CBSE Class 12 Applied Mathematics 2025 — Set 465/W1XZY/4

The rate of change of population P(t) with respect to time (t), where α, β are the constant birth and death rates, respectively, is (A) (dP)/(dt) = (α + β)P (B) (dP)/(dt) = (α - β)P (C) (dP)/(dt) = (α + β)/(P) (D) (dP)/(dt) = (α - β)/(P)

A pair of dice is thrown two times. If X represents the number of doublets obtained, then the expectation of X is (A) (1)/(6) (B) 1 (C) (1)/(3) (D) (11)/(36)

The mean of t-distribution is (A) 0 (B) 1 (C) 2 (D) not defined

The variations which occur due to change in climate, festivals or weather conditions are known as (A) secular variations (B) cyclic variations (C) seasonal variations (D) irregular variations

In a LPP, the maximum value of z = 3x + 4y subject to the constraints x + y ≤ 40, x + 2y ≤ 60, x, y ≥ 0 is (A) 120 (B) 140 (C) 150 (D) 130

The present value of a sequence of payments of ₹ 100 made at the end of every year and continuing forever, if the money is worth 5% compounded annually, is (A) ₹ 2,000 (B) ₹ 20,000 (C) ₹ 5,000 (D) ₹ 12,000

The demand function of a monopolist is given by p = 30 + 5x - 3x², where x is the number of units demanded and p is the price per unit. The marginal revenue when 2 units are sold, is (A) ₹ 28 (B) ₹ 23 (C) ₹ 1 (D) ₹ 14

If the cost function and revenue function of x items are respectively given as C(x) = 100 + 0.015x², R(x) = 3x, then the value of x for maximum profit is (A) 50 (B) 100 (C) 150 (D) 200

If a random variable X has the probability distribution P(X = x) = k, if x = 0 \2k, if x = 1 or 2 \0, otherwise, then the value of k is (A) (1)/(3) (B) (1)/(5) (C) (1)/(6) (D) (1)/(4)

The test statistic t for testing the significance of differences between the means of two independent samples is given by (A) t = dfracbarx - bary√(s) (B) t = dfracbarx - barys√((1)/(n₁) + (1)/(n₂)) (C) t = dfracbarx - baryfracs√(n-1) (D) t = dfracbarx + barys√((1)/(n₁) - (1)/(n₂))

The effective rate of interest equivalent to a nominal rate of 4% compounded semi-annually, is (A) 4.12\% (B) 4.04\% (C) 4.08\% (D) 4.14\%

The CAGR of an investment, whose starting value is ₹ 5,000 and it grows to ₹ 25,000 in 4 years, is : [Given (5)0.25 = 1.4953] (A) 49.53\% (B) 14.95\% (C) 495.3\% (D) 1.49\%

**Assertion (A) :** The area of the region bounded by the line y - 1 = x, the x-axis and the ordinates x = -1 and x = 1 is 2 square units. **Reason (R) :** The area of the region bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b is given by ∫ab f(x) dx. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

**Assertion (A) :** The differential equation representing the family of curves y = mx, m being an arbitrary constant, is x(dy)/(dx) - y = 0. **Reason (R) :** For a family of curves, the differential equation is obtained by differentiating the equation of family of curves with respect to x and then eliminating the arbitrary constant, if any. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

(a) The cost of Type I sugar is ₹ 25 per kg and Type II sugar is ₹ 35 per kg. If both Type I sugar and Type II sugar are mixed in the ratio 3:2, find the price per kg of the mixture. OR (b) Pipe A can fill a tank in 1 hour and Pipe B can fill it in 1(1)/(2) hours. If both the pipes are opened in the empty tank, how much time will they take to fill the tank ?

A boat goes 3.5 km upstream and then returns. Total time taken is 1 hour and 12 minutes. If the speed of the current is 1 km/h, then find the speed of the boat in still water.

A runs (3)/(2) times as fast as B. If A gives B a start of 40 m, how far must the winning post from the starting point be, so that A and B reach at the same time ?

Given A = 2 0 1 \3 4 5 \0 2 3 and B = 1 1 -5 \-5 1 -5 \1 -2 4 , find BA.

(a) If a fair coin is tossed 6 times, find the probability of getting atleast 4 heads. OR (b) Given that mean of a normal variate X is 9 and standard deviation is 3, then find : (i) the z-score of the data point 15 (ii) the data point if its z-score is 4.

Find the units digit in 7²⁹⁵.

Two numbers are selected at random (without replacement) from first six positive integers. Let X denotes the smaller of the two numbers obtained. Calculate the mathematical expectation of X.

(a) If the mean and variance of a binomial distribution are (4)/(3) and (8)/(9) respectively, then find P(x = 1). OR (b) The mortality rate for a certain disease is 0.007. Using Poisson distribution, calculate the probability for 2 deaths in a group of 400 people. [Use e-2.8 = 0.0608]

(a) There are two types of fertilizers F₁ and F₂. F₁ consists of 10% nitrogen and 6% phosphoric acid. F₂ consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that he needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for his crop. If F₁ costs ₹ 6 per kg and F₂ costs ₹ 5 per kg, how much of each type of fertilizer should be used so that the cost is minimum. Formulate a linear programming problem. OR (b) Solve the following linear programming problem graphically : Maximise z = 50x + 30y subject to 2x + y ≤ 18 3x + 2y ≤ 34 x, y ≥ 0.

A machinist is making engine parts with axle diameter of 0.7 cm. A random sample of 10 parts shows mean diameter 0.742 cm with a standard deviation of 0.04 cm. On the basis of this sample, find if you would say that the work is inferior. (Given t₉(0.05) = 2.262)

Calculate EMI under Flat-Rate System for a loan of ₹ 5,00,000 with 7.5% annual interest rate for 5 years.

(a) If A = 2 -3 5 \3 2 -4 \1 1 -2 , find A⁻¹ and hence solve the following system of linear equations : 2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y - 2z = -3 OR (b) Using properties of determinants, prove that Δ = (b+c)² a² a² b² (c+a)² b² c² c² (a+b)² = 2abc(a + b + c)³

If the supply function is p = 4 - 5x + x², then find the producer's surplus when price is 18.

(a) Compute the seasonal indices by 4-year moving averages from the given data of production of paper (in thousand tons) : | Year | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | |---|---|---|---|---|---|---|---|---|---|---| | Index number | 2450 | 1470 | 2150 | 1800 | 1210 | 1950 | 2300 | 2500 | 2480 | 2680 | OR (b) Fit a straight-line trend by method of least squares for the following data : | Year | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | |---|---|---|---|---|---|---| | Production (in tons) | 210 | 225 | 275 | 220 | 240 | 235 |

A machine costs ₹ 1,00,000 and its effective life is estimated to be 12 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 ₹ 5,000 only. 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 5% effective. [Use (1.05)¹² = 1.7958]

A man has an expensive square-shaped piece of golden board of side 36 cm. He wants to turn it into a box without top by cutting a square from each corner and folding the flaps. Let x cm be the side of square, which is cut from each corner. The figure shows a square sheet of side 36 cm with a small square of side x cm cut from each corner; folding up the four flaps forms an open box of height x. Based on the above information, answer the following questions : (i) Find the expression for the volume (V) of open box in terms of x. (ii) Find (dV)/(dx). (iii) Find the value of x for which the volume (V) is maximum. OR (iii) Find the maximum volume of the open box.

There are two factories located one at P and the other at Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 4, 4 and 6 units of the commodity while the production capacity of the factories at P and Q are 9 and 5 units respectively. The cost of transportation per unit is given as : | From / To | A | B | C | |---|---|---|---| | P | 160 | 100 | 150 | | Q | 100 | 120 | 100 | Based on the above information, answer the following questions : Let x units and y units of the commodity be transported from factory P to the depots at A and B respectively, then The flow network has Factory P (supply 9 units) at the top and Factory Q (supply 5 units) at the bottom, each supplying three outlets A (demand 4), B (demand 4) and C (demand 6); x units are sent from P to A and y units from P to B. (i) Find (in terms of x and y) how many units commodity be transported from factory P to depot C. (ii) Find how many units of commodity be transported from factory Q to A, B and C respectively. (iii) Using (i) and (ii), find the total transportation cost z. OR (iii) Using (i) and (ii), find the constraint inequalities for minimum cost z.

Ramesh borrowed a home loan amount of ₹ 7,00,000 from a bank at an interest of 12% per annum for 30 years, to be paid in monthly installments. Based on the above information, answer the following questions : (i) Write the formula for calculating EMI by reducing balance method. (ii) Write the values of P, i and n respectively. (iii) Find the EMI. [Use (1.01)⁻³⁶⁰ = 0.02781668] OR (iii) If the loan is to be returned in 20 years, find EMI. [Use (1.01)⁻²⁴⁰ = 0.09180584]