HSC Commerce (English Medium)
HSC Commerce: Marketing and Salesmanship
HSC Commerce (Marathi Medium)
Academic Year: 2024-2025
Date & Time: 3rd July 2025, 11:00 am
Duration: 3h
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General Instructions:
- All questions are compulsory.
- There are six questions divided into two sections.
- Write answers of Section-I and Section-II in the same answer book.
- Use of logarithmic tables is allowed. Use of calculator is not allowed.
- For L.P.P. and Time Series graph paper is not necessary. Only rough sketch of graph is expected.
- Start answer to each question on a new page.
- For each objective type of question (i.e. Q.1 and Q.4) only the first attempt will be considered for evaluation.
Martix B = `[(0, 3, 1),(-3, 0, -4),("P", 4, 0)]` is skew-symmetric then value of P is ______.
1
–1
0
–3
Chapter:
If y = elog x then `dy/dx` = ______.
`1/x`
`1/2`
`e^(log x)/x`
0
Chapter:
`int (1 - x)^-3 dx` = ______.
`1/2 (1 - x)^-2 + c`
`1/2 (1 + x)^-2 + c`
`1/2 (1 - x)^-2 + x/2 + c`
`1/2 (1 - x)^-2 - x/2 + c`
Chapter:
If `int_0^a 3x^2 dx = 8` then a = ______.
0
2
8
`4/3`
Chapter:
Area of the region bounded by the curve y = x2, the X-axis and the lines x = 1 and x = 3 is ______.
`3/26` sq. units
3 sq. units
26 sq. units
`26/3` sq. units
Chapter:
The order and degree of the differential equation `((d^2y)/(dx^2))^2 + ((dy)/(dx))^2 = a^x` are ______ respectively.
1, 1
1, 2
2, 2
2, 1
Chapter:
For `int (x - 1)/(x - 1)^3 e^ x dx = e^x * f(x) + c`, where f(x) = (x + 1)2.
Chapter:
The integrating factor (I.F.) of `dy/dx + y = e^-x` is ex.
Chapter:
Negation of “Some men are animal” is ______.
Chapter: [1] Mathematical Logic
If the average revenue is 45 and clasticity of demand is 3 then marginal revenue is ______.
Chapter:
To find the value of `int (10x^9 + 10^x * log 10)/(10^x + x^10) dx`, the proper substitution is ______.
Chapter:
Determine whether the following statement pattern is a tautology, contradiction, or contingency.
[(p ∧ q) ∨ (~p)] ∨ [p ∧ (~ q)]
Chapter: [1] Mathematical Logic
Find the area of the region bounded by the parabola y2 = 4x and the line x = 3.
Chapter: [7] Applications of Definite Integration
Find MPC, MPS, APC and APS, if the expenditure Ec of a person with income I is given as Ec = (0.0003) I2 + (0.075) I ; When I = 1000.
Chapter: [4] Applications of Derivatives
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Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Chapter: [8] Differential Equation and Applications
Express the following equations in matrix form and solve them by the method of reduction:
x + 2y + z = 8, 2x + 3y - z = 11, 3x - y - 2z = 5.
Chapter:
Write the converse, inverse, contrapositive of the following statement.
If a man is bachelor, then he is happy.
Chapter: [1] Mathematical Logic
Find the inverse of `[(3, 1, 5),(2, 7, 8),(1, 2, 5)]` by adjoint method.
Chapter:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Chapter: [3] Differentiation
Evaluate the following.
`int "x"^2 *"e"^"3x"`dx
Chapter: [5] Integration
Evaluate: `int_1^4 (root(3)(x + 6))/(root(3)(x + 6) + root(3)(11 - x))*dx`
Chapter:
Find the values of x, such that f(x) is increasing function
f(x) = 2x3 – 15x – 144x – 7
Solution:
Given: f(x) = 2x3 – 15x2 – 144 – 7
∴ f′(x) = 6x2 – 30x – 144
Now, f′(x) > 0, as f(x) is increasing
∴ 6x2 – 30x – 144 = 0
∴ x2 – 5x – 24 = 0
∴ (x – 8)(x + 3) > 0
Case (I) x – 8 > 0 and x + 3 > 0
x > 8 and x > –3
∴ x > `square`
Case (II) x – 8 < 0 and x + 3 < 0
x < 8 and x < –3
∴ x < `square`
∴ f(x) = 2x3 – 15x2 – 144 – 7 is increasing if and only if x ∈ (–∞, `square`) or x ∈ (`square`, ∞)
Chapter:
Solve the following differential equation, hence find the particular solution when x = 0, y = 1
`y^3-dy/dx=xdy/dx`
Solution:
`y^3=xdy/dx+dy/dx`
∴ y3 = (x + 1)·`square`
∴ (x + 1)dy = y3 dx
Separating the variables, we get
`1/y^3dy=1/(x+1)dx`
Now integrating, we get
∴ `1/y^3dy=1/(x+1)dx`
∴ `-1/(2y^2)=square+c` ...(i)
which is required general solution
put x = 0 and y = 1 in (i)
`-1/(2(1)^2)=log|0 + 1| + c`
∴ `square=c`
∴ `-1/(2y^2)=square-1/2`
is the particular solution.
Chapter:
The sum due is also called as ______.
Face value
Present value
Cash value
True discount
Chapter:
Following are different types of insurance.
- Life insurance
- Health insurance
- Liability insurance
Only I
Only II
Only III
All the three
Chapter:
|bxy + byx | ≥ ______.
|r|
2 |r|
r
2r
Chapter: [11] Linear Regression
If P01(L) = 120.4 and P01(P) = 130.6 then P01 (D – B) is ______.
25.1
60.2
125.5
65.3
Chapter:
F(x) is c.d.f. of discrete r.v. X whose distribution is
| Xi | – 2 | – 1 | 0 | 1 | 2 |
| Pi | 0.2 | 0.3 | 0.15 | 0.25 | 0.1 |
Then F(– 3) = ______.
0
1
0.2
0.15
Chapter:
Given p.d.f. of a continuous r.v.X as
f(x) = `x^2/(3)` for –1 < x < 2
= 0, otherwise
then F(1) = _______.
`(1)/(9)`
`(2)/(9)`
`(3)/(9)`
`(4)/(9)`
Chapter:
State whether the following statement is True or False.
The bankers discount is also called as commercial discount.
True
False
Chapter:
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The optimum value of the objective function of LPP occurs at the center of the feasible region.
True
False
Chapter: [14] Linear Programming
State whether the following statement is True or False.
If E(X) > Var (X) then X follows Binomial distribution.
Chapter:
The region represented by the inequality y ≤ 0 lies in _______ quadrants.
Chapter: [14] Linear Programming
The time required for printing of four books A, B, C and D is 5, 8, 10 and 7 hours while its data entry requires 7, 4, 3 and 6 hrs respectively. The sequence that minimizes total elapsed time is ______.
Chapter: [15] Assignment Problem and Sequencing
If X has Poisson distribution with parameter m and P(X = 3) = P(X = 4) then m = ____.
Chapter:
A person wants to create a fund of ₹ 6,96,150 after 4 years at the time of his retirement. He decides to invest a fixed amount at the end of every year in a bank that offers him interest of 10% p.a. compounded annually. What amount should he invest every year? [Given (1.1)4 = 1.4641]
Chapter: [10] Insurance and Annuity
Following are given information about advertising expenditure and sales.
| Advertisement expenditure (₹ in lakh) (X) |
Sales (₹ in lakh) (Y) |
|
| Arithmetic mean | 10 | 90 |
| Standard deviation | 3 | 12 |
Correlation coefficient between X and Y is 0.8.
- Obtain the regression of Y on X
- What is the likely sales when the advertising budget is ₹ 15 lakh?
Chapter:
Base year weights (W) and current year price relatives (I) are given in Problem. Calculate the cost of living index in:
Find y if the cost of living index is 200.
| Group | Food | Clothing | Fuel & Lighting | House Rent | Miscellaneous |
| I | 180 | 120 | 160 | 300 | 200 |
| W | 4 | 5 | 3 | y | 2 |
Chapter: [13] Index Numbers
A bill of ₹ 5,475 drawn on 19th January 2015 for 8 months was discounted on 28th February 2015 at 8% p.a. interest. What is the banker’s discount? What is the cash value of the bill?
Chapter:
Following table shows that all India Infant Mortality Rates (per '000) for years 1980 to 1986:
| Years | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 |
| IMR | 10 | 6 | 5 | 3 | 3 | 1 | 0 |
Fit a trend line to the above data by the method of least squares.
Chapter:
Four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost matrix is given below:
| Machines | Places | ||||
| A | B | C | D | E | |
| M1 | 4 | 6 | 10 | 5 | 6 |
| M2 | 7 | 4 | – | 5 | 4 |
| M3 | – | 6 | 9 | 6 | 2 |
| M4 | 9 | 3 | 7 | 2 | 3 |
Find the optimal assignment schedule.
Chapter: [15] Assignment Problem and Sequencing
Obtain 4 yearly centered moving averages for the following time series:
| Years | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 | 1987 | 1988 | 1989 | 1990 | 1991 |
| Number of crime ('000) | 40 | 42 | 43 | 42 | 44 | 44 | 43 | 46 | 47 | 45 | 46 |
Chapter:
Calculate Walsh’s Price Index Number for the following data:
| Commodity | Base Year | Current year | ||
| Price | Quantity | Price | Quantity | |
| L | 4 | 8 | 3 | 2 |
| M | 6 | 16 | 8 | 9 |
| N | 8 | 18 | 7 | 32 |
Chapter:
A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes.
Chapter:
The equations of two regression lines are 8x – 10y + 66 = 0 and 40x – 18y = 214. Find
- The mean values of X and Y
- Correlation coefficient between X and Y
Chapter:
Maximize Z = 60x + 50y Subject to x + 2y ≤ 40, 3x + 2y ≤ 60, x ≥ 0, y ≥ 0
Chapter: [14] Linear Programming
Find the sequence that minimizes the total elapsed time to complete the following jobs in the order AB. Find the total elapsed time and idle times for both the machines.
| Job | I | II | III | IV | V | VI | VII |
| Machine A | 7 | 16 | 19 | 10 | 14 | 15 | 5 |
| Machine B | 12 | 14 | 14 | 10 | 16 | 5 | 7 |
Solution:
Using the optimal sequence algorithm, the following optimal sequence can be obtained.
| `square` | `square` | IV | V | III | `square` | `square` |
Total elapsed time is obtained as follows
| Job | Machine A | Machine B | ||
| Time In | Time Out | Time In | Time Out | |
| `square` | 0 | 5 | 5 | 12 |
| `square` | 5 | 12 | 12 | 24 |
| IV | 12 | 22 | 24 | 34 |
| V | 22 | 36 | 36 | 52 |
| III | 36 | 55 | 55 | 69 |
| `square` | 55 | 71 | 71 | 85 |
| `square` | 71 | 86 | 86 | 91 |
∴ Total elapsed time T = 91 units
Idle time for machine A = `square` units
Idle time for machine B = `square` units
Chapter:
The probability distribution of X is as follows:
| x | 0 | 1 | 2 | 3 | 4 |
| P[X = x] | 0.1 | k | 2k | 2k | k |
Find:
- k
- P(X > 2)
- P(1 < X < 4)
Solution:
The table gives a probability distribution and therefore
∑P(X = x) = 1
P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1
0.1 + k + 2k + 2k + k = 1
6k = 1 – 0.1
6k = 0.9
k = `square`
P(X > 2) = P(X = 3) + P(X = 4)
= 2k + k
= `square`
P(1 < X ≤ 4 = PX = `square`) + P(X = 3) + PX = 4)
`square` + 2k + k
= 0.75
Chapter:
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Maharashtra State Board previous year question papers 12th Standard Board Exam Mathematics and Statistics with solutions 2024 - 2025
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