JEE Advanced 2023 PCM June 4 Paper 2 Question Paper Pdf - Check Solutions with Answer Key Pdf

Step 1: Define the probabilities for heads and tails.

P(H) = 1/3, P(T) = 2/3.

P = P(HH) + (P(THH) + P(TTHH) + ...).

P = 1/9 + 2/9 * 1/9 + 2²/9 * 1/9 + ...

P = (1/3 * 1/3) / (1 - 2/3 * 1/3) + (2/3 * 1/3 * 1/3) / (1 - 2/3 * 1/3) = 5/21

Step 1: Simplify using trigonometric identities.

tan⁻¹(6y / (9 - y²)) + cot⁻¹((9 - y²) / 6y) = π (Sum of complementary angles).

tan⁻¹(6y / (9 - y²)) = π/3.

6y / (9 - y²) = √3.

6√3y = -9 + y²

solve quadratic equation.

Sum of solutions = 3√3 - 6 + √3 = 4√3 - 6.

Step 1: Calculate the vectors PQ, PR, and PS.

We are given the following points:

\( \vec{a} = \hat{i} + 2\hat{j} - 5\hat{k} \), \( \vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k} \), \( \vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k} \), \( \vec{d} = \hat{i} - \hat{j} + 6\hat{k} \).

Step 2: Verify coplanarity using the scalar triple product.

The vectors PQ, PR, and PS are coplanar if the scalar triple product is zero:

            |  2   4   8  |
            | 12/5 6/5 12 |
            |  0  -3  11  |
        

Expanding along the first row and solving, we get a nonzero result, so the points are NOT coplanar.

Step 3: Verify if \( \frac{\vec{b} + 2\vec{d}}{3} \) divides PR internally.

\( \frac{\vec{b} + 2\vec{d}}{3} = \frac{(3 + 2)\hat{i} + (6 + 2(-1))\hat{j} + (3 + 2(6))\hat{k}}{3} \)

\( = \frac{5\hat{i} + 4\hat{j} + 15\hat{k}}{3} \)

Comparing with PR, it divides PR internally in the ratio 5:4. Hence, option (B) is correct.

Final Answer: Option B

Quick Tip: Use the scalar triple product to check coplanarity, and remember vector algebra for cross products and dividing line segments.

Step 1: Given Matrix

            M = | 1  1  1 |
                | 1  0  1 |
                | 0  1  0 |
        

Step 2: Determinant Calculation

The determinant of M is calculated as:

            |M| = -1 + 1 + 1 = 0
        

Since the determinant is zero, M is non-invertible. Hence, option A is incorrect.

Step 3: Solving for the Null Space of M

Solving MX = 0:

            | 1  1  1 |   | a1 |   | 0 |
            | 1  0  1 | * | a2 | = | 0 |
            | 0  1  0 |   | a3 |   | 0 |
        

From the equations:

            a1 + a2 + a3 = 0
            a1 + a3 = 0
            a2 = 0
        

Solving gives a1 = -a3, confirming option B.

Step 4: Checking for M - 2I Invertibility

            M - 2I =
            | -1  1  1 |
            |  1 -2  1 |
            |  0  1 -2 |
        

Calculating determinant:

            |M - 2I| = 0
        

Since the determinant is zero, M - 2I is not invertible, making option D incorrect.

Quick Tip: Use row reduction or determinant calculations to check invertibility.

Step 1: Define the piecewise function. The function f(x) is given as:

Step 2: Analyze continuity.

The function f(x) is defined piecewise with changes at x = 1/4, x = 1/2, and x = 3/4. To check continuity, evaluate the left-hand limit (LHL) and right-hand limit (RHL) at these points.

• At x = 1/4:
  LHL: lim_{x → 1/4⁻} f(x) = 0, RHL: lim_{x → 1/4⁺} f(x) = 0.
  Therefore, f(x) is continuous at x = 1/4.
• At x = 1/2:
  LHL: lim_{x → 1/2⁻} f(x) = 0,
  RHL: lim_{x → 1/2⁺} f(x) = 0.
  Thus, f(x) is continuous at x = 1/2.
• At x = 3/4:
  LHL: lim_{x → 3/4⁻} f(x) = 2 (3/4 - 1/4)² (3/4 - 1/2),
  RHL: lim_{x → 3/4⁺} f(x) = 3 (3/4 - 1/4)² (3/4 - 1/2).
  Since these values are not equal, f(x) is discontinuous at x = 3/4.

Step 3: Analyze differentiability.

For differentiability, calculate f'(x) within each interval and check at the boundary points x = 1/4, x = 1/2, and x = 3/4.

• f(x) in (1/4, 1/2):
  f(x) = (x - 1/4)² (x - 1/2).
  Differentiate:
  f'(x) = 2(x - 1/4)(x - 1/2) + (x - 1/4)².

Step 4: Find the minimum value of f(x).

The minimum value occurs where f'(x) = 0. Solve:

(x - 1/4)(2x - 1/2 + x - 1/4) = 0.

x = 1/4 or x = 5/12.

At x = 5/12:

f(x) = (5/12 - 1/4)² (5/12 - 1/2) = -1/432.

Final Answer:

The function f(x) is discontinuous at x = 3/4, non-differentiable at x = 1/2 and x = 3/4, and achieves its minimum value of -1/432 at x = 5/12.

Step 1: Analyze the condition d²f/dx² > 0.
The condition implies that f(x) is concave upward in the interval (-1, 1). The graph of f(x) must be a parabola-like curve opening upward.
Step 2: Analyze intersection with y = x.
The function f(x) intersects y = x at points where f(x) - x = 0.

• X_f = 0: Possible when f(x) lies entirely above y = x (e.g., f(x) = x² + 1).
• X_f = 2: Possible when f(x) intersects y = x at two points (e.g., f(x) = 2x²).

Final Answer: X_f is at most 2 for all functions in S. Option (D) is incorrect.

Step 1: Write the function and use the Fundamental Theorem of Calculus. The given function is:

f(x) = ∫₀^{x tan⁻¹(x)} (e^{-cos t} / (1 + t²⁰²³)) dt.

Using the Fundamental Theorem of Calculus for derivatives of integrals:

f'(x) = (e^{-cos(x tan⁻¹ x)} / (1 + (x tan⁻¹ x)²⁰²³)) × d/dx (x tan⁻¹ x).

Step 2: Differentiate x tan⁻¹ x.

Let g(x) = x tan⁻¹ x. Differentiate using the product rule:

g'(x) = tan⁻¹ x + x / (1 + x²).

Substitute g'(x) into the expression for f'(x):

f'(x) = (e^{-cos(x tan⁻¹ x)} / (1 + (x tan⁻¹ x)²⁰²³)) × (tan⁻¹ x + x / (1 + x²)).

Step 3: Analyze f'(x).

The term e^{-cos(x tan⁻¹ x)} is always positive, and 1 + (x tan⁻¹ x)²⁰²³ > 0 for all x. Thus, f'(x) = 0 only when:

tan⁻¹ x + x / (1 + x²) = 0.

Step 4: Solve tan⁻¹ x + x / (1 + x²) = 0.

For x = 0:

tan⁻¹(0) + (0 / (1 + 0²)) = 0.

Thus, x = 0 is the only critical point.

Step 5: Evaluate f(x) at x = 0.

Substituting x = 0 into f(x):

f(0) = ∫₀⁰ (e^{-cos t} / (1 + t²⁰²³)) dt = 0.

Step 6: Conclusion.

Since f'(x) > 0 for x > 0 and f'(x) < 0 for x < 0, x = 0 is the global minimum.

Final Answer: The minimum value of f(x) is:

0.

Quick Tip: The minimum or maximum value of an integral function occurs at critical points where f'(x) = 0.

Step 1: Rewrite the differential equation.

(dy/dx) - (2x / (x² - 5)) y = -2x(x² - 5).

This is a first-order linear differential equation.

Step 2: Find the integrating factor (I.F.).

I.F. = e^{∫ (-2x / (x² - 5)) dx} = e^{-ln(x² - 5)} = 1 / (x² - 5).

Step 3: Solve for y(x).

Multiply through by the I.F.:

(1 / (x² - 5)) (dy/dx) - (2x / (x² - 5)²) y = -2x.

This simplifies to:

d/dx (y / (x² - 5)) = -2x.

Integrate both sides:

(y / (x² - 5)) = -x² + c.

Solve for y:

y = -x²(x² - 5) + c(x² - 5).

Step 4: Apply initial condition y(2) = 7.

7 = -4(-1) + c(-1) ⇒ c = -3.

Step 5: Maximum value of y(x).

y = -(x⁴ - 2x² - 15) = -(x² - 1)² - 16.

The maximum value is 16.

Quick Tip: For first-order linear differential equations, use the integrating factor method to find the general solution.

Step 1: Count the total number of elements in X that are multiples of 5.

We are given subsets of X, and we need to calculate the number of elements divisible by 5 across different ranges.

Number of multiples of 5 in each subset:

• 4
• 12
• 4
• 6
• 12

Total number of multiples of 5: 4 + 12 + 4 + 6 + 12 = 38.

Step 2: Count the elements divisible by 5 but not by 20.

Among the 38 elements divisible by 5, we calculate those not divisible by 20 as follows:

• 1
• 3
• 3

Total multiples not divisible by 20: 1 + 3 + 3 = 7.

Step 3: Calculate the probability.

The probability p is the ratio of elements divisible by 5 and also divisible by 20 to the total elements divisible by 5:

p = (Total divisible by 5 - Not divisible by 20) / (Total divisible by 5).

Substituting the values:

p = (38 - 7) / 38 = 31/38.

Final Answer:

38p = 31

Quick Tip: For combinatorics problems, consider symmetries and constraints in the number construction.

Step 1: Represent point P and vertices of the octagon.

Let P = 2e^iθ, and the vertices of the octagon are given as:

A_k = 2e^i(2kπ/8) = 2e^i(kπ/4), for k = 0, 1, 2, ..., 7.

Step 2: Compute PA_k.

The distance between P and A_k is:

PA_k = |P - A_k| = |2e^iθ - 2e^i(kπ/4)| = 2 |e^iθ - e^i(kπ/4)|.

Using the identity |e^ix - e^iy| = 2 |sin((x - y) / 2)|, we get:

PA_k = 4 |sin((θ - kπ/4) / 2)|.

Step 3: Product of all distances PA_k.

The product of all distances is:

∏_k=0⁷ PA_k = ∏_k=0⁷ [4 |sin((θ - kπ/4) / 2)|].

Simplify:

∏_k=0⁷ PA_k = 4⁸ ∏_k=0⁷ |sin((θ - kπ/4) / 2)|.

Step 4: Symmetry and roots of unity.

The roots of unity identity implies symmetry in the angles:

∏_k=0⁷ sin((θ - kπ/4) / 2) = sin(4θ) / 8.

Thus:

∏_k=0⁷ PA_k = 4⁸ · (|sin(4θ)| / 8).

Simplify further:

∏_k=0⁷ PA_k = 2⁹ |sin(4θ)|.

Step 5: Maximum value of ∏_k=0⁷ PA_k.

The maximum value occurs when |sin(4θ)| = 1, which happens at:

4θ = π/2 ⇒ θ = π/8.

Substituting:

∏_k=0⁷ PA_k = 2⁹ = 512.

Final Answer:

The maximum value of ∏_k=0⁷ PA_k is:

512.

Quick Tip: Use symmetry and trigonometric simplifications when working with regular polygons inscribed in circles.

Step 1: Total number of matrices. Since \( a, b, c, d \) can each take 8 values:

Total matrices = \( 8^4 = 4096 \).

Step 2: Compute the determinant of \( R \).

\( \det(R) = 5(bc - ad) \).

For \( R \) to be invertible, \( bc - ad \neq 0 \).

Step 3: Calculate non-invertible cases.

• Case 1: If \( a, b, c, d \neq 0 \), then \( bc - ad = 0 \) leads to 91 cases.
• Case 2: If \( a = 0 \) or \( b = 0 \), \( bc = ad = 0 \), leading to 225 cases.

Step 4: Calculate invertible matrices.

Number of invertible matrices = \( 4096 - (91 + 225) = 3780 \).

Quick Tip: For matrix invertibility, ensure the determinant is non-zero by analyzing all possible cases.

Step 1: Write the equations of the circles. The equations of the given circles are:
C₁: x² + y² = 1,   C₂: (x - 4)² + (y - 1)² = r². Step 2: Find the radical axis of the two circles. The radical axis is obtained by subtracting the equations of the circles:
(x² + y² - 1) - [(x - 4)² + (y - 1)² - r²] = 0.
Expand  C₂ :
(x - 4)² + (y - 1)² = x² - 8x + 16 + y² - 2y + 1 = x² + y² - 8x - 2y + 17.
Substitute back:
(x² + y² - 1) - (x² + y² - 8x - 2y + 17 - r²) = 0.
Simplify:
8x + 2y - (18 - r²) = 0.
The radical axis is:
4x + y = (18 - r²)/2. Step 3: Determine point  B  on the radical axis. Point  B  is where the radical axis meets the  x -axis ( y = 0 ):
4x + 0 = (18 - r²)/2.
Solve for  x :
x = (18 - r²)/8.
Thus,  B = ((18 - r²)/8, 0). Step 4: Apply the distance condition. Given  AB = √5 , where  A = (4, 1)  is the center of  C₂ , compute the distance:
AB = √((4 - (18 - r²)/8)² + (1 - 0)²).
Simplify:
AB = √(((32 - (18 - r²))/8)² + 1).
AB = √(((14 + r²)/8)² + 1).
Equating to  √5 :
√(((14 + r²)/8)² + 1) = √5.
Square both sides:
((14 + r²)/8)² + 1 = 5.
Simplify:
((14 + r²)/8)² = 4.
Take the square root:
(14 + r²)/8 = 2   (only positive root is valid).
Solve for  r² :
14 + r² = 16 ⇒ r² = 2. Step 5: Verify the solution. Substitute  r² = 2  back into the equations. The distance  AB = √5  is satisfied, confirming the solution. Final Answer:
r² = 2. Quick Tip: For problems involving tangents and circles, use radical axes and distance formulas to derive key relationships.

Step 1: Analyze the problem. The triangle  ABC  is an obtuse-angled triangle with vertices on a circle of radius  R = 1 . The difference between the largest and smallest angles is  π/2 , and the sides are in arithmetic progression. Step 2: Use trigonometric relations. Let  A > B > C  be the angles, where  A - C = π/2 . Using the property of arithmetic progression:
a + c = 2b.
Since  A + B + C = π :
π/2 + B + C = π ⇒ B + C = π/2. Step 3: Calculate sine values. Using the given conditions:
sin C = (-1 + √7)/4,   sin A = (√7 + 1)/4,   sin B = √7/4. Step 4: Compute the area. The area of the triangle is:
a = 2R² sin A sin B sin C = 2(1)² · (√7 + 1)/4 · √7/4 · (-1 + √7)/4.
Simplify:
a = 2/64 (6√7)
Thus:
⇒ (64a)² = 1008. Quick Tip: For obtuse triangles in a circle, use sine rules and trigonometric identities to compute the area.

Step 1: Recall the formula for the inradius. The inradius  r  of a triangle is given by:
r = Δ/s,
where  Δ  is the area of the triangle and  s  is the semi-perimeter. Step 2: Express  Δ  and  s  in terms of given parameters. The area  Δ  of the triangle is given by:
Δ = 1/2 · base · height.
Let the sides of the triangle be  n, n+d,  and  n+d' , where  d  and  d'  are offsets. The semi-perimeter  s  is:
s = (n + (n+d) + (n+d'))/2 = (3n + d + d')/2. Step 3: Use the angle to simplify  Δ . The height can be expressed as:
height = (n + d) sinα,
where  α  is the included angle. Thus:
Δ = 1/2 · n · (n + d) sinα. Step 4: Substitute into the inradius formula. The inradius is:
r = Δ/s = (1/2 n (n + d) sinα)/((3n + d + d')/2).
Simplify:
r = (n (n + d) sinα)/(3n + d + d'). Step 5: Relate to trigonometric identities. If  sin(2α) = 2 cosα sinα , we can express  sinα  in terms of given parameters. Substitute this into  r :
r = sin(2α)/3. Step 6: Given  r = 1/4 , solve for  α . From:
sin(2α) = 1/4,
we have:
sin(2α) = 1/4. Quick Tip: The inradius depends on the area and the semi-perimeter, requiring correct computations of both.

Step 1: Construct the probability distribution. From the problem:
P(0) = 0,   P(1) = 0,   P(2) = 4/49,   P(3) = 20/49,   P(4) = 25/49. Step 2: Compute  E(X) .
E(X) = ∑_i=0⁴ i · P(i) = 0 · 0 + 0 · 0 + 2 · 4/49 + 3 · 20/49 + 4 · 25/49.
On Simplifying:
E(X) = (8 + 60 + 100)/49 = 168/49 = 24/7. Step 3: Calculating  7E(X) .
7E(X) = 7 · 24/7 = 24. Quick Tip: For expected value problems, multiply probabilities by outcomes and sum them up systematically.

Step 1: Compute total pairs. The total number of ways to choose two points is:
⁴⁹C₂ Step 2: Compute favorable pairs. Consecutive points are friends. There are  2 · 6 · 7 = 84  such pairs. Step 3: Compute probability  p .
p = 84/⁴⁹C₂. Step 4: Compute  7p .
Simplify:
7p = 0.5. Quick Tip: For combinatorics problems, carefully count total and favorable outcomes to calculate probabilities.

Step 1: Testing the dipole configuration.
The electric potential at  P  due to the initial dipole is given as:
V₀ = (kp₀/r²) cos 45°,
where  p₀  is the dipole moment. Step 2: With new dipole moment.
After the charges are moved, the new dipole moment is:
the Component P_0' along xaxis = -P₀/2 i
the Component P_0' along yaxis = P₀ j
Potential due to P_0 j
⇒ (kP₀ cos 45°)/r² = V₀
Potential due to -P₀/2 i
The new potential at  P  is:
(k(P₀/2) cos 135°)/r² = -V₀/2
Thus, the new potential is  V₀ - V₀/2 = V₀/2  Quick Tip: For electric dipoles, the potential at a point is proportional to the dipole moment and the cosine of the angle.

Step 1: Perform dimensional analysis for  Y . The dimensional formula of  Y  is:
[M L^-1 T^-2] = [c^αh^βG^γ],
where  c ,  h , and  G  are constants with specific dimensional formulas. Step 2: Write the dimensional formulas for  c ,  h , and  G . The dimensional formulas are:
[c] = [L T^-1],   [h] = [M L² T^-1],   [G] = [M^-1 L³ T^-2]. Step 3: Expand the dimensions of the right-hand side. Using the given dimensions:
[c^α] = [L^α T^-α],   [h^β] = [M^β L^2β T^-β],   [G^γ] = [M^-γ L^3γ T^-2γ].
The overall dimensions of  c^αh^βG^γ  are:
[M^β M^-γ] [L^α L^2β L^3γ] [T^-α T^-β T^-2γ].
Simplify:
[M^{β - γ}] [L^{α + 2β + 3γ}] [T^{-α - β - 2γ}]. Step 4: Match the dimensions of  Y . Equating the dimensions on both sides:
[M L^-1 T^-2] = [M^{β - γ} L^{α + 2β + 3γ} T^{-α - β - 2γ}].
Equate the powers of  M, L, T :
1. For  M :
β - γ = 1   ... (i).
2. For  L :
α + 2β + 3γ = -1   ... (ii).
3. For  T :
-α - β - 2γ = -2   ... (iii).
Step 5: Solve the equations. From equation (i):
β = 1 + γ   ... (iv).
Substitute  β = 1 + γ  into equations (ii) and (iii):
From (ii):
α + 2(1 + γ) + 3γ = -1 ⇒ α + 2 + 2γ + 3γ = -1.
α + 5γ = -3   ... (v).
From (iii):
-α - (1 + γ) - 2γ = -2 ⇒ -α - 1 - γ - 2γ = -2.
-α - 3γ = -1 ⇒ α + 3γ = 1   ... (vi).
Solve equations (v) and (vi):
From (vi):
α = 1 - 3γ   ... (vii).
Substitute  α = 1 - 3γ  into (v):
1 - 3γ + 5γ = -3 ⇒ 1 + 2γ = -3.
2γ = -4 ⇒ γ = -2.
Substitute  γ = -2  into (iv):
β = 1 + (-2) = -1.
Substitute  γ = -2  into (vii):
α = 1 - 3(-2) = 1 + 6 = 7.
Step 6: Final result. The values of  α, β, γ  are:
α = 7,   β = -1,   γ = -2.
Final Answer:
α = 7, β = -1, γ = -2. Quick Tip: Dimensional analysis is a powerful method to determine relationships between physical quantities.

Step 1: Write the velocity components. The velocity vector is given as:
V→ = α(yx ^ + 2xy ^).
From this:
v_x = αy,   v_y = 2αx. Step 2: Differentiate to find acceleration components. The acceleration vector is:
a→ = (dV→)/dt = d/dt(v_xx ^) + d/dt(v_yy ^).
For  v_x = αy , differentiate with respect to  t :
(dv_x)/dt = α (dy)/dt.
Substitute  (dy)/dt = v_y = 2αx :
(dv_x)/dt = α (2αx) = 2α²x.
For  v_y = 2αx , differentiate with respect to  t :
(dv_y)/dt = 2α (dx)/dt.
Substitute  (dx)/dt = v_x = αy :
(dv_y)/dt = 2α (αy) = 2α²y.
Thus, the acceleration vector is:
a→ = (2α²x)x ^ + (2α²y)y ^.
Step 3: Calculate the force. Using Newton's second law:
F→ = ma→.
Substitute  a→ :
F→ = m[(2α²x)x ^ + (2α²y)y ^].
Simplify:
F→ = 2mα² (xx ^ + yy ^).
Final Answer:
F→ = 2mα² (xx ^ + yy ^).
Quick Tip: For velocity-dependent motion, differentiate with respect to time to find the acceleration.

Step 1: Analyze internal energy. The internal energy is:
U_n = n/2RT   (proportional to  n ). Step 2: Analyze speed of sound. The speed of sound is:
v_n = √(γRT/M),
where  γ = 1 + 2/n . As  n  increases,  γ  decreases, and hence  v_n  decreases. Step 3: Conclusion. For  n = 5, 7 , we have:
v₅ > v₇,   U₅ < U₇. Quick Tip: Use degrees of freedom to calculate internal energy and speed of sound for ideal gases.

Step 1: Analyze the refractive index gradient. The refractive index increases linearly from  n₁  to  n₂  over a height  h . This causes the wavefront to tilt as the speed of light decreases with increasing refractive index. Step 2: Calculate the tilt of the wavefront. At time  t , the wavefront PQ is given by:
n₂d/c = n₁d/c + x/c.
Simplifying for  x :
x = d(n₂ - n₁). Step 3: Determine the angle of deflection. The angle of deflection  θ  is:
tan θ = x/h = (n₂ - n₁)d/h.
Thus:
θ = tan^-1[(n₂ - n₁)d/h]. Step 4: Validate the dependency on  n₂ - n₁ . The deflection angle depends only on the difference  n₂ - n₁ , as seen in the expression for  θ , and not on the individual values of  n₁  or  n₂. Quick Tip: In problems involving gradient refractive indices, the deflection angle can be found using basic trigonometric relations between wavefront displacement and the slab's geometry.

Step 1: Angular impulse-momentum relation. The angular impulse-momentum relation gives:
Jh = I_cmω,
where  J  is the impulse and  I_cm  is the moment of inertia about the center of mass. Step 2: Rolling condition. For rolling without slipping, the linear velocity  v_cm  and angular velocity  ω  are related by:
v_cm = bω. Step 3: Calculate  h_m . The critical height  h_m  is found by equating:
h_m = I_cm/Mb.
For different cases:
1. For  a → 0 ,  h_m = b/2.
2. For  a → b ,  h_m = b. Step 4: Additional observations.
1. For  h = 0  and  μ = 0 , rolling does not occur due to the absence of torque.
2. When  h = h_m , rolling without slipping occurs.
Quick Tip: In rolling motion, the relationship between linear velocity, angular velocity, and radius is key to determining rolling conditions.

Step 1: Calculate the wave speed in the dielectric medium. The wave number  k  is related to the wavelength  λ  as:
k = 2π/λ.
From the given data, the wave number is:
k = 10⁷/3.
The angular frequency  ω  is given as:
ω = 2π · 5 × 10¹⁴.
The speed of the wave  v  is:
v = ω/k.
Substitute the values of  ω  and  k :
v = (2π · 5 × 10¹⁴)/(10⁷/3) = (6 · 5 × 10¹⁴)/10⁷ = 1.5 × 10⁸ m/s. Step 2: Calculate the refractive index of the medium. The refractive index  n  is related to the speed of light  c  and the speed of the wave  v  as:
n = c/v.
Substitute  c = 3 × 10⁸ m/s  and  v = 1.5 × 10⁸ m/s :
n = (3 × 10⁸)/(1.5 × 10⁸) = 2. Step 3: Relate the magnetic field amplitude to the electric field amplitude. The amplitude of the magnetic field  B  is related to the electric field  E  and the speed of the wave  v  by:
B = E/v.
Substitute  E = 30 V/m  and  v = 1.5 × 10⁸ m/s :
B = 30/(1.5 × 10⁸) = 2 × 10^-7 T. Step 4: Express the magnetic field components. The magnetic field components are sinusoidal and can be expressed as:
B_x = -Bsin[2π(5 × 10¹⁴t - 10⁷z/3)],
B_y = 2B sin[2π(5 × 10¹⁴t - 10⁷z/3)].
Substitute  B = 2 × 10^-7 :
B_x = -2 × 10^-7 sin[2π(5 × 10¹⁴t - 10⁷z/3)],
B_y = 4 × 10^-7 sin[2π(5 × 10¹⁴t - 10⁷z/3)].
Final Answer: The magnetic field components are:
B_x = -2 × 10^-7 sin[2π(5 × 10¹⁴t - 10⁷z/3)],
B_y = 4 × 10^-7 sin[2π(5 × 10¹⁴t - 10⁷z/3)]. Quick Tip: In EM waves, the electric and magnetic fields are perpendicular to each other and the direction of propagation, and their magnitudes are related by the wave speed.

Step 1: Angular impulse and linear momentum. The impulse  J  is given as:
J = Mv_cm.
For angular motion:
J · R/2 = Iω,
where  I = MR²/4  is the moment of inertia about the center of mass. Step 2: Calculate  v_cm  and  ω . From the above equations:
v_cm = J/M,   ω = 2J/MR. Step 3: Time to return to the initial position. The time for the coin to return is:
T = 2v_cm/g = 2J/gM. Step 4: Total angle rotated. The angle rotated in time  T  is:
θ = ωT = 2J²/M²gR.
Substituting  J = √(π/2) × 10^-2 ,  M = 5 gm , and  R = 4/3 cm :
θ = 60π. Step 5: Number of rotations. The number of rotations is:
n = θ/2π = 60π/2π = 30. Quick Tip: In problems involving rotational motion, angular momentum conservation is key to relating linear impulse and rotational variables.

Step 1: Calculate the induced emf in the loop. The induced emf  ε  in the loop is generated due to the change in magnetic flux. Using the Biot-Savart law, the magnetic field at a distance  r  from the wire is:
B(r) = μ₀I/2πr.
For a small strip of width  b , the flux change due to motion of the loop with velocity  v  is:
ε = dΦ/dt.
The change in flux for two sides of the loop at distances  d  and  d + ℓ  (from the current-carrying wire) is:
ε = ∫_d^{d + ℓ} (μ₀Ivb/2πr)dr.
Simplify:
ε = (μ₀Ivb/2π)(1/d - 1/(d + ℓ)). Step 2: Current in the loop. The induced current  I₀  in the loop is given by Ohm's law:
I₀ = ε/R.
Substitute  ε :
I₀ = ((μ₀Ivb/2π)(1/d - 1/(d + ℓ)))/R.
Simplify:
I₀ = (μ₀Ivb/2πR)(ℓ/d(d + ℓ)). Step 3: Solve for  v . Rearrange to solve for  v :
v = 2πRI₀d(d + ℓ)/μ₀Ibℓ. Step 4: Substitute the given values. Given:
I₀ = 10 μA,   b = 2 cm = 0.02 m,   ℓ = 4 cm = 0.04 m,
d = 4 cm = 0.04 m,   R = 10 Ω,   I = 10 A.
Substitute:
v = (2π(10)(10 × 10^-6)(0.04)(0.04 + 0.04))/((4π × 10^-7)(10)(0.02)(0.04)).
Simplify:
v = (2π(10)(10^-6)(0.04)(0.08))/((4π × 10^-7)(0.02)(0.04)). v = (2 · 10 · 0.04 · 0.08)/(4 · 10^-7 · 0.02 · 0.04).
v = 0.064/(3.2 × 10^-8) = 2 × 10⁶ m/s.
Final Answer: The velocity of the loop is:
v = 2 m/s. Quick Tip: In electromagnetic induction, the induced emf depends on the rate of change of magnetic flux through the loop.

The frequency of vibration of a string is given by:
f = nv/2ℓ,
where:
-  n  is the harmonic number,
-  v  is the speed of the wave,
-  ℓ = 1 m  is the length of the string.
For two successive harmonics:
f_n = nv/2ℓ   and   f_n+1 = (n+1)v/2ℓ.
The difference in frequencies is:
f_n+1 - f_n = v/2ℓ.
Given:
f_n+1 - f_n = 1000 - 750 = 250 Hz.
Substitute values:
v/(2 · 1) = 250 ⇒ v = 500 m/s.
The speed  v  is related to tension  T  and linear mass density  μ  by:
v = √(T/μ),
where  μ = mass/length = (2 × 10^-5)/1 = 2 × 10^-5 kg/m . Substitute  v = 500 :
500 = √(T/(2 × 10^-5)).
Square both sides:
250000 = T/(2 × 10^-5) ⇒ T = 250000 × 2 × 10^-5 = 5 N. Quick Tip: The difference between successive harmonics is constant for a vibrating string and is determined by the wave speed and string length.

Step 1: Apply Boyle's Law for isothermal compression. The gas is compressed isothermally, so:
P₀V₀ = P_AV_A.
Substitute  V_A = 100/101 V₀ :
P₀V₀ = P_A(100/101 V₀).
Simplify to find  P_A :
P_A = P₀ 101/100. Step 2: Calculate the pressure transmitted to the liquid. The pressure  P_A  is transmitted to the liquid, so:
P_D = P_A = P₀ 101/100. Step 3: Capillary pressure equation. The pressure at the capillary is given by:
P_B = P_C + 2T/r,
where:
 T = 0.075 N/m  (surface tension),
 r = 0.1 mm = 0.0001 m  (capillary radius).
Rearranging:
P_C = P_B - 2T/r. Step 4: Use hydrostatic pressure to relate  P_D  and  P_C . The pressure difference between  P_D  and  P_C  causes the liquid to rise in the capillary:
P_D = P_C + ρgh.
Substitute  P_C = P_B - 2T/r :
P_D = (P_B - 2T/r) + ρgh. Step 5: Relate  P_D  to  P_B . From Boyle's law, we know  P_D = 101/100 P₀ , so:
101/100 P₀ = (P₀ - 2T/r) + ρgh.
Rearrange to solve for  h :
ρgh = 101/100 P₀ - P₀ + 2T/r.
Simplify:
ρgh = P₀/100 + 2T/r. Step 6: Solve for  h .
h = (P₀/100 + 2T/r)/ρg. Step 7: Substitute the given values. Given:
P₀ = 10⁵ Pa,   T = 0.075 N/m,   r = 0.0001 m,   ρ = 10³ kg/m³,   g = 10 m/s².
Substitute:
h = (10⁵/100 + (2 · 0.075)/0.0001)/(10³ · 10).
h = (1000 + 1500)/10000.
Simplify:
h = 2500/10000 = 0.25 m.
Convert to cm:
h = 25 cm.
Final Answer: The height of the liquid column is:
25 cm. Quick Tip: The height of the liquid column in a capillary depends on surface tension, pressure difference, and the radius of the capillary tube.

The activity of a radioactive source is proportional to the number of radioactive nuclei present:
A(t) = A₀(1/2)ⁿ,
where  A₀  is the initial activity, and  n  is the number of half-lives.
For  S₁  after 3 half-lives:
A₁ = A₀(1/2)³ = A₀/8.
For  S₂  after 7 half-lives:
A₂ = A₀(1/2)⁷ = A₀/128.
The ratio  A₁/A₂  is:
A₁/A₂ = (A₀/8)/(A₀/128) = 128/8 = 16. Quick Tip: In radioactive decay, the activity decreases exponentially with time. Each half-life reduces the activity to half its previous value.

Calculation of Work Done in Two Thermodynamic Cycles The work done in cycle I is given by:
W_I = 4P₀V₀ + 8P₀V₀ ln 2 - 6P₀V₀
The work done in cycle II is:
W_II = 4P₀V₀ ln 2 - P₀V₀
Step 1: Work Ratio Calculation
W_I/W_II = 2
Conclusion: The ratio of work done in cycle I to cycle II is 2, indicating that cycle I requires twice the work compared to cycle II. Quick Tip: For cyclic processes, work done is equal to the area enclosed by the  P-V  curve. Isothermal processes depend on  ln  terms for work calculations.

Step 1: Calculate  f₁  for  S₁ . The apparent frequency is given by:
f₁ = (v/(v - v_s))f₀,
where  v = 324 m/s ,  v_s = 4 m/s , and  f₀ = 656 Hz . Substituting the values:
f₁ = (324/(324 - 4)) 656 = 324/320 × 656 = 664.2 Hz. Step 2: Calculate  f₂  for  S₂ . Since  S₂  is stationary relative to the detector:
f₂ = f₀ = 656 Hz. Step 3: Calculate the beat frequency. The beat frequency is given by:
f_b = |f₁ - f₂| = |664.2 - 656| = 8.2 Hz. Quick Tip: The beat frequency is the absolute difference between the frequencies of two overlapping sound waves.

Step 1: Calculate air density. From the ideal gas law:
ρ = ρ₀(T_a/T) = 1.2 × 300/360 = 1 kg/m³. Step 2: Apply Bernoulli's equation. At points  1  and  2 :
P_a + 0 + 0 = P_a - ρgh + 0 + ρv²/2,
where  h = 9 m . Solving for  v :
v = √(2(ρ_a - ρ)gh/ρ) = √40 m/s. Step 3: Calculate mass flow rate. The mass flow rate is given by:
Q = ρAv = πd²/4 × v = π(0.1)²/4 × √40 × 1000 = 49.61 gm/s. Quick Tip: For streamline flow, apply Bernoulli's principle and calculate density changes using the ideal gas law.

Step 1: Write expressions for pressure at the surfaces. At the bottom surface inside the fluid, the pressure is:
P_inside = P_a + ρgh,
where:
P_a  is the atmospheric pressure,
ρ  is the fluid density,
g = 10 m/s²  is the acceleration due to gravity,
h = 10 m  is the depth.
At the top surface outside the fluid, the pressure is:
P_outside = P_a + ρ_agh,
where  ρ_a  is the density of air. Step 2: Compute the pressure difference. The pressure difference between the inside and outside is:
ΔP = P_inside - P_outside.
Substitute the expressions:
ΔP = (P_a + ρgh) - (P_a + ρ_agh).
Simplify:
ΔP = (ρ - ρ_a)gh. Step 3: Substitute the given values. Given:
ρ = 1 kg/m³,   ρ_a = 1.2 kg/m³,   g = 10 m/s²,   h = 10 m.
Substitute:
ΔP = (1.2 - 1) · 10 · 10.
Simplify:
ΔP = 0.2 · 100 = 20 N/m².
Final Answer: The pressure difference is:
20 N/m². Quick Tip: For pressure difference problems, consider the density variations and height difference using hydrostatic principles.

Step 1: Analyze each statement.
(I) Lyophobic colloids are indeed not formed by simple mixing. They require stabilizing agents.
(II) In emulsions, both the dispersed phase and the dispersion medium are liquids. This is correct.
(III) Micelles form at the critical micelle concentration (CMC) and specific temperatures, like the Kraft temperature. This makes the statement incorrect.
(IV) Tyndall effect requires different refractive indices of the dispersed phase and medium, so this statement is incorrect.
Quick Tip: Colloids exhibit unique properties such as the Tyndall effect, which depends on differences in refractive indices. Lyophobic colloids require stabilizers for formation.

Step 1: Analyze each reaction step.
-  P : Treating the compound with  Mg/dry ether  followed by hydrolysis ( H₂O ) yields a primary alcohol containing four carbon atoms.
-  Q : The reaction sequence produces a carboxylic acid, which undergoes Kolbe’s electrolysis to form an eight-carbon compound.
-  R : The reaction forms an aldehyde that does not participate in the Cannizzaro reaction.
-  S : The reaction results in the formation of a primary amine with six carbon atoms. Quick Tip: Understand the mechanism of Grignard reagents and the conditions required for reactions like Kolbe’s electrolysis and Cannizzaro reactions.

Step 1: Analyze the nature of sucrose. Sucrose is a non-reducing disaccharide because its anomeric carbons (C1 of glucose and C2 of fructose) are involved in a glycosidic bond, leaving no free aldehyde or keto group. Hence, sucrose does not react with bromine water. Step 2: Hydrolysis of sucrose. When sucrose is hydrolyzed by acid or the enzyme sucrase, it produces equimolar amounts of glucose and fructose:
sucrose + H₂O → glucose + fructose.
Glucose is dextrorotatory (+52.7°) and fructose is laevorotatory (-92.4°). The resulting solution exhibits a net laevorotation because fructose’s negative optical rotation dominates. Step 3: Conclusion.
The key properties of sucrose are:

• It is a non-reducing sugar and does not react with bromine water.
• On hydrolysis, it gives glucose and fructose, with the solution becoming laevorotatory.

Step 1: Analyze the given complexes. [Pt(NH₃)₂Br₂]: This complex is square planar and exhibits cis-trans isomerism.
(A) [Pt(en)(SCN)₂]: This complex does not exhibit isomerism because of the symmetrical arrangement of ligands.
(B) [Zn(NH₃)₂Cl₂]: Being a tetrahedral complex, it does not exhibit geometrical isomerism.
(C) [Pt(NH₃)₂Cl₄]: This octahedral complex shows geometrical isomerism due to the possibility of cis and trans arrangements of ligands.
(D) [Cr(en)₂(H₂O)(SO₄)]⁺: This complex is octahedral and exhibits geometrical isomerism due to different spatial arrangements of the ligands. Quick Tip: For geometrical isomerism, the arrangement of ligands in octahedral or square planar complexes is key. Tetrahedral complexes do not exhibit this type of isomerism.

Step 1: Analyze edge lengths and radii. For metal  x  (FCC):
L_x = 2√2r_x.
For metal  y  (BCC):
L_y = 4r_y/√3.
For metal  z  (SC):
L_z = 2r_z.
Step 2: Packing efficiencies. (i) For FCC (Z = 4) metal 'x', 4r_x = √2L_x
P.E. = (Z × 4/3π(r_x)³)/a_x³ = (4 × 4/3π(r_x)³)/(L_x)³ = (4 × 4/3π(r_x)³)/((4r_x/√2)³ = 0.24π
(ii) For BCC (Z = 2) metal 'y', 4r_y = √3L_y
P.E. = (Z × 4/3π(r_y)³)/a_y³ = (2 × 4/3π(r_y)³)/(L_y)³ = (2 × 4/3π(r_y)³)/((4r_y/√3)³ = 0.22π
(iii) For SC (Z = 1) metal 'z', 2r_z = L_z
P.E. = (Z × 4/3π(r_z)³)/a_z³ = (1 × 4/3π(r_z)³)/(L_z)³ = (1 × 4/3π(r_z)³)/(2r_z)³ = π/6 = 0.17π
Thus,  P.E._FCC > P.E._BCC > P.E._SC . Step 3: Density comparison.
4r_x = √2L_x,   4r_y = √3L_y,   2r_z = L_z
L_x = 4r_x/√2,   L_y = 4r_y/√3
L_y/L_x = (4r_y/√3)/(4r_x/√2) = √2r_y/√3r_x = (√2r_y)/(√3·(√3/2r_y)) = 4/3
Therefore, L_y > L_x
For option (C)
4r_x = √2L_x,   4r_y = √3L_y
L_x = 4r_x/√2,   L_y = 4r_y/√3
L_y/L_x = (4r_y/√3)/(4r_x/√2) = √2r_y/√3r_x = (√2 × 8/√3r_x)/(√3r_x) = 8√2/3
Therefore, L_x < L_y (incorrect)
For option (D)
d_x = (4 × M_x)/((4r_x/√2)³ × N_A)
d_y = (4 × M_y)/((4r_y/√3)³ × N_A)
r_y = 8/√3r_x,   M_x/M_y = 1/2
d_x/d_y = 512/2√2 = 256/√2
Therefore, d_x > d_y (correct)
ρ_x > ρ_y > ρ_z. Quick Tip: Packing efficiency decreases from FCC to BCC to SC. Use relations between edge length and radius to solve such problems.

Step 1: Analyze the reactions for  P, Q, R, and S .
P: Oxidation of the alkyl group results in the formation of a dicarboxylic acid.
Q: Hydrolysis of the acyl chloride produces the same dicarboxylic acid as in  P .
R: The product lacks an alpha-hydrogen, making it incapable of undergoing aldol condensation.
S: The product cannot undergo the Cannizzaro reaction since it is not a non-enolizable aldehyde. Quick Tip: Use the structural features of the products to analyze their reactivity towards condensation and Cannizzaro reactions.

The balanced half-reactions are:
For permanganate ion reduction:
MnO₄^- + 8H⁺ + 5e^- → Mn²⁺ + 4H₂O × 2.
For hydrogen sulfide oxidation:
H₂S → S + 2H⁺ + 2e^- × 5.
Combine the two half-reactions:
2MnO₄^- + 16H⁺ + 5H₂S → 2Mn²⁺ + 5S + 8H₂O + 10H⁺.
Simplify:
2MnO₄^- + 6H⁺ + 5H₂S → 2Mn²⁺ + 5S + 8H₂O.
Here:
x = 8 (moles of water),   y = 10 (moles of electrons transferred).
The sum is:
x + y = 8 + 10 = 18. Quick Tip: Balance redox reactions by ensuring conservation of both mass and charge.

The species with  sp³ -hybridized central atoms are as follows: [SiO₄]^4-: The central silicon atom exhibits tetrahedral geometry with  sp³  hybridization. SOCl₂: The central sulfur atom has tetrahedral geometry with  sp³  hybridization. Ni(CO)₄: The central nickel atom forms a tetrahedral structure with  sp³  hybridization. XeO₂F₂: The central xenon atom exhibits a distorted tetrahedral geometry with  sp³  hybridization. SF₄: The central sulfur atom has a see-saw geometry and  sp³  hybridization. Thus, the total number of species with  sp³ -hybridized central atoms is 5. Quick Tip: Determine the hybridization by considering the steric number (sum of bonded atoms and lone pairs).

Step 1: Determine the number of atoms in the zero oxidation state for each molecule. (i) Br₃O₈: In this molecule, one bromine atom is in the zero oxidation state. (ii) F₂O: Both fluorine atoms are in the zero oxidation state. (iii) H₂S₄O₆: Among the sulfur atoms, one is in the zero oxidation state. (iv) H₂S₅O₆: Two sulfur atoms are in the zero oxidation state. (v) C₃O₂: All three carbon atoms are in the zero oxidation state. Step 2: Add the counts.
1 (from Br₃O₈) + 2 (from F₂O) + 1 (from H₂S₄O₆) + 2 (from H₂S₅O₆) = 6.
Final Answer: The total number of atoms in the zero oxidation state is:
6. Quick Tip: Identify zero oxidation states by analyzing the bonding and oxidation state rules for each element.

The radius of an orbit is given by:
r_n = (52.9 · n²)/Z pm.
For  n₁ :
105.8 = (52.9 · n₁²)/2 ⇒ n₁² = 4 ⇒ n₁ = 2.
For  n₂ :
26.45 = (52.9 · n₂²)/2 ⇒ n₂² = 1 ⇒ n₂ = 1.
Energy difference:
1/λ = 109677 · Z² (1/n₁² - 1/n₂²).
Substituting  Z = 2 :
1/λ = 109677 · 4 (1/1 - 1/4) = 3 · 109677.
λ = 10⁷/(3 · 109677) = 30.3 nm = 30 nm. Quick Tip: For hydrogen-like atoms, use the radius formula and the Rydberg equation for energy transitions.