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

If A and B are square matrices of order 3, A is a non-singular matrix and AB = O, then the matrix B is ______.

unit matrix

scalar matrix

non-singular matrix

null matrix

If m and n are respectively the order and degree of the differential equation `d/dx (dy/dx)^3 = 0` then the value of (m – n) is ______.

0

1

2

3

The derivative of xy = c² with respect to x is ______.

`dy/dx = (2c - y)/x`

`dy/dx = c^2/x^2`

`dy/dx = (-y)/x`

`dy/dx = y/x`

Consider `Δ = |(2a, 2b, 2c),(2e, f, g),(2i, j, k)|`

Assertion: The value of `Δ = 4 xx |(a, b, c),(e, f, g),(i, j, k)|`

Reason: If all elements of one row or one column of a determinant are multiplied by a scalar, k then the value of the determinant is multiplied by k.

Which of the following is correct?

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.

Five numbers x₁, x₂, x₃, x₄, x₅ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order such that x₁ < x₂ < x₃ < x₄ < x₅. What is the probability that x₂ = 7 and x₄ = 11?

`26/51`

`3/104`

`1/68`

`1/34`

If `A = ((a, 0, 0),(0, a, 0),(0, 0, a))`, then Aⁿ equal to ______.

`((a^n, 0, 0),(0, a^n, 0),(0, 0, a^n))`

`((a, 0, 0),(0, a^n, 0),(0, 0, a))`

`((a^n, 0, 0),(0, a, 0),(0, 0, a^n))`

`((na, 0, 0),(0, na, 0),(0, 0, a^n))`

Observe the following graphs (a), (b), (c) and (d), each representing different types of functions.

(a)	(b)
(c)	(d)

Statement 1: A function which is continuous at a point may not be differentiable at that point.

Statement 2: Graph (c) is an example of a function that is continuous but not differentiable at the origin.

Which 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 f(x) = kx² + 7x – 4 and f′(5) = 97 then what is the value of k?

–4

0

4

9

Statement 1: If a relation R on a set A satisfies R = R^–1, then R is symmetric.

Statement 2: For a relation R to be symmetric, it is necessary that R = R^–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.

Assertion: The equality tan (cot^–1 x) = cot (tan^–1 x), is true for all x ∈ R.

Reason: The identity tan^–1 x + cot^–1 x = `π/2`, is true for all x ∈ R.

Which of the following is correct?

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.

Given two events A and B such that `P(A/B)` = 0.25 and P(A ∩ B) = 0.12. The value of P(A' ∩ B) is ______.

0.36

0.48

0.88

0.036

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of the determinant of C²?

If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.

The given function f: R → R is a many-to-one function. Give a reason.

There are three machines and 2 of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines?

If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.

Find the values of ‘a’ for which the function f(x) = b – ax + sin x is increasing on R.

Find a point on the curve y = (x − 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Show that the general solution of the differential equation:

`dy/dx` = y cot 2x is log y = `1/2 log|sin2x| + C`

If `int x^5 cos (x^6)dx = k sin (x^6) + C`, find the value of k.

Evaluate:

`int_0^5 sqrt(x)/(sqrt(5 - x) + sqrt(x)) dx`

A music streaming app uses the function f(x) = `tan^-1 (x/10)` to assign a mood score based on the number of hours a user listens to music per week. Let the listening times (in hours/week) of two users, User A and User B, be 6 and 8 respectively. Compute the combined mood score of user A and user B, that is, f(6) + f(8).

Solve:

sin^–1 (x) + sin^–1 (1 – x) = cos^–1 x

If y = (A + Bx)e^−2x, prove that `(d^2y)/(dx^2) + 4 dy/dx + 4y = 0`.

Solve the following differential equation:

`x dy/dx = y - x tan (y/x)`

Solve the following differential equation:

`(1 + x^2) dy/dx + 2xy = 4x^2`

Three friends go to a restaurant to have pizza. They decide who will pay for the pizza by tossing a coin. It is decided that each one of them will toss a coin and if one person gets a different result (heads or tails) than the other two, that person would pay. If all three get the same result (all heads or all tails), they will toss again until they get a different result.

1. What is the probability that all three friends will get the same result (all heads or all tails) in one round of tossing?
2. What is the probability that they will get a different result in one round of tossing?
3. What is the probability that they will need exactly four rounds of tossing to determine who would pay?

A school offers students the choice of three modes for attending classes:

• Mode A: Offline (in-person) – 40% of students 
• Mode B: Online (live virtual classes) – 35% of students 
• Mode C: Recorded lectures – 25% of students 

After a feedback survey:

• 20% of students from Mode A reported the class as “Excellent”.
• 30% from Mode B rated it as “Excellent”.
• 50% from Mode C rated it as “Excellent”.

A student is selected at random from the entire group, and it is found that they rated the class as “Excellent”.

(a) Represent the data in terms of probability. Define the events clearly.

(b) Using Bayes’ Theorem, find the probability that the student attended the Recorded lectures (Mode C), given that they rated the class as “Excellent”.

(c) Interpret your result. Which mode has the highest likelihood of being chosen if a student says “Excellent”?

Answer the following question:

1. Translate the problem into a system of equations.
2. Solve the system of equation by using matrix method.
3. Hence, find the cost of one paper bag, one scrap book and one pastel sheet.

Evaluate:

`int (x^2 + x + 1)/((x + 2)(x^2 + 1)) dx`

Evaluate:

`int_(π/4)^((3π)/4) (x)/(1 + sin x) dx`

A person has manufactured a water tank in the shape of a closed right circular cylinder. The volume of the cylinder is `539/2` cubic units. If the height and radius of the cylinder are h and r.

1. Express h in terms of radius r and given volume. 
2. Let the total surface area of the closed cylinder tank be S; express S in terms of radius r. 
3. If the total surface area of the tank is minimum, then prove that radius r = `7/2` units.
4. Find the height of the tank.

A Dolphin jumps and takes a path given by the equation h(t) = `1/2 (-7t^2 + 3t + 2)`, (t ≥ 0), where h(t) is the height of the Dolphin at any point in time.

1. Is the function differentiable for t ≥ 0? Justify.
2. Find the instantaneous rate of change of height at t = `1/14`.
3. h(t) is increasing in `(-∞, 3/14)`. Is this true or false? Justify.
4. Find the time at which the Dolphin will attain the maximum height. Also find the maximum height.

In a school, three subject teachers English, Math and Science sometimes give surprise tests on the same day. Based on past records:

• The English teacher gives a test 90% of the time.
• The Math teacher gives a test 80% of the time.
• The Science teacher gives a test 70% of the time.

Each teacher decides independently. If the average number of surprise tests is less than 2.3 then the teachers should coordinate better to increase the performance of the students. Otherwise, no action is needed.

Let X be the number of surprise tests a student gets on a given day. So, X ∈ {0, 1, 2, 3}.

1. Find the probability for each possible number of surprise tests. 
2. Use the probabilities to build a distribution table. 
3. Calculate the average number of surprise tests per day. 
4. Based on your calculations, decide: Should the teachers coordinate better? Or is the current plan acceptable?

Consider the following statements and choose the correct option:

Statement 1: If `veca` and `vecb` represent two adjacent sides of a parallelogram, then the diagonals are represented by `veca + vecb` and `veca - vecb`.

Statement 2: If `veca` and `vecb` represent two diagonals of a parallelogram, then the adjacent sides are represented by `2(veca + vecb)` and `2(veca - vecb)`.

Which 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.

A plane passes through three points A, B and C with position vectors `hati + hatj, hatj + hatk` and `hatk + hati` respectively. The equation of the line passing through the point P with position vector `hati + 2hatj + 2hatk` and normal to the plane is ______.

`vecr = (hati + 2hatj + 2hatk) + λ(hati + hatj + hatk), λ ∈ R`

`vecr = (hati + hatj + hatk) + λ(hati + 2hatj + 2hatk), λ ∈ R`

`vecr * (hati - hatj - hatk) = hati + 2hatj + 2hatk`

x – 1 = y = z

If the direction cosines of a line are `< 1/c, 1/c, 1/c >` then ______.

c > 0

0 < c < 1

c = `±sqrt3`

𝑐 > 2

If `veca` and `vecb` are unit vectors enclosing an angle θ and `|veca + vecb| < 1`, then find the values between which θ lies.

Shown below is a cuboid. Find `vec(BA).vec(BC)`.

A building is to be constructed in the form of a triangular pyramid ABCD as shown in the figure. Let the angular points be A(0, 1, 2), B(3, 0, 1), C(4, 3, 6) and D(2, 3, 2) and let G be the point of intersection of the medians of ∆BCD.

Using the above information, answer the following:

1. What will be the length of vector `vec(AG)`?
2. Find the area of ∆ABC.

What are the values of x for which the angle between the vectors? `2x^2hati + 3xhatj + hatk` and `hati - 2hatj + x^2hatk` are obtuse?

Given that B and C lie on the line `(x + 3)/5 = (y - 1)/2 = (z + 4)/3` and BC = 5 units, find the area of ΔABC.

Find the equation of the plane containing the line `x/(-2) = (y - 1)/3 = (1 - z)/1` and the point (–1, 0, 2).

Find the area bounded by the curve y = x(4 – x) and the x-axis from x = 0 to x = 5 as shown in the figure given above.

Which condition is true if Average Cost (AC) is constant at all levels of output?

MC > AC

MC = AC

MC < AC

MC = `1/2` AC

Which of the following statement(s) is/are correct with respect to regression coefficients?

Statement 1: It measures the degree of linear relationship between two variables.

Statement 2: It gives the value by which one variable changes for a unit change in the other variable.

Which 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.

Mean of x = 53, mean of y = 28 regression coefficient of y on x = −1.2, regression coefficient of x on y = −0.3. Find the coefficient of correlation (r).

The total revenue received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5. Find the marginal revenue when x = 5.

A manufacturing company finds that the daily cost of producing x items of product is given by C(x) = 210x + 7000. Find the minimum number that must be produced and sold daily if each item is sold for ₹ 280.

A real estate company is going to build a new residential complex. The land they have purchased can hold at most 500 apartments. Also, if they make x apartments, then the monthly maintenance cost for the whole complex would be as follows:

Fixed cost = ₹ 4000

Variable cost = ₹ (14x – 0.04x²)

How many apartments should the complex have in order to minimize the maintenance costs?

The demand function of a monopoly is given by x = 100 − 4p. Find the quantity at which the MR will be zero.

A survey of 50 families to study the relationships between expenditure on accommodation in (₹ x) and expenditure on food and entertainment (₹ y) gave the following results:

`sumx = 8500, sumy = 9600`, σ_x = 60, σ_y = 20, r = 0.6

Estimate the expenditure on food and entertainment when expenditure on accommodation is ₹ 200.

A linear programming problem is given by Z = px + qy, where p, q > 0 subject to the constraints x + y ≤ 60, 5x + y ≤ 100, x ≥ 0 and y ≥ 0.

1. Solve graphically to find the corner points of the feasible region.
2. If Z = px + qy is maximum at (0, 60) and (10, 50), find the relation of p and q. Also mention the number of optimal solution(s) in this case.

The feasible region for an L.P.P. is shown in the adjoining figure:

Based on the given graph, answer the following questions.

1. Write the constraints for the L.P.P.
2. Find the coordinates of the point B.
3. Find the maximum value of the objective function Z = x + y.