ISC Class XII Board Examination
ISC Mathematics — Specimen Question Paper 2026
Official CISCE specimen question paper, published with its answer key, ahead of the ISC Class XII Board Examination 2026.
Specimen2026Subject code 860
- Board
- ISC Class XII Board Examination
- Class
- Class 12
- Subject
- Mathematics
- Year
- 2026
- Total marks
- 80 marks
- Duration
- Not verified
Verification and rights
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ExamVault links to the paper published on cisce.org. The answer key is part of the same document. The document is CISCE copyright and is not reproduced here.
Official answer keyQuestions in this paper
Section A
- 1((i))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
If A and B are square matrices of order 3, A is non-singular matrix and AB = O,
then the matrix B is:1 mark - 1((iv))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
Consider ∆= |2e f g|
2i j k
a b c
Assertion: The value of ∆ = 4 × | 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? - 1((vi))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
a 0 0
If A = (0 a 0), then An equals to
0 0 a1 mark - 1((vii))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
Observe the following graphs1 mark - 1((ix))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
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?1 mark - 1((x))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
Assertion: The equality tan(cot -1 x) = cot(tan-1 x), is true for all x ∈ R
π
Reason: The identity tan-1 x + cot -1 x = 2 , is true for all x ∈ R
Which of the following is correct?1 mark - 1((xii))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
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 determinant of C2?1 mark - 1((xiii))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.1 mark - 1((xiv))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
The given function f: R → R is many to one function. Give reason.1 mark - 1((xv))In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
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?1 mark - 2((i))dy x-y
If x = e y , then prove that dx = x log x
OR
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ISC (CLASS XII) SPECIMEN QUESTION PAPER 2026 - 2((ii))Find the values of ′a′ for which the function f(x) = b - ax + sin x is increasing on
R. - 5((i))If ∫ x 5 cos( x 6 ) dx = k sin(x 6 ) + C, find the value of ‘k’.
OR - 9((i))Solve the following differential equation:
dy y
x dx = y - x tan ( x )
OR - 9((ii))Solve the following differential equation:
dy
(1 + x 2 ) + 2xy = 4x 2
dx - 10((i))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.
(a) What is the probability that all three friends will get the same result (all heads
or all tails) in one round of tossing?
(b) What is the probability that they will get a different result in one round of
tossing?
(c) What is the probability that they will need exactly four rounds of tossing to
determine who would pay?
OR - 10((ii))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”?
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ISC (CLASS XII) SPECIMEN QUESTION PAPER 2026 - 11((i))Translate the problem into a system of equations.
- 11((ii))Solve the system of equation by using matrix method.
- 11((iii))Hence, find the cost of one paper bag, one scrap book and one pastel sheet.
- 12((i))x 2 +x+1
Evaluate: ∫ (x+2 )(x 2 +1) dx
OR - 12((ii))3π
x dx
Evaluate: ∫π4 1+ sin x dx - 13((i))A person has manufactured a water tank in the shape of a closed right circular
cylinder. The volume of the cylinder is 2 cubic units. If the height and radius of the
cylinder be h and r. - 13((ii))A Dolphin jumps and taken a path given by the equation
h(t) = 2 (-7t 2 + 3t + 2), (t ≥ 0) , h(t) is the height of the Dolphin at any point of
time. - 14((i))Find the probability for each possible number of surprise tests.
- 14((ii))Use the probabilities to build a distribution table.
- 14((iii))Calculate the average number of surprise tests per day.
- 14((iv))Based on your calculations, decide: Should the teachers coordinate better? Or is the
current plan acceptable?
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ISC (CLASS XII) SPECIMEN QUESTION PAPER 2026
Section B
- 15((i))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
Consider the following statements and choose the correct option:
Statement 1: If a⃗ and b⃗⃗ represents two adjacent sides of a parallelogram then the
diagonals are represented by a⃗ + b⃗ and a⃗ - b⃗ .
Statement 2: If a⃗ and b⃗ represents two diagonals of a parallelogram then the adjacent
sides are represented by 2(a⃗ + b⃗ ) and 2(a⃗ - b⃗).
Which of the following is correct? - 15((ii))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
A plane passes through three points A, B and C with position vectors î + ĵ, ĵ + k̂ and
k̂ + î respectively. The equation of the line passing through the point P with position
vector î + 2ĵ + 2k̂ and normal to the plane is - 15((iii))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
If the direction cosines of a line are < 1 , 1 , 1 > then
c c c - 15((iv))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
If a⃗ and b⃗ are unit vectors enclosing an angle θ and |a⃗ + b⃗| < 1, then find the values
between which θ lies.
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ISC (CLASS XII) SPECIMEN QUESTION PAPER 2026 - 15((v))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
Shown below is a cuboid. Find ⃗⃗⃗⃗⃗⃗
BA . BC⃗⃗⃗⃗⃗⃗ - 16((ii))2] (i) 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 (a) What will be the length of vector ⃗⃗⃗⃗⃗⃗ AG ? (b) Find the area of ∆ABC. OR
What are the values of x for which the angle between the vectors?
2x 2 î + 3xĵ + k̂ and î - 2ĵ + x 2 k̂ is obtuse? - 17((ii))4] (i) x+3 y-1 z+4 Given, B and C lie on the line = = and BC = 5 units find the area of 5 2 3 ΔABC. OR ---------------------------------------------------------------------------------------------------------------------- ISC (CLASS XII) SPECIMEN QUESTION PAPER 2026 Visit CollegeDekho
Find the equation of the plane containing the line x = y-1 = 1-z and the
-2 3 1
point (-1, 0, 2).
Section C
- 19((i))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
Which condition is true if Average Cost (AC) is constant at all levels of output? - 19((ii))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
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? - 19((iii))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
Mean of x = 53, mean of y = 28 regression co-efficient y on x = -1 · 2, regression
co-efficient x on y = -0 · 3. Find coefficient of correlation (r). - 19((iv))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
The total revenue received from the sale of x unit of a product is given by
R(x) = 3x 2 + 36x + 5. Find the marginal revenue when x = 5. - 19((v))5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
A manufacturing company finds that the daily cost of producing x item of product is
given by C(x) = 210x + 7000. Find the minimum number that must be produced
and sold daily for break even, if each item is sold for ₹280. - 20((ii))2] (i) A real estate company is going to build a new residential complex. The land they have purchased can hold at the 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 ∙ 04x 2 ) How many apartments should the complex have in order to minimize the maintenance costs? OR
The demand function of a monopoly is given by x = 100 - 4p. Find the quantity at
which the Marginal Revenue will be zero. - 22((i))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
(a) Solve graphically to find the corner points of the feasible region.
(b) 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.
OR - 22((ii))The feasible region for an L.P.P. is shown in the adjoining figure:
Based on the given graph, answer the following questions.
(a) Write the constraints for the L.P.P.
(b) Find the co-ordinates of the point B.
(c) Find the maximum value of the objective function Z = x + y.
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ISC (CLASS XII) SPECIMEN QUESTION PAPER 2026
MATHEMATICS