Explanation

To find the probability that a pair of books will always be together, we need to count the number of arrangements where the pair is together and divide it by the total number of arrangements.

Let's consider the pair of books as a single unit. Then, we have 9 units (the pair and the remaining 8 books) to arrange on the shelf. The number of ways to arrange these 9 units is 9!.

However, within the pair, the two books can be arranged in 2! ways (either book can be on the left or right).

So, the total number of arrangements where the pair is together is 9! × 2!.

Now, we divide the number of favorable arrangements (where the pair is together) by the total number of arrangements:

Probability = (9! × 2!) / 10!

= (9! / 10!) × 2

= (1/10) × 2

= 1/5

Explanation

For the parabola y^2 = −4x, three normals can be drawn from a point (h, 2) to the parabola only if h < −2.

This condition ensures that the point lies in the correct region where three normals are possible.

Explanation

Track length = 1 km

Formula used:

Distance Speed x Time

Calculation:

"A" beats "B" by 50 m or 10 sec, means "B" covers 50 m in 10 sec

Speed of "B" = 50/10 = 5 m/s

Time to cover 1000 m = 1000/5 = 200 sec

"A" beats "B" by 10 sec means

"A" reach 10 sec early

Time taken by "A" to complete the race = 200 - 10 = 190 sec

∴ Time taken by "A" to complete the race is 190 sec.

Explanation

Principal amount (P) = Rs. 800

Rate of interest (R) = 5% per annum

Compound interest (CI) = Rs. 882 - Rs. 800 = Rs. 82

Using the compound interest formula:

A = P (1 + R/100)^n

where A is the amount after n years, we can plug in the values:

882 = 800 (1 + 5/100)^n

Simplifying the equation:

(1.05)^n = 882/800

(1.05)^n = 1.1025

Now, taking the logarithm of both sides (base 1.05):

n = log1.05(1.1025)

n ≈ 2

So, it will take approximately 2 years for Rs. 800 to amount to Rs. 882 at 5% p.a. compounded annually.

Explanation

• Fear of failure in examinations, fear of social situations like meeting with people in high offices or being lonely in the house, and fear of wild animals—are important fear-causing stimuli.
• These are common examples of fears that individuals may experience in various situations.

Explanation

3x² - 7x - 6 = 0

We can factor the equation:

(3x + 2)(x - 3) = 0

This gives us two possible solutions:

3x + 2 = 0 --> 3x = -2 --> x = -2/3

x - 3 = 0 --> x = 3

So, the roots of the equation are x = -2/3 and x = 3.

The correct answer is:

3, -2/3

Explanation

The formula for simple interest is:

Simple Interest (SI) = (Principal x Rate x Time) / 100

Rearrange the formula to solve for Rate (R):

R = (SI / Principal) x (100 / Time)

Plug in the values:

R = (70 / 150) x (100 / 10)

R = 0.4667 x 10

R = 14/3

R = 4.67 (approximately)

So, the rate of interest is approximately 4.67% or 14/3%.

Explanation

To find the number with the most divisors, let's analyze each option:

• 99 = 3 × 3 × 11 (3 divisors)
• 101 (prime number, only 2 divisors: 1 and 101)
• 182 = 2 × 7 × 13 (4 divisors)
• 176 = 2 × 2 × 2 × 2 × 11 (6 divisors)

The number with the most divisors is 176, with 6 divisors.

Explanation

Length of the train = 240 m

Time taken to pass the pole = 24 seconds

Speed of the train = Distance / Time = 240 m / 24 s = 10 m/s

Total distance to be covered = Length of train + Length of platform = 240 m + 650 m = 890 m

Time taken to pass the platform = Total distance / Speed = 890 m / 10 m/s = 89 seconds

Explanation

For proper adjustment in the environment, the following conditions must be fulfilled:

• Physical fitness and health: Good physical health and fitness are essential for adapting to the environment.
• Social acceptability of the person: Acceptance by others is crucial for feeling comfortable and adjusted in a social setting.
• Free from psychological diseases: The absence of mental health issues like anxiety, depression, or other psychological disorders is vital for adjusting to the environment.