(a) 3

(b) 4

(c) 5

(d) 6

Solution:

Given that,

A group of 630 children is seated in rows for a group photo session.

Total children seated = 630

Each row contains three less children then the row in front of it.

Let A be the number of children in 1^st row

Case I: If there are three rows

A, A - 3 and A - 6

A + A - 3 + A - 6 = 630 = 3A - 9 = 630 or A = 213

Thus it is possible

Case II: 4 rows

A + A - 3 + A - 6 + A - 9 = 630 or A = 162 which is possible

Case III: 5 rows

A + A - 3 + A - 6 + A - 9 + A - 12 = 630 or A = 132 which is possible

Case IV: 6 rows

A + A - 3 + A - 6 + A - 9 + A - 12 + A - 15  = 630 or A  = 112.5 (NOT POSSIBLE)

Hence option (d) is correct

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Q2. There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers? [CSAT 2017]

(a) 47

(b) 49

(c) 51

(d) 45

Solution:

Given that,

There are thirteen 2-digit consecutive odd numbers.

Let the number be 2x + 1, 2x + 3........2x + 25

If 39 is the mean of the first five such numbers

(2x + 1 + 2x + 3 + 2x + 5 + 2x + 7 + 2x + 9)/5 = 39

10x + 25 = 195

10x = 170

x = 17

Thus the 13 numbers are 35, 37, 39, 41......59

Sum of terms,

Sn = n/2 [first term + last term]

Sn = 13/2 [35+59]

      = 611

(35 + 37 + 39....+ 59)/13 = 611/13 = 47

Hence option (a) is correct

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Q3. From January 1, 2021, the price of petrol (in Rupees per litre) on m^th day of the year is 80 + 0.1m where m = 1, 2, 3 ..., 100 and thereafter remains constant. On the other hand, the price of diesel (in Rupees per litre) on n^th day of 2021 is 69 + 0.15n for any n. On which date in the year 2021 are the prices of these two fuels equal? [CSAT 2021]

(a) 21st May

(b) 20th May

(c) 19th May

(d) 18th May

Solution:

Given that,

The price of petrol (in Rupees per litre) on m^th day of the year is 80+0.1m where m = 1 2, 3 ..., 100 and thereafter remains constant.

On 100th day the price = 80 + 0.1 x 100 = Rs.90

thereafter remains constant

On the other hand, the price of diesel (in Rupees per litre) on n^th day of 2021 is 69+ 0.15n for any n.

For equal prices there are two cases

Case (i): before 100th day

80+0.1m = 69+ 0.15m (where m = n prices become equal)

m = 220 days

Thus it is not possible as m <100 days

Case (ii): m ≥ 100

As the price will remain constant

69+ 0.15m = 90

m = 140 days

Thus the date will be 20th May

Hence option (b) is correct

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Q4. For five children with ages a < b < c < d < e; any two successive ages differ by 2 years.

Question: What is the age of the youngest child?

Statement-1: The age of the eldest is 3 times the youngest.

Statement-2: The average age of the children is 8 years.

Which one of the following is correct in respect of the above Question and the Statements? [CSAT 2023]

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

(b) The Question can be answered by using either Statement alone.

(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

(d) The Question cannot be answered even by using both the Statements together.

Solution:

Given that,

For five children with ages a < b < c < d < e any two successive ages differ by 2 years

Let the ages be x

X, X + 2, X + 4, X + 6 and X + 8

5X + 20.....(i)

From statement I: The age of the eldest is 3 times the youngest.

X + 8 = 3 x (X)

X = 4

Thus the ages are 4, 6, 8, 10 and 12

Hence statement I alone is sufficient to answer the question

Statement-2: The average age of the children is 8 years.

(X + X + 2 + X + 4 + X + 6 + X + 8) = 40

5X = 20 or X = 4

Thus the ages are 4, 6, 8, 10 and 12

Hence statement II alone is sufficient to answer the question

Option (b) is correct

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Q5. A boy plays with a ball and he drops it from a height of 1.5 m. Every time the ball hits the ground, it bounces back to attain a height 4/5th of the previous height. The ball does not bounce further if the previous height is less than 50 cm. What is the number of times the ball hits the ground before the ball stops bouncing? [CSAT 2021]

(a) 4

(b) 5

(c) 6

(d) 7

Solution:

Given that,

Height from which ball is dropped = 1.5 m or 150 cm

Bounces 4/5th of the height every time

Does not bounce if the height is < 50cm

No. of drops	Height before	Bouncing Height
1	150	(4/5) x 150 = 120 cm
2	120	(4/5) x 120 = 96 cm
3	96	(4/5) x 96 = 76.8 cm
4	76.8	(4/5) x 76.8 = 61.44 cm
5	61.44	(4/5) x 61.44 = 49.152 cm
6	49.152 cm	No bounce

No. of bounces = 5

Hence option (b) is correct

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Q6. What is the sum of the first 28 terms in the following sequence?

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2,...  [CSAT 2024]

(a) 83

(b) 84

(c) 85

(d) 86

Solution:

Given that,

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2.......

Now,

Understanding the sequence

Leaving first term aside

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2.......

Reverse order

Sequence	Total sum
1	1
2, 1	3
3, 2, 1	6
4, 3, 2, 1	10
5, 4, 3, 2, 1	15
6, 5, 4, 3, 2, 1	21
Total terms = 21	Total sum = 56

Now for 28 terms

Sequence would be 7, 6, 5, 4, 3, 2

not taking 1 as it will be 29th term

Sum = 27 now adding the 1^st term 1 then 27+1 = 28

Total 28 term sum = 56 + 28 = 84

Hence option (b) is correct