CBSE / NCERT Class 10 Maths Practice Question Papers with Answers (2026-27)

Brain Grain · braingrain.in

Maths — Practice Paper · Set 1

Class: 10CBSE / NCERTMax Marks: 42

Name: ____________________Reg No: ____________

Part I — Multiple Choice Questions 6 × 1 = 6

Choose the correct answer. (Answer all questions.)

1.If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of $80^\circ$ , then $\angle POA$ is equal toA. $50^\circ$B. $60^\circ$C. $70^\circ$D. $80^\circ$[1]

2.Tick the correct answer in the following : Area of a sector of angle $p$ (in degrees) of a circle with radius $R$ isA. $\dfrac{p}{180}\times2\pi R$B. $\dfrac{p}{180}\times\pi R^2$C. $\dfrac{p}{360}\times\pi R^2$D. $\dfrac{p}{720}\times2\pi R^2$[1]

3.In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that $\angle POQ = 110^\circ$ , then $\angle PTQ$ is equal toA. $60^\circ$B. $70^\circ$C. $80^\circ$D. $90^\circ$[1]

4.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :A. 12 cmB. 13 cmC. 8.5 cmD. $\sqrt{119}$ cm[1]

5.Which of the following cannot be the probability of an event?A. $\dfrac{2}{3}$B. $-1.5$C. $15\%$D. $0.7$[1]

6.From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle isA. 7 cmB. 12 cmC. 15 cmD. 24.5 cm[1]

Part II — Short Answer Questions 18 × 2 = 36

Answer briefly. (Answer all questions.)

7.Which of the following are APs ? If they form an AP, find the common difference d and write three more terms. (i) 2, 4, 8, 16, . . . (ii) $2, \dfrac{5}{2}, 3, \dfrac{7}{2}, . . .$ (iii) - 1.2, - 3.2, - 5.2, - 7.2, . . . (iv) - 10, - 6, - 2, 2, . . . (v) $3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, . . .$ (vi) 0.2, 0.22, 0.222, 0.2222, . . . (vii) 0, - 4, - 8, -12, . . . (viii) $-\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}, . . .$ (ix) 1, 3, 9, 27, . . . (x) a, 2a, 3a, 4a, . . . (xi) $a, a^2, a^3, a^4, . . .$ (xii) $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, . . .$ (xiii) $\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, . . .$ (xiv) $1^2, 3^2, 5^2, 7^2, . . .$ (xv) $1^2, 5^2, 7^2, 7^3, . . .$[2]

8.The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.[2]

9.A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.[2]

10.The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.[2]

11.In Fig. 6.18, if LM || CB and LN || CD, prove that $\dfrac{AM}{AB} = \dfrac{AN}{AD}$ .[2]

12.The distribution below gives the weights of 30 students of a class. Find the median weight of the students. Weight (in kg): 40 - 45, 45 - 50, 50 - 55, 55 - 60, 60 - 65, 65 - 70, 70 - 75 Number of students: 2, 3, 8, 6, 6, 3, 2[2]

13.A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$ .[2]

14.Find the sum of the first 40 positive integers divisible by 6.[2]

15.Given 15 cot A = 8, find sin A and sec A.[2]

16.In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii). In (i), AD = 1.5 cm, DB = 3 cm and AE = 1 cm. In (ii), DB = 7.2 cm, AE = 1.8 cm and EC = 5.4 cm.[2]

17.A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig. 11.8). Find (i) the area of that part of the field in which the horse can graze. (ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use $\pi = 3.14$ )[2]

18.In a circle of radius 21 cm, an arc subtends an angle of $60^\circ$ at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord[2]

19.Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.[2]

20.The table below shows the daily expenditure on food of 25 households in a locality. Daily expenditure (in ₹): 100 - 150, 150 - 200, 200 - 250, 250 - 300, 300 - 350 Number of households: 4, 5, 12, 2, 2 Find the mean daily expenditure on food by a suitable method.[2]

21.Find the number of terms in each of the following APs : (i) 7, 13, 19, . . . , 205 (ii) $18, 15\dfrac{1}{2}, 13, . . . , - 47$[2]

22.A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see Fig. 12.16).[2]

23.A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red ? (ii) not red?[2]

24.In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that: (i) $\triangle ABC \sim \triangle AMP$ (ii) $\dfrac{CA}{PA} = \dfrac{BC}{MP}$[2]

🔑 Show Answer Key — Set 1

1. Correct option: (A) $50^\circ$ .
2. Correct option: (C) $\dfrac{p}{360}\times\pi R^2$ .
3. Correct option: (B) $70^\circ$ .
4. Correct option: (D) $\sqrt{119}$ cm.
5. Correct option: (B) $-1.5$ .
6. Correct option: (A) 7 cm.
7. (i) Not an AP. (ii) AP; $d = \dfrac{1}{2}$ ; next terms $4, \dfrac{9}{2}, 5$ . (iii) AP; $d = -2$ ; next terms $-9.2, -11.2, -13.2$ . (iv) AP; $d = 4$ ; next terms $6, 10, 14$ . (v) AP; $d = \sqrt{2}$ ; next terms $3 + 4\sqrt{2}, 3 + 5\sqrt{2}, 3 + 6\sqrt{2}$ . (vi) Not an AP. (vii) AP; $d = -4$ ; next terms $-16, -20, -24$ . (viii) AP; $d = 0$ ; next terms $-\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}$ . (ix) Not an AP. (x) AP; $d = a$ ; next terms $5a, 6a, 7a$ . (xi) Not an AP in general. (xii) AP; $d = \sqrt{2}$ ; next terms $\sqrt{50}, \sqrt{72}, \sqrt{98}$ . (xiii) Not an AP. (xiv) Not an AP. (xv) Not an AP.
8. The altitude is 5 cm and the base is 12 cm.
9. The height of the pedestal is $0.8(\sqrt3+1)$ m.
10. $x = 35$ .
11. $\dfrac{AM}{AB} = \dfrac{AN}{AD}$ .
12. The median weight is approximately $56.67$ kg.
13. The volume of the solid is $\pi$ cm $^3$ .
14. The sum is 4920.
15. $\sin A=\dfrac{15}{17}$ and $\sec A=\dfrac{17}{8}$ .
16. (i) $EC = 2\text{ cm}$ . (ii) $AD = 2.4\text{ cm}$ .
17. (i) The grazing area is $19.625$ m $^2$ . (ii) The increase in grazing area is $58.875$ m $^2$ .
18. (i) Length of the arc $=22$ cm. (ii) Area of the sector $=231$ cm $^2$ . (iii) Area of the segment $=\left(231-\dfrac{441\sqrt3}{4}\right)$ cm $^2$ (about $40.04$ cm $^2$ ).
19. The sum is 1661.
20. The mean daily expenditure on food is $₹211$ .
21. (i) 34 terms. (ii) 27 terms.
22. The volume of wood in the pen stand is approximately $523.53$ cm $^3$ .
23. (i) $\dfrac{3}{8}$ (ii) $\dfrac{5}{8}$
24. (i) $\triangle ABC \sim \triangle AMP$ (ii) $\dfrac{CA}{PA} = \dfrac{BC}{MP}$

Brain Grain · braingrain.in

Maths — Practice Paper · Set 2

Class: 10CBSE / NCERTMax Marks: 42

Name: ____________________Reg No: ____________

Part I — Multiple Choice Questions 6 × 1 = 6

Choose the correct answer. (Answer all questions.)

1.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :A. 12 cmB. 13 cmC. 8.5 cmD. $\sqrt{119}$ cm[1]

2.Which of the following cannot be the probability of an event?A. $\dfrac{2}{3}$B. $-1.5$C. $15\%$D. $0.7$[1]

3.From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle isA. 7 cmB. 12 cmC. 15 cmD. 24.5 cm[1]

4.If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of $80^\circ$ , then $\angle POA$ is equal toA. $50^\circ$B. $60^\circ$C. $70^\circ$D. $80^\circ$[1]

5.Tick the correct answer in the following : Area of a sector of angle $p$ (in degrees) of a circle with radius $R$ isA. $\dfrac{p}{180}\times2\pi R$B. $\dfrac{p}{180}\times\pi R^2$C. $\dfrac{p}{360}\times\pi R^2$D. $\dfrac{p}{720}\times2\pi R^2$[1]

6.In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that $\angle POQ = 110^\circ$ , then $\angle PTQ$ is equal toA. $60^\circ$B. $70^\circ$C. $80^\circ$D. $90^\circ$[1]

Part II — Short Answer Questions 18 × 2 = 36

Answer briefly. (Answer all questions.)

7.If $\cot \theta = \dfrac{7}{8}$ , evaluate : (i) $\dfrac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)}$ , (ii) $\cot^2\theta$[2]

8.State whether the following are true or false. Justify your answer. (i) sin (A + B) = sin A + sin B. (ii) The value of sin $\theta$ increases as $\theta$ increases. (iii) The value of cos $\theta$ increases as $\theta$ increases. (iv) sin $\theta$ = cos $\theta$ for all values of $\theta$ . (v) cot A is not defined for A = 0°.[2]

9.A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use $\pi = 3.14$ )[2]

10.The following table shows the ages of the patients admitted in a hospital during a year: Age (in years): 5 - 15, 15 - 25, 25 - 35, 35 - 45, 45 - 55, 55 - 65 Number of patients: 6, 11, 21, 23, 14, 5 Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.[2]

11.Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in ₹): 500 - 520, 520 - 540, 540 - 560, 560 - 580, 580 - 600 Number of workers: 12, 14, 8, 6, 10 Find the mean daily wages of the workers of the factory by using an appropriate method.[2]

12.A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.[2]

13.It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?[2]

14.A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.[2]

15.The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures. Number of students per teacher: 15 - 20, 20 - 25, 25 - 30, 30 - 35, 35 - 40, 40 - 45, 45 - 50, 50 - 55 Number of states / U.T.: 3, 8, 9, 10, 3, 0, 0, 2[2]

16.The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.[2]

17.Represent the following situations in the form of quadratic equations: (i) The area of a rectangular plot is $528\ \text{m}^2$ . The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. (ii) The product of two consecutive positive integers is 306. We need to find the integers. (iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age. (iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.[2]

18.A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of $\dfrac{1}{4}$ m and a tread of $\dfrac{1}{2}$ m. Calculate the total volume of concrete required to build the terrace.[2]

19.A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.[2]

20.The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.[2]

21.Find the LCM and HCF of the following integers by applying the prime factorisation method: (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25[2]

22.If $P(E) = 0.05$ , what is the probability of 'not E'?[2]

23.State whether the following quadrilaterals are similar or not: Fig. 6.8 shows quadrilateral PQRS with each side 1.5 cm and quadrilateral ABCD with each side 3 cm, with right angles marked in ABCD.[2]

24.To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs $\dfrac{1}{4}$ th the distance AD on the 2nd line and posts a green flag. Preet runs $\dfrac{1}{5}$ th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?[2]

🔑 Show Answer Key — Set 2

1. Correct option: (D) $\sqrt{119}$ cm.
2. Correct option: (B) $-1.5$ .
3. Correct option: (A) 7 cm.
4. Correct option: (A) $50^\circ$ .
5. Correct option: (C) $\dfrac{p}{360}\times\pi R^2$ .
6. Correct option: (B) $70^\circ$ .
7. (i) $\dfrac{49}{64}$ (ii) $\dfrac{49}{64}$
8. (i) False (ii) True for $0^\circ\le\theta\le90^\circ$ (iii) False (iv) False (v) True
9. (i) Area of the minor segment $=28.5$ cm $^2$ . (ii) Area of the major sector $=235.5$ cm $^2$ .
10. Mode $\approx36.82$ years and mean $=35.375$ years. The two values are close; the most common age is about $36.82$ years, while the average age is about $35.38$ years.
11. The mean daily wage is $₹545.20$ .
12. The volume of water left is $\dfrac{7920000}{7}$ cm $^3$ or approximately $1131428.57$ cm $^3$ .
13. The probability is $0.008$ .
14. (i) $\dfrac9{10}$ (ii) $\dfrac1{10}$ (iii) $\dfrac15$
15. Mode $=30.625$ students per teacher and mean $\approx29.21$ students per teacher. The most common ratio is about $31$ students per teacher, while the average is about $29$ students per teacher.
16. The height of the building is $\dfrac{50}{3}$ m.
17. (i) $2x^2 + x - 528 = 0$ , where $x$ is the breadth. (ii) $x^2 + x - 306 = 0$ , where $x$ is the smaller integer. (iii) $x^2 + 32x - 273 = 0$ , where $x$ is Rohan’s present age. (iv) $x^2 - 8x - 1280 = 0$ , where $x$ is the train’s speed in km/h.
18. The total volume of concrete required is $750\text{ m}^3$ .
19. The greatest diameter is 7 cm, and the surface area of the solid is $332.5$ cm $^2$ .
20. The radius of the circle is 3 cm.
21. (i) HCF $= 3$ , LCM $= 420$ . (ii) HCF $= 1$ , LCM $= 11339$ . (iii) HCF $= 1$ , LCM $= 1800$ .
22. The probability of 'not E' is $0.95$ .
23. The quadrilaterals are not similar.
24. The distance between the green and red flags is $\sqrt{61}$ m. Rashmi should post the blue flag at $(5,22.5)$ .

Brain Grain · braingrain.in

Maths — Practice Paper · Set 3

Class: 10CBSE / NCERTMax Marks: 42

Name: ____________________Reg No: ____________

Part I — Multiple Choice Questions 6 × 1 = 6

Choose the correct answer. (Answer all questions.)

3.In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that $\angle POQ = 110^\circ$ , then $\angle PTQ$ is equal toA. $60^\circ$B. $70^\circ$C. $80^\circ$D. $90^\circ$[1]

4.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :A. 12 cmB. 13 cmC. 8.5 cmD. $\sqrt{119}$ cm[1]

5.Which of the following cannot be the probability of an event?A. $\dfrac{2}{3}$B. $-1.5$C. $15\%$D. $0.7$[1]

6.From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle isA. 7 cmB. 12 cmC. 15 cmD. 24.5 cm[1]

Part II — Short Answer Questions 18 × 2 = 36

Answer briefly. (Answer all questions.)

7.In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?[2]

8.Find the coordinates of the points which divide the line segment joining A(- 2, 2) and B(2, 8) into four equal parts.[2]

9.The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure : Expenditure (in ₹): 1000 - 1500, 1500 - 2000, 2000 - 2500, 2500 - 3000, 3000 - 3500, 3500 - 4000, 4000 - 4500, 4500 - 5000 Number of families: 24, 40, 33, 28, 30, 22, 16, 7[2]

10.If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.[2]

11.12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.[2]

12.A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm $^3$ . Check whether she is correct, taking the above as the inside measurements, and $\pi = 3.14$ .[2]

13.A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that (i) She will buy it ? (ii) She will not buy it ?[2]

14.If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.[2]

15.The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.[2]

16.In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct. In Fig. 7.8, A(3, 4), B(6, 7), C(9, 4) and D(6, 1).[2]

17.Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method. Number of heartbeats per minute: 65 - 68, 68 - 71, 71 - 74, 74 - 77, 77 - 80, 80 - 83, 83 - 86 Number of women: 2, 4, 3, 8, 7, 4, 2[2]

18.Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (i) (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0) (ii) (-3, 5), (3, 1), (0, 3), (-1, - 4) (iii) (4, 5), (7, 6), (4, 3), (1, 2)[2]

19.Prove that the following are irrational: (i) $\dfrac{1}{\sqrt{2}}$ (ii) $7\sqrt{5}$ (iii) $6 + \sqrt{2}$[2]

20.Find the distance between the following pairs of points : (i) (2, 3), (4, 1) (ii) (- 5, 7), (- 1, 3) (iii) (a, b), (- a, - b)[2]

21.The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate. Literacy rate (in %): 45 - 55, 55 - 65, 65 - 75, 75 - 85, 85 - 95 Number of cities: 3, 10, 11, 8, 3[2]

22.200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?[2]

23.If the 3rd and the 9th terms of an AP are 4 and - 8 respectively, which term of this AP is zero?[2]

24.A die is thrown once. Find the probability of getting (i) a prime number; (ii) a number lying between 2 and 6; (iii) an odd number.[2]

🔑 Show Answer Key — Set 3

1. Correct option: (A) $50^\circ$ .
2. Correct option: (C) $\dfrac{p}{360}\times\pi R^2$ .
3. Correct option: (B) $70^\circ$ .
4. Correct option: (D) $\sqrt{119}$ cm.
5. Correct option: (B) $-1.5$ .
6. Correct option: (A) 7 cm.
7. 234 trees will be planted.
8. $(-1,\dfrac{7}{2})$ , $(0,5)$ and $(1,\dfrac{13}{2})$ .
9. The modal monthly expenditure is approximately $₹1847.83$ , and the mean monthly expenditure is $₹2662.50$ .
10. $S_n = n^2$ .
11. The probability is $\dfrac{11}{12}$ .
12. The calculated capacity is approximately $346.51$ cm $^3$ , so the child's measurement of $345$ cm $^3$ is approximately correct.
13. (i) $\dfrac{31}{36}$ (ii) $\dfrac{5}{36}$
14. $x=6$ and $y=3$ .
15. $n = 16$ and $d = \dfrac{8}{3}$ .
16. Champa is correct; ABCD is a square.
17. The mean heartbeats per minute is $75.9$ .
18. (i) A square. (ii) A quadrilateral with no special type indicated by equal/parallel side tests. (iii) A parallelogram.
19. In each part we use that $\sqrt{2}$ and $\sqrt{5}$ are irrational, and argue by contradiction. (i) If $\dfrac{1}{\sqrt{2}}$ were rational, say $\dfrac{1}{\sqrt{2}} = r$ , then $\sqrt{2} = \dfrac{1}{r}$ would be rational — a contradiction. So $\dfrac{1}{\sqrt{2}}$ is irrational. (ii) If $7\sqrt{5}$ were rational, say $7\sqrt{5} = r$ , then $\sqrt{5} = \dfrac{r}{7}$ would be rational — a contradiction. So $7\sqrt{5}$ is irrational. (iii) If $6 + \sqrt{2}$ were rational, say $6 + \sqrt{2} = r$ , then $\sqrt{2} = r - 6$ would be rational — a contradiction. So $6 + \sqrt{2}$ is irrational.
20. (i) $2\sqrt{2}$ (ii) $4\sqrt{2}$ (iii) $2\sqrt{a^2+b^2}$
21. The mean literacy rate is approximately $69.43\%$ .
22. The logs are placed in 16 rows, with 5 logs in the top row.
23. The 5th term is zero.
24. (i) $\dfrac12$ (ii) $\dfrac12$ (iii) $\dfrac12$

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