MHT CET 2024 16 May Shift 1 Question Paper : Download PCM Question Paper with Answers PDF

The line of intersection of the two planes is parallel to the cross product of their normal vectors. The normal vectors are: n1 = 3i - j + k and n2 = i + 4j - 2k.

The direction vector of the line is given by: d = n1 × n2.

Compute the cross product:
d = |i j k|
|3 -1 1|
|1 4 -2|

Expanding the determinant:
d = i((-1)(-2) - (1)(4)) - j((3)(-2) - (1)(1)) + k((3)(4) - (-1)(1))

Simplify:
d = i(2 - 4) - j(-6 - 1) + k(12 + 1)
d = -2i + 7j + 13k.

The direction vector is: -2i + 7j + 13k.

Let l, m, n represent the direction cosines of the line. From the first equation: l + m + n = 0 → n = -(l + m).

Substitute n = -(l + m) into the second equation:
2l² + 2m² - n² = 0 → 2l² + 2m² - (-(l + m))² = 0.

Simplify:
2l² + 2m² - (l² + 2lm + m²) = 0 → l² + m² - 2lm = 0.

Factorize:
(l - m)² = 0 → l = m.

Substitute l = m into l + m + n = 0:
2l + n = 0 → n = -2l.

The direction cosines of the two lines are proportional to: (l, m, n) = (1, 1, -2) and (-1, -1, 2). Since the direction cosines are negatives of each other, the lines are antiparallel, and the angle between them is 180°.

The expected value E(X) is given by:
E(X) = Σ [x * P(X = x)].

Step 1: Verify the total probability
The total probability must sum to 1:
P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 5/16 + k/48 + 1/4 + 1/4.

Simplify:
5/16 + k/48 + 12/48 + 12/48 = 1
Convert 5/16 to 48: 5/16 = 15/48.
15/48 + k/48 + 12/48 + 12/48 = 1
(15 + k + 12 + 12)/48 = 1 → k = 9.

Step 2: Find P(X = 1)
P(X = 1) = (kx)/48 = 9/48.

Step 3: Calculate E(X)
E(X) = 0 * (5/16) + 1 * (9/48) + 2 * (12/48) + 3 * (12/48).
E(X) = 0 + 9/48 + 24/48 + 36/48 = 69/48 = 1.4375.

Final Answer: 1.4375.

The surface area S of a sphere is given by:
S = 4πr², where r is the radius of the sphere.

The volume V of the sphere is given by:
V = (4/3)πr³.

Given: dS/dt = 2 cm²/sec, r = 6 cm.
We need to find dV/dt, the rate of increase of volume.

Step 1: Relating dS/dt and dr/dt
Differentiate S = 4πr² with respect to t:
dS/dt = 8πr * dr/dt.
Solve for dr/dt:
dr/dt = (dS/dt) / (8πr).
Substitute dS/dt = 2 and r = 6:
dr/dt = 2 / (8π * 6) = 1 / 24π.

Step 2: Relating dV/dt and dr/dt
Differentiate V = (4/3)πr³ with respect to t:
dV/dt = 4πr² * dr/dt.
Substitute r = 6 and dr/dt = 1 / 24π:
dV/dt = 4π * (6)² * (1 / 24π).
dV/dt = (4π * 36) / (24π) = 144 / 24 = 6 cm³/sec.

Final Answer: 6 cm³/sec.

To determine where f(x) is strictly decreasing, we analyze the derivative f'(x).

The derivative is:

f'(x) = d/dx (2x³ - 15x² - 144x - 7).

Step 1: Compute f'(x)

Differentiate term by term: f'(x) = 6x² - 30x - 144.

Step 2: Solve f'(x) = 0

Factorize f'(x) to find critical points:

6x² - 30x - 144 = 0.

Divide through by 6:

x² - 5x - 24 = 0.

Factorize:

(x - 8)(x + 3) = 0.

Thus, the critical points are:

x = -3, x = 8.

Step 3: Analyze the intervals

The critical points divide the real line into three intervals: (-∞, -3), (-3, 8), and (8, ∞).

Test the sign of f'(x) in each interval:

• For x in (-∞, -3), choose x = -4: f'(-4) = 72 > 0. f'(x) > 0, so f(x) is increasing.
• For x in (-3, 8), choose x = 0: f'(0) = -144 < 0. f'(x) < 0, so f(x) is decreasing.
• For x in (8, ∞), choose x = 9: f'(9) = 72 > 0. f'(x) > 0, so f(x) is increasing.

Conclusion:

f(x) is strictly decreasing in the interval (-3, 8).

Final Answer: (-3, 8).

Given: y = (sin x)^y.

Take the natural logarithm on both sides:

ln y = y ln(sin x).

Differentiate both sides with respect to x:

(1 / y)(dy/dx) = d/dx [y ln(sin x)].

Apply the product rule to the right-hand side:

(1 / y)(dy/dx) = (dy/dx) ln(sin x) + y (d/dx [ln(sin x)]).

The derivative of ln(sin x) is:

d/dx [ln(sin x)] = cot x.

Substitute this into the equation:

(1 / y)(dy/dx) = (dy/dx) ln(sin x) + y cot x.

Multiply through by y to eliminate the denominator:

dy/dx = y (dy/dx) ln(sin x) + y² cot x.

Rearrange to isolate dy/dx:

dy/dx (1 - y ln(sin x)) = y² cot x.

Solve for dy/dx:

dy/dx = y² cot x / (1 - y ln(sin x)).

Final Answer: y² cot x / (1 - y log(sin x)).

From the given equation: sin^-1(x) + cos^-1(y) = 3π/10.

Using the identity: sin^-1(x) + cos^-1(x) = π/2, we know: cos^-1(y) = π/2 - sin^-1(y).

Substitute cos^-1(y) = π/2 - sin^-1(y) into the equation: sin^-1(x) + (π/2 - sin^-1(y)) = 3π/10.

Simplify: sin^-1(x) + π/2 - sin^-1(y) = 3π/10.

Rearrange to find sin^-1(x) - sin^-1(y):
sin^-1(x) - sin^-1(y) = 3π/10 - π/2.

Simplify: sin^-1(x) - sin^-1(y) = 3π/10 - 5π/10 = -2π/10 = -π/5.

Now, calculate cos^-1(x) + sin^-1(y):
cos^-1(x) = π/2 - sin^-1(x).

Substitute: cos^-1(x) + sin^-1(y) = (π/2 - sin^-1(x)) + sin^-1(y).

Simplify: cos^-1(x) + sin^-1(y) = π/2 - (sin^-1(x) - sin^-1(y)).

Substitute sin^-1(x) - sin^-1(y) = -π/5:
cos^-1(x) + sin^-1(y) = π/2 - (-π/5).

Simplify: cos^-1(x) + sin^-1(y) = π/2 + π/5.

Convert to a common denominator: cos^-1(x) + sin^-1(y) = 5π/10 + 2π/10 = 7π/10.

Final Answer: 7π/10.

Step 1: Simplify sin^-1[sin(-600°)]

The range of sin^-1 is [-π/2, π/2]. To bring -600° within this range:

-600° + 720° = 120°.

Thus: sin(-600°) = sin(120°).

The value of sin(120°) is: sin(120°) = sin(180° - 60°) = sin(60°) = √3/2.

Since -600° lies in the third quadrant, sin^-1[sin(-600°)] is:

sin^-1(√3/2) = π/3.

Step 2: Simplify cot^-1(-√3)

The range of cot^-1 is [0, π]. For cot^-1(-√3), we note:

cot^-1(-√3) = π - cot^-1(√3).

The value of cot^-1(√3) is: cot^-1(√3) = π/6.

Thus: cot^-1(-√3) = π - π/6 = 5π/6.

Step 3: Add the two results

Now, sum the results: sin^-1[sin(-600°)] + cot^-1(-√3) = π/3 + 5π/6.

Simplify: π/3 + 5π/6 = 2π/6 + 5π/6 = 7π/6.

Final Answer: π/6.

For A · A^-1 = I (the identity matrix), we verify the values of a and c such that:

      [[0, 1, 2], 
       [1, 2, 3], 
       [3, a, 1]] 
       · 
       (1/2) [[1, -1, 1], 
              [-8, 6, 2c], 
              [5, -3, 1]] = I.
      

Step 1: Simplify for a

Consider the third row of A and the first column of A^-1:

(3)(1) + (a)(-8) + (1)(5) = 0.

Simplify:

3 - 8a + 5 = 0.

8 - 8a = 0 → a = 1.

Step 2: Simplify for c

Consider the second row of A and the third column of A^-1:

(1)(1) + (2)(2c) + (3)(1) = 0.

Simplify:

1 + 4c + 3 = 0.

4c + 4 = 0 → c = -1.

Final Answer: a = 1, c = -1.

The expected value E(X) is given by: E(X) = Σ [x · P(X = x)] for x = 1 to n.

Substitute P(X = x) = 2x / [n(n + 1)]:

E(X) = Σ [x · (2x / [n(n + 1)])] = (2 / [n(n + 1)]) Σ x².

Step 1: Use the sum of squares formula

The sum of squares of the first n natural numbers is: Σ x² = n(n + 1)(2n + 1) / 6.

Substitute this into the equation for E(X):

E(X) = (2 / [n(n + 1)]) · [n(n + 1)(2n + 1) / 6].

Simplify:

E(X) = 2(2n + 1) / 6 = (2n + 1) / 3.

Final Answer: (2n + 1) / 3.