HSC Commerce (English Medium)
HSC Commerce: Marketing and Salesmanship
HSC Commerce (Marathi Medium)
Academic Year: 2025-2026
Date & Time: 21st February 2026, 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.
If y = `root 3 (a^2 + x^2)` then `dy/dx` = ______
`2/3x(a^2 + x^2)^((−2)/3)`
`2/3x(a^2 + x^2)^(2/3)`
`2/3(a^2 + x^2)^((−2)/3)`
`2/3(a^2 + x^2)^(2/3)`
Chapter:
if y = log `((e^x)/(x^2)) then dy/dx` = ______.
`(2 − x)/x`
`(x − 2)/x`
`(e − x)/e^x`
`(x − e)/(ex)`
Chapter:
U: the set of all real numbers.
Q: the set of all rational numbers.
I: the set of all integers.
The above Venn diagram represents the truth value of which of the following statements?
Some integers are rational numbers.
All integers are rational numbers.
No integers are rational numbers.
All rational numbers are integers.
Chapter:
The equation of normal to the curve y = 3x2 − x + 1 at (1, 3) is ______.
x − 5y − 16 = 0
x + 5y − 16 = 0
x − 5y + 16 = 0
−5y − x − 16 = 0
Chapter:
The solution of `dy/ dx` = 1 is ______
x + y = c
xy = c
x2 + y2 = c
y − x = c
Chapter: [8] Differential Equation and Applications
If `∫_0^a 3x^2 dx = 8` then a = ______.
2
0
`8/3`
1
Chapter:
State whether the following statement is true or false:
If y = 20 + 15x + x2 then `dx/dy = 1/(15 + 2x)`
Chapter:
State whether the following statement is true or false:
If `∫ x/((1 + x) (2 + x)) dx = ∫ (A/(1 + x) + B/(2 + x))dx` then A =1, B = 2.
Chapter:
State whether the following statement is true or false:
Order and degree of a differential equation are always positive integers.
True
False
Chapter: [8] Differential Equation and Applications
Statement p ↔ q is true, when p and q have ______ truth values.
Chapter:
If y = `"e"^"ax"`, then `"x" * "dy"/"dx" =`______.
Chapter: [3] Differentiation
If f(x) = x. log x then its minimum value is ______.
Chapter:
Examine whether the following statement pattern is a tautology, a contradiction or a contingency.
(p ∧ ~ q) → (~ p ∧ ~ q)
Chapter: [1] Mathematical Logic
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Chapter: [3] Differentiation
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Chapter: [4] Applications of Derivatives
Evaluate the following integrals : `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx`
Chapter:
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If the demand function is D = 50 – 3p – p2. Find the elasticity of demand at p = 5 comment on the result.
Chapter: [4] Applications of Derivatives
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement:
p ↔ ~ q
Chapter: [1] Mathematical Logic
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
~ (p ∨ q)
Chapter: [1] Mathematical Logic
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
q → p
Chapter: [1] Mathematical Logic
Evaluate the following.
`int 1/(4"x"^2 - 20"x" + 17)` dx
Chapter: [5] Integration
Solve the following differential equation:
(x2 − y2)dx + 2xy dy = 0
Chapter:
Find the area of the region bounded by the curve x2 = 16y and the line y = 4.
Chapter:
Express the following equations in matrix form and solve them by the method of reduction:
x − y + z = 1, 2x − y = 1, 3x + 3y − 4z = 2
Chapter:
Obtain the differential equation from the relation Ax2 + By2 = 1, where A and Bare constants.
Chapter:
The difference between face value and present worth is called ______.
Banker’s discount
True discount
Banker’s gain
Cash value
Chapter:
Insurance companies collect a fixed amount from their customers at a fixed interval of time. This amount is called ______.
EMI
Installment
Contribution
Premium
Chapter:
Which of the following can’t be a component of a time series?
Seasonality
Cyclical
Trend
Mean
Chapter: [12] Time Series
Price Index Number by Simple Aggregate Method is given by ______.
`∑(p_1)/(p_0) × 100`
`∑(p_0)/(p_1) × 100`
`(∑p_1)/(∑p_0) × 100`
`(∑p_0)/(∑p_1) × 100`
Chapter:
If the corner points of the feasible region are (0, 10), (2, 2) and (4, 0) then the point of minimum z = 3x + 2y is ______.
(2, 2)
(0, 10)
(4, 0)
(2, 4)
Chapter:
State whether the following statement is True or False.
The banker’s discount is always lower than the true discount.
True
False
Chapter:
The cost of living Index Number using weighted relative method is given by `(∑IW)/(∑W).`
Chapter:
To convert an assignment problem into a minimization problem, the smallest element in the matrix is deducted from all other elements.
Chapter:
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The difference between the banker’s discount and the true discount is called ______.
Chapter:
If P01 (L) = 225, P01 (P) =144 then P01(F) = ______.
Chapter:
A dish washing machine holds up to 40 pieces of large crockery (x). This constraint is given by ______.
Chapter:
A shop is valued at ₹3,60,000 for 75% of its value. If the rate of premium is 0.9%, find the premium paid by the owner of the shop. Also, find the agents commission if the agent gets commission at 15% of the premium.
Chapter:
Given the following information about the production (X) and demand (Y) of a commodity, obtain the regression line of X on Y.
| Production (X) | Demand (Y) | |
| Mean | 85 | 90 |
| S.D. | 5 | 6 |
Coefficient of correlation between X and Y is 0.6. (3) Also, estimate the production when demand is 100.
Chapter:
The following table shows the production of pig-iron and ferro alloys ('000 metric tons)
| Years | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
| Production | 0 | 4 | 9 | 9 | 8 | 5 | 4 | 8 | 10 |
Find trend values for the above data by using 5 yearly moving averages.
Chapter:
Solve the following L.P.P. by graphical method:
Minimize: z = 8x + 10y
Subject to: 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.
Chapter: [14] Linear Programming
Solve the following assignment problem for minimization:
| Men | Tasks (in hours) | |||
| I | II | III | IV | |
| A | 7 | 25 | 26 | 10 |
| B | 12 | 27 | 3 | 25 |
| C | 37 | 18 | 17 | 14 |
| D | 18 | 25 | 23 | 9 |
Chapter:
The probability distribution of a discrete r.v. X is as follows:
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | k | 2k | 3k | 4k | 5k | 6k |
- Determine the value of k.
- Find P(X ≤ 4)
- P(2 < X < 4)
- P(X ≥ 3)
Chapter:
The true discount on a sum is `3/8` of the sum due at 12% p.a. Find the period of the bill.
Chapter:
Solve the following problem:
Following data shows the number of boxes of cereal sold in years 1977 to 1984.
| Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 |
| No. of boxes in ten thousand | 1 | 0 | 3 | 8 | 10 | 4 | 5 | 8 |
Fit a trend line to the above data by graphical method.
Chapter: [12] Time Series
The following is the p.d.f. of a r.v. X.
f(x) = `{(x/(8), "for" 0 < x < 4),(0, "otherwise."):}`
Find P(x < 1.5),
Chapter:
The following is the p.d.f. of a r.v. X.
f(x) = `{(x/(8), "for" 0 < x < 4),(0, "otherwise."):}`
Find P(1 < x < 2),
Chapter:
The following is the p.d.f. of a r.v. X.
f(x) = `{(x/(8), "for" 0 < x < 4),(0, "otherwise."):}`
Find P(x > 2)
Chapter:
For the following bivariate data obtain the equations of two regression lines:
| X | 1 | 2 | 3 | 4 | 5 |
| Y | 5 | 7 | 9 | 11 | 13 |
Chapter: [11] Linear Regression
Calculate (a) Laspeyre’s and (b) Paasche's Price Index Numbers for the following data:
| Commodity | Base year | Current year | ||
| Price | Quantity | Price | Quantity | |
| P | 12 | 20 | 18 | 24 |
| Q | 14 | 12 | 21 | 16 |
| R | 8 | 10 | 12 | 18 |
| S | 16 | 15 | 20 | 25 |
Chapter:
Determine the optimal sequence of jobs that minimizes the total elapsed time for the data given below (processing time on machines is given in hours). Also find the total elapsed time and the idle time for three machines.
| Jobs | I | II | III | IV | V | VI | VII |
| Machine A | 3 | 8 | 7 | 4 | 9 | 8 | 7 |
| Machine B | 4 | 3 | 2 | 5 | 1 | 4 | 3 |
| Machine C | 6 | 7 | 5 | 11 | 5 | 6 | 12 |
Chapter:
In a town, 10 accidents take place in the span of 50 days. Assuming that the number of accidents follows Poisson distribution, find the probability that there will be 3 or more accidents on a day.
(Given that e-0.2 = 0.8187)
Chapter: [16] Probability Distributions
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