Mathematics · Class 12 Science

Uttar Pradesh Upmsp 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: 8
Total Papers: 8
Real Board Papers: 0
Sample papers: 254
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

31 Q2026complete

31 Q2025complete

31 Q2024complete

31 Q2023complete

38 Q2022complete

—2021Exam cancelled (COVID-19)

33 Q2020complete

33 Q2019complete

26 Q2018complete

2021 — Exam cancelled (COVID-19): The UP Board (UPMSP) cancelled the Class-12 Intermediate examination in 2021 due to COVID-19; no annual question paper was conducted or printed that year, so none exists to publish. Results were declared using a formula based on earlier assessments.

UP Board (UPMSP) Intermediate 2026 · Set CX

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: 100
Questions: 9
Duration: 195 min
Sections: 5

The marks / questions / duration above are the official exam pattern. We currently have 31 of this paper’s questions (344% of the full paper), with 31 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
Q1Section Q1	Multiple-choice (objective), all compulsory	5	1	5
Q2Section Q2	Very short answer, all compulsory	5	1	5
Q3–Q4Section Q3–Q4	Short answer (2 marks), all compulsory	8	2	16
Q5–Q6Section Q5–Q6	Long answer (5 marks) — attempt any five of six parts in each	10	5	50
Q7–Q9Section Q7–Q9	Long answer (8 marks) — internal choice (do any one part)	3	8	24
Total		9		100

The question paper

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

Board Examination

Mathematics

UP Board (UPMSP) Intermediate 2026 · Set CX

Series/Set: CXRoll No. ________

Time Allowed: 3 hr 15 minMaximum Marks: 100

General Instructions

This question paper contains 9 questions divided into 5 sections — Q1, Q2, Q3–Q4, Q5–Q6, Q7–Q9.
Section Q1 comprises 5 questions of 1 mark each (Multiple-choice (objective), all compulsory).
Section Q2 comprises 5 questions of 1 mark each (Very short answer, all compulsory).
Section Q3–Q4 comprises 8 questions of 2 marks each (Short answer (2 marks), all compulsory).
Section Q5–Q6 comprises 10 questions of 5 marks each (Long answer (5 marks) — attempt any five of six parts in each).
Section Q7–Q9 comprises 3 questions of 8 marks each (Long answer (8 marks) — internal choice (do any one part)).

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

Section A

Q1.

The solution of

\dfrac{dy}{dx}=e^{x+y}

is:

(a) $e^{-y}=e^{x}+c$
(b) $e^{x}+e^{-y}=c$
(c) $e^{-x}-e^{-y}=c$
(d) $e^{-x}+e^{-y}=c$

[1]

Q2.

\sin(\tan^{-1}x)

|x|<1

is equal to:

(a) $\dfrac{x}{\sqrt{1+x^{2}}}$
(b) $\dfrac{x}{\sqrt{1-x^{2}}}$
(c) $\dfrac{1}{\sqrt{1+x^{2}}}$
(d) $\dfrac{1}{\sqrt{1-x^{2}}}$

[1]

Q3.

Interval in which the given function

f(x)=x^{2}-4x+6

is increasing, is:

(a) $(2,10)$
(b) $(2,\infty)$
(c) $(-2,\infty)$
(d) $(0,\infty)$

[1]

Q4.

The function

f(x)=2x

x\in R

is:

(a) one-one but not onto
(b) one-one and onto
(c) many-one and onto
(d) many-one but not onto

[1]

Q5.

3P(A)=P(B)=\dfrac{5}{13}

and

P(A/B)=\dfrac{2}{5}

, then

P(A\cup B)

will be:

(a) $\dfrac{20}{39}$
(b) $\dfrac{16}{39}$
(c) $\dfrac{11}{39}$
(d) $\dfrac{14}{39}$

[1]

Page 1 of 6

Section B

Q1.

If the function f:R→ R and g:R→ R are defined by f(x)=cos x and g(x)=3x² respectively, find gof.

[1]

Q2.

If a line makes 90°, 60° and 30° with x, y and z-axes in the positive direction respectively, then find direction cosines.

[1]

Q3.

Find the value of tan⁻¹√(3)-sec⁻¹(-2).

[1]

Q4.

Find the value of the integral ∫ x²tan(x³+2)dx.

[1]

Q5.

If for a unit vector veca, (vecx-veca)·(vecx+veca)=12, then find |vecx|.

[1]

Section C

Q1.

If veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk then find |veca×vecb|.

[2]

Q2.

Prove that the function defined by f:R→\x∈ R:-1<x<1\, where f(x)=(2x)/(1+|x|), x∈ R is one-one.

[2]

Q3.

If f(x)=sin x, x∈[0,(π)/(2)] and g(x)=cos x, x∈[0,(π)/(2)], prove that (f+g) is not one-one.

[2]

Page 2 of 6

Q4.

If the lines (x-1)/(-3)=(y-2)/(2k)=(z-3)/(2) and (x-1)/(3k)=(y-1)/(1)=(z-6)/(-5) are mutually perpendicular, find the value of k.

[2]

Q5.

Find the Cartesian equation of a line parallel to the vector 3hati+2hatj-8hatk passing through the point (5,2,-4).

[2]

Q6.

If x∈(0,(π)/(4)), then prove that cot⁻¹(dfrac√(1+sin x)+√(1-sin x)√(1+sin x)-√(1-sin x))=(x)/(2).

[2]

Q7.

Find the value of the determinant a b c b c a c a b .

[2]

Q8.

If P(A)=(7)/(13), P(B)=(9)/(13) and P(A∩ B)=(4)/(13), then find the value of P(A∪ B) and P(A/B).

[2]

Section D

Q1.

[5]

Page 3 of 6

Q2.

Find the maximum value of the objective function Z=5x+10y by graphical method under the following constraints: x+2y≤ 120, x+y≥ 60, x-2y≥ 0, x≥ 0, y≥ 0.

[5]

Q3.

For what value of λ, the function defined by f(x)= λ(x²-2x), if x≤ 0 \4x+1, if x>0 is continuous at x=0?

[5]

Q4.

If the position vectors of points A, B, C and D are respectively hati+hatj+hatk, 2hati+5hatj, 3hati+2hatj-3hatk and hati-6hatj-hatk, then find the angle between the lines AB and CD. Prove that AB and CD are collinear.

[5]

Q5.

Find the interval at which the function f(x) given by f(x)=(4sin x-2x-xcos x)/(2+cos x) is strictly increasing and decreasing.

[5]

Q6.

If A= 1 2 3 \3 -2 1 \4 2 1 , then show that A³-23A-40I=0.

[5]

Page 4 of 6

Q7.

Prove that if A and B are independent events, then the probability of happening of at least one of them in A or B is [1-P(A')P(B')].

[5]

Q8.

Prove that the height of a right circular cone with maximum volume inscribed in a sphere of radius r is (4r)/(3).

[5]

Q9.

Find the area of region surrounded by x=0 and x=2π and curve y=sin x.

[5]

Q10.

Find the value of ∫-13/2 |xsin π x|dx.

[5]

Page 5 of 6

Section E

Q1.

Suppose that A= 2 -1 \3 4 , B= 5 2 \7 4 , C= 2 5 \3 8 . Find the matrix D so that CD-AB=0. **OR** Solve the following system of equations by matrix method: (2)/(x)+(3)/(y)+(10)/(z)=4; (4)/(x)-(6)/(y)+(5)/(z)=1; (6)/(x)+(9)/(y)-(20)/(z)=2.

[8]

Q2.

Prove that the function y=Ae3xcos 4x+Be3xsin 4x is the solution of the differential equation dfracd²ydx²-6(dy)/(dx)+25y=0, where A and B are arbitrary constants. **OR** Find ∫[√(tan x)+√(cot x)]dx.

[8]

Q3.

Find the particular solution of the differential equation (dy)/(dx)+ycot x=4xcsc x, (x≠ 0) given that y=0 at x=(π)/(2). **OR** Solve the following: (i) ∫ dfracxsin⁻¹x√1-x²dx (4 marks); (ii) ∫ ex((1+sin x)/(1+cos x))dx (4 marks).

[8]

Page 6 of 6