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Gulf Indian High School, Dubai

QUESTION BANK 2017 - 2018

Grade – X MATHEMATICS _____________________________________________________________________________

REAL NUMBERS

1. After how many places of decimal the decimal expansion of rational number 𝟒𝟕

𝟓 .𝟐𝟐

will terminate?

2. Find the largest number which divides 71 and 126, leaving remainders 6 and 9

respectively,

3. If x = 23 × 3 × 52, y = 22 × 33 Find the HCF and LCM of x and y.

4. Show that any positive odd integer is of the form 4m + 1 or 4m + 3, where m is

some integer.

5. Show that 5 - √𝟑 is irrational.

6. Can the number 6n, n being a natural number, can end with digit 5? Give reasons.

7. Find the LCM of 120 and 70 by fundamental theorem of arithmetic.

8. Find the HCF (867 , 255 ) using Euclid`s division lemma.

9. Write down the decimal expansion of 𝟏𝟔

𝟑𝟏𝟐𝟓 without actual division.

10. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the

maximum capacity of a container which can measure the petrol of either tanker

in exact number of times.

11. A charitable trust donates 28 different books of Maths, 16 different books of

Social science and 12 diiferent books of science to the poor students.

12. Show that m3 – m is divisible by 6 for each natural number m.

13. Show that the square of any positive integer is of the form 5k, 5k +1 or 5k +4 for

any natural number k.

14. Find the number which when divided by 117 gives 41 as quotient and 23 as

remainder.

15. Find the maximum number of boxes into which 1134 apples and 1215 oranges be

distributed so that each box contains the same number of apples and oranges

16. In a school, the duration of a period in junior section is 40 minutes and in senior

section is 60 minutes. If the first bell for each section rings at 9 a.m., when will

the two bells ring together again?

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17. Find the largest positive integer that will divide 398,436 and 542 leaving

remainders 7, 11 and 15 respectively.

18. Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.

19. Find the smallest number which when increased by 17 is exactly divisible by

both 520 and 468.

20. Determine the number nearest to 100000 but greater than 100000 which is

exactly divisible by each 8, 15 and 21.

21. Use Euclid`s division algorithm to find the HCF of 135 and 225.

22.

23.

24.

25.

26.

27.

28.

29.

30. Two milk containers contain 398 l and 436 l of milk. The milk is to be transferred to

another container with the help of a drum. While transferring to another container 7 l

and 11 l of milk are left in both the containers respectively. What will be the maximum

capacity of the drum?

31.

32.

33.

34.

35.

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POLYNOMIALS

1. Find the sum and product of zeroes of 3x2 + 5x + 6

2. Find the zeroes of the quadratic polynomial 6y2 – 7y – 3.

3. Find the zeroes of the polynomials p(x) = 2 – x2.

4. If 𝜶, 𝜷 𝒂𝒏𝒅 𝜸 are the zeroes of the polynomial p(x) such that (𝜶 + 𝜷 + 𝜸) = 3,

(𝜶𝜷 + 𝜷 𝜸 + 𝜸𝜶) = -10 and 𝜶𝜷𝜸 = - 24 then find p (x).

5. If 𝜶 𝒂𝒏𝒅 𝜷 are the zeroes of the quadratic polynomials 5y2 – 7y + 2, find the sum of their

reciprocals.

6. If 𝜶 𝒂𝒏𝒅 𝜷 are the zeroes of the quadratic polynomials 4x2 + 3x + 7, then find the value

of 𝟏

𝜶 +

𝟏

𝜷.

7. If 𝜶 𝒂𝒏𝒅 𝜷 are the zeroes of the quadratic polynomials 6x2 + x – 2 = 0, then find

𝜶

𝜷 +

𝜷

𝜶.

8. If 𝜶 𝒂𝒏𝒅 𝜷 are the zeroes of the quadratic polynomials z2 – 4z + 3 then form the

quadratic polynomials whose zeroes are 𝟑𝜶 𝒂𝒏𝒅 𝟑𝜷.

9. If 𝜶 𝒂𝒏𝒅 𝜷 are the zeroes of the quadratic polynomials x2 – 6x + a; find the value of ‘a’,

if 3𝜶 + 𝟐𝜷 = 20.

10. If 𝜶 𝒂𝒏𝒅 𝜷 are the zeroes of the quadratic polynomial x2 + x – 2 then find the value of

(𝟏

𝜶 -

𝟏

𝜷 ).

11. Find a quadratic polynomial whose zeroes are 5 + √𝟐 and 5 - √𝟐

12. Sum of product of zeros of quadratic polynomial are 5 and 17 respectively. Find the

polynomial.

13. Form a quadratic polynomial one of whose zero is 2 + √𝟓 and the sum of zeroes is 4.

14. In a graph of y = p (x), find the number of zeroes of p(x)

15. If the sum of zeroes of a given polynomial f(x) = x3 – 3k x

2 – x + 30 is 6. Find the value of

k.

16.

17.

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18.

19.

20.

21.

22.

23.

24.

25. If 𝜶 𝒂𝒏𝒅 𝜷 𝒂𝒓𝒆 𝒕𝒉𝒆 𝒛𝒆𝒓𝒐𝒆𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 k x2 + 4 x + 4.

Show that 𝜶𝟐 + 𝜷𝟐 = 24. Find the value of k

26.

27.

28.

29.

30.

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LINEAR EQUATIONS IN TWO VARIABLES

1. Draw the graphs of pair of linear equations x – y + 2 = 0 and 4 x – y – 4 = 0. Calculate

the area of the triangle formed by the lines so drawn and the x-axis.

2. Solve the following system of linear equations graphically: x – y = 1; 2x + y = 8. Shade

the area bounded by these lines and y axis. Also determine this area.

3. Solve the linear equations graphically: 2x – y – 4 = 0 and x + y + 1 = 0. Find the points

where the lines meet y axis.

4. Dinesh is walking along the line joining (1, 4) and (0, 6), Naresh is walking along the line

joining (3, 4,) and (1, 0). Represent on graph and find the point where both of them cross

each other.

5. Draw the graph of the following pair of linear equations x + 3 y = 6 and 2 x – 3 y = 12.

Find the ratio of the areas of the two triangles formed by first line, x = 0, y = 0 and

second line x = 0 , y = 0

6. Solve the following system of equations graphically and find the vertices of triangle

formed by these lines and the y- axis. 3x + y – 5 = 0; 2x – y − 5 = 0

7. Solve graphically the pair of equations 2x + 3y = 11 and 2x – 4y = -24. Hence the value of

co-ordinate of the vertices of triangle so formed.

8. Solve the following system of equations graphically and find the vertices of triangle

formed by these lines and the x- axis.

(i) 4x – 3y + 4 = 0; 4x + 3y − 20 = 0 (ii) 2x – 3y - 4 = 0; x – y − 1 = 0

9. Draw the graph of the equations x − y = 1 and 2x + y = 8. Shade the region bounded by

these lines and y- axis. Find the area of the shaded region.

10. The sum of digits of two digit number and the number obtained by interchanging the

digits is 132. If the two digits are differing by 2, find the numbers.

11. The sum of the two numbers is 18. The sum of their reciprocals is ¼ . Find the numbers.

[Ans: 12, 6]

12. Father’s age is three times the sum of ages of his two children. After 5 years his age will

be twice the sum of ages of two children. Find the age of father. [Ans: 45 years]

13. The age of the father is 3 years more than 3 times the son’s age. 3 years hence, the age of

the father will be 10 years more than twice the age of the son. Find their present ages.

[Ans: 33 yrs; 10 yrs]

14. A plane left 30 minutes later than the scheduled time and in order to reach the

destination 1500 km away in time, it has to increase the speed by 250 km/ hr from the

usual speed. Find its usual speed [Ans: 750km/hr]

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15. If in a rectangle, the length is increased and breadth reduced each by 2 metres, the area

is reduced by 28 sq. metres. If the length is reduced by 1 metre and breadth increased by

2 metres, the area increases by 33 sq. metres. Find the length and breadth of the

rectangle. [Ans: 23; 11 metres]

16. A passenger train takes 2 hours less for a journey of 300 km, if its speed is increased by

5 km/hour from its usual speed. Find its usual speed. [Ans: 25 km/hr]

17. A person rowing a boat at the rate of 5km/hour in still water takes thrice as much time

in going 40 km upstream as in going 40 km downstream. Find the speed of the stream

[Ans:2.5 km/hr]

18. ̀ 9,000 were divided equally among a certain number of persons. Had there been 20

more persons, each would have got ` 160 less. Find the original number of persons.

[Ans: 25]

19. Some students planned a picnic. The budget for food was ` 500. But 5 of these failed to

go and thus the cost of food for each member for each member increased by ` 5. How

many students attended the picnic? [Ans: 20]

20. Students of a class are made to stand in rows. If 4 students are extra in a row, there

would be 2 rows less. If 4 students are less in a row, there would be 4 more rows. Find

the number of students in the class. [Ans: 96]

21. A part of monthly hostel charges is fixed and the remaining depends on the number of

days one has taken food in the mess. When a student, A takes food for 20 days, A has to

pay ` 1000 as hostel charges, whereas a student B, who takes food for 26 days, pays `

1180 as hostel charges. Find the fixed charge and the cost of food per day.

[Ans: 400; 30]

22. Jamila sold a table and a chair for ` 1050, thereby making a profit of 10% on the table

and 25% on the chair. If she had taken a profit of 25% on the table and 10% on the

chair she would have got ` 1065. Find the cost price of each.

[Ans: ` 500; ` 400]

23. After covering a distance of 30 km with a uniform speed there is some defect in a train

engine and therefore, its speed is reduced to 𝟒

𝟓 th

of its original speed. Consequently the

train reaches its destination late by 45 minutes. Had it happened after covering 18

kilometers more, the train would have reached 9 minutes earlier. Find the speed of the

train and the distance of Journey. [Ans: 30 km/hr; 120 km]

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24. 90% and 97% pure acid solutions are mixed to obtain 21 litres of 95% pure acid

solutions. Find the amount of each type of acid to be mixed to form the mixture.

[Ans: 6; 15]

25. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger

diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the

pool can be filled. How long would it take for each pipe to fill the pool separately?

[Ans: 20; 30]

26. Two candles of equal heights but different thickness are lighted. The first burns off in 6

hours and the second in 8 hours. How long after lighting the both, will the first candle be

half the height of the second? [Ans: 4.8]

27. On selling a tea-set at 5% loss and a lemon-set at 15% gain, a crockery seller gains ` 7.

If he sells the tea-set at 5% gain and the lemon-set at 10% gain, he gains ` 13. Find the

actual price of the tea-set and the lemon-set? [Ans: 100; 80]

28. P takes 3 hours more than Q to walk 30 km. But, if P doubles his pace, he is ahead of Q

by 1 ½ hours. Find their speed of walking. [Ans: 10/3 ; 5 km/hr]

29. In a competitive examination, one mark is awarded for each correct answer while ½

mark is deducted for each wrong answer. Jayanthi answered 120 questions and got 90

marks. How many questions did she answer correctly? [Ans: 100]

30. In a function if 10 guests are sent from room A to B, the no. of guests in room A and B

are same. If 20 guests are sent from B to A, the no. of guests in A is double the no. of

guests in B. Find no. of guests in both the rooms in the beginning.

31. At a certain time in a deer park, the number of heads and the number of legs of deer and

human visitors were counted and it was found there were 39 heads and 132 legs. Find

the number of deer and human visitors in the park [Ans: 27; 12]

32. Solve: x + y + 2z = 9 ; 2x – y + 2 z = 6 ; 3x + y + 4 z = 17 [ Ans: 1; 2; 3]

33. Find the value of m for which the pair of linear equations 2x + 3 y – 7 = 0 and

(m – 1)x + (m + 1)y = (3m – 1) has infinitely many solutions

34. For what value of p will the following system of equations has no solution:

(2p – 1) x + (p – 1) y = 2p + 1; y + 3 x – 1 = 0

35. Find the values of a and b for which the following pair of linear equations has infinitely

many solutions:

2 x + 3 y = 7; ( a + b) x + ( 2 a – b) = 21 (ii) (a – b) x + (a + b) y = 3 a + b - 2

36. Check whether the following pair of linear equations is consistent or not:

x + y = 14; x – y = 4

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37. For what value of k the following pair of linear equations has unique solution?

7 x + 8 y = k ; 9 x – 4 y = 12

38. Find the value of k for which k x + 3 y = k – 3; 12 x + k y = k represent coincident lines.

39. Solve 41 x + 53 y = 135 ; 53 x + 41 y = 147

40.

41.

42.

43. Solve the following equations for x and y : 𝒙

𝒂 +

𝒚

𝒃 = 2

ax – by = a2 – b

2 .

44. Solve for x and y : 𝒙

𝒂 =

𝒚

𝒃

ax + by = a2 + b

2 .

45. Solve the following system of equations by cross – multiplication method

(a) a x + b y = 1

b x + a y = (𝒂+𝒃)𝟐

𝒂𝟐+ 𝒃𝟐 – 1 Ans:

𝒂

𝒂𝟐+ 𝒃𝟐;

𝒃

𝒂𝟐+ 𝒃𝟐

(b) a ( x+ y) + b ( x – y ) = a2 – ab + b

2

a (x + y ) – b ( x – y) = a2 + ab + b

2 Ans: x = b

2 / 2a ; y =

𝟐 𝒂 𝟐+ 𝒃𝟐

𝟐 𝒂

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46. Solve : 𝒂

𝒙 -

𝒃

𝒚 = 0 ;

𝒂 𝒃𝟐

𝒙 +

𝒂𝟐𝒃

𝒚 = a

2 + b

2; where x ≠ 0 ; y ≠ 𝟎

47.

48.

49. Solve the following system of equations:

(a) 3 x – y = 3 ; 7 x + 2y = 20 (b) 3 x – 5 y = - 1 ; x – y = - 1 (c) x + y = 7 ; 5 x + 12 y = 7

50. Solve the following pair of equations:

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TRAIANGLES

1) In the adjoining figure find AE if DE || BC

2) In ∆PQR, DE || QR and DE = 𝟏

𝟒 QR . Find

𝒂𝒓 (∆ 𝐏𝐐𝐑)

𝒂𝒓 ( ∆𝐏𝐃𝐄 )

3) The ratio of the corresponding altitudes of two similar triangles is 3/5. Is it correct to

say that the ratio of their areas is 6/5? Why?

4) The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one side of

the first triangle is 12 cm, determine the corresponding side of the second triangle.

5) Prove that the equilateral triangles descried on the two sides of a right angled triangle

are together equilateral triangle on the hypotenuse in terms of their areas.

6) Prove that the area of an equilateral triangle described on the side of a right angled

isosceles triangle is half the area of the equilateral triangle described on its hypotenuse.

7) If the areas of two similar triangles are equal, prove that they are congruent

8) In figure, DEFG is a square and ∟BAC = 900. Show that DE

2 = BD x EC

9) In an equilateral ∆ABC, AD is altitude from A on BC. Prove that 3AB2 = 4 AD

2

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10) In a right triangle ABC right angled at B. Side BC is trisected at points D and E. Prove

that 8 AE2 = 3 AC

2 + 5 AD

2

11) ∆ABC is an isosceles triangle in which AB = AC and D is a point on BC.

Prove that AB2 – AD

2 = BD x CD.

12)

13)

14) In the given figure PQ || BA; PR || CA. If PD = 12 cm find BD x CD

15) In a right triangle ABC, right angled at C, P and Q are points of the sides CA and CB

respectively, which divide these sides in the ratio 2 : 1. Prove that 9 AQ2 = 9 AC

2 + 4

BC2

16) There is a staircase as shown in figure connecting points A and B. Measurements of

steps are marked in the figure. Find the straight distance between A and B.

(Ans: 10)

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17) ABC is an isosceles triangle in which AB = AC = 10 cm. BC = 12 cm. PQRS is a

rectangle inside the isosceles triangle. Given PQ = SR = y cm, PS = QR = 2 x. Prove that

x = 6 – 3y/4

18) ABCD is a rectangle. ∆ ADE and ∆ ABF are two triangles such that ∟𝑬 = ∟𝑭 as shown

in the figure. Prove that AD x AF = AE x AB

19) Amit is standing at a point on the ground 8 m away from a house. A mobile network is

fixed on the roof of the house. If the top and bottom of the tower are 17 m and 10 m

away from the point. Find the heights of the tower of the house.

20) A man goes 18 m due east and then 24 m due north. Find the distance from the starting

point.

21) In the figure ∆ ABC is isosceles with AB = AC. P is the mid point of BC. If

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22) In the figure BDEF is a rectangle. C is the mid point of BD. AF = 7 cm, DE = 9 cm and

BD = 24 cm. If AE = 25 cm then prove that ∟ACE = 900

23) In the figure altitude is drawn to the hypotenuse of a right angled triangle the lengths of

different line-segments are marked. Determine x, y , z

24) In the given figure, BC ┴ AB, AE ┴ AB and DE ┴ AC. Prove that DE.BC = AD.AB

25) O is any point inside a rectangle ABCD (shown in the figure).

Prove that OB2 + OD

2 = OA

2 + OC

2

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TRIGONOMETRY

Prove the following:

1. sin4 θ + cos

4 θ = 1 – 2 sin

2 θ cos

2 θ [ Hint a

4 + b

4 = (a

2 + b

2)2 – 2 a

2b

2]

2. (sin θ + cos θ )2 + (sin θ - cos θ )

2 = 2

3. (sin4 θ – cos

4 θ) = (sin

2 θ – cos

2 θ) = (2 sin

2 θ – 1) = (1 – 2 cos

2 θ)

[Hint: Use a2 – b

2 formula]

4. sec2 θ + cosec

2 θ = sec

2 θ cosec

2 θ

[Hint: convert in terms of sin and cos and proceed]

5. sec4 θ – sec

2 θ = tan

4 θ + tan

2 θ

[ Hint: Take sec2 θ in common]

6. 2 sec 2 θ – sec

4 θ – 2 cosec

2 θ + cosec

4 θ = cot

4 θ – tan

4 θ

[ Hint: take common terms out and apply the identity]

7. ( 1 + tan A tan B)2 + ( tan A – tan B )

2 = sec

2 A sec

2 B

8. (sin A + sec A)2 + ( cos A + Cosec A )

2 = ( 1 + sec A cosec A)

2

9. ( 1 + cot A – cosec A) ( 1 + tan A + Sec A) = 2

[Hint: Convert in terms of sin and cos ]

10. 2 (sin6 θ + cos

6 θ) – 3 ( sin

4 θ + cos

4 θ ) + 1 = 0

[Hint: a3 + b

3 = ( a +b)

3 – 3 ab(a + b)]

11. sin6 θ + cos

6 θ + 3 sin

2 θ cos

2 θ = 1

12. ( sin8 θ – cos

8 θ) = (sin

2 θ – cos

2 θ ) ( 1 – 2 sin2 θ cos

2 θ)

13. 𝒔𝒊𝒏 𝜽

𝟏 + 𝒄𝒐𝒔 𝜽 +

𝟏+ 𝒄𝒐𝒔 𝜽

𝒔𝒊𝒏 𝜽 = 2 Cosec θ

14. √𝟏+𝐜𝐨𝐬 𝜽

𝟏−𝐜𝐨𝐬 𝜽 = cosec θ + cot θ

15. ( cosec θ – cot θ)2 =

𝟏−𝐜𝐨𝐬 𝜽

𝟏+𝐜𝐨𝐬 𝜽

16. 𝟏+ 𝐬𝐢𝐧 𝜽

𝟏−𝐬𝐢𝐧 𝜽 = ( sec θ + tan θ)

2

17. √𝟏+ 𝐬𝐢𝐧 𝜽

𝟏−𝐬𝐢𝐧 𝜽 + √

𝟏− 𝐬𝐢𝐧 𝜽

𝟏+ 𝐬𝐢𝐧 𝜽 = 2 sec θ

18. 𝟏

𝟏+𝐬𝐢𝐧( 𝟗𝟎°− 𝜽) +

𝟏

𝟏− 𝐬𝐢𝐧( 𝟗𝟎°− 𝜽) = 2 cosec

2 θ

19. 𝒄𝒐𝒔 𝜽

𝟏 −𝒕𝒂𝒏 𝜽 +

𝒔𝒊𝒏 𝜽

𝟏 −𝒄𝒐𝒕 𝜽 = sin θ + cos θ

20. 𝑪𝒐𝒕 𝑨+𝒄𝒐𝒔𝒆𝒄 𝑨−𝟏

𝐜𝐨𝐭 𝑨−𝒄𝒐𝒔𝒆𝒄 𝑨+𝟏 =

𝟏+𝐜𝐨𝐬 𝑨

𝐬𝐢𝐧 𝑨

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21. 𝑰𝒇 𝒙 = 𝒂 𝐬𝐞𝐜 𝜽 + 𝒃 𝐭𝐚𝐧 𝜽 𝒂𝒏𝒅 𝒚 = 𝒂 𝐭𝐚𝐧 𝜽 + 𝒃 𝐬𝐞𝐜 𝜽 𝒕𝒉𝒆𝒏 𝑷. 𝑻 𝒙𝟐 − 𝒚𝟐 = 𝒂𝟐 − 𝒃𝟐

22. 𝑰𝒇 𝐭𝐚𝐧 𝜽 + 𝒔𝒊𝒏𝜽 = 𝒎 𝒂𝒏𝒅 𝐭𝐚𝐧 𝜽 − 𝐬𝐢𝐧 𝜽 = 𝒏 𝒕𝒉𝒆𝒏 𝑺. 𝑻 (𝒊)( 𝒎𝟐 − 𝒏𝟐)𝟐 = 𝟏𝟔 𝒎𝒏

(ii) 𝒎𝟐 − 𝒏𝟐= ± 𝟒 √𝒎 𝒏

23. 𝑰𝒇 𝟐 𝐜𝐨𝐬 𝜽 − 𝐬𝐢𝐧 𝜽 = 𝒙 𝒂𝒏𝒅 𝐜𝐨𝐬 𝜽 − 𝟑 𝒔𝒊𝒏𝜽 = 𝒚, 𝑷. 𝑻 𝟐 𝒙𝟐 + 𝒚𝟐 − 𝟐 𝒙𝒚 = 𝟓

24. If sec 𝜽 − 𝐭𝐚𝐧 𝜽 = 𝟒 𝒕𝒉𝒆𝒏 𝒑𝒓𝒐𝒗𝒆 𝒕𝒉𝒂𝒕 𝐜𝐨𝐬𝜽 = 𝟖

𝟏𝟕

25. If tan = a/b, show that 𝐚 𝐬𝐢𝐧 − 𝐛 𝐜𝐨𝐬

𝐚 𝐬𝐢𝐧 + 𝐛 𝐜𝐨𝐬 =

𝒂𝟐 – 𝒃𝟐

𝒂𝟐+ 𝒃𝟐

26. If √3 tan = 3 sin , find the value of ( 𝟏 − 𝐬𝐢𝐧 ) ( 𝟏 + 𝐬𝐢𝐧 )

(𝟏−𝐜𝐨𝐬 𝜽 ) ( 𝟏+𝐜𝐨𝐬 𝜽)

27. In a ∆ABC, right angled at B, AB = 5 cm and AC + BC = 25 cm.

Find the values of Sin A, Cos A .

28. If tan A = √2 – 1 show that sin A Cos A = √2/4

29. Evaluate a) 4 tan2 30 + 1/8 Cot

2 60 + sin

230 cos

2 45 + ½ Sin

2 90

b) 2 cos2 90 + 3 cosec

2 60 + 4 cos

245 + tan

260

2 cosec 30 + 3 sec 60 – 7/3 cot2 30

30. If cos 3x = cos 30 sin 60 – sin 30 cos 60 then find the value of x.

Prove the following,

31. √sec2 + cosec

2 = tan + cot

32. ( tan2 - sin

2 ) = tan

2 .sin

2

33. ( cot - tan ) = 𝟐 𝐜𝐨𝐬𝟐 − 𝟏

𝐬𝐢𝐧 𝜽 𝒄𝒐𝒔𝜽

34. 𝐜𝐨𝐬𝐞𝐜 + 𝐜𝐨𝐭

𝐜𝐨𝐬𝐞𝐜 − 𝐜𝐨𝐭 = ( 1 + 2 cot

2 + 2 cosec cot )

35. cos4 - cos

2 = sin

4 - sin

2

36. 𝐜𝐨𝐬𝐞𝐜 𝐀

𝐜𝐨𝐬𝐞𝐜 𝐀 – 𝟏 +

𝐜𝐨𝐬𝐞𝐜 𝐀

𝐜𝐨𝐬𝐞𝐜 𝐀 + 𝟏 = 2 + 2 tan

2 A

37. √𝟏 + 𝐜𝐨𝐬

√𝟏 – 𝐜𝐨𝐬

+ √ 𝟏 – 𝐜𝐨𝐬

√𝟏+ 𝐜𝐨𝐬 = 2 cosec

38. Sin 6 + cos

6 = 1 – 3 sin

2 cos

2

39. 1 + tan 2 = 1 – tan

2

1 + cot 2 1 – cot

40. cos 3 - Sin

3 + cos

3 + sin

3 = 2

cos - sin cos + sin

41. Evaluate

2 sin 68 - 2 cot 15 - 3 tan 45 tan 20 tan 40 tan 50 tan 70

Cos 22 5 tan 75 5

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42. Sin 2

20 + sin 2 70 + sin ( 90 - ) sin + cos ( 90 - )cos

Cos220 + cos

2 70 tan cot

43. tan cot ( 90 - ) + sec cosec ( 90 - ) + ( sin 2

75 + sin 2

15 )

Tan 20 tan 40 tan 45 tan 50 tan 70

44. Cos 2 20 + cos

2 70 + 2 cosec

2 58 – 2 cot 58 tan 32 – 4 tan13 tan37 tan 45 tan 53 tan 77

Sec 2 50 – cot

2 40

45. If x cos 𝜽 – y sin 𝜽 = a ; x sin 𝜽 + y cos 𝜽 = b. Prove that x2 + y

2 = a

2 + b

2

46.

47.

48. If cot θ = 𝟏𝟓

𝟖; Evaluate

( 𝟐 + 𝟐 𝒔𝒊𝒏 𝜽) ( 𝟏 – 𝒔𝒊𝒏 𝜽)

( 𝟏+𝐜𝐨𝐬 𝛉 ) ( 𝟐−𝟐 𝐬𝐢𝐧 𝛉)

49.

50. In Figure AD = 4 cm, BD = 3 cm and CB = 12 cm, then find the value of cot 𝜽

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STATISTICS

1. The mode of a distribution is 55 & the modal class is 45-60 and the frequency preceding

the modal class is 5 and the frequency after the modal class is 10.Find

the frequency of the modal class. (Ans:15)

2. In a frequency distribution mode is 7.88, mean is 8.32 find the median. (Ans: 8.17

3. The mean of 30 numbers is 18, what will be the new mean, if each observation is

increased by 2? (Ans:20)

4. What is the value of the median of the data using the graph in figure of less than ogive

and more than ogive

5. Find the value of p, if the mean of the following distribution is 18.

Variable (x) 13 15 17 19 20 + p 23

Frequency (f) 8 2 3 4 5p 6

6. Find the mean, median and mode for the following data

Classes 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

Frequency 5 8 15 20 14 8 5

7. If the median of the following data is 52. 5. Find the value of x and y, if the total

frequency is 100.

CI 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 –

90

90 –

100

F 2 5 x 12 17 20 y 9 7 4

8. Draw less than ogive and more than ogive for the following distribution and hence find

its median

Classes 20 – 30 30 – 40 40 – 50 50 – 60 60 - 70 70 - 80 80 - 90

Frequency 10 8 12 24 6 25 15

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9. Find the mean marks for the following data

Marks Below

10

Below

20

Below

30

Below

40

Below

50

Below

60

Below

70

Below

80

Below

90

Below

100

No. of

Students

5 9 17 29 45 60 70 78 83 85

10. Draw a less than type and more than type ogives for the following distribution on the

same graph and also find the median from the graph.

Marks 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99

No of

students

14 6 10 20 30 8 12

11. The mean of the following distribution is 57.6 and the sum of the observation is 50. Find

the missing frequencies f1 and f 2

Classes 0 – 20 20 – 40 40 – 60 60 – 80 80 –

100

100 –

120

Total

Frequency 7 f1 12 f2 8 5 50

12. Calculate the mean, median and mode of the following distribution

13. A class teacher has the following absentee record of 40 students of a class for the whole

term. Write the above distribution as less than type cumulative frequency distribution.

14. Find the median of the following data

15. If the mode of the following distribution is 57.5, find the value of x.

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APPLICATION OF TRIGONOMETRY

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