Correct Answer: (D) 997

Explanation:

A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. To determine the largest 3-digit prime number, we start with the largest 3-digit number, which is 999, and check downward until we find a prime number.

1. 999: Not a prime number because it is divisible by 3 (999 ÷ 3 = 333).

2. 998: Not a prime number because it is even (divisible by 2).

3. 997: Check divisibility by all prime numbers less than √997 (approximately 31.5). The prime numbers to check are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.

– 997 is not divisible by any of these primes, so it is a prime number.

Thus, the largest 3-digit prime number is 997.

Correct Answer: (A) 1

Explanation:

Let’s solve each inequality step by step.

1. Solving -n + 2 ≥ 0:
-n + 2 ≥ 0
-n ≥ -2
n ≤ 2

2. Solving 2n ≥ 4:
2n ≥ 4
n ≥ 2

Combining Both Inequalities:
From the solutions above:
n ≤ 2 and n ≥ 2

The only integer that satisfies both conditions is:
n = 2; Answer: 1

Correct Answer: (B) 10%

Explanation:

Step 1: Define the variables
Let the selling price (SP) of one pen be Rs. x.
The total selling price of 45 pens = 45x.
The problem states that the loss is equal to the selling price of 5 pens.
Thus, the total loss = 5x.

Step 2: Understand the cost price (CP)
The cost price (CP) of 45 pens can be calculated as:
CP = SP + Loss
Substitute the values:
CP = 45x + 5x = 50x

Step 3: Calculate the loss percentage
The formula for loss percentage is:
Loss % = (Loss / CP) × 100
Substitute the values:
Loss % = (5x / 50x) × 100
Loss % = (1/10) × 100 = 10%

Correct option: (B) 10%

Correct Answer: (A) ₹1200

Explanation:

To determine the principal amount based on the difference between compound interest (CI) and simple interest (SI) over 2 years at an annual interest rate of 10%, follow these steps:

1. Calculate Simple Interest (SI):
SI = P × Rate × Time = P × 10% × 2 = 0.2P

2. Calculate Compound Interest (CI):
CI = P × (1 + Rate)^2 – P = P × (1.1)^2 – P = P × 1.21 – P = 0.21P

3. Find the Difference Between CI and SI:
Difference = CI – SI = 0.21P – 0.2P = 0.01P

Given that the difference is ₹12:
0.01P = 12 ⇒ P = 12 / 0.01 = 1200

Answer: ₹1200 Option (A)

The correct answer is (B) 8.

Explanation:

The number of subsets of a set is determined by the formula:
Number of subsets = 2^n, where n is the number of elements in the set.
Here, the set A = {0, 1, 2} has n = 3 elements. Substituting n = 3 into the formula:

Number of subsets = 2^3 = 8

The subsets of A are:
1. ∅ (the empty set)
2. {0}
3. {1}
4. {2}
5. {0, 1}
6. {0, 2}
7. {1, 2}
8. {0, 1, 2}

Final Answer: (B) 8

Correct Answer: (C) 1823.

Explanation:

To find the number, we can use the formula:

Number = (Divisor * Quotient) + Remainder

Here, the divisor is 53, the quotient is 34, and the remainder is 21.

Now, substituting the values:

Number = (53 * 34) + 21 = 1802 + 21 = 1823

The answer is (C) 1823.

Correct Answer: (D) 4 km/hr

Explanation:

Average Speed = Total Distance / Total Time

1. First Half:

– Distance: Let’s assume the total distance is 2 km. So, the first half is 1 km.
– Speed: 6 km/hr
– Time = Distance / Speed = 1 / 6 hours ≈ 0.1667 hours

2. Second Half:
– Distance: 1 km
– Speed: 3 km/hr
– Time = 1 / 3 hours ≈ 0.3333 hours
Total Time: 0.1667 + 0.3333 = 0.5 hours
Total Distance: 2 km
Average Speed: 2 / 0.5 = 4 km/hr

Answer: (D) 4 km/hr

Correct Answer: (C) 208

Explanation:

1. Let the number of boys = x, so the number of girls = 300 – x.

2. Total money equation: 1(300 – x) + 0.50(x) = 196

3. Simplify:

300 – x + 0.50x = 196

300 – 0.50x = 196

-0.50x = -104

x = 208

4. Cross Checking:

– Boys = 208 → ₹0.50 × 208 = ₹104

– Girls = 92 → ₹1 × 92 = ₹92

– Total = ₹104 + ₹92 = ₹196 

Final Answer: (C) 208

Correct Answer: (B) 2x³ – 7x² – 5x + 4

Explanation:

To determine which polynomial has both x – 4 and 2x – 1 as factors, we can use the Factor Theorem. According to this theorem:

– If x – c is a factor of a polynomial P(x), then P(c) = 0.
– Similarly, if 2x – 1 is a factor, then P(1/2) = 0.

Option (A): 2x^3 – 7x^2 + 5x – 4

1. Check x = 4:
P(4) = 2(4)^3 – 7(4)^2 + 5(4) – 4 = 128 – 112 + 20 – 4 = 32 ≠ 0
Since P(4) ≠ 0, x – 4 is not a factor.

Option (B): 2x^3 – 7x^2 – 5x + 4

1. Check x = 4:
P(4) = 2(4)^3 – 7(4)^2 – 5(4) + 4 = 128 – 112 – 20 + 4 = 0
x – 4 is a factor.

2. Check x = 1/2:
P(1/2) = 2(1/2)^3 – 7(1/2)^2 – 5(1/2) + 4 = 1/4 – 7/4 – 5/2 + 4 = 0
2x – 1 is also a factor.

Option (C): 2x^3 + 7x^2 + 5x + 4

1. Check x = 4:
P(4) = 2(4)^3 + 7(4)^2 + 5(4) + 4 = 128 + 112 + 20 + 4 = 264 ≠ 0
x – 4 is not a factor.

Option (D): 2x^3 + 7x^2 – 5x – 4
1. Check x = 4:
P(4) = 2(4)^3 + 7(4)^2 – 5(4) – 4 = 128 + 112 – 20 – 4 = 216 ≠ 0
x – 4 is not a factor.

Conclusion: Only Option (B) satisfies both conditions P(4) = 0 and P(1/2) = 0.

Answer: (B) 2x³ – 7x² – 5x + 4

Correct Answer: (B) 2x(x² + 3y²)

Explanation:

To factor the expression (x + y)³ + (x – y)³, let’s expand and simplify it:

(x + y)³ + (x – y)³ = (x³ + 3x²y + 3xy² + y³) + (x³ – 3x²y + 3xy² – y³)

= x³ + 3x²y + 3xy² + y³ + x³ – 3x²y + 3xy² – y³

= 2x³ + 6xy²

= 2x(x² + 3y²)

So, the factored form of (x + y)³ + (x – y)³ is 2x(x² + 3y²).

Answer: Option (B) 2x(x² + 3y²)