ID : in-8-Logical-R easoning [1]

Class 8
Logical Reasoning
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Choose correct answer(s) from the given choices

1 1 1 1
(1) (99999 + ) (99999 + ) − (99999 − ) (99999 − )= ?
33333 33333 33333 33333

a. 6 b. 3
c. 12 d. 33333

(2) How many more squares need to be shaded to cover 75% of the total area?

a. 6 b. 42
c. 24 d. 48

3
(3) P (x) is a polynomial such that P (x + ) = P (x). If P (9) = 344, find the value of P (21).
2

a. 344 b. 348
3 1
c. 344 d. 344
2 2

(4) What is the missing number in the series: 1, 2, 4, 3, 9, 4, 16, __25, 6, 36, 7 ?
a. 5 b. 4
c. 25 d. 6

(5) Tina introduces Rajesh as the son of the brother of her mother. How is Tina related to Rajesh?
a. Cousin b. Nephew
c. Aunt d. Daughter

(6) If we use (x+) to indicate the following sum:

1 + 2 + 3 + ... + x
then find the value of k in the following equation:
(15+) - (14+) = (k+).
a. 6 b. 7
c. 5 d. 3

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 ID : in-8-Logical-R easoning [2]

Fill in the blanks

(7) If we use (x+) to indicate the following sum:

1 + 2 + 3 + ... + x

then the value of k in the following equation is .

(21+) - (20+) = (k+).

Answer the questions

(8) If N = 111112, find the 5th digit in the expansion of N from the right.

(9) During a day, the hour hand and the minute hand of a clock form a right angle, at multiple times.
For example, the two hands form a right angle at 9 am. How many times during a day (24 hours)
will the two hands form a right angle?

(10) Ashish likes 144 but not 134, 1600 but not 1700, and 5625 but not 5610. Which of the following
four numbers would he like: 629, 618, 625 or 622?

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 ID : in-8-Logical-R easoning [3]

Solutions
(1) c. 12

Step 1
1 1 1 1
(99999 + ) (99999 + ) − (99999 − ) (99999 − )
33333 33333 33333 33333

2 2
1 1
(
= 99999 + ) − (99999 − )
33333 33333

Step 2
Using algebraic identities (a + b)2 = a
2
+ b
2
+ 2ab and (a − b)2 = a
2
+ b
2
− 2ab
2 2
1 1
(99999 + ) − (99999 − )
33333 33333

2 2
1 1 1 1
=99999
2
+ ( ) + 2 × 99999 × − [99999 2
+ ( ) − 2 × 99999 × ]
33333 33333 33333 33333

2 2
1 1 1 1
=99999
2
+ ( ) + 2 × 99999 × − 99999
2
− ( ) + 2 × 99999 ×
33333 33333 33333 33333

=2 × 3 + 2 × 3

=12

Step 3
Therefore, the value of
1 1 1 1
(99999 + ) (99999 + ) − (99999 − ) (99999 − ) is 12.
33333 33333 33333 33333

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 ID : in-8-Logical-R easoning [4]

(2) b. 42

Step 1
The total number of squares = number of rows × number of columns
=8×8
= 64

Step 2
Let us now find the required number of shaded squares, which is 75% of the total number of
squares,
or 75% of 64
75
= 64 ×
100
= 48

Step 3
We can see in the picture that 6 squares are already shaded.

Step 4
Therefore, the number of more squares which need to be shaded
= The required number of shaded squares − The number of squares already shaded
= 48 − 6
= 42

Step 5
Hence, 42 more squares need to be shaded to cover 75% of the total area.

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 ID : in-8-Logical-R easoning [5]

(3) a. 344

Step 1
It is given that,
3
P (x + ) = P (x)
2

3
It means that value of polynomial P for any x is same as its value for (x + ).
2

Step 2
3
It also means that if we keep adding to the x, the value will remain unchanged.
2
3
i. e. P (x) = P (x + )
2

3 3 3
⟹ P (x) = P [(x + )+ ] = P (x + 2 × )
2 2 2

3 3 3
⟹ P (x) = P [(x + 2 × )+ ] = P (x + 3 × )
2 2 2

3 3 3
⟹ P (x) = P [(x + 3 × )+ ] = P (x + 4 × )
2 2 2

and so on..

3 3 3
⟹ P (x) = P [(x + 3 × )+ ] = P (x + n × ) … [ Where n is any natural number]
2 2 2

Step 3
Now we can write P (21) in terms of P (9) as following,
P (21) = P (9 + 12)

3
⟹ P (21) = P (9 + 8 × )
2

⟹ P (21) = P (9) [Using relation (1)]

Step 4
Since P (21) = P (9), value of P (21) = 344

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 ID : in-8-Logical-R easoning [6]

(4) a. 5

Step 1
1, 2, 4, 3, 9, 4, 16, __25, 6, 36, 7
If we look at the given series carefully, we notice that the given series is like:
1, n, n2, (n+1), (n+1)2, (n+2), (n+2)2........
(Where, n = 2, 3, 4.....). So, we can say that the missing number in the series 1, 2, 4, 3, 9, 4,
16, __25, 6, 36, 7 is 5.

Step 2
Hence, option a is the correct answer.

(5) a. Cousin

Step 1
According to the question, Tina introduces Rajesh as the son of the brother of her mother.

Step 2
Or, we can say that the brother of Tina's mother is the maternal uncle of Tina.

Step 3
Or, we can say that Rajesh is the son of Tina's maternal uncle.

Step 4
Or, we can say that Rajesh is the cousin of Tina.

Step 5
Hence, option a is the correct answer.

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 ID : in-8-Logical-R easoning [7]

(6) c. 5

Step 1
It is given that x+ = 1 + 2 + 3 + ... + x

Step 2
According to the question, (15+) - (14+) = (k+)
⇒ (1+2+3+ ... +15) - (1+2+3+ ... +14) = (1+2+3+ ... +k)

⇒ (1+2+3+ ... +14) + 15 - (1+2+3+ ... +14) = (1+2+3+ ... +k)

⇒ 15 = (1+2+3+...+k)

Step 3
Let us find the value of k for which sum of first k natural numbers is 15.
Let us start with k = 2
For k = 2, 1 + 2 = 3
For k = 3, 1 + 2 + 3 = 6
For k = 4, 1 + 2 + 3 + 4 = 10
For k = 5, 1 + 2 + 3 + 4 + 5 = 15

Step 4
Therefore, the value of k is 5. Hence, option c is the correct answer.

(7) 6

Step 1
It is given that x+ = 1 + 2 + 3 + ... + x

Step 2
According to the question, (21+) - (20+) = (k+)
⇒ (1+2+3+ ... +21) - (1+2+3+ ... +20) = (1+2+3+ ... +k)

⇒ (1+2+3+ ... +20) + 21 - (1+2+3+ ... +20) = (1+2+3+ ... +k)

⇒ 21 = (1+2+3+...+k)

Step 3
Let us find the value of k for which sum of first k natural numbers is 21.
Let us start with k = 2
For k = 2, 1 + 2 = 3
For k = 3, 1 + 2 + 3 = 6
For k = 4, 1 + 2 + 3 + 4 = 10
For k = 5, 1 + 2 + 3 + 4 + 5 = 15
For k = 6, 1 + 2 + 3 + 4 + 5 + 6 = 21

Step 4
Hence, the value of k is 6.

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 ID : in-8-Logical-R easoning [8]

(8) 5

Step 1
The square of 11 is equal to 121.

Step 2
The square of 111 is equal to 12321.

Step 3
We can see that the squares of such numbers follow a pattern. Based on the pattern the
square of 11111 will be 123454321.

Step 4

We can see that the 5th digit in the square of 11111 from the right is 5.

(9) 44

Step 1
Let us look at the clock at 9 am:

We can see that the hour hand and the minute hand are making a right angle.

Step 2
In a clock, while the minute hand moves, the hour hand also moves, although a lot slowly.
It is easy to see that in a 12 hour period, the minute hand make 12 revolutions while, the hour
hand makes one.

We can visualize the above statement this way:

If we hold the clock in our hands and always keep on rotating it slowly, such that the hour
hand always stay on the same position, the minute hand will make 12 - 1 = 11 revolutions.
In other words, the minute hand makes 11 revolutions around the hour hand in a 12 hour
period.

Step 3
For each revolution around the hour hand, the minute hand makes a right angle twice with it.
The total number of times we see the two hands making a right angle is 11 × 2 = 22.

Step 4
In 12 hours, the number of times the two hands make a right angle = 22
In 24 hours, the number of times the two hands make a right angle = 22 × 2 = 44

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 ID : in-8-Logical-R easoning [9]

(10) 625

Step 1
If we look at the pattern, we notice that Ashish likes only those numbers which have complete
squares.
144 = 122
1600 = 402
5625 = 752

Step 2
Now, observe the given options and see if there is any number which has complete square.
Among the given options, we notice that 625 = 252.
Hence, 625 has complete square of 25.

Step 3
Therefore, Ashish will like 625.

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