CBSE Class 12 Applied Mathematics 2025 — Set 465/S/WXYZ/4

If y = xy, then (dy)/(dx) is : (A) xy (log x + 1) (B) (y²)/(x(1 + y log x)) (C) xy (log x - 1) (D) (y²)/(x(1 - y log x))

The function f(x) = ax is increasing on R, if : (A) a > 0 (B) a > 1 (C) a < 0 (D) 0 < a < 1

A function f : R → R is defined as f(x) = x³ + 1. The function f has : (A) no maximum value (B) no minimum value (C) both maximum and minimum values (D) neither maximum nor minimum value

The relation between “Marginal Cost (MC)” and “Average Cost (AC)” of producing ‘x’ units of a product is : (A) (d)/(dx)(AC) = x(MC - AC) (B) (d)/(dx)(AC) = x(AC - MC) (C) (d)/(dx)(AC) = (1)/(x)(MC - AC) (D) (d)/(dx)(AC) = (1)/(x)(AC - MC)

For a random variable X, E(X) = 3 and E(X²) = 11. The variance of X is : (A) 8 (B) 5 (C) 2 (D) 1

If the mean and standard deviation of a binomial distribution are 12 and 2 respectively, then the value of the parameter p is : (A) (5)/(6) (B) (1)/(6) (C) (1)/(3) (D) (2)/(3)

If the variance of a Poisson distribution is 2, then P(X = 2) is : (A) 4e² (B) 2e² (C) (2)/(e²) (D) (4)/(e²)

Normal distribution is symmetric about : (A) Variance (B) Co-variance (C) Mean (D) Standard deviation

Using the flat rate method, the EMI to repay a loan of ₹ 20,000 in 2(1)/(2) years at an interest rate of 8% per annum is : (A) ₹ 100 (B) ₹ 700 (C) ₹ 800 (D) ₹ 1,000

The graph of the inequality 3x + 2y > 6 is the : (A) entire XOY plane (B) whole XOY plane excluding the points on the line 3x + 2y = 6 (C) half plane that contains the origin (D) half plane that neither contains the origin nor the points on the line 3x + 2y = 6

The straight line trend is represented by the equation : (A) y = a + bx (B) y = a - bx (C) y = na + b Σ x (D) y = na - b Σ x

If for the purpose of t-test of significance, a random sample of size (n) 34 is drawn from a normal population, then the degree of freedom (N) is : (A) 32 (B) 33 (C) 35 (D) 36

Assertion (A) : Solution set of inequality |3x - 2| ≤ (1)/(2), x ∈ R is [(1)/(2), (5)/(6)]. Reason (R) : |x - a| ≤ r Leftrightarrow x ≤ a - r or x ≥ a + r. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

Assertion (A) : Matrix A = 0 -6 7; 6 5 -1; -7 1 0 is a skew-symmetric matrix. Reason (R) : A matrix A is skew-symmetric if A' = -A. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

(a) Two pipes P and Q together can fill a tank in 10 minutes. If pipe P takes 15 minutes less than Q to fill the tank alone, then find the time taken by pipe Q to fill the tank alone. OR (b) In a 200 m race, A beats B by 35 m or 7 seconds. Find the time taken by A to complete the race.

Find the probability distribution of a number of successes in two tosses of a die, where a success is defined as getting a number greater than 4.

Suppose that a 95% confidence interval states that population mean is greater than 100 and less than 300. How would you interpret this statement ?

Fill in the blanks : (a) t-distribution curve is symmetrical about the line _____. (b) The variable t of t-distribution lies between _____. (c) The mean of the t-distribution is _____. (d) The variance of the t-distribution is _____.

(a) Find the present value of a perpetuity of ₹ 4,200 payable at the beginning of each year, if money is worth 5% compounded annually. OR (b) Find the present value of a perpetuity of ₹ 5,000 payable at the end of each year, if money is worth 5% compounded annually.

(a) Prove that x+y x x; 5x+4y 4x 2x; 10x+8y 8x 3x = x³ OR (b) Prove that y+z z y; z z+x x; y x x+y = 4xyz

Find the inverse (if it exists) of the matrix A = 1 2 -2; -1 3 0; 0 -2 1 .

(a) Prove that the function f(x) = x² - x + 1 is neither strictly increasing nor strictly decreasing on the interval (-1, 1). OR (b) Find (dy)/(dx), if yx + xy + xx = ab.

(a) A person invested ₹ 5,000 in a fund for 5 years. The value of the investment was ₹ 4,800 at the end of the second year, ₹ 6,000 at the end of the third year, ₹ 6,700 at the end of the fourth year and on maturity, the final investment sold at ₹ 8,000. Find the CAGR. [Use (1·6)(1)/(5) = 1·098] OR (b) The annual depreciation of an asset is ₹ 50,000 and its scrap value after useful life of 10 years is ₹ 60,000. Find the original cost of the asset, using linear depreciation method.

(a) Find : ∫ (dx)/((x+1)² (x²+1)) OR (b) Solve the differential equation : (dy)/(dx) = ex-y + x² e-y

Minimise Z = 5x + 10y, subject to the constraints x + 2y ≤ 120 x + y ≥ 60 x - 2y ≥ 0 x, y ≥ 0

Solve for x : 1 ≤ |x - 2| ≤ 3

(a) A given rectangular area is to be fenced off in a field whose length lies along a straight river. If no fencing is needed along the river, show that the least length of fencing will be required when the length of the rectangular area is twice its breadth. OR (b) Solve the differential equation : x (dy)/(dx) + 2y = x² log x

(a) Fit a straight line trend by the method of least squares to the following data and find the trend values. | Year | 2010 | 2012 | 2013 | 2014 | 2015 | 2016 | 2019 | |---|---|---|---|---|---|---|---| | Sales (in lakh ₹) | 65 | 68 | 70 | 72 | 75 | 67 | 73 | OR (b) Find the trend values by taking 4-yearly moving averages for the following data. | Year | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | |---|---|---|---|---|---|---|---|---| | Sales (in thousand ₹) | 108 | 112 | 110 | 120 | 140 | 120 | 100 | 135 |

A machine costs a company ₹ 52,000 and its effective life is estimated to be 25 years. A sinking fund is created for replacing the machine by a new model at the end of its life time, when its scrap realizes a sum of ₹ 2,500 only. The price of the new model is estimated to be 25% more than the price of the present one. Find what amount should be set aside at the end of each year out of the profits for the sinking fund, if it accumulates at 3·5\% per annum compound. [Given (1·035)²⁵ = 2·3632]

According to an educational board survey, it was observed that class XII students apply at least one to four weeks ahead of college application deadlines. Let X represent the week when an average student applies ahead of a college’s application deadline and the probability of the student to get admission in the college P(X = x) is given as follows : P(X = x) = (kx)/(6), when x = 0, 1 or 2; [2mm]((1-k)x)/(6), when x = 3; [2mm](kx)/(2), when x = 4; [2mm]0, when x > 4 where k is a real number. Based on the above information, answer the following questions : (i) Determine the value of k. (ii) What is the probability that Mahesh will get admission in the college, given that he applied at least 3 weeks ahead of application deadline ? (iii) (a) Calculate the mathematical expectation of number of weeks taken by a student to apply ahead of a college’s application deadline. OR (iii) (b) To promote early admissions, the college is offering scholarships to the students for applying ahead of deadline as follows : ₹ 50,000 for applying 4 weeks ahead ₹ 20,000 for applying 3 weeks ahead ₹ 12,000 for applying 2 weeks ahead and ₹ 9,600 for applying 1 week ahead Determine the expected scholarship offered by the college.

The feasible region for an LPP is shown in the graph given below : The graph shows the constraint line CD through C(0, 6) and D(12, 0), the line EF through E(0, 4) and F(5, 0), and the line through A(0, 12) and D(12, 0); the shaded feasible region is the quadrilateral with vertices C(0, 6), the interior intersection X, B (on the x-axis near x=6) and E(0, 4). Based on the above information, answer the following questions : (i) Determine the equation of CD. (ii) Determine the equation EF. (iii) (a) Determine all the constraints for the LPP. OR (iii) (b) Find the maximum value of the objective function Z = 600x + 400y.

A man took a home loan of ₹ 40,00,000 from a bank at the interest of 6·75\% per annum compounded monthly which is to be amortized by equal payments at the end of each month for 10 years. Based on the above information, answer the following questions : (i) Find the monthly instalment. [Use (1·005625)⁻¹²⁰ = 0·510120] (ii) Find the principal outstanding at the beginning of 61st month. [Use (1·005625)⁶⁰ = 1·400115] (iii) (a) Find the interest amount paid in the 61st instalment. OR (iii) (b) Find the principal amount paid in the 61st instalment.