Commerce (English Medium)
Science (English Medium)
Arts (English Medium)
Academic Year: 2025-2026
Date: मार्च 2026
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General Instructions:
Read the following instructions very carefully and strictly follow them:
- This Question paper contains 38 questions. All questions are compulsory.
- This Question paper is divided into five Sections - A, B, C, D and E.
- In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) with only one correct option and Questions no. 19 and 20 are Assertion-Reason based questions of 1 mark each.
- In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks each.
- In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
- In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
- In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
- There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E.
- Use of calculators is not allowed.
Identify the function shown in the graph.
sin–1 x
sin–1 (2x)
`sin^-1(x/2)`
2 sin–1 x
Chapter:
If for three matrices A = `[a_(ij)]_(m xx 4)`, B = `[b_(ij)]_(n xx 3)` and C = `[c_(ij)]_(p xx q)` products AB and AC both are defined and are square matrices of the same order, then the values of m, n, p and q are ______.
m = q = 3 and n = p = 4
m = 2, q = 3 and n = p = 4
m = q = 4 and n = p = 3
m = 4, p = 2 and n = q = 3
Chapter:
If the matrix A = `[(0, r, -2),(3, p, t),(q, -4, 0)]` is skew-symmetric, then the value of `(q + t)/(p + r)` is ______.
–2
0
1
2
Chapter:
If A is a square matrix of order 4 and |adj A| = 27, then A (adj A) is equal to ______.
3
9
3I
9I
Chapter:
The inverse of the matrix `[(3, 0, 0),(0, 2, 0),(0, 0, 5)]` is ______.
`[(0, 0, 3),(0, 2, 0),(5, 0, 0)]`
`[(1/3, 0, 0),(0, 1/2, 0),(0, 0, 1/5)]`
`[(-1/3, 0, 0),(0, -1/2, 0),(0, 0, -1/5)]`
`[(-3, 0, 0),(0, -2, 0),(0, 0, -5)]`
Chapter:
Value of the determinant `[(cos67°,sin67°),(sin23°,cos23°)]` is ______.
0
`1/2`
`sqrt(3)/2`
1
Chapter:
If a function defined by f(x) = `{(kx + 1"," x ≤ π),(cosx"," x > π):}` is continuous at x = π, then the value of k is ______.
π
`(-1)/π`
0
`(-2)/π`
Chapter:
If f(x) = x tan–1 x, then f'(1) is equal to ______.
`π/4 - 1/2`
`π/4 + 1/2`
`-π/4 - 1/2`
`-π/4 + 1/2`
Chapter:
A function f(x) = 10 – x – 2x2 is increasing on the interval ______.
`(-∞, -1/4]`
`(-∞, 1/4)`
`[-1/4, ∞)`
`[-1/4, 1/4]`
Chapter:
The solution of the differential equation xdx + ydy = 0 represents a family of ______.
straight line
parabolas
circles
ellipses
Chapter:
If f(a + b − x) = f(x), then `int_a^b x f(x) dx` is equal to ______.
`(a + b)/2 int_a^b f(b - x) dx`
`(a + b)/2 int_a^b f(a - x) dx`
`(b - a)/2 int_a^b f(x) dx`
`(a + b)/2 int_a^b f(x) dx`
Chapter:
If `∫ x^3 sin^4 (x^4) cos (x^4) dx = a sin^5 (x^4) + C`, then a is equal to ______.
`-1/10`
`1/20`
`1/4`
`1/5`
Chapter:
A bird flies through a distance in a straight line given by the vector `hati + 2hatj + hatk`. A man standing beside a straight metro rail track given by `vecr = (3 + λ)hati + (2λ − 1)hatj + 3λhatk` is observing the bird. The projected length of its flight on the metro track is ______.
`6/sqrt(14)` units
`14/sqrt(6)` units
`8/sqrt(14)` units
`5/sqrt(6)` units
Chapter:
The distance of the point with position vector `3hati + 4hatj + 5hatk` from the y-axis is ______.
4 units
`sqrt(34)` units
5 units
`5sqrt(2)` units
Chapter:
If `veca = 3hati + 2hatj + 4hatk, vecb = hati + hatj - 3hatk` and `vecc = 6hati - hatj + 2hatk` are three given vectors, then `(2veca.hati)hati - (vecb.hatj)hatj + (vecc.hatk)hatk` is the same as the vector ______.
`veca`
`vecb + vecc`
`veca - vecb`
`vecc`
Chapter:
A student of class XII studying Mathematics comes across an incomplete question in a book.
Maximise Z = 3x + 2y + 1
Subject to the constraints x ≥ 0, y ≥ 0, 3x + 4y ≤ 12,
He/She notices the below shown graph for the said LPP problem and finds that a constraint is missing in it:
Help him/her choose the required constraint from the graph.
The missing constraint is:
x + 2y ≤ 2
2x + y ≥ 2
2x + y ≤ 2
x + 2y ≥ 2
Chapter:
The feasible region of a linear programming problem is bounded but the objective function attains its minimum value at more than one point. One of the points is (5, 0).
Then one of the other possible points at which the objective function attains its minimum value is:
(2, 9)
(6, 6)
(4, 7)
(0, 0)
Chapter:
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A person observed the first 4 digits of your 6-digit PIN. What is the probability that the person can guess your PIN?
`1/81`
`1/100`
`1/90`
1
Chapter:
Assertion (A): Value of the expression `sin^-1 (sqrt(3)/2) + tan^-1 1 - sec^-1 (sqrt(2))` is `π/4`.
Reason (R): Principal value branch of sin–1 x is `[-π/2, π/2]` and that of sec–1 x is `[0, π] - {π/2}`.
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true but (R) is not the correct explanation of (A).
(A) is true but (R) is false.
(A) is false but (R) is true.
Chapter:
Assertion (A): Given two non-zero vectors `veca` and `vecb`. If `vecr` is another non-zero vector such that `vecr xx (veca + vecb) = vec0`. Then `vecr` is perpendicular to `veca xx vecb`.
Reason (R): The vector `(veca + vecb)` is perpendicular to the plane of `veca` and `vecb`.
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true but (R) is not the correct explanation of (A).
(A) is true but (R) is false.
(A) is false but (R) is true.
Chapter:
If `y = log tan (π/4 + x/2)`, then prove that `(dy)/(dx) - secx = 0`.
Chapter:
Find out the area of the shaded region in the enclosed figure.
Chapter:
If f(x + y) = f(x) f(y) for all x, y ∈ R and f(5) = 2, f′(0) = 3, then using the definition of derivatives, find f′(5).
Chapter:
The two vectors `hati + hatj + hatk` and `3hati - hatj + 3hatk` represent the two sides OA and OB, respectively, of a ∆OAB, where O is the origin. The point P lies on AB such that OP is a median. Find the area of the parallelogram formed by the two adjacent sides as OA and OP.
Chapter:
If xy = ex – y prove that `dy/dx = logx/((log(xe))^2` and hence find its value at x = e.
Chapter:
If x = a(θ – sin θ), y = a(1 – cos θ) find `(d^2y)/(dx^2)`.
Chapter:
A spherical ball of ice melts in such a way that the rate at which its volume decreases at any instant is directly proportional to its surface area. Prove that the radius of the ice ball decreases at a constant rate.
Chapter:
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Sketch the graph y = |x + 1|. Evaluate\[\int\limits_{- 4}^2 \left| x + 1 \right| dx\]. What does the value of this integral represent on the graph?
Chapter: [8] Applications of the Integrals
Using integration, find the area of the region {(x, y) : x2 – 4y ≤ 0, y – x ≤ 0}.
Chapter:
Find the distance of the point (2, –1, 3) from the line `vecr = (2hati - hatj + 2hatk) + μ(3hati + 6hatj + 2hatk)` measured parallel to the z-axis.
Chapter:
Find the point of intersection of the line `vecr = (3hati + hatk) + μ(hati + hatj + hatk)` and the line through (2, –1, 1) parallel to the z-axis. How far is this point from the z-axis?
Chapter:
Solve graphically:
Maximise Z = 2x + y subject to
x + y ≤ 1200
x + y ≥ 600
`y ≤ x/2`
x ≥ 0, y ≥ 0
Chapter:
Two students Mehul and Rashi are seeking admission in a college. The probability that Mehul is selected is 0.4 and the probability of selection of exactly one of the them is 0.5. Chances of selection of them is independent of each other. Find the chances of selection of Rashi. Also find the probability of selection of at least one of them.
Chapter:
For two matrices A = `[(3, -6, -1),(2, -5, -1),(-2, 4, 1)]` and B = `[(1, -2, -1),(0, -1, -1),(2, 0, 3)]`, find the product AB and hence solve the system of equations:
3x – 6y – z = 3
2x – 5y − z + 2 = 0
–2x + 4y + z = 5
Chapter:
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Chapter: [7] Integrals
Solve the differential equation:
`y + d/dx (xy) = x (sinx + x)`
Chapter:
Find the particular solution of the differential equation:
`2y e^(x/y) dx + (y - 2x e^(x/y)) dy = 0` given that y(0) = 1
Chapter:
The two lines `(x - 1)/3` = −y, z + 1 = 0 and `(-x)/2 = (y + 1)/2` = z + 2 intersect at a point whose y-coordinate is 1. Find the coordinates of their point of intersection. Find the vector equation of a line perpendicular to both the given lines and passing through this point of intersection.
Chapter:
|
A city’s traffic management department is planning to optimize traffic flow by analyzing the connectivity between various traffic signals. The city has five major spots labelled A, B, C, D, and E.
|
The department wants to represent and analyze this data using relations and functions. Use the given data to answer the following questions:
- Is the traffic flow reflexive? Justify. (1)
- Is the traffic flow transitive? Justify. (1)
-
- Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation. (2)
OR - Does the traffic flow represent a function? Justify your answer.
- Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation. (2)
Chapter:
|
LED bulbs are energy-efficient because they use significantly less electricity than traditional bulbs while producing the same amount of light. They convert more energy into light rather than heat, reducing waste. Additionally, their long lifespan means fewer replacements, saving resources and money over time. A company manufactures a new type of energy-efficient LED bulb. The cost of production and the revenue generated by selling x bulbs (in an hour) are modelled as C(x) = 0.5x2 − 10x + 150 and R(x) = −0.3x2 + 20x respectively, where C(x) and R(x) are both in ₹.
|
To maximize the profit, the company needs to analyze these functions using calculus. Use the given models to answer the following questions:
- Derive the profit function P(x). (1)
- Find the critical points of P(x). (1)
-
- Determine whether the critical points correspond to a maximum or a minimum profit by using the second derivative test. (2)
OR - Identify the possible practical value of x (i.e., the number of bulbs that can realistically be produced and sold) that can maximize the profit if the resources available and the expenditure on machines allows to produce minimum 10 but not more than 18 bulbs per hour. Also calculate the maximum profit.
- Determine whether the critical points correspond to a maximum or a minimum profit by using the second derivative test. (2)
Chapter:
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Excessive use of screens can result in vision problems, obesity, sleep disorders, anxiety, low retention problems and can impede social and emotional comprehension and expression. It is essential to be mindful of the amount of time we spend on screens and to reduce our screen time by taking regular breaks, setting time limits, and engaging in non-screen-based activities.
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- What is the total percentage of students who suffer from anxiety and low retention issues in the class? (2)
- A student is selected at random, and he is found to suffer from anxiety and low retention issues. What is the probability that he/she spends screen time more than 4 hours per day? (2)
Chapter:
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