CUET Mathematics Question Paper 2024 Set D (Available) : Download Solutions and Answer Key pdf

Probability of rolling greater than 4: P(greater than 4) = 2/6 = 1/3.
Probability of rolling less than 4: P(less than 4) = 3/6 = 1/2.
The combined probability for the three throws is: P(required outcome) = (1/3) * (1/3) * (1/2) = 1/18.
 

To maximize Z at the given points, you need to solve for α and β by considering the slopes of the objective function and the boundary lines of the feasible region.

First, simplify y = ln(t²) as follows:
Since t = e^(2x), we have: y = 2 * ln(t) = 4x.
First derivative: dy/dx = 4
Second derivative: d²y/dx² = 0
Thus, d²y/dx² is 0.


 

For an n × n matrix, |kA| = k^n * |A|.
Here, |2A| = 2^4 * 4 = 64.

Given the matrices [A]₃×₂, [B]ₓ×ᵧ, and [C]₃×₁, for matrix multiplication to be defined, x must equal 2.
The resulting product [A][B] will have dimensions 3 × y, which must match [C]₃×₁, so y = 1.

Each die has a probability of 1/6 of showing a four. The expectation of X (the number of fours) is the sum of the expectations for each die: E(X) = E(X1) + E(X2).
E(X1) = 1/6, E(X2) = 1/6.
Thus, E(X) = 1/6 + 1/6 = 2/6 = 1/3.
 

Differentiate f(x) = 2x³ − 9x² + 12x − 5 to find f'(x) = 6x² − 18x + 12.
Solve f'(x) = 0 to find critical points within the interval [0, 3].
Evaluate f(x) at the endpoints x = 0 and x = 3, and at the critical points, to determine the absolute maximum and minimum values.
 

We need to determine which function’s second derivative equals 5x.
For (1), (2), and (3), the second derivatives do not yield 5x.
For (4), d²/dx² of 5x * (ln(5))² = 5x.
 

This integral involves solving the definite integral using substitution and evaluating at limits to reach the final result.

This is a standard integral that can be solved using substitution and evaluating the limits.

The cross product of ⃗a + ⃗b and ⃗a − ⃗b gives the perpendicular vector. After computing the cross product, normalize it to find the unit vector: −1/√6 î + 2/√6 ĵ − 1/√6 k̂.

Differentiate both sides with respect to x: cos y * dy/dx = sin(a + y) + x cos(a + y) * dy/dx.
Isolate dy/dx and simplify to get: dy/dx = sin²(a + y) / sin a.

Use the formula for the distance between two skew lines: d = |(⃗d₁ × ⃗d₂) · (⃗r₂ − ⃗r₁)| / |⃗d₁ × ⃗d₂|. After calculating the cross product and dot product, the distance is √328/7.

f(x) = 2 tan⁻¹(ex) − π/4 is an odd function and is strictly increasing over the entire real line. Thus, the correct answer is (3).

Using Bayes' theorem, we calculate the probability that the ball was not drawn from Bag-2, given that it is not black. The final result is 2/7.

The direction ratios of the line x − 3 / 2 = 2 − y / 3 = z + 4 / −1 are 2, −3, −1. Option (3) does not match the correct direction ratios, as the second direction ratio should be negative.

The relation R is reflexive (a line is parallel to itself), symmetric (if l₁ is parallel to l₂, then l₂ is parallel to l₁), and transitive (if l₁ is parallel to l₂, and l₂ is parallel to l₃, then l₁ is parallel to l₃).

By using the formula \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \) and substituting the given information, we find \( |\vec{b}| = 3 \).

By using the identity cot⁻¹(y) = π/2 − tan⁻¹(y), we can solve the equation and find that there are two solutions for x, one positive and one negative.

The determinant of each matrix is zero because they are singular. Solving the equations formed by the determinants, we find that the value of abc is 45.

This is an integral involving logarithmic functions, which can be simplified using standard integration techniques for logarithms.

(A) |x − 1| + |x − 2| is continuous everywhere. Match: (A) → (II).
(B) x − |x| is differentiable at x = 1. Match: (B) → (I).
(C) x − [x] is not differentiable at x = 1. Match: (C) → (III).
(D) x|x| is differentiable at x = 1. Match: (D) → (IV).

 

The equation of the line is x + 7√3a + y/b = 4.
The intercepts are: when x = 0, y = 4b, and when y = 0, x = 28√3a.
The area of the triangle formed is given by: Area = 1/2 × base × height.
The base is 28√3a and the height is 4b, so the area is: Area = 1/2 × (28√3a) × (4b) = 56√3ab.

 

We are given that A² = A.
Using this, simplify the expression A(I − 2A)³ + 2A³.
First, simplify (I − 2A)³ = I − 6A.
Then, the expression becomes A(I − 6A) + 2A.
Simplify to get A.


 

Analyze each differential equation and find the corresponding integrating factor:
(A) 1/x, (B) x³, (C) x², (D) x.

The integrand is odd with respect to x = π/4, and since the integral is symmetric about π/4, the value of the integral is 0.

For a square matrix Aₙₓₙ, the determinant of the adjugate of A is given by: |adj A| = |A|ⁿ⁻¹.
This property confirms that (A) is correct.
For the inverse of a matrix: |A⁻¹| = 1 / |A|.
This property confirms that (D) is correct.
(C) is valid, but it is not relevant to the determinant properties.
(B) is incorrect because |A| ≠ |adj A|ⁿ⁻¹.



 

The matrix is a scalar matrix because all diagonal elements are equal and non-zero.
It is also a diagonal matrix since all non-diagonal elements are zero.
This matrix is symmetric because A = Aᵀ.
It is not skew-symmetric because all diagonal elements are non-zero.

The curves 4x² = y and y = 4 intersect at y = 4.
We find the area between these curves from y = 0 to y = 4.
Using integration: Area = 2 * ∫ from 0 to 4 of √y dy = 16/3 sq. units.

The integrand simplifies to: e^x√x + 1 / (2√x) dx.
After substitution and simplification, the result is e^x√x + C.

For continuity at x = π, the left-hand limit, right-hand limit, and the value of the function at x = π must all be equal.
Equating the left-hand limit and right-hand limit gives k = -2/π.

Compute the product PQ: the result is a 3x3 matrix.
Take the transpose of PQ to get (PQ)ᵀ = [-2 4 2, 4 −8 −4, −1 2 1].

The variance is calculated using the formula Variance = E(X²) - (E(X))², where E(X) is the expected value and E(X²) is the expected value of X².

The amount accumulated in the sinking fund is calculated using the formula for compound interest. The surplus is the difference between the accumulated amount and the bond amount.

This is a matrix equation where we solve for the unknown \( n \) by performing matrix multiplication and solving the resulting equation.

To find the remainder, divide 351 by 7 and calculate the remainder. 351 ÷ 7 = 50 with a remainder of 6.

Solve the system of linear equations to find the values of x and y, and substitute them into the equation 5x + 3y.

The total number of possible outcomes is 15, and the favorable outcomes where the sum is greater than 3 are calculated. The probability is the ratio of favorable outcomes to total outcomes.

The point estimate of the population mean is the sample mean, which is calculated as the sum of the sample values divided by the number of values. Solving for x gives 12.

The effective rate of interest is calculated using the formula Effective Rate = (1 + i/n)n - 1, where i = 0.1 and n = 2.

Using the properties of matrix multiplication and the given equations, we find that \( Y^2 + 2Y = I + 3Y \).

The probability of getting exactly 3 heads or 7 heads can be computed using the binomial distribution formula, and the probability of 8 tails is derived from that.

The size of the sample is calculated using the formula for the confidence interval, where the margin of error and standard deviation are used.

Let the rate of Pipe B be x. Then the rate of Pipe A is 2x. The combined rate of filling the tank is given by 1/40, and solving for x gives the time for Pipe A to fill the tank.

The determinant of a 2x2 matrix [a, b; c, d] is calculated using the formula ad - bc. The determinant of the fourth option is even.

The derivatives of the functions are matched with their corresponding options based on standard derivative rules and simplifications.

By using the properties of probability distributions and calculating the values for each condition, we match the correct answers.

CAGR is typically used in finance for growth analysis over time, not for analyzing donations in non-profit sectors.

The depreciation formula is \( \text{Final cost} = \text{Initial cost} - \text{Depreciation} \times \text{Years} \). Solving gives 17 years for the price to reduce to ₹6,000.

Use the relation \( \text{time} = \frac{\text{distance}}{\text{speed}} \) and account for the effect of the stream speed to find the distance.

Differentiating the equation e^y = x^x twice with respect to x, and solving for the second derivative, we get ^d₂y / dx² - ^dy / dx = 0.

Use the formula \( 1 - P(\text{no hit})^n > 0.9 \), where \( P(\text{no hit}) = \frac{1}{4} \), and solve for \( n \).

According to the Central Limit Theorem, a sample distribution approaches a normal distribution as the sample size increases. The rest match the definitions of statistics terms.

The formula for simple interest \( I = P \times r \times t \) is used, where \( P = 100,000 \), \( I = 8,000 \), and \( t = 1 \). Solving gives \( r = 2.5\% \).

This linear programming problem maximizes the return function subject to constraints on investments in Plan A and Plan B and their relationship. The correct set of constraints ensures that the investment in Plan A is less than or equal to the investment in Plan B.

The moving average is calculated by taking the average of every 3 consecutive values. For example, the first moving average is \( \frac{12 + 15 + 18}{3} = 15 \), and so on.

The EMI for each property is calculated using the EMI formula for compound interest: \( EMI = \frac{P \times r \times (1+r)^n}{(1+r)^n-1} \), where \( P \) is the loan amount, \( r \) is the monthly interest rate, and \( n \) is the number of payments.

To maximize Z at the given points, you need to solve for α and β by considering the slopes of the objective function and the boundary lines of the feasible region.

In a skew-symmetric matrix, all diagonal elements must be zero. By equating the elements to their corresponding negatives and solving for x and y, we find 5x - y = 10.

The demand function is linear, so we find the equation of the demand curve and use the marginal revenue formula: MR = d(TR)/dx, where TR is the total revenue.

To maximize the area of the trapezium, we use the formula for the area of a trapezium with equal non-parallel sides and calculate the maximum when the height is maximized.

The probabilities are calculated using the binomial distribution formula for drawing defective and non-defective bulbs with replacement. Each probability is computed for the respective scenarios.