Solution:

Step 1: Start with the given equation: √3 sec(x) = -2

Step 2: Solve for sec(x):

sec(x) = -2 / √3 = -2√3 / 3

Step 3: Recall that sec(x) = 1 / cos(x), so:

cos(x) = -√3 / 2

Step 4: Determine the principal solutions where cos(x) = -√3 / 2. The cosine function is negative in the second and third quadrants:

x = 5π/6, 7π/6

Step 5: Among the given options, 5π/6 is present.

Solution:

Step 1: Observe the integral and try substitution. Let:

u = e^(3x) + 1, so that du = 3e^(3x) dx

Thus, e^(3x) / (e^(3x) + 1) dx = (1/3) du / u

Step 2: The integral becomes:

∫ (1/3) du / u = (1/3) ln|u| + C

Substitute back u = e^(3x) + 1:

(1/3) ln(e^(3x) + 1) + C

Thus, the solution is:

(1/3) ln(e^(3x) + 1) + C

Solution:

Step 1: Begin with the given differential equation:

(x dy/dx - y) sin(y/x) = x³ e^x

Rearrange to:

x dy/dx - y = x³ e^x / sin(y/x)

Step 2: Use an appropriate method to solve this equation. Since it is a linear form, we can apply the integrating factor method. After simplifications, integrate both sides with respect to x.

Step 3: The general solution obtained is:

e^x(x - 1) + cos(y/x) + C = 0

Solution:

Step 1: The equation of the parabola is:

y² = 4ax

The latus rectum of the parabola is the vertical line passing through the focus of the parabola. The focus of the parabola y² = 4ax is at (a, 0), so the latus rectum is x = a.

Step 2: The area bounded by the parabola and the latus rectum is given by:

Area = 2 ∫₀ᵃ √(4ax) dx

Step 3: Solve the integral:

Area = 2 ∫₀ᵃ 2√(ax) dx = 4√a ∫₀ᵃ √x dx

Using the formula for the integral of √x:

∫ √x dx = (2/3) x³/²

Area = 4√a [(2/3) x³/²]₀ᵃ = 4√a × (2/3) a³/²

Area = (8/3) a²

The total area is (1/2) × (8/3) a² = a²/2

Solution:

Step 1: Analyze the logical statements:

• Statement 1: p ∧ q = False, which implies at least one of p or q is False.
• Statement 2: p → q = False, which implies p is True and q is False.

Step 2: From Statement 2, since p → q is False, we conclude:

p = True and q = False

Conclusion: The truth values of p and q are True and False, respectively.

Solution:

We are given the matrix:

                A = [ 1  0  0 ]
                    [ 3  3  3 ]
                    [ 5  2 -1 ]
                

and we need to find its inverse.

Step 1: Calculate the determinant of A.

det(A) = 1 * | 3 3 | | 2 -1 |

det(A) = 1 * [(3 * -1) - (3 * 2)] = 1 * (-3 - 6) = -9.

Step 2: Calculate the adjoint of A.

The adjoint matrix is found by calculating the cofactor matrix and taking its transpose.

Step 3: Using the formula for the inverse:

A⁻¹ = (1/det(A)) * adj(A)

Substitute det(A) = -9 and adj(A):

                A⁻¹ = -1/3 *
                [ -3   0   0 ]
                [  3  -1   0 ]
                [ -9  -2   3 ]
                

Conclusion: The correct inverse matrix is:

                A⁻¹ = -1/3 *
                [ -3   0   0 ]
                [  3  -1   0 ]
                [ -9  -2   3 ]
                

Solution:

When the output of an AND gate is passed through a NOT gate, the AND operation is inverted. Mathematically:

Output = NOT (A AND B) = A NAND B

Thus, the equivalent logic gate is a NAND gate.

Solution:

The fringe spacing (β) in Young's double slit experiment is given by:

β = λL / d

where:

• λ is the wavelength of light
• L is the distance from the slits to the screen
• d is the distance between the slits

If d is doubled (d' = 2d), while λ and L remain constant:

β' = λL / (2d) = β / 2

Thus, the fringe spacing is halved.

Solution:

In a series LCR circuit, the total impedance (Z) is given by:

Z = √(R² + (X_L - X_C)²)

where:

• R is the resistance
• X_L = ωL is the inductive reactance
• X_C = 1 / (ωC) is the capacitive reactance

At resonance, the inductive reactance equals the capacitive reactance (X_L = X_C), so:

Z = √(R² + (X_L - X_C)²) = √(R²) = R

The impedance is minimized, which maximizes the current according to Ohm's Law:

I = V / Z

Hence, at resonance, the current is maximum because the inductive and capacitive reactances cancel each other.

Solution:

Butter is an example of a colloidal system where the liquid phase (water or milk) is dispersed in a solid matrix (fat). This type of colloid is referred to as "liquid in solid." In butter, the fat (solid phase) forms a continuous phase, while the water droplets are dispersed in it.

Conclusion: Therefore, butter is a "Liquid in Solid" colloid.

Solution:

Transmission Electron Microscopy (TEM) is a powerful technique used to visualize the size, shape, and morphology of nanoparticles at the nanometer scale. Unlike UV-Vis Spectroscopy, FTIR, and AAS, which provide information about optical properties, molecular vibrations, and elemental composition respectively, TEM offers direct imaging capabilities, allowing for precise determination of nanoparticle dimensions and structural characteristics.

Solution:

Hydrochloric acid (HCl) is a strong acid and dissociates completely in water:

HCl → H⁺ + Cl⁻

Therefore, the concentration of hydrogen ions [H⁺] is equal to the concentration of HCl, which is 0.01 M.

The pH is calculated using the formula:

pH = -log[H⁺]

Substituting the values:

pH = -log(0.01) = 2

Thus, the pH of the solution is 2.