Explanation

 The phrase "as if" is followed by the past tense (subjunctive mood) to express something imaginary or unreal.

Explanation

Original: 1 teacher per 15 students → 1200 ÷ 15 = 80 teachers

New ratio: 3 teachers per 40 students → (1200 ÷ 40) × 3 = 30 × 3 = 90 teachers

Teachers joined = 90 - 80 = 10

Explanation

Let apple = x, orange = y

5x + 4y = 340

Try options:

B) 5×40 + 4×20 = 200 + 80 = 340 

Explanation

limₓ→0 (sin(ax) / sin(bx))

Divide the numerator and the denominator by x:

= limₓ→0 [(sin(ax)/x) ÷ (sin(bx)/x)]  

= (a/b) × limₓ→0 [sin(ax)/(ax)] ÷ [sin(bx)/(bx)]

As x → 0, sin(kx)/(kx) → 1, so:

Answer: a / b

Explanation

 The function that is to be maximized or minimized is called the objective function

 It represents the goal of the optimization process and is typically a mathematical expression that relates to the quantity being optimized, such as cost, profit, or efficiency. 

Explanation

cosecθ − cotθ = (1 - cosθ) / sinθ

Explanation

Let current ages be:

• Ahsan = $A$

• Shakeel = $S$
  

Condition 1 (Five years ago):

$$A-5=7(S-5)\text{(Equation 1)}$$

Condition 2 (Three years later):

$$A + 3 = 3(S + 3) quad text{(Equation 2)}$$

Solving the equations:

From Equation 1:

$$A - 5 = 7S - 35 Rightarrow A = 7S - 30 quad text{(Equation 3)}$$

Substitute into Equation 2:

$$(7S - 30) + 3 = 3(S + 3) Rightarrow 7S - 27 = 3S + 9 Rightarrow 7S - 3S = 9 + 27 Rightarrow 4S = 36 Rightarrow S = 9$$

Now, plug back to find Ahsan's age:

$$$$A = 7S - 30 = 7(9) - 30 = 63 - 30 = 33

Explanation

Eccentricity of ellipse is 0 < e < 1

If e = 0, ellipse becomes a perfect circle

Explanation

Explanation

Total students = 35 → Total height = 35 × 161 = 5635 cm

14 girls = 14 × 151 = 2114 cm

So, 15 girls = (2114 + x) → x = height of 1 girl = assume x = 151

Total height of 15 girls ≈ 15 × 151 = 2265

Total height of boys = 5635 - 2265 = 3370

Mean height of 20 boys = 3370 ÷ 20 = 169 cm