Answer: (b) Real number

Explanation: The numbers which can be found on the number line and include both rational and irrational numbers are known as real numbers, e.g., -1.5,√2,0,1,2,3,π.Almost any number which you can imagine is a real number.

Answer: (c) Infinite rational numbers

Explanation: There are infinite rational numbers in between any two numbers.

Answer: (a) 3xyz

Explanation: It is given that

x + y + z = 0

On cubing both sides, we will get

(x + y + z)³ = 0

x³ + y³ + z³ – 3xyz = 0

So, the value of x³ + y³ + z³ = 3xyz

Answer: (b) 195

Explanation: From 1 – 99, the digit 1 occurs 20 times, and from 100 – 199, the digit 1 occurs 120 times.

So, from 1 to 199, the digit 1 occurs –

20 + 120 = 140 times

According to question 1 is occurring only 136 times, which means we need to remove 196, 197, 198, and 199. So, the required number of pages will be 195.

Answer: (c) 6

Explanation: Pick up the unit digit of each number and multiply them;

3 in 853

2 in 452

6 in 226

6 in 1346

∴ 3 x 6 x 2 x 6 = 216 (consider the unit digit in the product)

So, the unit digit in the product of 853 * 1346 * 452 * 226 is 6.

Answer: (d) 14400

Explanation: Number of odd numbers up to 240 = 240/2 = 120

Sum of first n odd numbers = n²

n = 120

Required sum = 120² = 14400

Answer: (c) 65889

Explanation: A number is divisible by 9 if the sum of its digits is divisible by 9.

The Sum of the digits of the given numbers are;

6+7+5+7+8= 33

5+6+7+8+5= 31

4+5+6+7+8= 30

6+5+8+8+9= 36

The sum of the digits of the number 65889 is 36, which is divisible by 9, so the correct answer is 65889.

Answer: (c) 5

Explanation: On dividing 9805 by 8, the remainder is 5. So, 5 is the smallest number which should be subtracted from 9805 to make it divisible by 8.

Answer: (c) 0

Explanation: Pick up the unit digit of each number and multiply them;

Unit digit in (341)⁴⁹¹

= Unit digit in (1)⁴⁹¹ = 1

Unit digit in (625)³¹⁷

= Unit digit in (5)³¹⁷ = 5

Unit digit in (6374)¹⁷⁹³

= (4)¹⁷⁹³

= unit digit in [(4²)⁸⁹⁶ x 4]

= unit digit in 6 x 4 = 4

So, on multiplying the unit digits, we will get –

1 x 5 x 4 = 20 (consider the unit digit in the product)

So, the unit digit in the product of {(341)⁴⁹¹ x (625)³¹⁷ x (6374)¹⁷⁹³} is 0.

Answer: (b) 306990

Explanation: A number that can be divisible by 3, 5, and 9 is also divisible by 45.

So, we check the divisibility of given numbers by the following rules –

A number is divisible by 3 if the sum of its entire digits is divisible by 3.

The number which ends with 0 or 5 is divisible by 5.

A number is divisible by 9 if the sum of its entire digits is divisible by 9.

The number 306990 fulfills all the requirements, so the answer is 306990.

Answer: (b) 13.6

Explanation: Let the number be X.

According to the question,

2/3 * 3/4 * X = 34

6/12 * X = 34

Or,

1/2 * X = 34

So, X = 68

Now, 20% of 68 is = 68 * 20/100 = 13.6

Answer: (b) 3

Explanation: The given number is divisible by 6, so it would be divisible by 2 and 3. As the last digit is 2, whatever be the value of X, it would be divisible by 2.

Now, 7 + X + 2 = 9 + X, must be divisible by 3.

So, X = 3 makes the number divisible by 3, so 3 is the required digit.

Answer: (d) 9966

Explanation: The largest four-digit number is = 9999, and on dividing it with 66, we will get 33 as the remainder.

So, the largest 4-digit number divided by 66 = 9999 – 33 = 9966

Answer: (d) 9966

Explanation: The largest four-digit number is = 9999, and on dividing it with 66, we will get 33 as the remainder.

So, the largest 4-digit number divided by 66 = 9999 – 33 = 9966

Answer: (c) 1

Explanation: We can write 6³⁶/215 as

(6³)¹²/215

Or, we can say 216¹²/215

We will always get remainder 1 on dividing 216 by 215

= 1¹²/215

So, the remainder will be 1.

Answer: (a) 245

Explanation: First we need to find the least number, so we have to find out the LCM of 8, 12, 16, and 20.

8 = 2 x 2 x 2

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

20 = 2 x 2 x 5

LCM = 2 x 2 x 2 x 2 x 3 x 5 = 240

240 is the least number that is exactly divisible by 8, 12, 16, and 20.

So, the required number that will leave remainder 5 is –

240 + 5 = 245

Answer: (d) 0.12

Explanation: Let’s examine each option individually,

√0.0004 = 0.02

√0.0121 = 0.11

(0.1)² = 0.1

0.12 = 0.12

From the above examination, we can see 0.12 is the largest among others.

Answer: (b) 7

Explanation: We have 2153¹⁶⁷, and the unit digit is = 3¹⁶⁷

As we know,

3¹ = 3 = unit digit is 3

3² = 9 = unit digit is 9

3³ = 27 = unit digit is 7

3⁴ = 81 = unit digit is 1

The cycle will continue. On the dividing the power of 3 by 4, we will get –

167/4 = remainder is 3

So, 3³ gives us the unit digit 7. So, the unit digit of 2153¹⁶⁷ is 7.

Answer: (a) a/b

Explanation: Suppose the numbers are P and Q

So, according to the question –

P + Q = a

P * Q = b

Sum of reciprocals of P and Q is = 1/P + 1/Q

= Q + P/PQ

= a/b

Answer: (a) 14/33

Explanation: Suppose the numbers are P and Q

So, according to the question –

P + Q = a

P * Q = b

Sum of reciprocals of P and Q is = 1/P + 1/Q

= Q + P/PQ

= a/b

Answer: (a) 7

Explanation: A number is divisible by 11 if the difference of the sum of digits on odd places and the sum of digits on even places is zero or divisible by 11. So, we have the number A381.

Hence, (A + 8) – (3 + 1) = either 0 or multiple of 11

If we put A = 7, then we get

(7 + 8) – (3 + 1) = 15 – 4 = 11 (Which is divisible by 11).

Answer: (a) 6

Explanation: A number is divisible by 9 if the sum of its digits is divisible by 9, e.g., 117. So, we have the number 381A.

Hence, 3 + 8 + 1 + A = multiple of 9

If we put A = 6, then we get

3 + 8 + 1 + 6 = 18 (which is a multiple of 9)

Answer: (b) 4

Explanation: According to the question when ‘n’ is divided by 5, we will get the remainder 2

n/5 = remainder 2

On putting n =7, and on dividing it with 5 we will get remainder 2

So, n = 7, and n² = 49

n²/5 = 49/5 = remainder 4

Answer: (a) 9

Explanation: Let the number be X.

So according to the question –

7X – 3X = 36

4X = 36

Or,

X = 36/4 = 9

So, the required number is 9.

Answer: (a) 11/5

Explanation: Given, 5^{(x + 3)} = 25^{(3x – 4)}

We can write it as –

5^{(x + 3)} = 5^{2 x (3x – 4)}

Or,

x + 3 = 2(3x – 4)

x + 3 = 6x – 8

or, 5x = 11

So, x = 11/5

Answer: (a) 12/35

Explanation: Suppose the numbers are P and Q

So, according to the question –

P + Q = 12

P * Q = 35

Sum of reciprocals of P and Q is = 1/P + 1/Q

= Q + P/PQ

= 12/35

Answer: (c) -126

Explanation: Given a = -4 and b = -2

On putting the values in the given equation, we will get –

(-4)³ – 3 x (-4)² + 3 x (-4) + 3 x (-2) + 3 x (-2)² + (-2)³

= -64 – 48 – 12 – 6 + 12 – 8

= -126

Answer: (a) 1

Explanation: Suppose the number is a.

So, according to the question –

a + 1/a = 2

(a² + 1)/a = 2

a² + 1 = 2a

Or, a² + 1 -2a = 0

We can say that (a – 1)² = 0

So, a =1

Hence the required number is 1, whose sum and reciprocal are equal to 2.

Answer: (a) 45

Explanation: Given ration 7 : 9

So, let the numbers be 7a and 9a. According to the question –

7a * 9a = 1575

63a² = 1575

a² = 1575/63

Or a² = 25

And a =5

So, the two numbers will be –

7a = 7 * 5 = 35

And 9a = 9 * 5 = 45

Hence, the greater number is 45 between the given numbers.

Answer: (d) 2 : 1

Explanation: Let the numbers be ‘a’ and ‘b’. According to the question –

(a + b) = 3(a – b)

(a + b) = 3a – 3b

2a = 4b

a = 2b

So, a/b = 2 : 1

Answer: (c) x⁶ / x³

Explanation: We can write x⁶ / x³ as,

x^{6 – 3} = x³

Answer: (a) 184

Explanation: Let a be the smallest even number

So, according to the question –

(a) + (a + 2) + (a + 4) + (a + 6) = 748

4(a) + 12 = 748

4a = 736

Or, a = 184

So, the smallest number be 184

Alternative solution

We can use an alternative solution of it

Middle term = 748/4 = 187

So, the numbers will 184, 186, 188, and 190

And the smallest number between them is 184.

Answer: (b) 140

Explanation: Here, we can apply the formula of a² – b² = (a + b) (a – b)

So, after applying the formula, we will get

(64 + 36) (64 – 36) = 20 * x

100 * 28 = 20 * x

x = 100 * 28/ 20 = 140

Answer: (d) 25

Explanation: According to the identity,

a² – b² = (a + b) (a – b)

It is given that (a² – b²)/(a + b) = 25

So, it can be written as (a + b) (a – b)/(a + b) = 25

Hence, (a – b) = 25

Answer: (a) 1951609

Explanation: Given 1397 x 1397

We can also write it as (1397)²

Or, (1400 – 3)²

On expanding we get,

= (1400)² + (3)² – (2 x 1400 x 3)

= 1960000 + 9 – 8400

= 1960009 – 8400

= 1951609

Answer: (b) 4

Explanation: Pick up the unit digit of each number and multiply them;

0² in (1570)² = 0

1² in (1571)² = 1

2² in (1572)² = 4

3² in (1573)² = 9

On adding the unit digits, we will get,

0 + 1 + 4 + 9 = 14 (unit digit = 4)

So, on adding (1570)² + (1571)² + (1572)² + (1573)², the unit digit will be 4.

Answer: (a) 3xyz/abc

Explanation: Given x = a(b – c), y = b(c – a), z = c(a – b)

x = a(b – c)

or, x/a = (b – c) …..(i)

y = b(c – a)

or, y/b = (c – a) …..(ii)

z = c(a – b)

or, z/c = (a – b) …..(iii)

Add equation (i), (ii), and (iii), we will get

x/a + y/b + z/c = b – c + c – a + a – b

x/a + y/b + z/c = 0

So, (x/a)³ + (y/b)³ + (z/c)³ = 3 (x/a) * (y/b) * (z/c) [because if a + b + c = 0, then a³ + b³ + c³ = 3abc]

= 3xyz/abc

Answer: (a) 8/7

Explanation: Suppose the numbers be a, b, and c. So, according to the question,

a + b + c = 2

and, a = ½b ….(i)

c = ¼b ….(ii)

From equation (i),

a/b = ½

and from equation (ii)

b/c = 4/1

Ratio between a, b, c is

a : b : c = 2x : 4x : 1x

(2x + 4x + 1x) = 2

7x = 2

So, x = 2/7

Then a = 2x = 2 * 2/7 = 4/7

b = 4x = 4 * 2/7 = 8/7

c = x = 2/7

So, the second number is 8/7.

Answer: (b) 33

Explanation: Suppose the three consecutive positive numbers be x, x + 1, and x + 2

According to the question,

(x)² + (x + 1)² + (x + 2)² = 365

On expanding, we will get,

x² + x² + 1 + 2x + x² + 4 + 4x = 365

3x² + 6x = 360

Or, x² + 2x – 120 = 0

=> (x – 10) (x + 12) = 0

So, x = 10

First number x = 10

Second number x + 1 = 11

Third number x + 2 = 12

So, sum of the numbers is = 10 + 11 + 12 = 33

Answer: (b) 42

Explanation: Suppose the three numbers be a, b, and c

According to the question,

a = 3b or, a/b = 3/1

c = 2/3a or, a/c = 3/2

Ratio between a, b, c is

a : b : c = 3x : x : 2x

(3x + x + 2x) = 252

6x = 252

So, x = 252/6 = 42

Then, first number a = 3x = 3 * 42 = 126

Second number, b = x = 42

Third number, c = 2x = 42 * 2 = 84

So, the second number is 42.

Answer: (c) 3(x – y) (y – z) (z – x)

Explanation: Suppose a = (x – y), b = (y – z), and c = (z – x)

On adding a, b, and c we will get

a + b + c = x -y + y – z + z – x

=> a + b + c =0

So, a³ + b³ + c³ = 3abc [because if a + b + c = 0, then a³ + b³ + c³ = 3abc]

We can say that (x – y)³ + (y – z)³ + (z – x)³ = 3 (x – y) (y – z) (z – x)

Answer: (c) 39 or 93

Explanation: We can solve it via two ways. Let’s see the ways.

Shortcut method:

We can also get the answer using the specified options. We can check the digits of the numbers mentioned in the options. As in the option ‘C’, there are 39 or 93, and on adding their digits, we will get 12, and on subtracting the digits, we will get 6.

Detailed method:

Suppose the digits of the two-digit number be ‘a’ and ‘b’.

So, according to the question

a + b = 12 ….(i)

a – b = 6 ……(ii)

Add equation (i) and (ii), we get

2a = 18

Or, a = 9

Put a = 9 in eqⁿ (i), we will get

b = 3

So, the two-digit number is 93 (when a > b)

Or 39 (when b > a)

Answer: (b) 7 : 6

Explanation: Let the numbers be a and b.

So, according to the question,

a * 3/5 = 70% of b

3a/5 = 70b/100

a/b = (70 * 5)/(100 * 3) = 7/6

So, the ration between a and b is 7 : 6.

Answer: (b) 5

Explanation: Given ratio is 1 : 2 : 3

So, the numbers will be 1a : 2a : 3a

Then, a³ + 8a³ + 27a³ = 4500

36a³ = 4500

Or, a³ = 4500/36 =125

So, a = 5

The smallest number is 5.

Answer: (b) 13

Explanation: Let the numerator be ‘a’ so, denominator will be ‘a + 3’.

According to the question,

(a + 7)/ ((a + 3) – 2) = 2/1

(a + 7)/(a + 1) = 2/1

2a + 2 = a + 7

=> a = 5

So, the fraction is a/a + 3 = 5/ 5 + 3

= 5/8

And the sum of the numerator and denominator of the fraction is 13.

Answer: (c) 2265025

Explanation: Given: 1505 x 1505

= 1505²

= (1500 + 5)²

Apply formula; (a + b)² = a² + b² + 2ab

= 1500² + 5² + 2 *1500* 5

= 2250000 + 25 + 15000

= 2265025

Answer: (b) 10

Explanation: The required number = Increase in result/(wrong multiplier – correct multiplier)

= 200/45 – 25

= 10

Answer: (a) c > a > b

Explanation: a = (0.4)² = 0.16

b = 0.04

c = 2/5 = 0.4

So, the relationship is c > a > b.

Answer: (a) 1

Explanation: Let the numerator of the fraction be x, so the denominator will be 11 – x

So, the fraction = x/11 – x

On adding 2 in both numerator and denominator, according to the question, the fraction will be –

(x + 2)/(11 – x + 2) = (x)/(11 – x) + 1/24

(x + 2)/(13 – x) – (x)/(11 – x) = 1/24

[11x + 22 – x² -2x – 13x + x²]/(13 – x) (11 – x) = 1/24

After solving the above equation, we will get

=> 528 – 96x = 143 – 24x + x²

x² + 72x – 385 = 0

(x + 77) (x – 5) = 0

So, x =5

So, numerator = 5

and, denominator = 11 – 5 = 6

Difference between both is = 6 – 5 = 1

Answer: (a) a + b + 2ab

Explanation: None