Mathematics Test 275

Result

Mathematics Test 275

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Question 1
4 / -1

Two numbers x and y are chosen at random without replacement from the first 30 natural numbers. The probability that x² - y² is divisible by 3 is

A
3/29

B
3/55

C
5/29

D
47/87

Solution

Total number of ways of choosing two numbers out of 30 is ³⁰C₂ We rewrite the first 30 natural numbers in three rows as follows:

Row I: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28
Row II: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
Row III: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

For x² - y² to be divisible by 3, either both x and y must be chosen from the same row or exactly one of x, y from Row I and the other from Row II.

Thus, the number of favourable ways

= 3 (¹⁰C₂) + 10 x 10 = 235

∴ probability of the required even
Question 2
4 / -1

Three numbers are chosen at random from the first 20 natural numbers. The probability that the selected numbers form a G.P. is

A
1/20

B
1/30

C
1/50

D
1/45

Solution
Question 3
4 / -1

The probability that in a group of 3 people, at least two will have the same birthday is

A

B

C

D

Solution

Let A be the event that at least two people have the same birthday.

Then,
Question 4
4 / -1

Past records show that on an average, 1 house out of 1000 in a certain district has a fire during a year. If there are 2000 houses in that district, the probability that 5 houses will have a fire is

A
0.036

B
0.084

C
0.131

D
None of these

Solution
Question 5
4 / -1

Alex is taking readings of an experiment. He calculated the algebraic sum of deviation of 25 observations which were measured from 40 was found out to be 5. What is the mean of the readings?

A
42.8

B
41.4

C
40.7

D
40.2

Solution
Question 6
4 / -1

A scientist was weighing each of 30 fishes. Their mean weight worked out is 30 g and standard deviation is 2 g. Later, it was found that the measuring scale was misaligned and always under-reported every fish's weight by 2 g. The correct mean and standard deviation (in g) of fishes, are respectively

A
32 and 2

B
32 and 4

C
28 and 2

D
28 and 4

Solution

Since mean (X + b) = E( X + b) = E(X) + b and Var (X + b) = Var X,

We get correct mean as 30 + 2 = 32 g and (s.d.) standard deviation as 2 g.
Question 7
4 / -1

If the standard deviation of x₁, x₂, …, x_n is 3.5, then the standard deviation of – 2x₁ – 3, – 2x₂ – 3, …, – 2x_n – 3 is:

A
-7

B
-4

C
7

D
0.75

Solution

The standard deviation of a set remains unchanged if each data is increased or decreased by a constant.

However, it changes similarly when data is multiplied or divided by a constant.

∴ The SD for the new data set will be = −2 × 3.5 = −7
Question 8
4 / -1

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are:

A
10, 11

B
3, 18

C
8, 13

D
1, 20

Solution
Question 9
4 / -1

Consider three observations a, b and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

A
b₂ = 3(a² + c²) + 9d²

B
b² = a² + c² + 3d²

C
b² = 3(a² + c² + d²)

D
b² = 3(a² + c²) - 9d²

Solution
Question 10
4 / -1

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is:

A
4 : 9

B
6 : 7

C
5 : 8

D
10 : 3

Solution