Mathematics: CUET Mock Test - 5 - CUET MCQ

The zero-length interval property is one of the properties used in definite integrals and they are always positive. The zero-length interval property is .

Concept:

EMI in reducing balance method, EMI = P × where P = Principle. i = interest rate, n = no. of payments

Calculation:

Given, Total cost of AC = Rs. 45,000 and Down payment = Rs. 5,000

⇒ Principal amount (P) = Total cost of AC - Down payment

⇒ P = Rs. 45,000 - Rs. 5,000

⇒ P = Rs. 40,000

Now, Monthly interest rate (i) = Annual interest rate / 12 months

⇒ i = 6% / 12 = 0.06 / 12 = 0.005

Number of monthly installments (n) = 5 years × 12 months = 5 × 12 = 60

∴ EMI = P ×

= 40,000 ×

= 40,000 ×

= 40,000 × 0.0194

= 776

∴ Monthly EMI is Rs. 776.

Calculations:

The depreciation of the machine is calculated using the given depreciation rate of 10% per annum, which means the value of the machine reduces to 90% (or 0.9) of its current value each year.

Given that the machine's initial cost is Rs. 20,000, the machine's value after 10 years is calculated by multiplying the initial cost by (0.9)^10. As provided, (0.9)^10 is approximately 0.35. So:

Value of machine after 10 years = Initial cost × (0.9)¹⁰ = Rs. 20,000 × 0.35 = Rs. 7,000.
So, the scrap value of the machine after 10 years, when its value depreciates at 10% p.a, is Rs. 7,000.

The correct option is 2 Rs. 7,000

Both population mean and sample mean can behave differently with changes in sample size.

• The population mean: it remains constant regardless of the sample size, assuming that the sample is a random sample from the population. This is because the population mean is a fixed value that represents the average of all the values in the population, and it does not change with the size of the sample.
• The sample mean: It can vary with changes in sample size due to the effect of sampling variability. The sample mean is an estimate of the population mean, and it becomes more precise as the sample size increases. This is because larger samples provide more information about the population, and thus the sample mean is more likely to be closer to the population mean.

As the sample size increases, the sample mean becomes a better estimator of the population mean, and the sampling variability decreases. This means that the sample mean is more likely to be closer to the population mean, and there is less uncertainty in the estimate. However, even with large sample sizes, there is still some variability in the estimate due to random sampling, and the sample mean may not be exactly equal to the population mean.
⇒ |x̅ − μ| decreases when increasing the size of the samples.

• The stages of drawing statistical inferences are:
• Stage I: (B) Identification of population: The population refers to the entire group of individuals or objects that possess a particular characteristic or feature of interest to the researcher. For example, if a researcher is interested in studying the average height of all people in a country, the population would be all people in that country.
• Stage II: (A) Sample: The group of the population may be too large or too costly to study in its entirety. so researchers often select a smaller subset of the population, known as a sample, to gather data from. In the above example, the researcher might take a sample of 1,000 people from the country to estimate the average height of the population.
• Stage III: (D) Data tabulation: Tabulation is another important stage where the sample data is arranged in a systemic manner that helps researchers to analyze in a better way.
• Stage IV: (E) Data Analysis: Data is analyzed in this stage. Data analysis refers to the process of systematically examining and interpreting data using various statistical and analytical methods in order to derive insights, identify patterns, and make informed decisions.
• Stage V: (C) Making Inference: This is the final and last stage of drawing an inference from available information or data to draw conclusions or make predictions about a particular situation or phenomenon that is not directly observed or measured. Inference is a key aspect of scientific research, as it allows researchers to draw meaningful conclusions from data and make predictions about future outcomes.

Hence, B, A, D, E, and C is the correct order of the stages for drawing statistical inferences

Concept:

Objective function: Linear function Z = ax + by, where a, b are constants, which has to be maximized or minimized is called a linear objective function.
In the above example, Z = ax + by is a linear objective function. Variables x and y are called decision variables.

By putting values of variables (coordinates of the point) in linear objective function we get the value of the point.

Calculations:

Given, Objective function for all LPP is z = 2x + 3y

Putting coordinates of points in the equation we get value of the point

e.g for corner point (0, 2)

z = 2x + 3y = 2 × 0 + 3 × 2 = 6

The difference of the maximum and minimum values of z is = 36 - 6 = 30

Given:

Line and point (3, 4, 5)

Concept:

If two vectors are perpendicular to each other then dot product of both is zero.

Calculation:

Let point A = (3, 4, 5)

and

Then point B = (k, 2k + 1, 3k + 2) on the line

Now, the line AB = B - A = (k - 3, 2k - 3, 3k - 3) .

DRs of given line (1, 2, 3)

We know that the Line AB is perpendicular to the given line

Then

(k - 3, 2k - 3, 3k - 3) ⋅ (1, 2, 3) = 0

⇒ k - 3 +4k - 6 + 9k - 9 = 0

⇒ 14k = 18 ⇒

Then line

The perpendicular length from point A on given line is the magnitude of AB.

Hence option (2) is correct.

Concept:

The vector equation of a line passing through a point with position vector and parallel to the vector is given by:

Calculation:

Let A = (3, 4, 5) and

Now the position of A is

As we know that the vector equation of a line passing through a point with position vector and parallel to the vector is given by:

∴ The equation of required line is:

l2 + m2 + n2 = 1
The cosines l, m, and n of the angles made by a vector with the coordinate axes are also known as the direction cosines of a vector.
For any vector, the sum of the squares of these direction cosines always equals one i.e l² + m² + n² = 1
This is a consequence of the generalization of the Pythagorean theorem known as the squared Euclidean norm (for 3D vectors).
Hence, Option 2 is Correct.

Given:

L is the line with direction ratios < 3, -2, 6 >, passing through (1, -1, 1)

Formula Used:

Equation of a line which passed through (x1,y1,z1) is given by

Where, l, m, n are the Direction Ratios

Distance of (x₁, y₁, z₁) and (x₂, y₂, z₂) is given by

Calculation:

Equation of L,

(let)

(x, y, z) = (3k + 1, -2k - 1, 6k + 1)

Distance of (3k + 1, -2k - 1, 6k + 1) & (1, -1, 1) is 2 units

⇒

⇒

⇒

⇒

⇒

Putting the value of k in (3k + 1, -2k - 1, 6k + 1) we get,

and

Concept:

The solution of the linear differential equation is given by

y × I.F =

Where I.F =

Explanation -

we have ⇒

Now integrating factor is I.F. =

Now the solution of the differential equation is -

⇒

⇒

⇒

⇒

Hence the option (i) is true.

= 3(4) – 4
= 8

= – 2(-3)
= 6

= 2(4) – 2
= 6

= 7(e⁶ – e²)

= – 4

 comes under the reverse integral property.
In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is 

In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is .

The adding intervals property of definite integrals is .

In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is 

The number of elements for a matrix with the order m × n is equal to mn, where m is the number of rows and n is the number of columns in the matrix.

The given matrix  has 3 rows and 2 columns. Therefore, the order of the matrix is 3×2.

According to Rolle’s theorem, if f : [a,b] → R is a function such that

• f is continuous on [a,b]
• f is differentiable on (a,b)
• f(a) = f(b) then there exists at least one point c ∈ (a,b) such that f’(c) = 0

According to Rolle’s theorem, if f : [a,a+h] → R is a function such that

• f is continuous on [a,a+h]
• f is differentiable on (a,a+h)
• f(a) = f(a+h) then there exists at least one θ c ∈ (0,1) such that f’(a+θh) = 0

A square matrix is a matrix in which the number of rows(m) is equal to the number of columns(n). Therefore, the matrix which follows the condition m=n is a square matrix.

The matrix in which the number of columns is greater than the number of rows is called a vertical matrix. There the matrix which follows the condition m>n is a vertical matrix.

Rolle’s theorem is just a special case of Lagrange’s mean value theorem when f(a) = f(b) and Lagrange’s mean value theorem is also called the mean value theorem.

According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that f is differentiable on (a,b) then the formula for Lagrange’s theorem is f’(c) = .

According to Rolle’s theorem, if f : [a,b] → R is a function such that

• f is continuous on [a,b]
• f is differentiable on (a,b)
• f(a) = f(b) then there exists at least one point c ∈ (a,b) such that f’(c) = 0

According to Rolle’s theorem, if f : [a,b] → R is a function such that

• f is continuous on [a,b]
• f is differentiable on (a,b)
• f(a) = f(b) then there exists at least one point c ∈ (a,b) such that f’(c) = 0