UPMSP High School Mathematics study guide
UPMSP High School Mathematics · Work through your exam, one topic at a time.
The UPMSP High School Mathematics examination is the Class 10 mathematics paper of the Uttar Pradesh Board of Secondary Education (UPMSP, commonly called UP Board). It is a school-leaving board examination taken by students who have completed Class 10 in schools affiliated to the board. The UPMSP High School Mathematics study guide has 12 sections and about 3,246 words.
01Number Systems and Real Numbers
This domain tests Euclid's division lemma, the fundamental theorem of arithmetic, HCF and LCM by prime factorisation, proofs that numbers such as √2 and √3 are irrational, and the decimal-expansion rule for rational numbers. Euclid's lemma: for integers a and b 0 there exist unique q and r with a = bq + r, 0 ≤ r < b.
287 words, about 1 min02Polynomials and Rational Expressions
The exam checks zeros of polynomials, the link between zeros and coefficients, the division algorithm, and simplification of rational expressions with stated restrictions. A zero of p(x) is a value c with p(c) = 0; graphically these are the x-intercepts.
299 words, about 1 min03Linear Equations in Two Variables
This domain tests solving pairs of linear equations graphically and algebraically, deciding consistency from coefficients, and translating word problems into two-variable equations. Standard form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0; each graph is a straight line.
258 words, about 1 min04Quadratic Equations
The exam tests recognising quadratic equations, solving by factorisation and the quadratic formula, using the discriminant to describe roots, and forming equations from given roots. Standard form ax² + bx + c = 0 with a ≠ 0.
271 words, about 1 min05Arithmetic Progressions
Questions cover identifying an AP, finding the nth term and the sum of n terms, and modelling situations such as savings, seating rows and salaries that change by a fixed amount. An AP has a constant difference d = aₙ − aₙ₋₁.
284 words, about 1 min06Coordinate Geometry
The exam tests the distance formula, the section and midpoint formulas, area of a triangle from coordinates, and collinearity, all on the Cartesian plane. Distance between (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²].
244 words, about 1 min07Triangles and Similarity
This domain tests the basic proportionality theorem, the AA, SSS and SAS similarity criteria, the ratio of areas of similar triangles, and Pythagoras' theorem with its converse. Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
264 words, about 1 min08Circles and Tangents
The exam tests the tangent–radius perpendicularity, equal tangent lengths from an external point, the number of tangents from different positions, and angle relations involving two tangents. A tangent touches the circle at exactly one point, the point of contact.
280 words, about 1 min09Constructions
This domain tests ruler-and-compass constructions: dividing a segment in a given ratio, constructing a triangle similar to a given triangle with a given scale factor, and drawing tangents to a circle from an external point. To divide AB in ratio m : n, draw a ray AX making an acute angle with AB, mark m + n equal arcs, join the last point to B, and draw a parallel through the m-th point.
302 words, about 2 min10Trigonometry and Its Applications
The exam tests trigonometric ratios, values at 0°, 30°, 45°, 60° and 90°, standard identities, complementary angles, and heights-and-distances problems using elevation and depression. In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; reciprocals are cosec, sec and cot.
243 words, about 1 min11Mensuration (Areas, Surface Areas, Volumes)
This domain tests areas of sectors and segments, surface areas and volumes of cylinders, cones, spheres, hemispheres and frustums, and solids formed by combining these shapes. Cylinder: curved surface 2πrh, total 2πr(r + h), volume πr²h.
236 words, about 1 min12Statistics and Probability
The exam tests mean, median and mode for grouped data, ogive reading, and classical probability of simple events. Mean by direct method: Σfᵢxᵢ/Σfᵢ; step deviation uses assumed mean a and class width h: x̄ = a + h(Σfᵢuᵢ/Σfᵢ).
278 words, about 1 min