Mathematics Test 281

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Mathematics Test 281

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Weekly Quiz Competition

Question 1
4 / -1

A line passes through the points (6, –7, –1) and (2, –3, 1). If the angle which the line makes with the positive direction of the x-axis is acute, then the direction cosines of the line are

A
2/3, –2/3, –1/3

B
2/3, 2/3, –1/3

C
2/3, –2/3, 1/3

D
2/3, 2/3, 1/3

Solution
Question 2
4 / -1

The equation of the line through the point (2, 1, 1) and the intersection of the lines 2x – y – 4 = 0 = y + 2z and x + 3z – 4 = 0 = 2x + 5z – 8 is

A
x + y + z = 2, x + 2y = 0

B
x + y + z = 2, x + 2z = 4

C
x + 2y + z = 1, x + 2z = 4

D
x + 2y + z = 1, x + 2y = 0

Solution
Question 3
4 / -1

The equation of a plane containing the line of intersection of the planes 2x - y - 4 = 0 and y + 2z - 4 = 0 and passing through the point (1, 1, 0) is:

A
x + 3y + z = 4

B
x - y - z = 0

C
x - 3y - 2z = -2

D
2x - z = 2

Solution

The required plane is:

(2x - y - 4) + λ(y + 2z - 4) = 0 ... (1)

It passes through (1, 1, 0).

⇒(2−1−4) +λ(1 − 4) = 0

⇒ −3 − 3λ = 0 ⇒ λ = −1

Substitute the value of λ=−1 in equation (1).

The required equation of plane is:

x−y−z=0
Question 4
4 / -1

Four persons can hit a target correctly with probabilities 1/2, 1/3, 1/4 and 1/8. If all hit at the target independently, then the probability that the target would be hit is

A
25/192

B
1/192

C
25/32

D
7/32

Solution
Question 5
4 / -1

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is

A
3/10

B
1/10

C
3/20

D
1/5

Solution
Question 6
4 / -1

Out of 11 consecutive natural numbers, if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:

A
5/101

B
10/99

C
5/33

D
15/101

Solution

Out of 11 consecutive natural numbers, there are either 6 even and 5 odd numbers or 5 even and 6 odd numbers.

When 3 numbers are selected at random, then total cases = ¹¹C₃

Since these 3 numbers are in A.P., let the numbers be a, b, c.

So, favourable cases = ⁶C₂ + ⁵C₂

= 15 + 10 = 25

P(3 numbers are in A.P.) = 25/(¹¹C₃) = 25/165 = 5/33
Question 7
4 / -1

Let A and B be two non-null events such that A ⊂ B. Then, which of the following statements is always correct?

A
P(A | B) = 1

B
P(A | B) = P(B) - P(A)

C
P(A | B) ≤ P(A)

D
P(A | B) ≥ P(A)

Solution
Question 8
4 / -1

A random variable has the following probability distribution:

The value of k is

A
0.1

B
-0.1

C
-1

D
1

Solution

Random variate

∑P(x) = 1

0 + k + 2k + 2k + 3k + k² + 2k² + 7k² + k = 1

10k² + 9k = 1

10k² + 10k - k - 1 = 0

10k(k + 1) - 1(k + 1) = 0

(k + 1)(10k - 1) = 0

k = -1, k = 1/10

∴ k ≠ -1

The probability of a variable cannot be negative, so

k = 0.1.
Question 9
4 / -1

If (1 + 3p)/3, (1 – p)/4 and (1 – 2p)/2 are the probabilities of three mutually exclusive events, then the set of all values of p is

A
1/3 ≤ p ≤ 1/2

B
1/4 ≤ p ≤ 1/3

C
–1 ≤ p ≤ 1/5

D
–2 ≤ p ≤ 1/3

Solution
Question 10
4 / -1

The mean and variance of a binomial distribution are 4 and 2, respectively. What is the probability of two successes?

A
37/256

B
219/256

C
1/2

D
7/64

Solution