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MATHEMATICS BOOKLET – SA1Chap.

No.

Name of the chapter Page

No.

1 Real numbers 1-8

2 Polynomials 8-14

3 Pair of linear equations in two variables 14-18

8 Introduction to trigonometry 18-22

14 Statistics 22-29

3 Triangles 30-37



Chapter – 1 Real Numbers1 Mark Questions

1. If two positive integers ‘m’ and ‘n’ can be expressed as m = ab2

and n= a3b; a,b being prime numbers, then LCM (m,n) is :

2. If the least factor of ‘a’ is 3, the least prime factor of ‘b’ is 7, then

the least prime factor of (a + b) is :

3. After how many places, the decimal form of will terminate ?

4. Write the condition to be satisfied by ‘q’ so that rational number

p/q has terminating decimal expansion.

5. HCF of two number is 23 and their LCM is 1449. If one of the

numbers is 161, then the other number is:

6. If a = bq + r in Euclid’s division lemma, then r must satisfy

7. A series of well defined steps which gives a procedure for solving

a type of problem is called _________

8. The unit’s digit of 73 is 3 then what will be the unit’s digit of 711?

9. Find the HCF of the smallest composite number and the smallest

prime number.



10. If HCF of 65 and 117 is expressible in the form 65m – 117.

Then the value of ‘m’ is :

2 Mark Questions

11. Find the HCF of 336 and 54 by Euclid’s division algorithm.

12. What is the smallest number by which (√5 − √3) be

multiplied to make it a rational number? Also find the number so

obtained.

13. Find the largest number which divides 70 and 125, leaving

remainder 5 and 8, respectively.

14. Explain why 3 × 5 × 7 + 7 + 2 × 7 is a composite number.

15. Given that HCF (135,225) = 45. Find LCM (135,225).

16. Show that every positive odd integer is of the form 4q + 1 or

4q + 3

17. Prove that there is no natural number for which 8n ends with

digit zero.

18. Show that 5√2 is irrational

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

exactly divisible by 520 and 468.



20. Find the HCF of LCM of 30, 72 and 432 by prime factorization

method.

21. The number 525 and 3000 are both divisible only by 3, 5, 15,

25 and 75. What is HCF (525, 3000)? Justify your answer.

22. There is a circular path around a sports field. Ankit takes 18

minutes to drive one round of the field, while Ankita takes 12

minutes for the same. Suppose they both start at the same point,

and go in the same direction. After how many minutes will they

meet again at the starting point ?

23. Prove that every positive even integer is of the form 2q and

that every positive odd integer is of the form 2q + 1, where q is

some integer.

24. Write the condition for terminating of a rational number.

And hence, find whether the rational number (13/3125) has a

terminating decimal or non-terminating repeating decimal.

25. What is the digit at unit’s place of 9n ?

3 Marks Questions



26. If the HCF of 210 and 55 is expressible in the form 210 × 5 –

55y, find the value of ‘y’.

27. Show that square of an odd positive integer is of the form

8m + 1, for some whole number m.

28. Prove that (√2 + √3) is irrational.

29. If n is an odd integer, then show that n2 – 1 is divisible by 8.

30. Prove that any number of the form 4x + 2 can never be a

perfect square.

31. Prove that ( √ ) is an irrational number.

32. Using Euclid’s division, find the largest number that divides

70 and 125, leaving remainder 5 and 8, respectively.

33. Without actually performing the long division, find if

987/10500 will have terminating or non-terminating repeating

decimal expansion. Give reasons for your answer.

34. Show that there is no positive integer ‘n’ for which √ − 1 +√ + 1 is rational.

35. State the following:

(i) Euclid’s Division Lemma with boundary conditions.



(ii) Fundamentals Theorem of Arithmetic.

36. Using Euclid’s lemma find the length of the longest tape

needed to measure a room of length, breadth, height of 520 cm,

480 cm, 750 cm respectively.

37. Find the HCF and LCM of 72, 120, 360 using prime

factorization method.

38. Prove that product of three consecutive positive integers is

divisible by 6.

39. If √√ = √ , then determine whether ‘x’ is rational

irrational.

4 Mark Questions

40. Prove that one of every three consecutive positive integers

is divisible by 3.

41. Show that one and only one of n,, + 2, n + 4 is divisible by 3.

42. Prove that square of any positive integer is either of the

form 3m or 3m + 1 for some integer m.



43. Prove that square of any positive integer is of the form

5q,5q+1, 5q + 4 for some integer, q.

44. Prove that √2 is irrational.

45. Three sets of English, Hindi and Mathematics books have to

be stacked in such a way that all the books are stored topic wise

and height of each stack is the same. The number of English books

is 96, the number of Hindi books is 240 and the number of

Mathematics books is 336. Assuming that books are of same

thicknss, determine the number of stacks of English, Hindi and

Mathematics books respectively.

46. Prove that √7 is irrational.

47. Prove that if x and y are odd positive integers then x2 + y2 is

even but not divisible by 4.

48. Prove that n3 – n is divisible by 6, for every positive integer

‘n’.

49. Use Euclid’s division lemma to show that the cube of any

positive integer is of the form 9m, 9m + 1 of 9m + 8.

Hots



50. Prove that if x and y are both positive integers then x2 + y2 is

even but not divisible by 4. (3 Marks)

CHAPTER – 2 POLYNOMIALS

1 Mark Questions

1. If a and b are the zeroes of 2x2 + 5x – 10, then the value of ab is

____ .

2. If one of the zeroes of the quadratic polynomial (k – 1)x2 + 2kx + 3

is 1, then the value of k is _____ .

3. For what values of k is the polynomial f(x) = 2x3 – kx2 + 5x + 9

exactly divisible by (x + 2) ?

4. p(x) and g(y) are any two polynomial with g(x) ≠ 0, then we can

find polynomials q(x) and r(x) such that p(x) = q(x)g(x) + r(x). Here

write the condition that r(x) should satisfy.

5. A polynomial of the form ax5 + bx3 + cx2 + dx + e has at most

______ zeroes.

6. The sum and product of zeroes of 2(x2- 1) + 3x – 9 are :

7. Find a quadratic polynomial, the sum and product of its zeroes are

1 and -6 respectively.



8. Find a quadratic equation, whose roots are(1 + √5) and (1-√5).

9. If (x + 2)(2x – 1)(3x – 2) = 0, find the zeroes of the polynomial.

10. If the sum of the zeroes of the polynomial f(x) = 2x3 – 3kx2 +

4x – 5 is 6, then find the value of k.

2 Mark Questions

11. Find the value of p for which x = b is a zero of the polynomial

x2 – (a + b) x + p.

12. Find the zeroes of the polynomial a2 – a – 12 and verify

relationship between the zeroes and the coefficients.

13. If one zero of the polynomial f(x) = 15x2 + 14x – k is

reciprocal of the other, then what will be the value of k?

14. If and are the roots of the equation ax2 – bx + c = 0, then

find the value of + .

15. If and are the roots of the equation 25x2 – 10x + 1 = 0,

find the value of 2+ 2 .

16. Find a quadratic polynomial, the sum of whose zeroes is -1

and the sum of their reciprocals is 1/6.



17. If and are the zeroes of the polynomial p(x) = x2 -16x +

63, then find the value of 4 3 + 3 4.

18. If the sum of the squares of the zeroes of a quadratic

polynomial x2- 18x + p is 180, find the value of p.

19. If one root of the quadratic equation 2x2 + px + 4 = 0 is ‘2’,

then find the other root and also find the value of ‘p’.

20. Find the value of the quadratic equation 2x2 – 3x – 2 at x = -

2.

3 Mark Questions

21. Find a quadratic polynomial whose zeroes are -2/√3 and

(√3)/4.

22. For what value of ‘k’, will the equation = have

roots reciprocal to each other ?

23. If and are zeroes of p(x) = x2 + px + q, then find a

polynomial having 1/ and 1/ as its zeroes: q ≠ 0.

24. What must be subtracted from 4x4 + 2x3 – 8x2 + 3x – 7 so

that it may be exactly divisible by 2x2 + x – 2 ?



25. On dividing x3 + 3x + 2by a polynomial g(x), the quotient and

the remainder are x – 2 and 16 respectively. Find g(x).

26. If and are the roots of a quadratic polynomial 3x2 – 2x –

1 = 0, find the value of + , without finding the values of and

.

27. If and are the zeroes of a quadratic polynomial such that

+ = 24 and – = 8, then find the quadratic polynomial.

28. If (Z – 3) is a factor of Z3 + aZ2 + bZ + 18 and a + b = - 7, find a

and b.

29. If and are the zeroes of the quadratic polynomial such

that g(x) = x2 – (a + 12)x + 3(3a + 4), such that ( + ) = ( ).

Then find the value of a.

30. Check whether the polynomial l(x) = x2 – 5x + 1 is a factor of

the polynomial m(x) = 4x4 – 13x3 – 31x2 + 35x – 10 by dividing m(x)

by l(x).

31. Divide – x3 + 3x2 – 3x + 5 by – x2 + x – 1 and verify the

division algorithm.



32. If and are the zeroes of 2x2 – 6x + 3, then what is the

value of + + 3 + + 2

33. Find the zeroes of the quadratic polynomial f(x) = abx2 + (b2

– ac)x – bc and verify the relationship between the zeroes and the

coefficients of the polynomial.

34. Use the division algorithm find the quotient q(y) and the

remainder r(y) when f(y) = 12y3 + 17y2 – 20y – 10 is divided by g(y)

= 3y2 + 2y – 5.

35. Find the G.C.D and L.C.M. of the following polynomials: p(x)

= 6(x – 2)(x2 + x – 6) and, q(x) = 3(x2 + 4x – 12).

36. Give examples of polynomials p(x), g(x), q(x) and r(x), which

satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg

q(x) = deg r(x) (iii) deg r(x) = 0

4 MARK QUESTIONS

37. Find the other zeroes of the polynomial x4- 5x3+ 2x2+10x – 8

if it is given that two of its zeroes are - √2 and √2.



38. If and are the zeroes of the quadratic polynomial 2x2 –

5x + 7, find a polynomial whose zeroes are 2 + 3 and 3 + 2 .

39. Divide (3x2 – x3+7x + 13) by (x – 2 – x2)and verify the division

algorithm.

40. On dividing 3x3+ x2 + 2x + 5 by a polynomial g(x), the

quotient and remainder are 3x – 5 and 9x + 10 respectively. Find

g(x).

41. If the polynomial x4 + 2x3+8x2+12x +18 is divided by another

polynomial x2+ 5, the remainder comes out to be px + q. find the

values of p and q.

42. It is given that √2 and -√2 are two zeroes of the polynomial

f(y) = 2y4-3y3-3y2+6y – 2, find all the zeroes of f(y).

HOTS

43. If and are zeroes of p(x) = x2- 2x + 3, find a polynomial

whose zeroes are ( − )/( + ) and ( − 1).44. If the polynomial f(x) = x4- 6x3 + 16x2 – 25x + 10 is divided by

another polynomial x2 – 2x + k, the remainder comes out to be x +

a, find k and a.



45. What must be added to the polynomial 3x4+5x3-7x2+5x+3 so

that the resulting polynomial is exactly divisible by (x2+3x +1)? The

degree of the polynomial to be added must be less than degree of

(x2+3x+1).

CHAPTER-3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

1 MARK QUESTIONS

1. General form of a linear equation in two variable is _______ .

2. Geometrical representation of a linear equation in two variables is

_______ .

3. If a pair of linear equations is consistent, then the lines will be

________ .

4. Find the number solutions for the pair of linear equations x + 2y +

5 = 0 and – 4x – 8y + 1 = 0.

5. The pair of equations y = 2 and y = 7 has ______ solutions/s.

6. Find the value of ‘k’ for which x + 2y = 5, 3x + ky + 15 = 0 are

inconsistent.

7. The point of intersection of line -3x + 7y = 3 with x – axis is

_________ .



8. Find the value of x and y which satisfy the equation x – y = 0; 2x –

y = 2 simultaneously.

9. If 4 x-y = 16 and x – 2y = 8 are system of the equations, then the

value of x + y is:

10. Find the value of ‘x’ in the following pair of linear equations:

4/x + 5y = 7 and 3/x + 5y = 5.

11. Sum of two numbers in 48 and their difference is 20. Find

the numbers

12. If 5x + 7y = 3 and 15x + 21y = k represent coincident lines,

then find the value of k.

13. The conditions so that a1x + b1y = c1; a2x + b2y = c2 have

unique solution is ______ .

14. If a pair of equations are consistent, the lines will be

________ .

4 MARK QUESTIONS

15. A boatman rows his boat 35km upstream and 55km

downstream is 12 hours. He can row 30 km upstream and 44km

downstream is 10 hours. Find the speed of the stream and that of



the boat in still water. Hence find the total time taken by the

boatman to row 50km upstream and 77 km downstream.

16. The students of a class are made to stand in a rows. If 3

students are extra in a row, there would be 1 row less. If 3

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

number in the class.

17. Draw the graphs of the following system of linear equations

: 4x + 3y – 24 = 0; y + 4 = 0. Obtain the vertices of the triangle so

obtained. Also determine its area.

18. (a) Solve: 217x + 131y = 913 and 131x + 217y = 827

(b) For what value of u the system of linear equations and

find unique solution.

19. Draw the graphs of the following system of linear equations

and find unique solutions: 4x – y – 8 = 0; 2x – 3y + 6 = 0. Also

determine vertices of triangle formed by the lines and the x – axis.

20. A boat covers 32 km upstream and 36 km downstream in 7

hours. Also, it covers 40 km upstream and 48 km downstream in 9

hours. Find the speed of the boat in still water and that of the

stream.



21. Solve the following system of linear equations graphically: 2x

– y – 4 = 0 x + y + 1 = 0 find the points where the line meets the y

axis.

22. A lending library has a fixed charge for the first three days

and an additional charge for each day thereafter. Saritha paid ₹ 27

for a book kept for seven days while Susy paid ₹ 21 for the book

she kept for five days. Find the fixed charge and the charges for

each extra day.

23. By the graphical method, find whether the pair of equations

3x + y + 4 = 0 and 6x – 2y + 4 = 0 are consistent or not. If

consistent then find its solution.

24. The sum of a two-digit number and the number obtained by

reversing the digits is 66. If the digits of the number differ by 2,

find the number. How many such numbers are there ?

HOTS

25. Two pipes running together can fill a cistern in 10/3 minutes.

If one pipe takes 3 minutes more than the other to ill the cistern,

find the time in which each would fill the cistern.

26. Solve the following :



(a + b)x + (a – b)y = a2 – b2 ; (a - b)x + (a + b)y = a2 + b2

CHAPTER-8 INTRODUCTION TOTRIGONOMETRY

1 MARK QUESTIONS

Q.1. If sec = 5/4, then what is the value of tan ?

Q.2. Express sinA in terms of cotA.

Q.3. The value of √√ = _______ .

Q.4. Sin2 25°+ sin265°= _______ .

Q.5. In a right triangle ABC right angled at B, if sin (A-C) = ½, then anglesA,B and C respectively are ______ .

Q.6. If x = a cos and y = b sin , then the value of b2x2 + a2y2 is.

Q.7. The value of sin2 5 + sin2 10° + ………. + sin2 85° + sin2 90° is _____ .

Q.8. In right triangle PQR right angled at Q, PQ = 8cm, QR = 6cm and PR= 10 cm. Find the value of 25(sin2 + 2 cos2 - tan ).

Q.9. The value of tan 1° tan 2° tan 3°…… tan 89° is :

Q.10. Find the value of ‘x’ if tan3x = sin 45° . cos 45° + sin 30°



2 MARK QUESTIONS

Q.11. If sin + sin2 = 1, check the validity of expression: cos2 + cos4 =1.

Q.12. If A, B and C are interior angles of ∆ ABC, then show that cot

= tan .

Q.13. In right triangle ABC, right angled at B, AB = y cm, BC = 10 cm, AC= x cm and < c = 30°. Find the values of x and y.

Q.14. ∆ ABC is right angled at B and and angle A = angle C, Is cosA =cosC ? Justify your answer.

Q.15. Solve the equation when 0°< < 90° ; 3 tan2 - 1 = 0.

Q.16. If 7sin2A + 3cos2 A = 4, show that tan A = 1/√3.

Q.17. If sin = a/b, then find sec + tan in terms of a and b.

Q.18. Simplify: sec2x(1 – sin2x) + secB(sinB/tanB).

Q.19. Evaluate (cos 60° + sin 30° - cot 30°)/(tan60° + sec 45° - cosec45°).Q.20. State whether the following expression is True or False. Justifyyour answer.



sin6 + cos6 = 1 – 3 sin2 . cos2

Q.21. If A,B are acute angles and sinA = cosB, then find the value of A +B.

Q.22. If 2x = secA and 2/y = tanA, then find the value of 2(x2-1/y2).

Q.23. In ∆ABC, right angled at B,BC = 3 and AC = 6. Determine angleBCA and angle BAC.

3 MARK QUESTIONS

Q.24. If cosec A = √10, then find other five trigonometric ratios.

Q.25. Prove that : cot2A/(1 + cosec A )=(1- sin A)/sinA

Q.26. If 7 cosecA – 3cotA = 7, prove that 7cotA – 3cosecA = 3.

Q.27. Prove that (SinA + CosecA)2 + (CosA + SecA)2 = 7 + tan2A + cot2A.

Q.28. Evaluate cos (40° + ) – sin(50° - ) + (cos2 40°+ cos2 50°) / (sin2

40° + sin250°).Q.29. Sec6x(secxtanx)- sec4x(secxtanx) = sec5xtan3x.

Q.30. Prove that (cot A – cos A) / (cot A + cos A) = (cosec A – 1) / (CosecA + 1).

Q.31. Prove that - = - .

Q.32. If cot = √7, show that = .



Q.33. If tanA + sinA = m and tanA –sinA = n, prove that (m2- n2)2 =16mn.

Q.34. Show that sin1/2 xcosx – sin5/2xcosx = cos3xsin1/2x.

Q.35. Prove that: secx + tanx = cosx/(1-sinx).

Q.36. Without using trigonometric table, evaluate the following :

2 °° + °° - 3 ° ° + ° °° ° ° ° °Q.37. Given 3 cos A – 4 sin A = 0. Evaluate without using table

.

4 MARK QUESTIONS

Q.38. If tan A = n. tan B and sin A = m . sin B, then prove that cos2A =(m2- 1) / (n2 – 1).

Q.39. Prove that (tan + sin ) / (tan – sin ) = (sec + 1) / (sec – 1) =(1 + cos ) / (1 – cos )= tan2 / (sec - 1)2.

Q.40. If (a2- b2) sin + 2 ab cos = a2 + b2, find the value of tan .

Q.41. Prove that + = sec . cosec - 2 sin . cos

Q.42. Prove that + = 1 + cosecAsecA



Q.43. If tanA + sinA = m and tanA – sinA = n, prove that (m2- n2)2=16mn.

Q.44. If x = r sinAcosC, y = r sinA sinC and z = r cos A, prove that r2 = x2+y2 + z2.

CHAPTER-14 STATISTICS1 MARK QUESTIONS

1. Construction of a cumulative frequency table is useful indetermining the _________ .

2. ̅ is the mean of x1, x2, x1,…………, xn, then the mean of , , ‘’’’,where k ≠ 0 is ________ .

3. The mean of n observations is ̅. If the first term is increased by 1,

second term by 2 and so on, then the new mean is _______ .

4. The relationship between mean, median and mode for a

moderately skewed distribution is _______ .

5. Median and mode of distribution are 21.2 and 21.4 respectively.

Then, its mean is : _____.

6. The meadian of a frequency distribution is found graphically with

the help of ________ .

7. Find the mode for the following series :



7.5, 7.3, 7.2, 7.2, 7.4, 7.7, 7.5, 7.3,7.2, 7.6, 7.2

8. If ̅ is the mean of n observation x1, x2, x1………………, xn then ∑ (xi - ̅) ,where I = 1 to n, is equal to _______ .

9. For what value of k, the mode of the following data is 7?

3.5,6,7,4,6,9,7,6,2k – 1, 10, 7, 12

2 Mark QUESTIONS

10. The mean of five number is 18. If one number is excluded, their

mean is 16, then what will be the excluded number ?

11. if median = 15 and mean = 16, the mode is _____ .

12. find the mean by direct method for the following data :

Classes 10-29 20-30 30-40 40-50 50-60 60-70 70-80

Frequency 4 8 10 12 10 4 2

13. The arithmetic mean of the numbers 7, 11, x, 3, y and 2 is 11. Find

the arithmetic mean of x and y.

14. Find the arithmetic mean of the 10 prime numbers.



15. The average mark scored by girls is 68 and that of the boys is 62.

The average mark of the whole class is 64. Find the ratio of the girls &

boys in the class.

16. The mean, of x-5y, x-3y, x-y, x+y, x+3y & x+5y is 12. Find the value of

x.

17. Find the median class of the following data:

Marks

Obtained

0-10 10-20 20-30 30-40 40-50 50-60

Frequency 8 10 12 22 30 18

3 MARK QUESTIONS

18. Find the unknown entries a, b, c, d, e, f in the following distribution

of heights of students in a class.

Height (in cm) frequency Cumulative frequency

135-140

140-145

145-150

150-155

4

b

18

d

a

11

c

40



155-160

160-165

e

5

46

F

19. Consider the following :

Class

interval

0-5 5-10 10-15 15-20 20-25

Frequency 10 15 12 20 9

20. The total number of marks scored by class in test is given below.

Find the mean.

Below 20 4

Below 40 12

Below 60 30

Below 80 44

Below 100 50

21.Find the mode for the following frequency distribution of marks

obtained by 80 students :



Marks 0-10 10-20 20-30 30-40 40-50

No. of

students

6 10 12 32 20

22. The mode of the following frequency distribution is 32. Find the

missing frequency in it:

Class

interval

0-10 10-20 20-30 30-40 40-50 50-60 60-70

frequency 8 10 x 16 12 6 7

23. Find the median from the following distribution.

Class

interval

4-6 6-8 8-10 10-12 12-14

Frequency 5 4 10 7 4

23. If the mean of the followimg data is 20, find the value of p.

x: 15 17 19 21 23

f: 2 3 4 5p 6



4 Mark QUESTIONS

25. The mean of the following frequency distribution is 62.8 and the

sum of II frequency is 50. Compute the missing frequencies f1 and f2.

Class 0-20 20-40 40-60 60-80 80-100 100-120 Total

frequency 5 f1 10 f2 7 8 50

26. The table below gives the distribution of villages under different

heights fom sea level in a certain region. Compute the mean of the

region :

Height (in

meters)

200 600 1000 1400 1800 2200

No. of

villages

142 265 560 271 89 16

27. Draw less than and more than ogives for the following distribution

on the same graph and hence find the median :



Marks 30-29 40-49 50-59 60-69 70-79 80-89 90-99

No. of

students

14 6 10 20 30 8 12

28.Find the mean and mode of the following data :

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

Frequency 3 8 10 15 7 4 3

29. During the medical check-up of 35 students of a class, their

weights were recorded as follows. Draw less than type ogive for

the given data. Hence, obtain the median weight from the graph

and verify the result by using the formula.

Wt. in kg Less

than

38

Less

than

40

Less

than

42

Less

than

44

Less

than

46

Less

than

48

Less

than

50

Less

than

52

No. of

students

0 3 5 9 14 28 32 35



30.The median of the following data is 28.5. find the values of x and

y, if the total frequency is 60.

Class

interval

0-10 10-20 20-30 30-40 40-50 50-60

No. of

students

5 x 20 15 y 5

31.The frequency distribution of scores obtained by 230 canditates

in an engineering entrance test is as follows :

Scores 400-

450

450-

500

500-

550

550-

600

600-

650

650-

700

700-

750

750-

800

No. of

candidates

20 35 40 32 24 27 18 34

Draw cumulative frequency curves by less than and more than method

on the same axes. Also, draw the two types of cumulative frequency

polygons.



CHAPTER-3 TRIANGLES

1 MARK QUESTIONS

1. In the adjoining figure, LM II AB. If AL = x – 3, AC = 2x, BM = x – 2

and BC = 2x + 3, the value of x is :

c

L N

A B

2. Let ∆ ~∆ and their areas be respectively 64 cm2 and 121

cm2. If EF = 15.4 cm, then the value of BC is _____ .

3. In ∆ LMN, <L = 50° and <N = 60°. If ∆LMN ~ ∆PQR, then <Q is equal

to :

4. If ∆ABC & ∆DEF are similar such that 2 AB = DE, and BC = 8 cm, then

EF = ______ cm.

5. Area of an equilateral triangle with sides 4 cm = ____ _____ .



2 MARK QUESTIONS

6. In the adjoining figure, = and <TQS = <PRS, show that ∆PQS~∆ TQR

T

P

Q S R

7. In the given figure AD BC. Prove that AB2 + CD2= BD2 + AC2.

D C

B

A

8. ABC is an isoceles triangle, right-angled at B. Equilateral triangles

ACD and ABE are constructed on sides AC and AB. Find the ratio

between the areas of ∆ ABE and ∆ ACD.



A D

E

B C

9. In the adjoining figure, ∆ABC is right- angled at C and DE AB, AD

= 3 cm, DC = 2cm, BC = 12 cm. Prove that ∆ABC ~ ∆ADE and

hence, find the lengths of AE and DE.

A

E

D

C B

10.In ∆PQR, DE II QR intersecting PQ and QR in points D and E,

respectively. If DE = QR, then show that PD = DQ.



11. In the given figure, E is a point CB produced of an isoceles triangle

ABC with AB = AC. If AD BC and EF AC, Prove that ∆ABD ~ ∆ ECF.

A

E

E B D C

12. The perimeters of two similar triangles are 36 cm and 48 cm

respectively. If one side of the first triangles is 9 cm, what is the

corresponding side of the other triangle ?

13. In ∆ABC, D and E are points on the sides AB and AC respectively

such that DE ll BC. If AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x –

3, then find x.

14. If ∆ABC ~ ∆DEF, AB = 4cm, DE = 6cm, EF = 9cm and FD = 12cm. Find

the perimeter of ∆ABC.



15. ABC is a right triangle right-angled at B. let D and E be any points on

AB and BC respectively. Prove that AE2 + CD2 = AC2 + DE2.

16. Any point X inside the ∆ DEF is joined to its vertices. From a point P

in DX, PQ is drawn parallel to DE meeting XE at Q and QR is drawn

parallel to EF meeting XF in R. Prove that PR ll DF.

17. In the adjoining figure ∆ ABC and ∆ AMP are 2 right angled triangles

at B and M respectively. Prove that

(1) ∆ABC ~∆AMP (2) =

C

M

P B P

18. In the adjoining figure, D and E are points on the sides CA and CB

respectively of a ∆ABC, right angled at C. Prove that AE2 + BD2 = AB2 +

DE2.



A

D

R E C

19. 2 Isosceles triangles have equal vertical angles and their areas are in

the ration 16:25. Find the ratio of their corresponding heights.

20. ABC is an isosceles triangle is which AB = AC = 10 cm and BC = 12.

PQRS is a rectangle triangle. Given PQ = SR= Y cm, PS = QR = 2x. Prove

that x = 6 – (3/4)y.

A

Q R

B P L S C

21. In the given figure, < AEF = <AFE and E is the mid-point of CA.

Prove that BD/CD = BF/CE.



B

G

F

C A

E

D

22. ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c

and let P be the length of perpendicular form C on AB prove that

(i) cp = ab

(ii) = +

4 MARK QUESTIONS

23. State and prove Basic proportionality Theorem. In a ∆ABC,D and E

are points on sides AB and AC respectively, such that BD = CE, <C, then

show that DE ll BC using converse of the above theorem.

24. In a right triangle ABC, right angled at C,P and Q are points of the

sides CA and CB respectively that divide these sides in the ratio 2 :1.



(i) 9AQ2 = 9AC2 + 4BC2

(ii) 9BP2= 9BC2 + 4AC2

(iii) 9 (AQ2 + BP2) = 12AB2

25. State and prove converse of Pythagoras theorem. Using the

theorem, prove that ∆ABC is right angled in the given figure.

A

24 cm 6 cm

D 8 cm

B C

26. O is any point in the interior of the ∆ABC. OD BC, OE AC and OFAB. Show that AF2 + BD2 + CE2 = AE2+ CD2 + BF2.27. In the adjoining figure, PA, QB and RC are each perpendicular to AC.

Prove that + = . P

R

x Q

z

A B C



ANSWERS

Chapter-1 Real numbers

1 Mark Questions

1) a3b2

2) 23) Four places4) Prime factorization of ‘q’ must be of the form 2m X 5n, where m

and n are non-negative integers.5) 2076) 0 ≤ r < b7) an algorithm8) 39) 210) 2

2 Marks Question11) 612) (√5 + √3) ; 213) 1314) Given no. is a multiple of 715) 67516) Prove17) 5 does not occur in the prime factorization of 8.



18)

19) 4663 (Hint: LCM (520, 468)-17)

20) 2160

21) 75

22) 36 minutes

23) .

24) Dr. is of the gorm 2m X 5n; terminating decimal.

25) Even power -1 , odd power -9.

3 Marks Questions

26) 19

27) .

28) .

29) Any positive odd integer is of the form n = 4q + 1 or 4q + 3

30) .

31) .

32) HCF of 65 and 117 = 13

33) Terminating decimal expansion.



34) .

35) .

36) HCF = 10 cm

37) HCF = 24; LCM = 360

38) .

39) .

4 MARKS QUESTIONS

40) take consecutive integers as n, n+1, n+2.

41) .

42).

43).

44).

45) English = 2, Hindi = 5, Mathematics = 7

46) .

47) .

48) .

49).



Hots

50) Odd + ve integers x = 2m + 1 and y = 2n + 1.

CHAPTER -2 POLYNOMIALS

1 MARK QUESTIONS

1) -52) -2/33) -17/44) r(x) = 0 or deg r(x) deg q(x)5) 5 zeroes6) -3/2, -11/27) K[x2-x-6]8) X2-2x-4 = 09) X = -2 or, x = ½ or, x = 2/310) K = 4

2 Marks Questions

11) p = ab12) zeroes are -3 and 413) k = -1514) –b/c15) 2/2516) K(x2 + x – 6)17) 4000752



18) P = 7219) 1,p = -620) P(1) = -3, p(-2) = 12

3 MARKS QUESTIONS

21) p(x) = 4√3 x2 + 5x - 2√3.

22)

23) qx2+ px + 124) x2 + 2x + 725) 5x – 1126) -10/327) X2-24x + 12828) A = -4, b= -329) A = 430) Remainder = 28x – 10 ≠ 0, not a factor31) Quotient = x – 2, remainder = 332) 1333) –b/a and c/b34) q(y)= 4y + 3 and r(y) = 5 – 6y35) GCD = 3 (x-2), LCM = 6(x+3)(x+6)(x-2)2

36) (i) p(x) = 6x2+3x + 2, g(x) = 2x2+x, r(x) = 2

(ii)p(x) = 8x3+6x2-x+7,g(x) = 2x2+1, q(x) = 4x + 3, r(x) = -5x + 4.

(iii)P(x) = 9x2+ 6x + 5, g(x)= 3x+2, q(x) = 3x, r(x) = 5



4 Marks Questions

37) 1,4

38) K(x2- x + 41)

39) .40) X2 + 2x + 141) px + q = 2x + 3, p = 2, q = 342) √2 , -√2, 1 and ½

Hots

43) 3x2 – 2x + 144) K = 5, a = -545) -3x – 1, degree of added polynomial = 1

CHAPTER- 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

1 MARK QUESTIONS

1) ax + by + c = 0, where a,b,c are real numbers, and a and b are notboth zero.

2) Straight line3) Intersecting or coincident4) No solution.5) No solution6) K = 6

7) (-1,0)



8) X = 2, y = 2

9) -1010) X= ½11) X = 34 and y = 1412) K = 9

13) ≠14) Intersecting or coincident

4 Marks Questions

15) Speed of the boat in still water = 8km/hr. speed of thestream = 3 km/hr.

16) No. of students = 36.17) A(3,4), B(_1,-4), c(9,-4);

Area = 40 sq. units18) (a) x=3, y = 2 (b) u ≠ 619) X = 3, y = 4; (3,4),(2,0),(-3,0)20) The speed of boat in still water = 10/km/hr and speed of

stream = 2km/hr21) (0,-4) for the first equation and (0,-1)for the second

equation.22) Rs. 15 and Rs.323) X= -1, y = -124) The two numbers are 42, 24.



Hots25) 5 minutes, 8 minutes.26) X = (a2+b2)/2a, y = (a2 + b2)/ 2a.

Chapter-8 Introduction to Trigonometry

1 Mark Questions

1) ¾

2) Sin A = √3) cosec – cot4) 15) 60°, 90° and 30°6) a2b2

7) 9.58) 2/39) 110) 15°2 Marks Questions

11) .12) .

13) X = √ , y = √14) Yes15) = 30°



16) .17) √(b+a) / √(b – a)18) 219) (√3-3)/320) False. (sin 80°- sin 10° = positive, as increases, value of sin

increases )21) A + B = 90°22) ½23) 60° and 30°3 Marks Questions

24) Sin A = 1/√10, cos A = 3/√10, tan A = 3, cot A = 3, sec A =√10√325) .26) .27) .28) 129) .30) .31) .32) .33) .34) .35) .



36) 1

37) Take = = K, Ans : 11/9

4 Marks Questions

38) .39) .

40)

41) .42) .43) .44) .45) .

CHAPTER-14 STATISTICS

1 Mark Questions

1) Median

2) ̅3) ̅ +4) Mode = 3 median – 2 mean5) 21.16) Ogives7) Mode = 7.28) 0



9) K = 4

2 Marks Questions

10) 2611) 1312) Mean = 42.213) 21.514) 12.915) 1:216) X = 1217) 30-40 is the median class.

3 Marks Questions

18) A = 4, b = 7, c= 29, d= 11, e=6, f= 5119) 10+15 = 2520) Mean 6021) 36.2522) X = 1523) 9.224) P=1

4 Marks Questions

25) f1= 8 and f2 = 1226) mean height = 984.5127) 69.5



28) Mean = 32.8, mode = 33.8529) Median = 46.530) X = 8, y = 7

CHAPTER – 3 TRIANGLES

1 MARK QUESTIONS

1) X= 92) 11.2 cm3) 70°4) EF = 4cm5) Area = 4√3.

2 Marks Questions

6) .7) .8) ½9) DE = 36/13cm and AE =15/13cm10) .11) .12) 12cm13) X=114) Perimeter = 18cm15) .16) .



3 Marks Questions

17) .18) .19) .20) .21) .22) .

4 Marks Questions

23) .24) .25) AB = 10 cm26) BC = 11.2 cm27) .

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