Mathematics · Class 12 Science

Uttarakhand Ubse Class 12 Mathematics — Real Previous-Year Papers

with complete answers

Real previous-year board papers, year by year — the official exam pattern, the full question paper, and every question solved the concept-first way. Distinct from the chapter-wise textbook bank.

2018–2026
Years of papers: 7
Total Papers: 7
Real Board Papers: 0
Sample papers: 233
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

—2026Paper not yet available

33 Q2025complete

35 Q2024complete

33 Q2023complete

33 Q2022complete

—2021Exam cancelled (COVID-19)

—2020Not available

33 Q2019complete

33 Q2018complete

2026 — Paper not yet available: This year’s exam was held, but neither the Uttarakhand Board’s official site nor any source we check has published the question paper yet. We publish only a paper we can verify against a real printed original — this one will appear here once it is.

2021 — Exam cancelled (COVID-19): The Uttarakhand Board cancelled the Class-12 board exam for this year nationwide due to COVID-19; no annual question paper was ever conducted or printed, so none exists to publish.

Uttarakhand Board Intermediate (Class 12) 2026 · Set ANNUAL

Real board examination

About this paper

The real Class-12 board examination held in 2026. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 80
Questions: 24
Duration: 180 min
Sections: 5

The marks / questions / duration above are the official exam pattern. We currently have 33 of this paper’s questions (138% of the full paper), with 33 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
Q1–Q7Section Q1–Q7	All compulsory (Q1 = MCQ, several parts)	16	1	16
Q8–Q12Section Q8–Q12	All compulsory	5	2	10
Q13–Q18Section Q13–Q18	All compulsory	6	4	24
Q19–Q23Section Q19–Q23	All compulsory	5	5	25
Q24Section Q24	Case/Source-based (3 sub-parts)	1	—	—
Total		24		80

The question paper

The questions we hold for this paper, laid out by section. Solutions are on the Answers tab.

Board Examination

Mathematics

Uttarakhand Board Intermediate (Class 12) 2026 · Set ANNUAL

Series/Set: ANNUALRoll No. ________

Time Allowed: 3 hoursMaximum Marks: 80

General Instructions

This question paper contains 24 questions divided into 5 sections — Q1–Q7, Q8–Q12, Q13–Q18, Q19–Q23, Q24.
Section Q1–Q7 comprises 16 questions of 1 mark each (All compulsory (Q1 = MCQ, several parts)).
Section Q8–Q12 comprises 5 questions of 2 marks each (All compulsory).
Section Q13–Q18 comprises 6 questions of 4 marks each (All compulsory).
Section Q19–Q23 comprises 5 questions of 5 marks each (All compulsory).
Section Q24 comprises 1 question (Case/Source-based (3 sub-parts)).

Above is the official exam pattern. The questions printed below are those we currently hold for this paper.

Section Q1–Q7

All compulsory (Q1 = MCQ, several parts) · 1 mark each · 16 of 16 shown

Q1.

The principal value of tan^{-1}(-\sqrt{3}) is:

(a)
(i) \pi/3
(b)
(ii) -\pi/3
(c)
(iii) \pi/6
(d)
(iv) -\pi/6

[1]

Q2.

The order of the matrix A = [[2, 5, 19, -7], [35, -2, 5, 12]] is:

(a)
(i) 4x2
(b)
(ii) 8
(c)
(iii) 2x3
(d)
(iv) 2x4

[1]

Q3.

Differentiation of cos(\sqrt{x}) with respect to 'x' is:

(a)
(i) sin(\sqrt{x})
(b)
(ii) -\frac{1}{2}\sqrt{x} sin(\sqrt{x})
(c)
(iii) \frac{-sin(\sqrt{x})}{2\sqrt{x}}
(d)
(iv) -sin(\sqrt{x})

[1]

Q4.

The rate of change of the area of a circle with respect to its radius r at r=6 cm is:

(a)
(i) 8\pi
(b)
(ii) 10\pi
(c)
(iii) 11\pi
(d)
(iv) 12\pi

[1]

Q5.

The value of \int \frac{sec^2 x}{cosec^2 x} dx is:

(a)
(i) tan x - x + c
(b)
(ii) tan x + x + c
(c)
(iii) cot x - x + c
(d)
(iv) log cosec x + c

[1]

Q6.

\int x^2 e^{x^3} dx equals:

(a)
(i) \frac{e^{x^3}}{3} + c
(b)
(ii) 3e^{x^3} + c
(c)
(iii) \frac{e^{x^2}}{3} + c
(d)
(iv) \frac{1}{2}e^{x^2} + c

[1]

Page 1 of 6

Q7.

If a line has the direction ratios -18, 12, -4, then the direction cosines of it will be: (a) (i) (-18)/(11), (12)/(11), (-4)/(11) (b) (ii) (-9)/(11), (6)/(11), (-4)/(11) (c) (iii) (-9)/(11), (6)/(11), (-2)/(11) (d) (iv) -9, 6, -4

[1]

Q8.

The number of arbitrary constants in the general solution of a differential equation of third order will be: (a) (i) 0 (b) (ii) 1 (c) (iii) 2 (d) (iv) 3

[1]

Q9.

Assertion (A): For vector veca = hati + 2hatj and vector vecb = 2hati + hatj, |veca| = |vecb|. Reason (R): Vector veca and vector vecb are equal vectors. (a) (i) Both A and R are correct and R is the correct explanation of A. (b) (ii) Both A and R are correct but R is not the correct explanation of A. (c) (iii) A is correct but R is incorrect. (d) (iv) Both A and R are incorrect.

[1]

Q10.

Assertion (A): Let Z be the set of integers. A function f: Z -> Z defined as f(x)=2x-3, ∀ x ∈ Z is bijective. Reason (R): A function is a bijective if it is both injective and surjective. (a) (i) Both A and R are correct and R is the correct explanation of A. (b) (ii) Both A and R are correct but R is not the correct explanation of A. (c) (iii) A is correct but R is incorrect. (d) (iv) Both A and R are incorrect.

[1]

Q11.

If x=a sec θ and y=b tan θ, then find (dy)/(dx).

[1]

Q12.

Find the second derivative for the function y=sin x + e2x.

[1]

Q13.

Find the integral: ∫ (√(x) - frac1√(x))² dx

[1]

Q14.

Find the general solution of the differential equation (dy)/(dx) = (1 + x²)(1 + y²).

[1]

Q15.

Find the direction cosine of the vector veca = 2hati - 3hatj + 4hatk.

[1]

Q16.

If probabilities P(A) = (5)/(12), P(B) = (1)/(2) and P(A ∩ B) = (1)/(3), evaluate P(A/B).

[1]

Page 2 of 6

Section Q8–Q12

All compulsory · 2 marks each · 5 of 5 shown

Q1.

Draw the graph of the function y=cos⁻¹ x and y=tan⁻¹ x.

[2]

Q2.

Find the interval in which the function f(x) = -2x³ - 9x² - 12x + 1 is strictly increasing.

[2]

Q3.

Find ∫ √(3 - 2x - x²) dx.

[2]

Q4.

If veca = hati + hatj + hatk, vecb = 2hati - hatj + 3hatk and vecc = hati - 2hatj + hatk, find a unit vector parallel to the vector 2veca - vecb + 3vecc.

[2]

Q5.

An international seminar is attended by 1000 scientists, out of which 400 are from Asia. It is known that 80 of these 400 Asian scientists are from India. What is the probability that a randomly chosen scientist is from India, if given that the chosen scientist is from Asia.

[2]

Section Q13–Q18

All compulsory · 4 marks each · 6 of 6 shown

Q1.

Let A=R-3 and B=R-1. Consider the function f:A->B defined by f(x)=((x-2))/((x-3)). Is the function f one-one onto? Justify your answer.

[4]

Page 3 of 6

Q2.

If A = [[0, 6, 7], [-6, 0, 8], [7, -8, 0]], B = [[0, 1, 1], [1, 0, 2], [1, 2, 0]], C = [[2], [-2], [3]]; Verify that A(B+C) = AB + AC.

[4]

Q3.

Determine function f defined by: f(x) = x² sin(1/x), if x ≠ 0 ; 0, if x = 0 is a continuous function?

[4]

Q4.

Evaluate the following definite integral: ∫₁⁴ (|x-1| + |x+2| + |x-3|) dx.

[4]

Q5.

Find the area bounded by the curve y=cos x between x=0 and x=2π.

[4]

Q6.

Find the vector equation of a line passing through the point (1, 2, -4) and perpendicular to the two lines: (x-8)/(3) = (y+19)/(-16) = (z-10)/(7) and (x-15)/(3) = (y-29)/(8) = (z-5)/(-5).

[4]

Page 4 of 6

Section Q19–Q23

All compulsory · 5 marks each · 6 of 5 shown

Q1.

Toshit, Kuldeep and Kanishk bought vegetables from a shop. All three bought potatoes, onion and tomatoes in different quantities. Toshit bought 2 kg potatoes, 3 kg onion and 1 kg tomatoes, Kuldeep bought 3 kg potatoes, 2 kg onion and 2 kg tomatoes and Kanishk bought 4 kg potatoes, 2 kg onion and 1 kg tomatoes. Toshit, Kuldeep and Kanishk paid bills of Rs. 180, Rs. 220 and Rs. 190 respectively. Find the price of potatoes, onion and tomatoes per kg using matrix method.

[5]

Q2.

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

[5]

Q3.

Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are collinear and find the ratio in which point B divides line AC.

[5]

Q4.

Minimise and maximise the objective function z = 2000 + 10x - 70y under the following constraints by graphical method: x + y <= 8 x + y >= 4 x <= 5 y <= 5 x, y >= 0.

[5]

Page 5 of 6

Q5.

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

[5]

Q6.

A farmer wants to construct an open cuboidal shape water reservoir for his field. The reservoir must have a depth of 2 meters and hold a total volume of 32 cubic meters of water. The construction cost for the cemented base is Rs. 2000 per m² while the cost of constructing the four walls is Rs. 1500 per m². If the cost of constructing the reservoir is to be kept to a minimum, then determine- (i) What should be the length and width of the reservoir? [3] (ii) What should be the minimum cost of constructing a reservoir? [2]

[5]

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