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Paper pattern

Part I · all 14 · 1 mark each · multiple choice14

Part II · answer 10 of 14 · 2 marks each · short answers, one question compulsory20

Part III · answer 10 of 14 · 5 marks each · worked answers, one question compulsory50

Part IV · answer 2 of 4 · 8 marks each · one graph and one construction, either-or choice16

Board paper total100

Tamil Nadu SSLC Mathematics paper: 100 marks, 3 hours, pass at 35. Check your school's model papers for the latest blueprint.

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2 marks · short answer

If A = {0, 2} and B = {1, 3, 5}, find A × B and B × A. Show that n(A × B) = n(B × A).Unit I If n(A) = 3 and n(B) = 4, how many relations are there from A to B?Unit I Let A = {1, 2, 3, 4} and B = {2, 4, 6, 8}. Is R = {(1, 2), (2, 4), (2, 6), (4, 8)} a function from A to B? Give two reasons.Unit I Let A = {1, 2, 3, 4} and B = {6, 9, 12}. Is f = {(1, 6), (2, 9), (3, 6), (4, 12)} one-one? Is it onto? Give reasons.Unit I Name the type of each function: f(x) = x³ − 2x, g(x) = 5, h(x) = 1/x (x ≠ 0) and p(x) = 4 − 3x.Unit I If f(x) = 3x + 1 and g(x) = x², find (f∘g)(2) and (g∘f)(2). Are they equal?Unit I Using Euclid's division algorithm, find the HCF of 252 and 594.Unit II Is 11 × 13 × 17 + 17 a composite number? Justify your answer.Unit II Find the least positive value of x such that 45 + x ≡ 2 (mod 8).Unit II Find the first four terms of the sequence aₙ = (n + 1)(n − 2).Unit II Find the sum of the infinite GP 25 + 5 + 1 + ....Unit II Find the value of 1³ + 2³ + 3³ + ... + 12³.Unit II Find the LCM of x³ − 27 and (x − 3)².Unit III Find the excluded values of (x + 3)/(x² − 5x + 6).Unit III Find the square root of 9x² − 24xy + 16y².Unit III Solve 6x² − x − 2 = 0 by factorisation.Unit III Determine the nature of the roots of 3x² − 5x + 4 = 0.Unit III Construct a 2 × 2 matrix A = [aᵢⱼ] whose elements are given by aᵢⱼ = 2i − j.Unit III ΔABC ~ ΔDEF with AB = 4 cm and DE = 6 cm. If the area of ΔDEF is 54 cm², find the area of ΔABC.Unit IV In ΔABC, DE ∥ BC with D on AB and E on AC. If AD = 2.5 cm, DB = 5 cm and AE = 3 cm, find EC.Unit IV In ΔPQR, the bisector of ∠P meets QR at S. If PQ = 5 cm, PR = 7 cm and QR = 6 cm, find QS and SR.Unit IV Check whether the triangle with sides 12 cm, 35 cm and 37 cm is right-angled.Unit IV The tangent from a point 17 cm away from the centre of a circle is 15 cm long. Find the radius of the circle.Unit IV Using Ceva's theorem, show that the medians of a triangle are concurrent.Unit IV Show that the points (−2, 5), (0, 2) and (2, −1) are collinear.Unit V Find the slope of the line joining (4, −2) and (−1, 3), and say whether its inclination is acute or obtuse.Unit V What is the slope of a line perpendicular to the line joining (1, 2) and (4, 8)?Unit V Find the equation of the line with slope −3 and y-intercept 5.Unit V Find the equation of the line whose x-intercept is −2 and y-intercept is 3.Unit V Find the x-intercept and the y-intercept of the line 4x − 5y − 20 = 0.Unit V Prove that (1 + tan²θ)(1 − sin²θ) = 1.Unit VI If tan θ = 20/21 and θ is acute, find sec θ using an identity.Unit VI A ladder makes an angle of 60° with the ground, and its foot is 2.5 m from the wall. Find the length of the ladder.Unit VI From the top of a 25 m tower, the angle of depression of a car on the road is 45°. How far is the car from the foot of the tower?Unit VI Prove that 1/(cosec θ − cot θ) = cosec θ + cot θ.Unit VI The radius and height of a cylinder are in the ratio 2 : 3 and its curved surface area is 1848 cm². Find its radius and height. (π = 22/7)Unit VII Find the slant height and curved surface area of a cone of radius 8 cm and height 15 cm. (π = 3.14)Unit VII The radii of two spheres are in the ratio 3 : 5. Find the ratio of their surface areas and of their volumes.Unit VII Find the slant height of a frustum whose end radii are 12 cm and 4 cm and whose height is 15 cm.Unit VII A tent is a cylinder of radius 3.5 m and height 3 m with a cone of slant height 5 m on top. Find the area of canvas needed. (π = 22/7)Unit VII How many spheres of radius 1 cm can be made by melting a solid sphere of radius 5 cm?Unit VII Find the range and coefficient of range of 31, 18, 27, 45, 22, 39.Unit VIII Find the standard deviation of the first 11 natural numbers.Unit VIII The mean of a data set is 18 and its standard deviation is 2.7. Find the coefficient of variation.Unit VIII Three coins are tossed together. Write the sample space and the event of getting exactly two heads.Unit VIII Find the probability that a leap year chosen at random has 53 Sundays.Unit VIII If P(A) = 0.6 and P(A ∩ B) = 0.25, find P(A ∩ B̄) and P(Ā).Unit VIII

5 marks · answer

Let A = {x ∈ N | x ≤ 2}, B = {2, 4} and C = {4, 6}. Verify that A × (B ∪ C) = (A × B) ∪ (A × C).Unit I Let A = {1, 2, 3, 4} and B = {2, 5, 7, 10, 17, 20}. R = {(x, y) | y = x² + 1} is a relation from A to B. Write R in roster form, find its domain, co-domain and range, and draw its arrow diagram. Is R a function?Unit I A function f: [1, 7) → R is defined by f(x) = 4x − 1 for 1 ≤ x < 3, f(x) = x² − 2 for 3 ≤ x < 5 and f(x) = 3x + 2 for 5 ≤ x < 7. Find (i) f(2) (ii) f(4) (iii) f(6) (iv) f(3) + f(5) (v) 2f(1) − f(6).Unit I f: N → N is defined by f(x) = 5x − 2. (i) Find the images of 1, 2 and 3. (ii) Find the pre-images of 23 and 48. (iii) Identify the type of function.Unit I A candle 30 cm long burns down 1.5 cm every hour. (i) Write its length L(t) after t hours as a function. (ii) Name the type of function. (iii) Find L(6). (iv) After how many hours does the candle burn out?Unit I If f(x) = 2x + 5 and g(x) = 3x − k, and f∘g = g∘f, find k.Unit I Find the HCF of 1716, 2002 and 3003 using Euclid's division algorithm.Unit II (i) Today is Friday. What day of the week will it be 2026 days from today? (ii) Find the remainder when 3⁴⁰ is divided by 10.Unit II In an AP, the 6th term is 23 and the 13th term is 51. Find the AP, its general term and the sum of its first 20 terms.Unit II How many terms of the AP 30, 27, 24, ... must be taken so that the sum is 135? Explain why there are two answers.Unit II The product of three consecutive terms of a GP is 1000 and their sum is 35. Find the three terms.Unit II Find the sum to n terms of the series 4 + 44 + 444 + ....Unit II A shopkeeper stacks soap boxes in a square pyramid with 15 layers. The top layer has 1 box, the next 2 × 2 boxes, and so on, down to a 15 × 15 bottom layer. How many boxes are in the stack?Unit II Solve: x + 2y − z = −1, 2x − y + z = 9, x + y + z = 4.Unit III Simplify 1/(x² − 3x + 2) + 1/(x² − 4x + 3).Unit III Find the values of a and b if 4x⁴ + 12x³ + 25x² + ax + b is a perfect square.Unit III The product of Arul's age 2 years ago and his age 3 years from now is 84. Find his present age.Unit III If α and β are the roots of 2x² − 3x − 1 = 0, form the quadratic equation whose roots are 1/α and 1/β.Unit III If A = [1, −1; 2, 3], B = [2, 1; 0, −1] and C = [1, 0; 3, 2], verify that A(B + C) = AB + AC.Unit III State and prove Thales theorem (basic proportionality theorem).Unit IV State and prove the angle bisector theorem.Unit IV State and prove Pythagoras theorem.Unit IV State and prove the alternate segment theorem.Unit IV A boy 1.5 m tall stands 7 m from the foot of a street lamp that is 5 m high. Find the length of his shadow.Unit IV In ΔABC, the cevians AD, BE and CF are concurrent. AF = 3 cm, FB = 2 cm, CE = 4 cm, EA = 3 cm and BC = 7.5 cm. Find BD and DC.Unit IV Find the value of k if the area of the triangle formed by (k, 0), (4, 0) and (0, 2) is 4 sq. units.Unit V Find the area of the quadrilateral whose vertices are P(−3, −2), Q(3, −3), R(4, 2) and S(−1, 4).Unit V Using slopes, show that A(1, 0), B(5, 2), C(4, 4) and D(0, 2) are the vertices of a rectangle.Unit V In triangle ABC, A(3, −1), B(−2, 2) and C(4, 5). Find the equation of the altitude from A.Unit V Find the equations of the lines through (2, 3) whose intercepts on the two axes add up to 10.Unit V Find the equation of the line through (4, −3) that is perpendicular to 2x + 7y − 5 = 0.Unit V If sec θ + tan θ = 7, find the values of sec θ, tan θ and sin θ.Unit VI Prove that sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2 cosec θ.Unit VI A girl 1.4 m tall stands 33.6 m from an overhead water tank and sees its top at an angle of elevation of 45°. Find the height of the top of the tank above the ground.Unit VI From the top of a lighthouse 90 m above sea level, the angles of depression of two ships on the same side are 30° and 45°. Find the distance between the ships. (√3 = 1.732)Unit VI A 6 m flagpole is fixed on top of a building. From a point on the ground, the angles of elevation of the top and the bottom of the flagpole are 60° and 45°. Find the height of the building. (√3 = 1.732)Unit VI A hollow copper pipe is 42 cm long. Its outer and inner radii are 5 cm and 4 cm. Find its curved surface area, total surface area and the volume of copper in it. (π = 22/7)Unit VII A sector of angle 216° is cut from a circle of radius 25 cm and folded into a cone. Find the radius, height and volume of the cone. (π = 3.14)Unit VII A hollow sphere has outer radius 7 cm and inner radius 3.5 cm. Find the volume of metal in it. (π = 22/7)Unit VII A flower pot open at the top is a frustum 15 cm deep. Its top and bottom radii are 12 cm and 4 cm. Find the area of material used and the volume of soil it holds. (π = 22/7)Unit VII A toy is a cone fixed on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find its volume and total surface area. (π = 22/7)Unit VII A well of inner diameter 2.8 m is dug 10 m deep. The earth taken out is spread evenly to form a platform 8 m long and 5.5 m wide. Find the height of the platform. (π = 22/7)Unit VII Find the standard deviation of 18, 25, 30, 22, 35, 20 using the assumed mean method.Unit VIII The marks of 40 students are 0-10: 5, 10-20: 8, 20-30: 12, 30-40: 10, 40-50: 5. Find the standard deviation.Unit VIII The scores of two batsmen in 6 innings are A: 38, 42, 50, 46, 54, 40 and B: 20, 65, 48, 30, 72, 35. Find the coefficient of variation of each and say who is more consistent.Unit VIII Two dice are rolled. Find the probability of getting (i) a sum of 9 (ii) a doublet (iii) a sum of at most 4.Unit VIII A number is chosen at random from 1 to 50. A: it is a multiple of 4; B: it is a multiple of 6. Find P(A ∪ B), P(A ∩ B̄) and the probability that it is neither.Unit VIII A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is a spade or a face card.Unit VIII

8 marks · graph or construction

Draw the graph of y = x² + x − 6 and hence solve x² + x − 6 = 0.Unit III Draw the graph of y = x² + 2x − 3 and use it to solve x² + x − 2 = 0.Unit III Draw the graph of y = x² − 4x + 4 and state the nature of the solutions of x² − 4x + 4 = 0.Unit III A water pump fills 15 litres every minute. Draw the graph of water filled against time. From the graph, find (i) the water filled in 5½ minutes (ii) the time taken to fill 120 litres.Unit III Draw the graph of xy = 18 for x > 0, y > 0. From the graph, find (i) y when x = 4 (ii) x when y = 1.5.Unit III ΔABC has BC = 6 cm, AB = 5 cm and AC = 4 cm. Construct a triangle similar to it with sides equal to 2/3 of the corresponding sides of ΔABC.Unit IV ΔPQR has QR = 5 cm, PQ = 4 cm and PR = 4.5 cm. Construct a triangle similar to it with sides equal to 5/3 of the corresponding sides of ΔPQR.Unit IV Construct ΔPQR with QR = 7 cm, ∠P = 70° and the median PM from P to QR = 4.5 cm. Measure the altitude from P to QR.Unit IV Construct ΔABC with BC = 6 cm, ∠A = 55° and the altitude from A to BC = 4 cm.Unit IV Draw ΔABC with BC = 6.2 cm and ∠A = 45°. The bisector of ∠A meets BC at D, with BD = 3.5 cm.Unit IV Draw a circle of radius 3.5 cm. From a point 9 cm from its centre, draw the pair of tangents. Measure their lengths and verify by calculation.Unit IV Draw a circle of radius 4.2 cm and mark a point P on it. Construct the tangent at P without using the centre, by the alternate segment theorem.Unit IV

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