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100+ Free UPMSP Intermediate Mathematics Practice Questions

Prepare for the Uttar Pradesh UPMSP Intermediate (Class 12) Mathematics — Code 131 exam with instant access — no signup required.

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2026 Statistics

Key Facts: UPMSP Intermediate Mathematics Exam

3h 15m

Duration of UPMSP Class 12 Mathematics board exam

UPMSP Official Scheme

100 Marks

Total maximum marks for Class 12 Mathematics paper

UP Board Exam Blueprint

33%

Minimum passing percentage requirement

UPMSP Passing Rules

100 MCQs

Verified practice questions in this question bank

OpenExamPrep

Prepare for UPMSP Class 12 Mathematics (Code 131) with 100 step-by-step practice MCQs covering Matrices, Determinants, Calculus, Vectors, 3D Geometry, LPP, and Probability.

Sample UPMSP Intermediate Mathematics Practice Questions

Try these sample questions to test your UPMSP Intermediate Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Which of the following binary relations defined on the set A = {1, 2, 3} is reflexive?
A.R = {(1, 1), (2, 2)}
B.R = {(1, 1), (2, 2), (3, 3), (1, 2)}
C.R = {(1, 2), (2, 3), (3, 1)}
D.R = {(1, 1), (2, 3)}
Explanation: A relation R on a set A is reflexive if (a, a) belongs to R for every element a in A. For set A = {1, 2, 3}, a reflexive relation must contain (1, 1), (2, 2), and (3, 3). The relation R = {(1, 1), (2, 2), (3, 3), (1, 2)} includes all three required identity pairs along with (1, 2).
2A binary relation R on a set A is defined as an equivalence relation if and only if it satisfies which of the following combined properties?
A.Reflexive and symmetric only
B.Reflexive, symmetric, and transitive
C.Symmetric and transitive only
D.Reflexive and transitive only
Explanation: By standard definition, an equivalence relation on a non-empty set A must simultaneously satisfy three fundamental properties: reflexivity, symmetry, and transitivity. If any of these three properties fails, the relation cannot be an equivalence relation.
3What is the total number of bijective functions that can be defined from set A to set B, given that n(A) = 4 and n(B) = 4?
A.16
B.24
C.64
D.256
Explanation: For two finite sets A and B with n(A) = n(B) = n, the total number of one-one onto (bijective) functions is given by n!. Here n = 4, so the total number of bijective functions is 4! = 4 x 3 x 2 x 1 = 24.
4Find the range of the function f: R -> R defined by f(x) = x / (1 + x^2).
A.[-1, 1]
B.[-1/2, 1/2]
C.(-infty, infty)
D.[0, 1/2]
Explanation: Let y = x / (1 + x^2). Rearranging gives y x^2 - x + y = 0. For real x, the discriminant D = 1 - 4 y^2 >= 0, which implies 4 y^2 <= 1 or -1/2 <= y <= 1/2. Thus the range of f(x) is [-1/2, 1/2].
5Find the principal value of sin^(-1)(-1/2).
A.-pi/6
B.5pi/6
C.-pi/3
D.7pi/6
Explanation: The principal value branch of sin^(-1)(x) is [-pi/2, pi/2]. Since sin(pi/6) = 1/2, sin(-pi/6) = -1/2. Because -pi/6 lies within [-pi/2, pi/2], the principal value is -pi/6.
6Which of the following represents the principal value branch of the inverse cosine function, cos^(-1)(x)?
A.[-pi/2, pi/2]
B.[0, pi]
C.(0, pi)
D.(-pi/2, pi/2)
Explanation: The inverse cosine function cos^(-1)(x) is defined for domain [-1, 1] with the principal value branch [0, pi].
7Evaluate the value of the trigonometric expression tan^(-1)(sqrt(3)) - sec^(-1)(-2).
A.pi/3
B.-pi/3
C.2pi/3
D.-2pi/3
Explanation: tan^(-1)(sqrt(3)) = pi/3 because tan(pi/3) = sqrt(3). For sec^(-1)(-2), since sec^(-1)(-x) = pi - sec^(-1)(x) and sec^(-1)(2) = pi/3, we get sec^(-1)(-2) = pi - pi/3 = 2pi/3. Therefore, pi/3 - 2pi/3 = -pi/3.
8Find the value of cos(sin^(-1)(3/5) + sin^(-1)(5/13)).
A.33/65
B.63/65
C.56/65
D.16/65
Explanation: Let A = sin^(-1)(3/5) and B = sin^(-1)(5/13). Then sin A = 3/5, cos A = 4/5, sin B = 5/13, cos B = 12/13. Using cos(A + B) = cos A cos B - sin A sin B = (4/5)(12/13) - (3/5)(5/13) = 48/65 - 15/65 = 33/65.
9Simplify the expression tan^(-1)((cos x - sin x) / (cos x + sin x)) for -pi/4 < x < 3pi/4.
A.pi/4 + x
B.pi/4 - x
C.x - pi/4
D.pi/2 - x
Explanation: Divide numerator and denominator inside by cos x: (1 - tan x)/(1 + tan x) = tan(pi/4 - x). Taking tan^(-1)(tan(pi/4 - x)) gives pi/4 - x since pi/4 - x falls in the principal domain (-pi/2, pi/2).
10If sin^(-1)x + sin^(-1)y + sin^(-1)z = 3pi/2, what is the value of x^100 + y^100 + z^100 - 9 / (x^101 + y^101 + z^101)?
A.0
B.1
C.3
D.-3
Explanation: Since the maximum value of sin^(-1)t is pi/2, the sum sin^(-1)x + sin^(-1)y + sin^(-1)z = 3pi/2 holds if and only if sin^(-1)x = sin^(-1)y = sin^(-1)z = pi/2. This forces x = y = z = 1. Substituting x = y = z = 1 yields (1 + 1 + 1) - 9 / (1 + 1 + 1) = 3 - 9/3 = 0.

About the UPMSP Intermediate Mathematics Exam

The Uttar Pradesh UPMSP Intermediate (Class 12) Mathematics — Code 131 examination tests students on advanced secondary mathematics topics. Key domains include Relations and Functions, Inverse Trigonometry, Matrices and Determinants, Differential and Integral Calculus, Differential Equations, Vector Algebra, 3D Geometry, Linear Programming, and Probability. This 100-question practice set offers rigorous, step-by-step verified math problems formatted to help students excel in UP Board examinations.

Assessment

State board examination consisting of objective multiple-choice items, short calculation questions, and detailed multi-step analytical problems evaluating Class 12 NCERT-aligned mathematics.

Time Limit

3 hours 15 minutes (195 minutes)

Passing Score

33% aggregate pass mark

Exam Fee

₹600.75 for institutional (regular) Intermediate candidates and ₹806 for private candidates for the 2026 examination, plus ₹206 per additional subject. UPMSP charges one registration fee per candidate, not per subject paper. (Uttar Pradesh Madhyamik Shiksha Parishad (UPMSP))

UPMSP Intermediate Mathematics Exam Content Outline

15%

Relations, Functions & Algebra

Types of relations, reflexive, symmetric, transitive, one-one and onto functions, inverse trigonometric functions, matrix algebra, transpose, determinant properties, adjoint, matrix inverse, and system of linear equations.

25%

Differential Calculus

Continuity, differentiability, chain rule, implicit functions, logarithmic differentiation, parametric equations, second-order derivatives, rate of change, tangents, normals, increasing/decreasing functions, and maxima-minima.

25%

Integral Calculus & Differential Equations

Indefinite and definite integrals, integration by substitution, partial fractions, integration by parts, definite integral properties, area under simple curves, area between curves, order/degree of differential equations, variable separation, homogeneous and linear differential equations.

18%

Vector Algebra & 3D Geometry

Vectors, scalar product, vector product, projection, direction cosines and ratios, line equations in 3D, shortest distance between skew lines, plane equations, angle between planes, and distance of a point from a plane.

17%

Linear Programming & Probability

Formulation of linear programming problems, graphical optimization, feasible regions, conditional probability, multiplication rule, independent events, Bayes' Theorem, random variables, probability distribution, mean, variance, and binomial distribution.

How to Pass the UPMSP Intermediate Mathematics Exam

What You Need to Know

  • Passing score: 33% aggregate pass mark
  • Assessment: State board examination consisting of objective multiple-choice items, short calculation questions, and detailed multi-step analytical problems evaluating Class 12 NCERT-aligned mathematics.
  • Time limit: 3 hours 15 minutes (195 minutes)
  • Exam fee: ₹600.75 for institutional (regular) Intermediate candidates and ₹806 for private candidates for the 2026 examination, plus ₹206 per additional subject. UPMSP charges one registration fee per candidate, not per subject paper.

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

UPMSP Intermediate Mathematics Study Tips from Top Performers

1Master core calculus formulas including differentiation of composite/trigonometric/logarithmic functions and integration techniques.
2Practice matrix inversion using determinant properties and solving 3x3 systems of linear equations via matrix inversion method.
3Understand vector dot and cross product applications such as calculating work done, torque, vector projections, and shortest distances between skew lines.
4Solve linear programming problems graphically by identifying corner points of feasible regions and evaluating objective functions.
5Utilize Bayes' Theorem and Binomial distribution formulas for probability calculation problems.

Frequently Asked Questions

What is UPMSP Code 131 Mathematics?

Code 131 is the official subject code for Intermediate (Class 12) Mathematics prescribed by the Uttar Pradesh Board (UPMSP).

Is the UPMSP Class 12 Mathematics syllabus aligned with NCERT?

Yes, the UP Board Class 12 Mathematics curriculum follows the standard NCERT syllabus covering Algebra, Calculus, Vectors/3D Geometry, Linear Programming, and Probability.

What is the passing mark for UPMSP Class 12 Mathematics?

Students must achieve at least 33% aggregate marks to pass the UPMSP Class 12 Mathematics examination.

How many questions are included in this practice question bank?

This bank contains exactly 100 verified multiple-choice practice questions with full step-by-step mathematical explanations and wrong-option feedback.

What topics carry the highest weightage in Class 12 UPMSP Math?

Calculus (Differential and Integral Calculus combined with Differential Equations) carries the largest overall weightage (~50%), followed by Vectors & 3D Geometry (~18%), Algebra (~15%), and Probability & LPP (~17%).