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)

Correct Answer: (B) 20

Explanation:

1. Define Variables:

– Let the work rate of B be r (work units per day).
– Since A’s capacity is three times more than B’s, the work rate of A is 3r.

2. Combined Work Rate:

– Together, A and B have a combined work rate of 3r + r = 4r.

3. Total Work:

– If A and B together complete the work in 15 days, the total work W can be expressed as:
W = Combined Rate × Time = 4r × 15 = 60r

4. Work Rate of A Alone:
– The work rate of A alone is 3r.
– To find the time T A alone takes to complete the work:

T = W / A’s Rate = 60r / 3r = 20 days

Answer: (B) 20

Correct Answer: (D) 33.33%

Explanation:

1. Given:
– The cost price (CP) of 40 oranges is equal to the selling price (SP) of 30 oranges.
– We need to find the percentage profit.

2. Let’s Assume:
– Let the cost price per orange be C.
– Therefore, the total cost price for 40 oranges is 40C.
– Let the selling price per orange be S.
– Therefore, the total selling price for 30 oranges is 30S.

3. Setting Up the Equation:
40C = 30S
⇒ S = 40C / 30 = 4C / 3

4. Calculating Profit:
– Profit per orange is:
S – C = (4C / 3) – C = (4C – 3C) / 3 = C / 3
– Profit Percentage is:
(Profit / Cost Price) × 100% = (C / 3) / C × 100% = (1 / 3) × 100% ≈ 33.33%

Answer: (D) 33.33%

The correct answer is: (A) 3 times

Explanation:

– Original area = s^2

– New side length = 2s, so new area = (2s)^2 = 4s^2

– Increase in area = 4s^2 – s^2 = 3s^2

Thus, the area increases by 3 times.

The correct answer is: (B) Standard deviation

Explanation:

1. Median: The median is the middle value of a data set when arranged in order. If a constant number is added to all the scores, the position of the middle value remains the same, but the value itself will increase by the constant. Hence, the median changes.

2. Standard deviation: The standard deviation measures the spread or dispersion of the data around the mean. Adding a constant to all the scores shifts the entire data set by that constant, but it does not affect the spread of the data. Therefore, the standard deviation remains unchanged.

3. Mean: The mean is the average of all the scores. If a constant is added to all the scores, the total sum of the scores increases by that constant multiplied by the number of scores, which in turn increases the mean. Hence, the mean changes.

4. Mode: The mode is the most frequently occurring value in a data set. If a constant is added to all the scores, the most frequent value will also increase by that constant. Hence, the mode changes.

Thus, the standard deviation is the only measure that does not change when a constant is added to all the scores in a series.

Correct Answer: (D) It quadruples

Explanation:

The area of a circle is given by the formula:

Area = π r^2

Where r is the radius of the circle. If the radius is doubled, the new radius becomes 2r. Substituting this into the formula for the area:

New Area = π (2r)^2 = π (4r^2) = 4 × (π r^2)

This shows that the new area is 4 times the original area. Therefore, when the radius is doubled, the area quadruples.

Thus, the correct answer is (D) It quadruples.