Mathematics Official Board Paper 2025-2026 ISC (Commerce) Class 12 Question Paper Solution

If A IS a square matrix of order 3 and its determinant is |A| = −3, then the value of |−4A| is ______.

202

192

−212

−192

Consider the function ‘f’ given by f(x) log x, x > 0, then the function ‘f’ is ______.

differentiable and continuous at x = 1.

differentiable but not continuous x = 1.

continuous but not differentiable at x = 1.

Neither differentiable nor continuous at x = 1.

If events A and B are mutually exclusive, such that P(A) = `1/5` and P(B) = 2/3, then the value of P(A∪B) is ______.

`11/15`

`3/15`

`14/15`

`13/15`

Assertion: f(x) = `{(1 + x",", x ≤ 2), (5 - x",", x > 2):}` at x = 2 is not differentiable.

Reason: A function is said to be differentiable at x = a if Left hand derivative is equal to Right hand derivative.e., Lf'(a) = Rf'(a).

Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.

Assertion is true and Reason is false.

Assertion is false and Reason is true.

The value of `∫_0^(3"/"2)`|x| dx is ______.

`1/8`

`9/8`

`9/4`

`3/4`

Statement 1: If 0 < x < then the value of tan⁻¹ (cot x) = `x/2 − x`

Statement 2: tan⁻¹(tan x) = x, ∀ x ∈ R

Statement 1 is true and Statement 2 is false.

Statement 2 is true and Statement 1 is false.

Both the staterments are true.

Both the statements are false.

How many possible matrices can be formed of order 3 × 3 if each entry is either 0 or 1?

64

256

512

216

Observe the graph given below and answer the question that follows.

Statement 1: f(x) increases in (−∞,-1) and (1, ∞)

Statement 2: f(x) decreases in (−∞, 0) and (1, ∞)

Which one of the following is correct?

Statement 1 is true and Statement 2 is false.

Statement 2 is true and Statement 1 is false.

Both the statements are true.

Both the statements are false.

If set A contains four elements and set B contains five elements, then the number of one-one and onto mapping from A → B is ______.

120

0

720

20

The solution of `dy/dx` − y = 1, y(0) = 1 is given by ______.

y = −e^x + 1

y = −e ^{− x − 1}

y = −1 + ex

y = 2e^х − 1

Assertion: The system of three linear equations in three unknown variables can be written in the matrix form as AX = B. It has a unique solution X = A⁻¹B.

Reason: Matrix A is non-singular.

Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.

Assertion is true and Reason is false.

Assertion is false and Reason is true.

If `x = e^(y + e^(y + e^(y + ...∞)))` x > 0 then find `dy/dx`.

Solve for x: `|(1,−2,5 ),(2,x,−1),(0,4,2x)|` = 86

Find the principal value of `sec^-1 (- sqrt(2))`.

A relation R on the set A − {a, b, c} is defined by R = {(a, b), (b, a)}. Is the relation R symmetric? Justify.

Using properties of the determinant, show that:

`|(b-c,c-a,a-b),(c-a,a-b,b-c),(2(a-b), 2(b-c), 2(c-a))|`

Let f(x) = 4 − (x − 7)³ be an invertible function, then find f⁻¹(x).

Find the range of the function f(x) = `1/(3−2 sinx)`

Evaluate:

`∫ e^x/(1 + e^(2x))dx`

A die marked 1, 2, 3 in red and 4, 5, 6 in green is thrown. Let A be the event ‘Number appearing is odd’ and B be the event ‘Number appearing is green’.

Prove that the events A and B are not independent.

The surface of a spherical balloon is increasing at the rate of 4 cm²/seс. Find the rate of change of volume when its radius is 12 cm.

Find the equation of the normal at (1, 2) to the curve x² = 4y.

If tan⁻¹ `((sqrt(1 + x^2) − sqrt(1 − x^2))/(sqrt(1 + x^2) + sqrt(1 − x^2)))` = α, prove that sin 2 α = x².

Solve for x: 2 tan⁻¹`(1/3)` + sec⁻¹`((5sqrt2)/7)` = tan ⁻¹ x

If y = `x^3 log(1/x)`, then prove that `x(d^(2)y)/(dx^2) − 2 dy/dx + 3x^2 = 0`

A sports store owner conducts a game ‘weekend-surprise’ every Friday for his customers.

He fills two bags with cricket balls of red and white colours. The first bag has 4 white and 4 red balls, while the second bag contains 3 white and 5 red balls. The rules of the game are:

Children of a society practise building human pyramids for 16 days to participate in the pyramid building competition during the Janmashtami festival. During practice sessions, the number of pyramids successfully formed in a day are X = 0, 1, 2, 3, 4. The data of the practice sessions is given in the following table:

1. Find the number of days on which the children made 3 pyramids.
2. Form a probability distribution table for the number of pyramids made per day. Verify if it is a valid probability distribution table.
3. Calculate the average number of pyramids formed.

A van is carrying a large amount of money in cash to deposit it in two ATM machincs on a hill station. The location of these machines is at the turning points of the path traced by the van, given by the equation h(x) = 2x³ − 18x² + 48x + 3, (x ≥ 0) where h(x) is the height of the hill (in 100 m) at any point.x.

1. Prove that the van is at the height of 300 m when it starts moving.   [1]
2. Find the location of the two ATM machines.   [2]
3. Calculate the difference between the heights of the location of the two ATM machines.   [1]
4. If the difference in the height of the location of the two ATM machines is greater than 1 km, then an extra armed security guard will be required. Based on the difference calculated in subpart (iii), determine if an extra armed guard will be required to protect the van.   [1]
5. Find the absolute maxima and absolute minima for h(x) in [0,4].    [1]

Evaluate:

`∫(sin x  dx)/(cos x (1 − sin x))`

Using properties of the definite integral, calculate the value of: 

`∫_0^(x/2) sin^2 x/(1 + sinx cos x) dx`

Solve the following differential equation:

x²dy + (xy + y²) dx = 0

Find the particular solution for the following differential equation:

`sqrt(1 − y^2) dx = (sin^(−1) y − x)dy,` given that y(0) = 0

A scalar is multiplied by a unit vector, then the resultant is:

Statement 1: A vector with the magnitude of the scalar.

Statement 2: A vector with unit magnitude.

Statement 1 is true and Statement 2 is false.

Statement 2 is true and Statement 1 is false.

Both the statements are true.

Both the statements are false.

The projection of `hati + 2hatj − 3hatk` on 2`hati − hatj + hatk` is ______.

`sqrt3/sqrt2`

`(−sqrt3)/sqrt2`

`(−3)/2`

`(−sqrt3)/2`

The direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4) are ______.

`(3/sqrt11, 1/sqrt11, 1/sqrt11)`

`(1/sqrt11, 1/sqrt11, 1/sqrt11)`

`(3/sqrt11, (−1)/sqrt11, (−1)/sqrt11)`

`((−3)/sqrt11, 1/sqrt11, (−1)/sqrt11)`

Find the area of a parallelogram whose adjacent sides are given by the vectors:

`veca = veci − vecj + 3veck and vecb = 2veci − 7vecj + 4veck`

Three drone cameras A, B and C are recording a hockey match. Their positions with respect to a control tower are given by the following coordinates:

A(1, 4, 6), B(3, 4, 5) and C(5, 4, 4).

Using vector method, show that the drone cameras A, B and C are moving in a straight path while recording the match.

If `veca` and `vecb` are mutually perpendicular vectors, `|veca + b|` = 13 and `|vecd|` = 5, then find the value of `|vecb|`.

The paths traced by two hot air balloons are: 

`(x − 1)/2 = (y − b)/3 = (z − 3)/4 and (x − 4)/5 = (y − 1)/2 = z/1`

Find the value of ‘b’ to be avoided so that the two hot air balloons do not collide.

A school is preparing the stage for its annual day function. They want to place a hanging mic and a hanging light on the stage.

• They decide to position the mic at the point (3,2,1) such that it is equidistant from a plain backdrop and the hanging light, as shown below.
  
• The equation of the surface of the plain backdrop is 2x − y + z + 1 = 0

(a) Find the distance between the mic and the plain backdrop.

(b) Calculate the coordinates of the position of the hanging light.

Find the area of the region bounded by y = `sqrt(4 − x^2)` and x axis using integration.

Statement 1: Two regression coefficients cannot have the same sign.

Statement 2: Both the regression coefficients can be numerically greater than unity.

Which one of the following is correct?

Statement 1 is true and Statement 2 is false.

Statement 2 is true, and Statement 1 is false.

Both the statements are true.

Both the statements are false.

Which one of the following statements is true about Marginal Revenue?

It is always constant for all firms.

It is always equal to the average revenue.

It is the revenue gained from decreasing output by 1 unit.

MR at x = a is the additional revenue obtained by increasing the output from a to a + 1.

The cost of manufacturing x units of a commodity is 27 + 12x + 3x².

Find the output for which Average Cost is decreasing.

For two variables x and y, if σx = 5, r = `(−1)/2, byx = (−2)/7` then find the value of σy.

The total cost function for production and marketing of a product is given by C(x) = `(3x^2)/4 − 7x + 3,` where x is the number of units produced.

Find the level of output (number of units produced) for which MC = AC.

A company produces a commodity with 36,000 as a fixed cost. The variable cost is estimated to be 25% of the total revenue earned. The selling price of the product is 20 per unit.

Find the following:

1. Cost function
2. Profit function

A school is organising an art and craft exhibition. The management has decided to donate the profit earned from the sale of exhibition items to an NGO.

• Total cost function for organising the exhibition is: `C(x) = −x^2 + 11x + 50`
• Each item is sold for 6.

Find the condition for the number of items to be sold to earn profit.

Consider the following data of a bivariate distribution:

• The mean of the variablesx and y are 25 and 30 respectively.
• The regression coefficient of x on y is 0.4 and the regression coefficient of y on x is 1.6.

(a) Find the lines of best fit for the bivariate distribution.   [2]

(b) Estimate the value of y when x = 60.   [1]

(c) What is the coefficient of correlation between x and y?   [1]

If the regression lines of a bivariate distribution are 4x − 5y + 33 = 0 and 20x − 9y − 107 = 0, then

1. Calculate the arithmetic mean of x and y.
2. Estimate the value of x when y = 7.
3. Find the variance of y when σx = 3.

Raunak is a small-scale entrepreneur who sells sewing machines in a rural market. He wants to expand his business but has two main constraints: Capital and Storage.

He has a total capital of ₹ 5,760 to invest. The godown can store a maximum number of 20 sewing machines.

Raunak sells two types of machines:

• Electronic sewing machine, each costs him 360.
• Manually operated sewing machine, each costs him 240.

His wife Radhika suggests selling an electronic machine ata profit of ₹ 22 and a manually operated sewing machine at a profit of 18.

Using the concept of Linear Programming Problem, find the number of sewing machines of each type that Raunak should sell to maximise his profit