day work = 1/2a

∴ B’s one day work = 1/[2(3.75 – a)]

Amount of work done by A and B in 1 day = 1/1.8 = 5/9

⇒ 1/2a + 1/[2(3.75 – a)] = 5/9

⇒ 1/a + 1/(3.75 – a) = 10/9

⇒ (3.75 – a + a)/(a)(3.75 – a) = 10/9

⇒ 9 × 3.75 = 10 × (a)(3.75 – a)

⇒ 33.75 = 37.5a – 10a²  

⇒ 10a² – 37.5a + 33.75 = 0

⇒ (a – 2.25)(a – 1.5) = 0

⇒ a = 2.25, 1.5

As B is more efficient, A will take more days to complete the work

a = 2.25

⇒ B’s one day work = 1/[2(3.75 – a)] = 1/3

∴ B will take 3 days to complete the task himself.

Distance between (4, 5) and (4, 2) = 5 – 2 = 3

Distance between (4, 2) and (8, 2) = 8 – 4 = 4

Distance between (4, 5) and (8, 2)

= \(\sqrt{(4\,8)^2\,(5\,2)^2}\) = \(\sqrt{4^2\,3^2}\) = 5

3, 4, 5 are sides of a right angled triangle, since 3² + 4² = 5²

The coordinates of circumcentre is (6 ,3, 5)

Radius of circumcircle = 2.5 unit

Given:

Milena and Mashuka can together complete a work in 36 days

Mashuka completed 2/5 part work alone and rest Milena completed alone and total time taken by both of them is 78 days

Mashuka is more efficient than Milena

Formula Used:

Work = efficiency × time

Calculation:

Let efficiency of Mashuka be x

Total work = 36 days 

⇒ Efficiency of Milena and Mashuka = 36/36 = 1 

⇒ Efficiency of Milena = 1 – x

Mashuka's work = (2/5) × 36 

Milena's work = (1 – 2/5) × 36 = (3/5) × 36

Total time = \(\frac{{\frac{2}{5} × 36}}{x} + \;\frac{{\frac{3}{5} × 36}}{{\left( {1 - x} \right)}}\)

\( ⇒ 78 = \;\frac{{\frac{2}{5} × 36}}{x} + \;\frac{{\frac{3}{5} × 36}}{{\left( {1 - x} \right)}}\)

\( ⇒ 39 × 5\; = \;\frac{{\left( {1 - x} \right) × 36 + 54\;\left( x \right)}}{{x\;\left( {1 - x} \right)}}\)

\( ⇒ 36 - 36x + 54x\; = 195\left( {x - {x^2}} \right)\)

\(⇒ 65\;{x^2} - 59x + 12 = 0\)

⇒ x = 4/13 and 3/5 

Here x = 4/13 is not valid because it is given that x > (1 – x)

⇒ x = 3/5

Efficiency of Milena = 1 – 3/5 = 2/5

Time taken by Milena to complete the work alone = \(\frac{{36}}{{\frac{2}{5}}}\,\)days

⇒ 18 × 5 days

⇒ 90 days

∴ 90 days taken by Milena to complete the work alone.

Take AB as x axis and AD as y axis, a and b are the sides of the rectangle. 

The coordinates of the points are marked on the picture.

PA² + PC² + x² + y² + (x – a)² + (y – b)²

PB² + PD² = (x – a)² + y² + x² + (y – b)²

PA² + PC² = PB² + PD²

Distance between (-6, 8) and (2, P)

= \(\sqrt{(-6-2)^2+(8-P)^2}\)

They are equal

8² + (8 - P)² = 6² + (6 - P)²

64 + 64 - 16P + P²

= 36 + 36 - 12P + P²

4P = 56

\(P=\frac{56}{4}=14\)

Distance between (0,0) and (8,6)

= \(\sqrt{8^2-6^2}=10\)

Area = 10² = 100

According to the principle of ‘Division of Work’, the work should be divided into small jobs. This is because of the fact that any complex task can be performed efficiently if it is divided into smaller units and performed by specialists.

Correct option is (b) Division of work

(b) children’s garden

Given : 

(10x + 5) : (42x + 8) = 5 : 8 

Concept used :

x: y = a : b 

Triplicate ratio for a : b will be 

x : y = a³ : b³ 

Calculations :

(10x + 5) : (42x + 8) = 5³ : 8³ (triplicate ratio for 5 : 8)

⇒ (10x + 5) : (42x + 8) = 125 : 512 

⇒ x = 12 

⇒ (x)³ = 12³

⇒ 1728 

∴ the value of x will be 1728

Given:

The number series:

10, 5, ?, 10, 40, 320

Concept Used:

Number series following a fixed pattern.

Calculation;

The given number series follows the pattern:

10 × 2^-1 = 5

5 × 2⁰ = 5

5 × 2¹ = 10

10 × 2² = 40

40 × 2³ = 320

∴ The number coming in place of “?” in the given number series is 5

Given:

The number series:

10, 15, 25, 40, ?, 85

Concept Used:

Number series following a fixed pattern.

Calculation;

The given number series follows the pattern:

10 + (5 × 1) = 15

15 + (5 × 2) = 25

25 + (5 × 3) = 40

40 + (5 × 4) = 60

60 + (5 × 5) = 85

∴ The number in place of “?” in the given number series is 60

Given:

60, 40, 50, ?, 180, 460

Calculation:

The series follows the following pattern

60 × 0.5 + 10 = 40

40 × 1 + 10 = 50

50 × 1.5 + 10 = 85

85 × 2 + 10 = 180

180 × 2.5 + 10 = 460

∴ The Required term in the series will be 85

Given:

The number series:

100, 90, ?, 40, 0, - 50

Concept Used:

Number series following a fixed pattern.

Calculation;

The given number series follows the pattern:

100 – (10 × 1) = 90

90 – (10 × 2) = 70

70 – (10 × 3) = 40

40 – (10 × 4) = 0

0 – (10 × 5) = - 50

∴ The number in place of “?” in the given number series is 70

Calculation:

The given number series follows the pattern as:

⇒ 22 + 7 = 29

⇒ 29 + 9 = 38

⇒ 38 + 11 = 49

⇒ 49 + 13 = 62

⇒ 62 + 15 = 77

∴ 77 will come in place of the question mark (?) 

Given:

The number series:

1, 2, 6, 20, 88, 445

Concept Used:

Number series following a fixed pattern.

Calculation;

The given number series follows the pattern:

1 × 1 + 1 = 2 

2 × 2 + 2 = 6

6 × 3 + 3 = 21

21 × 4 + 4 = 88

88 × 5 + 5 = 445

∴ The Wrong Number in the given number series is 20

The number series follows logic as:

3

(3 × 2)  = 6

(6 × 2)  = 12

(12 × 2) = 24

(24× 2)  = 48

∴ the missing number is 48.

Considering the given series

12, 13, 28, 88, 352, 1765

The logic of the given series can be explained as

12 × 1 + 1 = 13

13 × 2 + 2 = 28

28 × 3 + 3 = 87

87 × 4 + 4= 352

340 × 5 + 5 = 1765

∴ Wrong term in given number series is 88.

Calculation:

The series follows following pattern

⇒ 128 × 0.5 = 64

⇒ 64 × 0.5 = 32

⇒ 32 × 0.5 = 16

⇒ 16 × 0.5 = 8

⇒ 8 × 0.5 = 4

∴ The value of ? is 16

Given:

Ratio = 3 : 5 : 2

a - c = 900

Calcultions:

Let, the ratio is 3x, 5x and 2x

⇒ a - c = 900

⇒ 3x - 2x = 900

⇒ x = 900

⇒ Total profit = 3x + 5x + 2x = 10x

⇒ Total profit = 10 × 900

⇒ Total profit = Rs. 9000

∴ The total profit is Rs. 9000.

The pattern of the given series is:

⇒ 64 × 1/2 = 32

⇒ 32 × 1/2 = 16

⇒ 16 × 1/2 = 8

⇒ 8 × 1/2 = 4 

⇒ 4 × 1/2 = 2

The missing number is 4.

Given:

The number series:

12, 13, 28, ?, 352, 1765

Concept Used:

Number series following a fixed pattern.

Calculation;

The given number series follows the pattern:

12 × 1 + 1 =  13

13 × 2 + 2 =  28

28 × 3 + 3 =  87

87 × 4 + 4 =  352

352 × 5 + 5 =  1765

∴ The number in place of “?” in the given number series is 87