Mathematics and Statistics Board Question Paper 2025-2026 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

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)`

if y = log `((e^x)/(x^2)) then dy/dx` = ______.

`(2 − x)/x`

`(x − 2)/x`

`(e − x)/e^x`

`(x − e)/(ex)`

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.

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

The solution of `dy/ dx` = 1 is ______

x + y = c

xy = c

x² + y² = c

y − x = c

If `∫_0^a 3x^2 dx = 8` then a = ______.

2

0

`8/3`

1

State whether the following statement is true or false:

If y = 20 + 15x + x² then `dx/dy = 1/(15 + 2x)`

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.

State whether the following statement is true or false:

Order and degree of a differential equation are always positive integers.

True

False

Statement p ↔ q is true, when p and q have ______ truth values.

If y = `"e"^"ax"`, then `"x" * "dy"/"dx" =`______.

If f(x) = x. log x then its minimum value is ______.

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(p ∧ ~ q) → (~ p ∧ ~ q)

If e^x + e^y = e^x+y, then show that `"dy"/"dx" = -e^(y - x)`.

Evaluate: `∫(1 + x)/"x + e"^(- x) dx`

Evaluate: `∫(1 + x)/"x + e"^(- x) dx`

The consumption expenditure E_c of a person with the income x. is given by E_c = 0.0006x² + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.

Evaluate the following integrals : `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx`

If the demand function is D = 50 – 3p – p². Find the elasticity of demand at p = 5 comment on the result.

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement:

p ↔ ~ q

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

~ (p ∨ q)

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

q → p

Evaluate the following.

`int 1/(4"x"^2 - 20"x" + 17)` dx

Solve the following differential equation:

(x² − y²)dx + 2xy dy = 0

Find the area of the region bounded by the curve x² = 16y and the line y = 4.

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

Obtain the differential equation from the relation Ax² + By² = 1, where A and Bare constants.

The difference between face value and present worth is called ______.

Banker’s discount

True discount

Banker’s gain

Cash value

Insurance companies collect a fixed amount from their customers at a fixed interval of time. This amount is called ______.

EMI

Installment

Contribution

Premium

If b_yx > 1 then b_xy is ______.

> 1

< 0

= 0

< 1

Which of the following can’t be a component of a time series?

Seasonality

Cyclical

Trend

Mean

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`

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)

State whether the following statement is True or False.

The banker’s discount is always lower than the true discount.

True

False

The cost of living Index Number using weighted relative method is given by `(∑IW)/(∑W).`

To convert an assignment problem into a minimization problem, the smallest element in the matrix is deducted from all other elements. 

The difference between the banker’s discount and the true discount is called ______.

If P₀₁ (L) = 225, P₀₁ (P) =144 then P₀₁(F) = ______.

A dish washing machine holds up to 40 pieces of large crockery (x). This constraint is given by ______.

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.

Given the following information about the production (X) and demand (Y) of a commodity, obtain the regression line of X on Y.

Coefficient of correlation between X and Y is 0.6. (3) Also, estimate the production when demand is 100.

The following table shows the production of pig-iron and ferro alloys ('000 metric tons)

Find trend values for the above data by using 5 yearly moving averages.

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.

Solve the following assignment problem for minimization:

The probability distribution of a discrete r.v. X is as follows:

1. Determine the value of k.
2. Find P(X ≤ 4)
3. P(2 < X < 4)
4. P(X ≥ 3)

The true discount on a sum is `3/8` of the sum due at 12% p.a. Find the period of the bill.

Solve the following problem:

Following data shows the number of boxes of cereal sold in years 1977 to 1984.

Fit a trend line to the above data by graphical method.

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),

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),

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)

For the following bivariate data obtain the equations of two regression lines:

Calculate (a) Laspeyre’s and (b) Paasche's Price Index Numbers for the following data:

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.

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)