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Formula

Step-by-Step Derivation

1. The equations of the parabolas are $P_1: y = 4x^2$ and $P_2: y = x^2 + 27$.
First, find the points of intersection of $P_1$ and $P_2$:
4. $$ 4x^2 = x^2 + 27 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = \pm 3 $$
6. The area $A_1$ enclosed by $P_1$ and $P_2$ is:
8. $$ A_1 = \int_{-3}^{3} (x^2 + 27 - 4x^2) \, dx = 2 \int_{0}^{3} (27 - 3x^2) \, dx $$
10. $$ A_1 = 2 \left[ 27x - x^3 \right]_{0}^{3} = 2(81 - 27) = 108 $$
12. Next, find the points of intersection of $P_1$ and the line $y = \alpha x$:
13. $$ 4x^2 = \alpha x \implies x = 0 \text{ or } x = \frac{\alpha}{4} $$
14. The area $A_2$ enclosed by $P_1$ and $y = \alpha x$ is:
15. $$ A_2 = \int_{0}^{\alpha/4} (\alpha x - 4x^2) \, dx = \left[ \alpha \frac{x^2}{2} - \frac{4x^3}{3} \right]_{0}^{\alpha/4} $$
16. $$ A_2 = \alpha \left( \frac{\alpha^2}{32} \right) - \frac{4}{3} \left( \frac{\alpha^3}{64} \right) = \frac{\alpha^3}{32} - \frac{\alpha^3}{48} = \frac{\alpha^3}{96} $$
17. We are given that $A_1 = 6 A_2$:
18. $$ 108 = 6 \left( \frac{\alpha^3}{96} \right) \implies 18 = \frac{\alpha^3}{96} \implies \alpha^3 = 1728 $$
19. Taking the cube root, we get $\alpha = 12$.

Final Answer

Therefore, the power of the lens is -2.0D (Option D).

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Rust is a hydrated iron(III) oxide with the formula Fe₂O₃ · xH₂O, formed by the reaction of iron, oxygen, and water.

Pendulum Formula

Where:

• T = Time period (s)
• l = length of string (m)
• g = acceleration due to gravity (m/s²)

The formula holds true under small angular displacements.

Acceleration $a = \frac{v - u}{t} = \frac{35 - 10}{25} = 1 \text{ m/s}^2$.

Force $F = ma = 5 \times 1 = 5 \text{ N}$.

Work done $W = qV = 2 \times 12 = 24 \text{ J}$.

This is the energy transferred when a charge moves through a potential difference.

Total electrons $= 2 + 8 + 7 = 17$. The element is Chlorine (Cl).

It belongs to Group 17 (halogens) and has 7 valence electrons, making it highly electronegative.

$\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$

$\cos\theta = \frac{4}{5}$ (taking positive value for acute angle).

A neutral solution has equal concentrations of $\text{H}^+$ and $\text{OH}^-$ ions.

At 25°C, $[\text{H}^+] = 10^{-7}$ M, so pH $= -\log(10^{-7}) = 7$.

The Moon's gravitational acceleration is about $\frac{1}{6}$th of Earth's.

$g_{\text{moon}} \approx \frac{9.8}{6} \approx 1.63 \text{ m/s}^2 \approx 1.6 \text{ m/s}^2$.

Mitochondria are called the "powerhouse of the cell" because they generate most of the cell's supply of adenosine triphosphate (ATP), used as a source of chemical energy.

They perform oxidative phosphorylation through the electron transport chain located in their inner membrane.

$f'(x) = 3x^2 - 6x$

$f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3$.

The Ampere (A) is the SI base unit of electric current, named after André-Marie Ampère.

1 Ampere = 1 Coulomb of charge flowing per second. Volt is the unit of potential difference, Coulomb is the unit of charge, and Ohm is the unit of resistance.

Saturn has the most confirmed natural satellites of any planet in our solar system, surpassing Jupiter with over 140 known moons as of recent surveys.

Its largest moon, Titan, is the only moon in the solar system with a dense atmosphere.