Explanation

Given points P(2,1) and Q(8,1) lie on the same horizontal line (same y = 1)  

So, any valid rectangle vertex must form a right angle and match in x or y.

Check options:

- A) (2, 8) → shares x with P → possible 

- B) (8, 2) → shares x with Q but different y → possible 

- C) (2, -8) → shares x with P → possible 

- D) (-2, 8) → does not share x or y with either point → cannot form right angle with both

Explanation

Total marbles = 5  

You remove 4 marbles (randomly)  

We want the probability yellow is among the 4 removed.

There are 5 marbles, and you’re removing 4, so the chance that yellow is among them is:

= Number of favorable outcomes / Total outcomes  

Yellow is in 4 out of 5 possible selections →  

P(yellow is removed) = 4/5  

Explanation

Every minute:  

Climbs 20 yards, slips 5 yards → Net gain = 15 yards/min

In 5 minutes, net gain = 15 × 5 = 75 yards  

In the 6th minute, he climbs a full 20 yards to reach 95 yards, but we only need to reach 80 yards  

→ So in the 6th minute, he climbs from 75 to 80 yards in (5/20 = 0.25) min = 15 seconds

Explanation

Total = 6 + 2 + 2 + 3 + 5 + 8 + 2 = 28  

Number of values = 7  

Mean = 28 ÷ 7 = 4  

Explanation

We simplify the expression:

x(x⁵)⁴ / x⁴

Step 1: Simplify the exponent  

(x⁵)⁴ = x^(5×4) = x²⁰

Now the expression becomes:  

x × x²⁰ / x⁴

Step 2: Combine the exponents  

Numerator: x × x²⁰ = x²¹  

Now divide: x²¹ / x⁴ = x^(21−4) = x¹⁷

Explanation

We use the algebraic identity:

(a² − b²) = (a − b)(a + b)

Let:  

a = 1,000,001  

b = 999,999

Now:

(1,000,001)² − (999,999)² = (1,000,001 − 999,999)(1,000,001 + 999,999)  

= (2)(2,000,000) = 4,000,000

Explanation

Step 1: Solve for x  

-2x - 5 = 48  

Add 5: -2x = 53  

Divide by -2: x = -26.5

Now find: 2x + 57  

= 2(-26.5) + 57  

= -53 + 57  

= 4

Explanation

: Average weight = 14 pounds, Number of turkeys = 25

Total weight = 25 × 14 = 350 pounds

Explanation

Let the number of English books Danish read in 2000 = x  

Then Urdu books in 2000 = 2x (twice as many)

Total books:

- In 1999:  

  English = 10, Urdu = 7 → Total = 17

- In 2000:  

  English = x, Urdu = 2x → Total = x + 2x = 3x

Total books in 2 years = 17 + 3x  

Total Urdu books = 7 + 2x  

Given: 60% of total books were Urdu  

So:

(7 + 2x) / (17 + 3x) = 0.6

Multiply both sides:

7 + 2x = 0.6(17 + 3x)  

7 + 2x = 10.2 + 1.8x  

2x - 1.8x = 10.2 - 7  

0.2x = 3.2  

x = 16

So, in 2000:

- English = 16  

- Urdu = 2×16 = 32  

- Total = 48

Explanation

Given:

- Average of 6 tests = 75  

  → Total marks = 6 × 75 = 450

- After dropping lowest grade, average of 5 tests = 85  

  → Total of best 5 tests = 5 × 85 = 425

So, the lowest grade = 450 - 425 = 25