The question shapes that recur most on CBSE boards — drawn from 10 years of papers, the official blueprint, and examiner-pattern analysis. Master these patterns first.
These are important question patterns to prioritise — not predictions of the exact 2027 paper. Full preparation still matters.
Question stacks
Showing all important questions across all topics.
Real Numbers
High-ROI4 Q - 12 marks1 competency Q
High-value pattern stack for focused 2027 revision. Includes 1 competency-style question.
Section B2 marks
Using the Fundamental Theorem of Arithmetic, find the prime factorisation of 392.
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Section C3 marks
Prove that √2 is an irrational number.
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Section C3 marks
Prove that √5 is irrational.
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Section E4 marksCase-based
A school is planning to distribute prizes. It has 240 pens and 168 notebooks. They are to be packed into the greatest possible number of identical groups, so that every group gets the same number of pens and the same number of notebooks, with nothing left over.
(i) Find the HCF of 240 and 168 using prime factorisation.
(ii) What is the greatest number of groups that can be formed?
(iii) How many pens and how many notebooks will each group receive?
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Pair of Linear Equations in Two Variables
Good-to-do8 Q - 19 marks4 competency Qs
Useful pattern stack for completing topic coverage. Includes 4 competency-style questions.
Section C3 marks
Solve the pair of linear equations: 2x + 3y = 11 and 3x − 2y = 4 using the elimination method.
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Section A1 mark
For the system 5x − 2y = 1 and 10x − 4y = 2, the lines are:
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Section B2 marks
Solve: x + y = 8 and 2x − y = 4.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The pair of equations 3x + 4y = 12 and 6x + 8y = 25 has no solution.
Reason:For no solution, the ratios of coefficients satisfy a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
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Section C3 marks
Two numbers have a sum of 45. Five times the smaller number exceeds the larger by 15. Form a pair of linear equations and find the numbers.
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Section E4 marksCase-based
A shop sells pencils at ₹5 each and pens at ₹10 each. On a certain day, 50 items were sold for a total of ₹400. (a) If the number of pens sold is ten more than the number of pencils sold, form the equations. (b) Find the number of pencils and pens sold.
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Section E4 marksCase-based
Aman bought 3 pens and 2 pencils for Rs 60. Beena bought 2 pens and 5 pencils for Rs 62.
(i) Form a pair of linear equations for the given situation.
(ii) Solve the equations to find the cost of one pen and one pencil.
(iii) What is the cost of 4 pens and 4 pencils?
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The system x + y = 5 and 2x + 2y = 10 has infinitely many solutions.
Reason:If a1/a2 = b1/b2 = c1/c2, the pair of linear equations is consistent and dependent.
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Polynomials
Must-crack4 Q - 9 marks2 competency Qs
High-priority pattern stack for 2027 board revision. Includes 2 competency-style questions.
Section B2 marks
If a polynomial p(x) = (x − 2)(x + 3), find all zeroes of p(x).
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Section B2 marks
For the quadratic polynomial p(x) = 2x² − 5x + 3, find the sum and product of its zeroes using coefficient–root relations.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:A quadratic polynomial can have at most two zeros.
Reason:The degree of a polynomial determines the maximum number of zeros it can have.
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Section E4 marksCase-based
A rectangular garden has its area represented by the polynomial p(x) = 6x² + 11x − 10.
(a) Find the zeros of p(x).
(b) If x represents a length in metres, which zero is valid? Why?
(c) Verify the relationship between zeros and coefficients.
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Quadratic Equations
High-ROI7 Q - 18 marks3 competency Qs
High-value pattern stack for focused 2027 revision. Includes 3 competency-style questions.
Section C3 marks
Solve the quadratic equation 2x² − 3x − 5 = 0 using the quadratic formula.
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Section A1 mark
What is the value of the discriminant for 4x² + 4x + 1 = 0 and what does it tell about the roots?
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Section B2 marks
Solve: 2x² − 7x + 3 = 0.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:If the discriminant of a quadratic equation is negative, the equation has no real roots.
Reason:The square root of a negative number is imaginary.
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Section C3 marks
The product of two consecutive odd integers is 99. Form a quadratic equation and find the integers.
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Section E4 marksCase-based
A rectangular board has an area of 96 cm². Its length is 2 cm more than three times its breadth.
(a) Set up a quadratic equation to represent the situation.
(b) Find the length and breadth of the board.
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Section E4 marksCase-based
A ball is thrown upwards and its height h (in metres) after t seconds is given by h = −5t² + 20t.
(i) After how many seconds does the ball hit the ground?
(ii) At what times is the ball at a height of 15 m?
(iii) For what value of k will −5t² + 20t = k have equal roots? For this value of k, find the time at which the ball is at height k.
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Trigonometry
Must-crack8 Q - 18 marks4 competency Qs
High-priority pattern stack for 2027 board revision. Includes 4 competency-style questions.
Section B2 marks
If sin θ = 3/5 and θ is acute, find cos θ using the identity sin²θ + cos²θ = 1.
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Section A1 mark
If cos θ = 5/13 for an acute angle θ, what is sin θ?
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Section B2 marks
Prove that sec² θ − tan² θ = 1.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The value of sin θ increases as θ increases from 0° to 90°.
Reason:In a unit circle, the y‑coordinate of a point on the circumference increases as the angle increases from 0° to 90°.
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Section C3 marks
A ladder 10 m long leans against a vertical wall making an angle of 60° with the ground. Find (i) the height at which the ladder touches the wall and (ii) the distance of the foot of the ladder from the wall.
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Section E4 marksCase-based
A kite is flying at a height of 30 m above the ground. The string makes an angle of 60° with the horizontal. The person’s hand is 1.5 m above the ground.
(a) Find the length of the string between the person’s hand and the kite.
(b) Find the horizontal distance of the kite from the person.
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Section E4 marksCase-based
From the top of a 45 m high lighthouse, the angles of depression of two ships on the same side are 30° and 60°.
(a) Draw a rough figure for the situation.
(b) Find the distance of the nearer ship from the base.
(c) Find the distance between the two ships.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:sin²θ + cos²θ = 1 for all values of θ.
Reason:In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
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Arithmetic Progressions
Good-to-do6 Q - 12 marks3 competency Qs
Useful pattern stack for completing topic coverage. Includes 3 competency-style questions.
Section A1 mark
Which of the following is an arithmetic progression (AP)?
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Section B2 marks
The 4th term of an AP is 11 and the 9th term is 26. Find the first term and common difference.
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Section C3 marks
Find the sum of first 30 terms of the AP 7, 10, 13, ...
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The sum of the first 20 terms of the AP: 1, 4, 7, 10, ... is 590.
Reason:Sₙ = n/2 [2a + (n−1)d] where a is the first term and d is the common difference.
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Section E4 marksCase-based
Riya saves ₹50 in the first week, ₹55 in the second week, ₹60 in the third week, and so on.
(i) Is this an AP? If yes, find the common difference.
(ii) How much will she save in the 15th week?
(iii) Find her total savings after 20 weeks.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The sum of the first n natural numbers is n(n+1)/2.
Reason:Natural numbers 1, 2, 3, ..., n form an AP with first term 1 and common difference 1.
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Triangles
Must-crack4 Q - 6 marks1 competency Q
High-priority pattern stack for 2027 board revision. Includes 1 competency-style question.
Section A1 mark
If two angles of one triangle are equal to two angles of another triangle, then the triangles are:
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Section C3 marks
In ΔABC, DE ∥ BC with D on AB and E on AC. If AD = 3 cm, DB = 5 cm and AE = 4.5 cm, find AC.
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Section A1 mark
In ΔPQR and ΔXYZ, if ∠P = ∠X, ∠Q = ∠Y and PQ = XY, which similarity criterion verifies ΔPQR ∼ ΔXYZ?
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:Two triangles with equal corresponding angles are similar.
Reason:AAA similarity criterion states that if all three angles of one triangle are equal to the three angles of another triangle, the triangles are similar.
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Statistics
High-ROI6 Q - 15 marks3 competency Qs
High-value pattern stack for focused 2027 revision. Includes 3 competency-style questions.
Section A1 mark
For the grouped data: class 10–20 (4), 20–30 (9), 30–40 (6), identify the modal class.
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Section B2 marks
Find the mean of the data: values 10, 20, 30 with frequencies 1, 3, 1 respectively.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The mean of a dataset is always affected by extreme values.
Reason:The mean uses every observation in its calculation.
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Section C3 marks
Calculate the mode of the following grouped data using the formula:
Class: 0–10 (3), 10–20 (8), 20–30 (12), 30–40 (7).
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Section E4 marksCase-based
The lifetimes (in hours) of 30 electric bulbs are recorded and grouped as follows:
Class: 50–60 (5), 60–70 (8), 70–80 (10), 80–90 (7).
(a) Identify the median class.
(b) Compute the median using the formula.
(c) Comment on the distribution of lifetimes.
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Section E4 marksCase-based
The following table shows the marks distribution of 50 students:
Marks: 0-10(5), 10-20(8), 20-30(12), 30-40(15), 40-50(10)
(a) Find the mean marks using the direct method.
(b) Find the median class.
(c) Calculate the mode of the data.
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Probability
High-ROI6 Q - 11 marks3 competency Qs
High-value pattern stack for focused 2027 revision. Includes 3 competency-style questions.
Section A1 mark
A card is drawn from a standard deck of 52 cards. What is the probability of drawing the queen of hearts?
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Section B2 marks
A bag contains 4 red, 5 black and 3 white balls. One ball is drawn at random. What is the probability that it is black?
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:P(E) + P(not E) = 1 for any event E.
Reason:The complement of an event and the event together cover all possible outcomes.
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Section B2 marks
Two dice are rolled simultaneously. Find the probability that the sum of the numbers obtained is 8.
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Section E4 marksCase-based
A box contains 5 red, 4 green, and 3 blue balls.
(a) A ball is drawn at random. Find the probability that it is red.
(b) Find the probability that the ball drawn is not blue.
(c) Find the probability that the ball is either green or blue.
(d) If 2 more red balls are added, what is the new probability of drawing a red ball?
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The sum of probabilities of an event and its complement is 1.
Reason:Probability of an impossible event is 0.
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Coordinate Geometry
High-ROI2 Q - 5 marks2 competency Qs
High-value pattern stack for focused 2027 revision. Includes 2 competency-style questions.
Section E4 marksCase-based
On a city map, three bus stops are located at A(2, 3), B(8, 7) and C(14, 11). A new bus stop D is to be built at the midpoint of A and C.
Based on the above, answer the following:
(i) Find the coordinates of D. [1 mark]
(ii) Show that B divides AC in the ratio 1 : 1. [1 mark]
(iii) Find the distance between A and C. [2 marks]
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The midpoint of the line segment joining (2, 3) and (4, 7) is (3, 5).
Reason:Midpoint formula is ((x₁+x₂)/2, (y₁+y₂)/2).
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Circles
Must-crack2 Q - 2 marks2 competency Qs
High-priority pattern stack for 2027 board revision. Includes 2 competency-style questions.
Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The lengths of tangents drawn from an external point to a circle are equal.
Reason:A tangent to a circle intersects the circle at exactly one point.
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Section A1 markAssertion–Reason
Assertion–Reason: refer to assertion and reason below.
Assertion:The lengths of tangents drawn from an external point to a circle are equal.
Reason:A tangent to a circle is perpendicular to the radius at the point of contact.
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Areas Related to Circles
Good-to-do1 Q - 4 marks1 competency Q
Useful pattern stack for completing topic coverage. Includes 1 competency-style question.
Section E4 marksCase-based
A designer creates a table mat in the shape of a sector of a circle with radius 28 cm and central angle 90°. (Use π = 22/7.)
(i) Find the area of the sector.
(ii) Find the length of the arc of the sector.
(iii) The mat is further cut along the radius bisecting its angle into two equal parts. Find the central angle and the area of each part.
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Surface Areas and Volumes
Must-crack1 Q - 4 marks1 competency Q
High-priority pattern stack for 2027 board revision. Includes 1 competency-style question.
Section E4 marksCase-based
A toy is in the shape of a cone mounted on a hemisphere. The radius of the hemisphere is 7 cm and the total height of the toy is 17 cm.
(a) Find the height of the cone.
(b) Find the slant height of the cone.
(c) Find the total surface area of the toy.
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