Mathematics (গণিত)
26 chapters · 312 practice questions with full solutions — all free.
🎯 Full Mathematics (গণিত) section test
40 questions across the whole section · 40 min · one timed test
Quadratic Equation in One Variable (একচলবিশিষ্ট দ্বিঘাত সমীকরণ)
A quadratic equation in one variable has the standard form ax² + bx + c = 0 with a ≠ 0, and it has at most two roots. Roots are found by factorisation, by expressing the equation as a perfect square, or by Sridhar Acharya's formula; the discriminant b² − 4ac tells the nature of the roots even before solving.
Exam tip: Expect a 1-mark MCQ, fill-in or true/false (the condition a ≠ 0, the value of k when one root is given, or the nature of roots from b² − 4ac), 2-mark short answers on forming an equation or finding roots, and a long answer on a real-life word problem or Sridhar Acharya's formula.
12 questionsPractice testSimple Interest (সরল সুদকষা)
Simple interest is the extra money paid for using a principal for some time, and it is always calculated on the original principal. With principal p, rate r% per annum and time t years, I = prt/100 and Amount = Principal + Interest; any one of p, r, t and I can be found when the other three are known.
Exam tip: Simple interest often gives the Arithmetic 1-mark MCQ, fill-in or true/false (rate at which a sum doubles, interest of 1 paisa per rupee per month), 2-mark short answers finding p, r or t, and a long Arithmetic word problem; always write I = prt/100 with units (₹, years, %).
12 questionsPractice testTheorems related to Circle (বৃত্ত সম্পর্কিত উপপাদ্য)
This chapter links the chords of a circle with its centre. Equal chords cut off equal arcs and make equal angles at the centre (and the converse holds), exactly one circle passes through three non-collinear points, and the perpendicular from the centre to a chord bisects it, while the line joining the centre to the midpoint of a chord is perpendicular to it.
Exam tip: Geometry 1-mark MCQs, fill-ins and true/false often ask how many circles pass through three points or a chord length from the radius and distance; 2-mark answers use r² = d² + (l/2)²; a long answer may ask you to prove that the perpendicular from the centre bisects a chord, or its converse — learn both proofs with the figure and the RHS/SSS congruence step.
12 questionsPractice testRectangular Parallelopiped or Cuboid (আয়তঘন)
A cuboid (rectangular parallelopiped) is a solid bounded by six rectangular faces, like a brick or a matchbox; a cube is a cuboid whose length, breadth and height are all equal, like a ludo dice. For a cuboid of length a, breadth b and height c, total surface area = 2(ab + bc + ca), volume = abc and diagonal = √(a² + b² + c²).
Exam tip: The Mensuration 1-mark MCQ, fill-in or true/false often asks for the number of edges, vertices or diagonals, or how the volume or surface area changes when an edge changes; 2-mark questions ask for a diagonal or the surface area, and long answers are real-life problems on tanks, rooms or bricks.
12 questionsPractice testRatio and Proportion (অনুপাত ও সমানুপাত)
A ratio compares two quantities of the same kind and has no unit; four quantities are in proportion when the ratio of the first two equals the ratio of the last two. This chapter uses the types of ratio (greater and lesser inequality, inverse, compound), continued proportion, the mean, third and fourth proportionals, and the properties of proportion (invertendo, alternendo, componendo, dividendo, componendo-dividendo, addendo) to solve algebraic problems.
Exam tip: Expect a 1-mark objective item (MCQ, fill in the blank or true/false) on the mean, third or fourth proportional or a compound or inverse ratio, 2-mark short answers on finding a ratio or a proportional, and a written proof or equation solved with componendo-dividendo or addendo — name the property you use at each step.
12 questionsPractice testCompound Interest (up to 3 years) and Uniform Rate of Increase or Decrease (চক্রবৃদ্ধি সুদ ও সমহার বৃদ্ধি বা হ্রাস)
In compound interest the interest of each period is added to the principal, and the next period's interest is calculated on this new amount; so for the same sum, rate and time (more than one year) compound interest is more than simple interest. The same formula is used for any quantity that grows or falls at a uniform rate every year — population, production or the value of a machine (depreciation).
Exam tip: In the objective section Arithmetic gets one MCQ, one fill in the blank and one true/false, and this chapter is a frequent source: the CI − SI difference, the rate for a half-year or a quarter, or a growth/depreciation formula. Written questions ask you to find the amount, principal, rate or time, or to solve population and machine-value problems — write the formula first, then substitute.
12 questionsPractice testTheorems related to Angles in a Circle (বৃত্তস্থ কোণ সম্পর্কিত উপপাদ্য)
An arc of a circle subtends only one angle at the centre but infinitely many angles at points on the rest of the circle. This chapter proves three theorems — the angle at the centre is twice the angle in the circle on the same arc, angles in the same segment are equal, and the angle in a semicircle is a right angle — and uses the condition for four points to be concyclic.
Exam tip: In the objective section Geometry gets one MCQ, one fill in the blank and one true/false — often on the angle at the centre, the angle in a semicircle or angles in the same segment. Short answers ask you to find unknown angles, and the proof of any of the three theorems can come as a long answer, so practise each proof with a neat figure.
12 questionsPractice testRight Circular Cylinder (লম্ব বৃত্তাকার চোঙ)
A right circular cylinder is the solid generated when a rectangle revolves about one of its sides; it has two equal, parallel circular plane faces and one curved surface, and its axis is perpendicular to the bases. With base radius r and height h, this chapter uses the formulas for curved surface area, total surface area and volume (taking π = 22/7) to solve real-life problems on pipes, pillars, drums, tanks and rollers.
Exam tip: In the objective section Mensuration gets one MCQ, one fill in the blank and one true/false — for the cylinder, typically a formula, the number of plane and curved surfaces, or the effect of changing r or h. Written problems on pipes, pillars, tanks and painting cost are common — write the formula, keep all units the same, and use π = 22/7 unless told otherwise.
12 questionsPractice testQuadratic Surd (দ্বিঘাত করণী)
A quadratic surd is √a where a is a positive rational number that is not the square of a rational number, so every quadratic surd is irrational. Surds are classified as pure or mixed and as like or unlike; expressions with surds are simplified by combining like surds and by rationalising the denominator with a rationalising factor (usually the conjugate surd).
Exam tip: Quadratic surd belongs to the Algebra part: expect a 1-mark MCQ, fill-in or true/false (type of surd, rationalising factor, a quick product or quotient) and 2-mark short answers on simplifying or rationalising. Long answers often give x = a + √b and ask for x + 1/x, x² + 1/x² or a similar value — rationalise 1/x first.
12 questionsPractice testTheorems related to Cyclic Quadrilateral (বৃত্তস্থ চতুর্ভুজ সংক্রান্ত উপপাদ্য)
A quadrilateral whose four vertices lie on one circle is a cyclic quadrilateral. Its opposite angles are supplementary (proved using the angle at the centre), and conversely, if a pair of opposite angles of a quadrilateral is supplementary, its four vertices are concyclic. These two results, together with angles in the same segment and the angle in a semicircle, solve most angle problems in this chapter.
Exam tip: The theorem 'opposite angles of a cyclic quadrilateral are supplementary' is a regular Geometry long-answer proof, so learn the figure and every step. 1-mark MCQ, fill-in or true/false items ask for an opposite or exterior angle, and riders (short or long answers) ask you to prove that a parallelogram, rhombus or trapezium inscribed in a circle is a rectangle, square or isosceles trapezium.
12 questionsPractice testConstruction: Circumcircle and Incircle of a Triangle (সম্পাদ্য : ত্রিভুজের পরিবৃত্ত ও অন্তর্বৃত্ত অঙ্কন)
The circumcircle of a triangle passes through all three vertices; its centre (the circumcentre) is the point where the perpendicular bisectors of the sides meet. The incircle touches all three sides; its centre (the incentre) is the point where the internal bisectors of the angles meet, and its radius is the perpendicular distance from the incentre to any side.
Exam tip: In Madhyamik, constructions are asked only as long answers, usually with an internal choice between two constructions. Draw neatly with ruler and compasses, keep all construction marks, and measure and state the radius if asked. 1-mark MCQ or true/false items can ask where the circumcentre or incentre lies, or which lines meet there.
12 questionsPractice testSphere (গোলক)
A sphere is a solid bounded by a single curved surface, every point of which is at the same distance (the radius) from the centre; a plane through the centre cuts it into two hemispheres. Learn the surface-area and volume formulas for both solids and apply them to real-life problems such as balls, domes, bowls and melting one solid to make others.
Exam tip: Sphere belongs to the Mensuration part: expect 1-mark MCQ, fill-in or true/false items on formulas, ratios and the effect of changing the radius, 2-mark short answers on a direct formula, and a long-answer word problem (melting and recasting, a dome or a bowl, painting or making cost). Take π = 22/7 unless told otherwise and always write the unit.
12 questionsPractice testVariation (ভেদ)
Variation describes how one variable changes when another changes. If x varies directly as y, x = ky; if x varies inversely as y, xy = k; and if x varies jointly as y and z, x = kyz, where k is a non-zero constant called the variation constant. Find k from one given pair of values first, then use it to find the unknown value.
Exam tip: Variation is in the Algebra part: expect a 1-mark MCQ, fill-in or true/false (x ∝ y ⇒ x = ky, xy = constant in inverse variation, x ∝ y and y ∝ z ⇒ x ∝ z), 2-mark short answers finding k or a missing value, and a longer problem asking you to prove a variation relation or solve a real-life sum; always begin by writing the relation with the variation constant k.
12 questionsPractice testPartnership Business (অংশীদারি কারবার)
In a partnership business two or more people invest money (capital) together and share the profit or loss. In a simple partnership all capitals are invested for the same time, so the profit or loss is shared in the ratio of the capitals; in a mixed (compound) partnership the capitals are invested for different times, so it is shared in the ratio of capital × time.
Exam tip: Partnership is in the Arithmetic part: the 1-mark MCQ, fill-in or true/false usually tests the sharing rule (ratio of capitals, or of capital × time), 2-mark short answers ask for one partner's share, and the long Arithmetic word problem often has a partner joining late or withdrawing capital — write each partner's capital × time clearly before forming the ratio.
12 questionsPractice testTheorems related to Tangent to a Circle (বৃত্তের স্পর্শক সংক্রান্ত উপপাদ্য)
A tangent is a straight line that meets a circle at exactly one point (the point of contact), while a secant cuts it at two points. The key theorems are: the tangent is perpendicular to the radius through the point of contact; the two tangents from an external point are equal and subtend equal angles at the centre; and when two circles touch, their centres and the point of contact are collinear.
Exam tip: Theorem proofs (tangent ⊥ radius, equal tangents from an external point, collinearity of the centres and the point of contact) come only as long answers; the 1-mark MCQ, fill-in or true/false usually asks the number of tangents or common tangents, the tangent–radius angle (90°) or the distance between centres, and 2-mark short answers use √(d² − r²) or ∠APB + ∠AOB = 180°.
12 questionsPractice testRight Circular Cone (লম্ব বৃত্তাকার শঙ্কু)
A right circular cone is the solid formed when a right-angled triangle is revolved about one of the sides containing the right angle; it has one curved surface and one plane circular base. With base radius r, height h and slant height l = √(r² + h²): curved surface area = πrl, total surface area = πr(l + r) and volume = (1/3)πr²h.
Exam tip: Cone is in the Mensuration part: the 1-mark MCQ, fill-in or true/false often asks the number of plane or curved surfaces, the cone–cylinder volume relation (1 : 3) or the effect of doubling the radius, 2-mark short answers find l, the curved surface area or the volume, and the longer Mensuration problem is usually a tent or conical-heap real-life sum — always find l = √(r² + h²) first.
12 questionsPractice testConstruction: Tangent to a Circle (সম্পাদ্য : বৃত্তের স্পর্শক অঙ্কন)
This chapter teaches how to construct tangents to a circle with ruler and compasses: the tangent at a given point on the circle, and the two tangents from an external point. Both constructions rest on one fact — the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Exam tip: Constructions are asked only as a long answer (one construction with an internal choice), never as a 1-mark item. But the facts behind them — tangent ⊥ radius at the point of contact, how many tangents pass through a point, equal tangents from an external point — often appear in the Geometry MCQ, fill-in and true/false. In the construction answer, keep every arc visible and write the steps briefly.
12 questionsPractice testSimilarity (সদৃশতা)
Two figures are similar when they have the same shape but not necessarily the same size: corresponding angles are equal and corresponding sides are proportional. For triangles either condition alone is enough, and the chapter uses the parallel-line (basic proportionality) theorem, the equiangular, SSS and SAS conditions and the right-triangle result to find unknown lengths.
Exam tip: Geometry 1-mark items often ask for a missing length using DE ∥ BC or ΔABC ~ ΔDEF, or a true/false such as 'all squares are similar'. The theorem 'the perpendicular from the right-angled vertex to the hypotenuse makes two triangles similar to each other and to the whole triangle' is the one proof from this chapter that can come as a long answer.
12 questionsPractice testReal-life Problems related to Different Solid Objects (বিভিন্ন ঘনবস্তু সংক্রান্ত বাস্তব সমস্যা)
This chapter applies the surface-area and volume formulas of the cuboid, right circular cylinder, sphere, hemisphere and right circular cone to real-life problems such as melting and recasting, filling containers, cutting small solids out of a bigger one, and painting or covering surfaces. The key idea is that when a solid is melted and recast, its volume stays the same.
Exam tip: Mensuration has one MCQ, one fill-in and one true/false item in the paper — usually a formula or a ratio of volumes. Short and long answers are word problems on melting and recasting, filling containers or counting pieces: write the volume equation first, cancel π early and keep fractions until the last step.
12 questionsPractice testTrigonometry: Concept of Measurement of Angle (ত্রিকোণমিতি : কোণ পরিমাপের ধারণা)
Trigonometry ('measurement of triangles') lets us find heights and distances that cannot be measured directly. This chapter treats an angle as the rotation of a ray — anticlockwise rotation gives a positive angle and clockwise a negative one — and introduces two systems of measuring angles: the sexagesimal system (degree, minute, second) and the circular system (radian), linked by πᶜ = 180°.
Exam tip: Trigonometry has one MCQ, one fill-in and one true/false item in the paper; from this chapter they usually test πᶜ = 180°, the value of 1 radian, degree–radian conversion or the sign of an angle. 2-mark answers ask conversions like 40°20′ to radians, the third angle of a triangle, or the angle traced by clock hands — always show the × π/180 or × 180/π step.
12 questionsPractice testConstruction: Determination of Mean Proportional (সম্পাদ্য : মধ্যসমানুপাতী নির্ণয়)
If a : x = x : b, then x is called the mean proportional of a and b, so x² = ab and x = √(ab). This chapter constructs the mean proportional of two given line segments with the help of a semicircle, and uses the same construction to draw a square equal in area to a given rectangle or a given triangle.
Exam tip: Madhyamik asks this chapter only as a long-answer construction, usually as one of two choices: draw the mean proportional of two given segments, or a square equal in area to a given rectangle or triangle. Usually only the construction marks are required, so draw to exact measure with a sharp pencil and keep every arc visible; the idea x² = ab also helps in 1-mark questions on mean proportional.
12 questionsPractice testPythagoras Theorem (পিথাগোরাসের উপপাদ্য)
Pythagoras theorem states that in a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Its converse says that if this relation holds in a triangle, the angle opposite the longest side is a right angle. Both proofs are in the syllabus, and the results are used to find diagonals, heights and distances.
Exam tip: Pythagoras theorem and its converse are among the theorems set for proof in the long-answer section, so learn the textbook figure with the perpendicular AD and the similar-triangle steps. The Geometry 1-mark items (MCQ, fill in the blank, true/false) and 2-mark short answers test triples, diagonals, ladders and rhombus sides.
12 questionsPractice testTrigonometric Ratios and Trigonometric Identities (ত্রিকোণমিতিক অনুপাত এবং ত্রিকোণমিতিক অভেদাবলি)
For an acute angle θ of a right-angled triangle, the six trigonometric ratios compare its sides: sin, cos, tan and their reciprocals cosec, sec, cot. The ratios depend only on the angle, are linked by the identities sin²θ + cos²θ = 1, sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1, and have standard values at 0°, 30°, 45°, 60° and 90°. They are used to find sides of right triangles and to eliminate θ from a pair of equations.
Exam tip: The Trigonometry branch gets one MCQ, one fill in the blank and one true/false (1 mark each), often on standard values or on limits such as whether sin θ = 4/3 is possible. 2-mark short answers evaluate expressions or use one identity, and long answers ask you to prove identities or eliminate θ, so write the identity you use at every step.
12 questionsPractice testTrigonometric Ratios of Complementary Angle (পূরক কোণের ত্রিকোণমিতিক অনুপাত)
Two angles are complementary when their sum is 90°, so the complement of θ is (90° − θ); the two acute angles of a right-angled triangle are complementary. Each trigonometric ratio of (90° − θ) equals the paired ratio of θ (sin with cos, tan with cot, sec with cosec), which reduces many expressions to 0 or 1 and helps find unknown angles.
Exam tip: 1-mark items (MCQ, fill in the blank, true/false) very often ask the value of expressions like sin 25°/cos 65° or tan 15° · tan 75°. 2-mark questions ask you to find θ from equations like sin 3θ = cos(θ − 6°) or to express a ratio in terms of angles between 0° and 45°, and long answers mix these with identities.
12 questionsPractice testApplication of Trigonometric Ratios: Heights and Distances (ত্রিকোণমিতিক অনুপাতের প্রয়োগ : উচ্চতা ও দূরত্ব)
Heights and distances uses trigonometric ratios in right-angled triangles to find the heights of towers, trees and poles, and the distances between objects, without measuring them directly. The angle of elevation (looking up) and the angle of depression (looking down) are always measured from the horizontal line through the observer's eye.
Exam tip: Heights and distances regularly gets the 5-mark long problem (towers, ladders or two angles of elevation); it can also supply one of the Trigonometry 1-mark MCQ, fill-in or true/false items (shadow equal to height means 45°, angle of elevation = angle of depression) or a 2-mark short answer. Draw a neat right-angled triangle, mark the angle from the horizontal, and leave the answer in √3 form unless √3 = 1.732 is given.
12 questionsPractice testStatistics: Mean, Median, Ogive, Mode (রাশিবিজ্ঞান : গড়, মধ্যমা, ওজাইভ, সংখ্যাগুরুমান)
Mean, median and mode are measures of central tendency — single values that represent a whole set of data. For grouped data the mean is found by the direct, short-cut (assumed mean) or step-deviation method, the median by its formula or from an ogive, and the mode by the modal-class formula; the three are linked by the empirical relation Mode = 3 Median − 2 Mean.
Exam tip: All the paper's Statistics questions come from this chapter: one 1-mark MCQ, one fill-in and one true/false (median or mode of a few numbers, Mode = 3 Median − 2 Mean, what the intersection of the two ogives gives), a 2-mark short answer, and two 4-mark long answers (out of three) on the mean, median or mode of a grouped distribution or on drawing an ogive. Set out the table with fᵢxᵢ and cumulative-frequency columns neatly.
12 questionsPractice test