PC LMR 2011

2011

[email protected]

PRASHANT CLASSES

19/01/11

Last Minute Revision L.M.R.

About Mr Prashant Gohil

• Mr Prashant Gohil is a Mathematics Examiner for IB Mathematics since Jan 2007.

• He corrects IB Diploma Mathematics Answer Scripts of a number of International Schools.

• He is working as the IB Diploma Coordinator for a prestigious School in South Mumbai. He is

also the Head of Mathematics Department at this School.

• He has attended a number of Workshops conducted by the IBO. He has attended Workshops at

Seoul (South Korea), Ching Mai (Thailand) and Hongkong.

• He conducts Graphic Display Calculator Workshops for Cranes Software (Authorized distributors

for Texas Instruments). He has conducted a GDC workshop at Navrachna School, Baroda and

Dhirubhai Ambani International School, Mumbai.

• He has also completed a training workshop for IGCSE Mathematics (Cambridge).

• His earlier assignments include:

o IB Mathematics Teacher at BD Somani International School,

o ISC Physics and Calculus AP Teacher at Cathedral and John Connon School

o ICSE Mathematics Teacher at Bombay International School

o ICSE Mathematics and Physics teacher at Palm Beach School

• He has been tutoring students, since 1990, at various University, Pre-university and High School

Level. He has trained students for

o Mathematics and Statistics module at University Level (London School of Economics,

Nottingham University, University of Illinois etc.)

o IB Mathematics Higher Level and Standard Level ( Dhirubhai Ambani International

School, Ecole Mondiale, B.D Somani, Jamnabai Narsee, Fazlani, D.Y. Patil etc)

o HSC Mathematics(Std 11 and 12),

o ICSE Mathematics and Physics (Std 9 and 10)

• He has also tutored students for a number of entrance tests such as

o SAT

o SAT Subject Tests- Mathematics and Physics

o GMAT

• He was a Merit List holder and received a National Scholarship for his performance at B.Sc.

Examinations at University of Mumbai.

• Presently, he is pursuing M.Ed from IGNOU.

PRASHAANT CLASSES Name:_________________ for IB, IGCSE, ICSE, 11th, 12th,- MATHS, PHYSICS. Call 2380 00 92.

Shop No. 12, Mahavir Apartments, Opp Swati Snacks, Tardeo Road, Mumbai 7.

Last Minute Revision 2011

1. The list price of an article is Rs.4,750. A rebate of

5% is given on it. After the rebate sales tax at the

rate of 6% is charged on it. Find the total cost of

the article to be paid by the buyer.

2. If x + 2y − z

3a − b=

y + 2z − x

3b − c=

z + 2x − y

3c − a

show that each fraction = x + y + z

a + b + c.

3. In the figure [not drawn to scale,] LM is parallel

to BC. BA=12 cm, AL=4 cm and AC=18 cm.

Calculate : (i) the length of CM. (ii)the value

of ratio and then

4. In the given figure AC is the diameter of the semi

circle. BM is perpendicular to AC. If AM = 16cm

and the diameter of the circle is 25 cm, calculate 1

BM 2 Area of the shaded part. use =3.142.

5. Solve the following inequation and represent the

solution set on a number line. 30 − 4(2x−1) > −8

and x ∈ { positive integers}

6. On what sum of money will the difference between

the simple interest and the compound interest for 2

years at 5% p.a will be Rs 75?

7. A part of Rs 5110 was invested in 8% Rs 100

shares quoted at Rs 98 and the rest in 9 % shares

quoted at Rs 105. If the total dividend received

from both is Rs 430, find the sum invested in 8%

shares.

8. Mr Soni invested Rs 9000 in 15% (Rs 10) shares

selling at Rs.45 After a year he sold these shares

at Rs 40 each and invested the proceeds in 10%

(Rs20) shares selling at Rs25. Calculate: -

(i) the number of shares purchased,

(ii) the income from the first investment,

(iii) the income from the second investment;

(iv) percentage return on his original investment in

the second investment.

9. Car A travels x km for every litre of petrol, While

car B travels (x+5) km for every litre of petrol. (i)

Write down the number of litres of petrol used by

car A and car Bin covering a distance of 400 km.

(ii)iIf car A uses 4 litres of petrol more than car B

in covering the 400 km, write down an equation in

x and solve it to determine the number of litres of

petrol used by car B for the journey.

10. A certain sum of money amounts to Rs 6600 in 1

year and to Rs 7986 in 3 years, at compound

interest. Find the sum and the rate of interest.

11. Given P = { x : 5 < 2x − 1 ≤ 11, x ∈ R}

Q ={ x : −1 ≤ 3 + 4x < 23, x ∈ I}

where R = {real numbers}, I = { integers}

Represent P and Q on number lines. Write down

the elements of P ∩ Q.

12. P is a point on the line joining A(4,3) and B(−2,6)

such that 5AP=2BP.Find the co−ordinates of P.

PRASHANT CLASSES --TUITIONS FOR IB, IGCSE- MATHS & PHYSICS.Experienced IB teacher and IB examiner.

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PRASHAANT CLASSES Name:_________________ for IB, IGCSE, ICSE, 11th, 12th.- MATHS, PHYSICS. Call 2380 00 92.


13. At what rate per cent per annum C.I. will Rs

2,000 amount to Rs 2,315.25 in 3 years?

14. The scale of a map is 1: 210000. A plot of land of

area 83.79 km2 is to be represented on the map.

Find :

(i) The number of kilometres on the ground which

is represented by 1 centimetre on the map.

(ii) The area in km2 that can be represented by 1

cm2.

(iii) The area on the map that represents the plot of

land.

15. Calculate the annual income of five hundred, 12%

shares bought at Rs.150 at a premium of 25%.

When the market value of these shares rose to a

premium of 50%, he sold some shares, just enough

to raise Rs.18,000. Calculate the number of shares

remaining.

16. A trader bought a number of articles for Rs 1200.

Ten were damaged and he sold each of the rest at

Rs 2 more than what he paid for it, thus getting a

profit of Rs 60 on the whole transaction . Taking

the number of articles he bought a x, form an

equation in x and solve it.

17. Solve the inequation: .

Represent your solution on a number line.

18. Find the ratio in which the line joining A(−2,7)

and B(6,1) is divided by the line y=2. Hence find

the co−ordinate of the point where line AB cuts the

line y=2.

19. Use graph paper for this question. Plot the points

A (8,2) and B(6, 4). These two points are the

vertices of a figure which is symmetrical about x =

6 and y−2=0. Complete the figure o the graph .

Write down the geometrical name of the figure.

20. The weights of 60 boys are given in the following distribution table, find the median, lower quartile,upper

quartile, interquartile range.

Weight in kg 37 39 38 41 40

No. of boys 6 10 14 12 18

21. Furniturewalla buys wood for Rs.17,325 including

5 % VAT. He sells the chair made from that wood

for Rs 24,640 including 12 % VAT.

a) What is the tax paid by him on purchase?

b) What is the tax to be collected by him?

c) What VAT will he have to pay to the

government?

22. Points (3, –4 ) and (9 , –4) are invariants points

under reflection in the line L1. Points (–6 ,5) and

(–6, –8) are invariant points on reflection in line

L2.

(i)Name or write equations for the lines L1 and L2.

(ii)Write down the images of points P (3 ,4 ) on

reflection in L1. Name the images as P′. (iii)Write

down the images of Q(-5,-2) on reflection in L2.

Name the images as Q′.

23. A person invested 20%, 30% and 25% of his

savings in buying shares of three different

companies A, B and C which declared dividends

of 10%, 12% and 15% respectively. If his total

income on account of dividends be Rs 4675, find

his saving and the amount which he invested

buying shares of each company.

24. Find the values of a and b so that the polynomial

x3 −4x2 + ax +b is exactly divisible by x2 + x

− 2.

25. If (x+2) and (x-3) are the factors of x3 + ax + b,

find the values of a and b. With these values of a

and b, factorise the given expression.

26. The product of the digits of a two digit number is

24. If its units digit exceeds twice its tens digit by

2; Find the number.


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27. Given that a,b are the roots of x2−5x + 6 = 0, a>

b, frame the equation whose roots are 2a and

3b.

28. Rs125 is divided equally among a certain number

of children; if there were 25 children more, each

would have received 25 paise less. Find the

number of children.

29. List the solution set of 30 − 4 [ 2x − 1] < 30,

given that x is a positive integer.

30. Find the equation of the perpendicular dropped

from the point (−1,2) onto the line joining the

points (1,4) and (2,3).

31. Find the values of x which satisfy the inequation:

Graph the solution set on the real number line.

32. In the figure alongside,find: (i)Co−ordinates of

A,B and E. , (ii)Slope of CD(// to AE) ,

(iii)Equation of CD , (iv)Co−ordinates of C , D ,

(v)The ratio CB : BD.

33. Find the third proportional to

34. Two circles touch externally. The sum of their

areas is 130 cm2 and the distance between their

centres is 14 cm. Find the radii of the circles.

35. A publisher buys paper for Rs. 28,090 including

6% VAT. He makes 1000 notebooks and sells

them at Rs.50 each. He has to charge VAT at 10%

on the finished goods. What VAT is he liable to

pay to the government?

36. A man borrowed Rs 2048 at 6.25% p.a compound

interest. At the end of the first year he repays Rs.

176 and at the end of the second year he repays

Rs253. How much should he pay (i) at the

beginning of the 3rd year; (ii) at the end of the

third year to settle the loan.

37. The line y = 3x − 2 bisects the join of (a,3) and (2,

−5), find the value of a.

38. Solve the given inequation and graph the solution

on the number line.

2y − 5 < y + 2 ≤ 8y + 9 ; y ∈ R.

39. If A = . Find : i At.A ii A.At, where

At is the transpose of matrix A.

40. Miss Mina purchased the following items for her

mother and her neighbour:

Mother : 5 eggs 4 pies

Neighbour: 4 eggs 3 pies

If eggs are 80 p. each, pies 60p. write the above

information as a matrix product, hence find the

total cost of the bill for her mother and neighbour.

41. A person invests Rs 4368 and buys some

hundred-rupee shares at 91. Shares worth Rs 2400

face value are sold at Rs 95 and the remainder

when they have fallen to 85. Find his gain or loss

on the total transaction.

42. From a solid cylinder whose height is 8 cm and

radius 6 cm, a conical cavity of height 8 cm and

of base radius 6 cm is hollowed out. Find the

volume and the total surface area of the remaining

solid.


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43. Framing quadratic equations from the given roots

3 + , 3 −

44. Solve

45. The catalogue price of a data projector is

Rs.69,000 The shopkeeper gives a discount of 9%

on the listed price. He gives a further off-season

discount of 15% on the balance. However, sales

tax at 13% is charged on the remaining amount.

Find: (i) The amount of sales tax a customer has

to pay.

(ii) The final price he has to pay for the

data projector. (Give all your answers to the

nearest rupee)

46. Find the equation of the line AB, through (3,2) and

perpendicular to the line 2y=3x+5. Thus line AB

meets the x−axis at point P and y−axis at point Q.

Find the co−ordinates of points P and Q. Also,

find the area of triangle OPQ, where O is the

origin.

47. The side AB of an equilateral triangle ABC is

parallel to x-axis. Find the slopes of all its sides.

48. The compound interest on a certain sum of money

for 2 years is Rs 1890 and the simple interest for 2

years at the same rate is Rs 1800. Calculate the

sum and the rate of interest.

49. A solid right circular cone of height 20 cm and

base radius 15 cm is melted and casted into

smaller cones of equal sizes with height 5 cm and

base radius 1.5 cm. Find how many cones are

made.

50. Without solving, comment upon the nature of

roots of each of the following equations:

6x2 − 13x + 4 =0

51. The mean lengths of 25 pencils in a box is 20 cm.

It was later discovered that 5 pencils were not a

part of that box and were removed. The lengths of

the pencils that were removed was 12cm, 10 cm,

8cm, 11cm and 9 cm respectively. Calculate the

average length of the remaining pencils.

52. The line 4x−3y+12=0 meets x−axis at point A.

Find the co−ordinates of A. Further, find the

equation of the line through point A and

perpendicular to 4x−3y+12=0.

53.

54. Find the equation of the line which passes through

the point (−2, 3) and is perpendicular to the line 2x

+ 3y + 4 = 0.

55. Five year’s ago, a woman’s age was the square of

her son’s age. Ten years hence her age will be

twice that of her son’s age. Find:

(i) the age of the son five years ago

(ii) the present age of the woman.

56. The roots of x2 − 6px + 45= 0 are p and 5p. Find

the values of p.

57. If x: y be the sub duplicate ratio of x−a : :y−a ,

prove that

58. (1,5) and (−3,−1) are the co−ordinates of vertices

A and C respectively of rhombus ABCD. Find the

equations of the diagonals AC and BD.

59. Draw an ogive for the following distribution:

Income in Rs 120−140 140−160 160−180 180−200 200−220 220−240

No of

employees

30 72 90 80 70 28

Use the ogive i the median income, iithe number of employees whose income exceeds Rs190.


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60. P is the solution set of 7x − 2 > 4x + 1 and Q is

the solution set of 9x − 45 5 (x − 5); where x

R. Represent: (i) , (ii) P − Q , (iii)

on different number lines.

61. where Mt is transpose of M.

62. The figure alongside

given represents the lines y=x+1 and y= x−1.

Write down the angles that the lines make with the

positive direction of the x−axis. Hence determine

63. Spherical marbles of diameter 1.4 cm are dropped

into cylindrical beaker containing some water and

are fully submerged. The diameter of the beaker is

7 cm. Find how many marbles have been dropped

in it if the water rises by 5.6 cm.


for the fourth year is Rs.1452 and for the fifth

year is Rs. 1597.20. Find:

a) the rate of interest,

b) the compound interest for the sixth year,

c) the compound interest for the third year.



7x2 − 9x + 2 =0

66. The following table gives the height of the plants in centimeter. if the mean height of the plants is 60.95 cm, find

the value of ‘f’.

Height [cm] 50 55 58 60 65 70 71

No. of plants 2 4 10 f 5 4 3

67. If x3 + ax2 +bx + 6 has x − 3 as a factor and

leaves a remainder 8 when divided by x +1.

Find a and b.

68. A man sells 60, Rs 15 shares of a company paying

12 % dividend, at Rs.21 each and invests the

proceeds in Rs 6 shares of another company at Rs

9 each. Find his change in income, if the second

company pays a dividend of 8 %.

69. A company declares 12% dividend to the

shareholders. If a man receives Rs. 5400 as his

dividend, find the nominal value of his shares.

70. If A= ; find the matrix X

such that 2AX=B.


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71. Describe and draw the following loci

i ) A and B are two fixed points, the locus of a

point P such that angle APB = 90'.

ii) AB is a chord of a circle, the locus of a point in

the circle so that it is equidistant from A and

B.

iii) The locus of a point, which is equidistant from

two fixed points.

iv) The locus of a point, intersecting straight in

which is equidistant from two intersecting

lines.

v ) The locus of the mid-points of all equal chords

in a circle.

vi) The locus of mid-points of all parallel chords

in a circle

vii) The locus of a point equidistant from two

concentric circles.

viii) The locus of the centre of a given circle of

which rolls around the outside of a second

circle and is always touching it.

ix) The locus of the centres of all circles that are

tangent to both sides of a given angle.

x) The locus of the centres of all circles passing

through two fixed points.

xi) The locus of points equidistant from three

given non-collinear points.

xii) The locus of vertices of all isosceles triangles

having a common base.

xiii) The locus of the vertices of all triangles with

a given base and a given altitude.

xiv) The locus of a point P so that AB2 = AP2 +

BP2 where A and B are two fixed points.

xv) The locus of a point P so that ∆ ABP is

constant, where A and B are two fixed points.

xvi) The locus of a point in rhombus ABCD so

that it is equidistant from

72. Solve, 2 significant figures: x2 + 9x + 5 = 0

73. Find the 2×2 matrix X which satisfies the equation

74. At what rate will a sum of money become 1.96

times of itself in 2 years at compound interest?

75. Points (5 , 1) and (−2, 1) are invariants points

under reflection in the line L1. Points (2, −6) and

(2 ,3) are invariant points on reflection in line L2.

Give the equation of line L1 and L2.Write down the

images of P(2,4) and Q (-3,8) on reflection in L2.

Name the images as P′′ and Q′′respectively.

76. Do not use a graph paper. Points A (2, −1) and B

(2 , 4) are invariant when reflected in line L1.

Points C (−3 , 5) and D (4, 5) are invariant when

reflected in line L2. (i) Find the equation of line L1

and L2; (ii) Reflect point A in line L2 (iii) Reflect

point C in line L1.

77. If the speed of a car is increased by 10 km per hr,

it takes 9 minutes less to cover a distance of 18

km. Find the speed of the car.

78. A shopkeeper buys a certain number of books for

Rs.720. If the cost per book was Rs.5 less, the

number of books that could be bought for Rs.720

would be 2 more. Taking the original cost of each

book to be Rs. x, write an equation in x and solve

it.


80. In an auditorium, the number of rows was to the

number of seats in each row. When the number of

rows was doubled and the number of seats in each

row was reduced by 10, the total number of seats

increased by 300. How many rows were there?

81. Three consecutive natural numbers are such that

the square of the middle number exceeds the

difference of the squares of the other two by 32.

Assume the middle number to be x and form a

quadratic equation satisfying the above statement.

Hence; find the three numbers.


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82. Mr. Mistry has 120 shares of nominal value of Rs

100 and he decides to sell them when they are at a

premium of 60% . He invests the proceeds in

shares of nominal value of Rs. 50, quoted at 4%

discount, paying 15% dividend annually. Calculate

: (i) the sale proceeds; (ii) the number of shares

he buys; and (iii) the annual dividend from these

shares.

83. The hypotenuse of a right−angled triangle is 17

cm and the sum of other two sides is 23 cm. Find

the lengths of its sides.

84. Find the equation of the mirror line in each of the cases.

Sr. No Point Image Equation of the

mirror line

1 (2 ,4) (2 , 8)

2 (-6 , 8) (4 , 2)

3 (3 , -2) (3 , 0)

4 (2 , 3) (6 , 3)

5 (-3 , 7) (-4 , 0)

6 (2 , 5) (- 1 , 8)

85. Which is a better investment : 12 % at Rs 120 or

8% at Rs 90 ?

86. Calculate the amount and compound interest on

Rs.14500 at 6 % p.a. for 12 months compounded

half yearly.

87. If q is the mean proportional between p are r,

prove that:p2−q2 + r2 = q4

88. The mean age of a class of 40 students is 13

years. The mean age of 25 boys in the class is 13

years 3 months, what is the mean age of the girls

in the class.

89. The roots of x2−4kx + 12 = 0 are k and 3k. Find

the value of k.

90. An article is sold for Rs 810. It was sold after a

discount of 10% on the list price and then adding

sales tax at a rate of 20%. Calculate the list price

of the article.

91. If A and B are square matrices, is ( A+ B)2 = A2 +

2 AB + B2? Explain.

92. The equation of a line is x − y = 4. Find its slope

and y−intercept. Also, find its inclination.

93. In what ratio is the join of (4,3 and (2,−6 divide by

x−axis. also find the co−ordinates of the point of

intersection.

94. A sum of money is lent out at compound interest

for 2 years at 20 % p.a., compound interest being

reckoned yearly. If the same sum of money was

lent out at compound interest at the same rate per

cent per annum, compound interest being reckoned

half-yearly, it would have fetched Rs 482 more by

way of interest. Calculate the sum lent out.

95. The abscissa of a point is four times its ordinate.

Find the co−ordinates of the points if its distance

from (6,2) is units.

96. Lines 2x−by+5=0 and ax+3y=2 are parallel to

each other. find the relation connecting a and b.


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97. The daily expenditure of 50 students is given below:

Expenditure [Rs.] 10−20 20−30 30−40 40−50 50−60 60−70

No. of students 5 14 a b 11 8

The mean daily expenditure per student is Rs.40.20 p. Find the values of a and b.

98. The points (K, 3), (2, −4) and (−K + 1, −2) are

collinear, Find K.

99. A merchant buys an article for Rs. 44,000/- and

sells it to a customer for Rs.68,000/-. If the VAT

rate is 11%; find the VAT paid by the merchant.

100. In , given A (2 , 4), B (−4, −6) and C (0 ,

8). Find :(1) equation of median AM (2) altitude

BN (3) perpendicular bisector of AB.

101. Find the equation of the line passing through (5,

−3) and parallel to x−3y = 4.

102. Find the positive integers p and q such that

p qÈ

ÎÍÍÍÍ

˘

˚˙̇˙̇

p

q

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙̇˙̇˙̇˙̇˙̇˙̇

= 25[ ]

103. Two solid spheres have surface areas of 5 cm2 and

45 cm2 respectively and the mass of the smaller

sphere is 3 kg. Find the mass of the larger sphere.

104. Given A(x+2,−2) and B (11,6). Find x if AB=17.

105. The centre of a circle of radius 13 units is the

point (3,6). P (7,9) is a point inside the circle.

APB is a chord of the circle such that AP=PB.

Calculate the length of AB.

106. Attempt this on a graph paper. (i)Plot A (2 ,2), B (

0 , − 2) and C (4 , − 2). Locate M, the midpoint of

AC. (ii)Complete the figure ABDC, so that it is

symmetrical about the line BC. Write the

coordinates of D. write the geometrical name of

the figure ABDC. (iii)Is the figure ABDC

symmetrical about AD ? Locate , the image of

M on reflection in BC. Write the coordinates of

.

107. Find the equation of the line passing through

(−5,7) and parallel to (i)x − axis (ii)y − axis.

108.

The diagram represents two inequations A and B on real number

lines. (i) Write

down A and B in set builder notation. (ii) Represent A ∩ B,

A ∩ B′ on two different number lines.

109. Salman purchased a pair of shoes for Rs.7920,

which included sales tax. If the rate of sales tax is

10%, find the list price of shoes.

110. Factorise: x3 − 37x − 84.

111. The centre of a circle is (2k −1, 3k +1) and it

passes through the point (−3,−1). Find the value or

values of k, if the diameter of the circle is of length

20 units.


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112. An article is bought at a certain price and a VAT

at a rate of 10% was paid. It was sold for

Rs.1,600 more for which he has to charge 11 %

VAT. If he deposits Rs. 816 as VAT with the

government, find the cost price of the article.

113. If x =a + 3b + a − 3b

a + 3b − a − 3b, prove that 3bx

2 -

2ax + 3b = 0.

114. Draw two intersecting lines to include an angle of

30°. Use ruler and compasses only to locate

points which are equidistant from these lines and

also 2 cm away from their point of intersection.

How many such points exist?

115. Use ruler and compasses only for this question.

Draw a circle of radius 4 cm and mark two chords

AB and AC of the circle of length 6 cm and 5 cm

respectively.

(i) Construct the locus of points, inside the circle,

that are equidistant from A and C. Prove your

construction.

(ii)Construct the locus of points, inside the circle,

that are equidistant from AB and AC.

116. Draw a histogram for the frequency distribution and use it to find the mode. Also, calculate the Arithmetic mean.

Marks 31−40 41−50 51−60 61−70 71−80 81−90 91−100

frequenc

y

5 9 12 18 15 6 3

117. Points A and B have co-ordinates (–1, –5 ) and

(–1 , –1) respectively. Find:

(i)The equation of the mirror line so that A is

image of B.

(ii) the equation of the line in which both the

points are invariant.

118. Find two numbers such that the mean proportional

between them is 12 and the third proportional to

them is 96.

119. Ms. Bachan goes to a shop to buy a walkman,

costing Rs.3638. The rate of sales tax is 7%. She

tells the shopkeeper to reduce the price of the radio

to such an extent that she has to pay Rs 3638

inclusive of sales tax. Find the reduction needed in

the price of the radio.

120.

121. Draw a cumulative frequency table, plot an ogive for the data given below and form the cumulative frequency

curve, find : (i) the median marks, (ii) the lower quartile, (iii) the upper quartile , (iv)Find passing marks if 80%

of students passed, (v)If students scoring more than 75% mark are awarded scholarship, how many get?

Mar

ks

1−10 11−20 21−30 31−40 41−5

0

51−60 61−7

0

71−80 81−

90

91−

100

No.o

f

pupil

s

4 6 7 16 25 21 12 5 3 1


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122. Solve

123. Points (−2,−1) and (1,8) lie on the line y = mx + c.

Find the numerical values of m and c.

124. A map of a country is drawn to the scale 1 :

400000. Calculate: (i) the distance on the map

which will represent an actual country side

distance of 40 km. (ii)the area in cm2 on the map

which would represent an actual area of 64 km2.

125. What point on x−axis is equidistant from points

(7, 6) and (−3,4).

126. Solve :

127. Two numbers are in the ratio 2:5 and if 3 be added

to each, they are in the ratio 5:11. Find the

numbers.

128. Match the equations A, B, C, D with the lines L1 ,

L2 , L3, L4 , whose graphs are roughly drawn in

the adjoining diagram.

Hence, find the point of intersection of B and C.

129. Find the equation of the perpendicular bisector of

the line segment obtained on joining the points

(6,−3) and (0,3).

130. A triangle whose area is 12 cm2 is transformed

under enlargement about a point in space. If the

area of its image is 108 cm2, find the scale factor

of the enlargement.

131. A person invested Rs 16,000 and Rs 20,000 in

buying shares of two companies which later on

declared dividends of 12% and 8% respectively.

He collects the divided and sells out all his shares

at a loss of 2% and 3% respectively on his

investment. Find his total earning from the above

transaction.

132. Solve for matrix X if

133. If (p − 1) x + 2y = 4 is perpendicular to x + y = 3

find p.


10 [email protected]



134. Calculate the mean, median and mode:

X 0 2 4 6 8 10

F 2 3 5 7 2 1




Write down the images of points P (3 ,4) and Q (

−5 , −2) on reflection in L1. Name the images as P′

and Q′ respectively.

136. If 6x2 − 3y2: x2 + y2 is equal to 6:25, find the

value of x:y.

137. If

138. Plot the points A (2,1), B (2,8), C(5,5) and D

(5,−2) on graph paper. (i) Name the figure ABCD;

(ii) Write down the equation of its line of

symmetry. (iii) If P, Q are the images of A and D

when reflected in the line CD. Write down the

co−ordinates of P and Q. Name the completed

figure ABPQD.

139. Mukesh deposited Rs.220/- per month in a bank

for 24 months under the Recurring Deposit

Scheme. What will be the maturity value of his

deposits, if the rate of interest is 7 % per annum

and interest is calculated at the end of every

month?

140. By purchasing Rs 25 auto shares for Rs 40 each a

man gets 4% profit on his investment. What rate

per cent is company paying? What is his dividend

if he buys 120 shares?

141. Complete the following:

Sr. No. P P′ Mirror line Line in which

points are

invariant

1 (2 , 0) (- 4 , 0)

2 (0 , 3) (0 , - 6)

3 (3 , 1) (9 , 1)

4 (2 , 7) (2 , -1)

142. Solve, 2 significant figures: x2 − 10x + 6 = 0





143. Mr. Pascal invested Rs 8000 in 8% Rs 100 shares

selling at Rs80. After a year he sold these shares

at Rs 75 each and invested the proceeds in Rs 90

shares selling at Rs 100 with a dividend of 12%.

Calculate : (i.)his income from the first

investment; (ii) his income from the second

investment, (iii) the increased percentage return on

his original investment.



25 x2 −10x + 1 = 0

145. A owns 560 shares of a company. The face value

of each share is Rs 25. The company declares a

dividend of 9%. Calculate : (i) the dividend A

would receive; (ii) the rate of interest on his

investment considering that A bought these shares

at Rs 30 per share in the market.

146. In the figure given alongside AB and CD are the

lines 2x−y+6=0 and x−2y=4 respectively.(i) Write

down the co−ordinates of A,B,C and D; (ii) Prove

that triangles OAB and ODC are similar; (iii) Is

the figure ABCD cyclic? Give reasons for your

answer.

147. Prove that : If two triangles are similar the ratio of

their areas equals ratio of the square of their

corresponding sides.

148. A spherical ball of lead has been melted and made

into identical small balls each with radius equal to

half the radius of the original one. How many such

balls can be made?

149. If b is the mean proportional between a and c.

Prove:





150. Fill in the blanks:

Sr.no. point Reflect in Image

1 (2 , 6) x-axis

2 (3 , −4) y-axis

3 (−2 , 4) y = 0

4 (6 , −2) x = 0

5 (−2 , −4) origin

6 (−3 , 2) (−3, −2)

7 (−3 , 7) (3 , 7)

8 (3 , 4) x = −1

9 (−2 , 3) y = 5

10 (−2 , 8) (−2 , 4)

11 (4 ,−3) (4 , 5)

151. A girl goes to her friends house, which is at a

distance of 12 km. she covers half the distance at a

speed of x km/hr. and the remaining at a speed of

x+2 km/hr. If she takes 2 hrs. 30 minutes to cover

the whole distance, find ‘x’.

152. Draw a triangle ABC in which AB = 6 cm, BC =

4.5 cm and AC = 5 cm.

(i) Draw and label the locus of the centres of all

circles which touch AB and AC,

(ii)the locus of the centres of all the circles of

radius 2 cm which touch AB.

Hence, construct the circle of radius 2 cm which

touches AB and AC.

153. A =0 4

3 0

6

−1

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙̇˙̇˙̇˙̇˙̇˙̇

and

B =

sin00

−cot450

−5

tan450

cose c 300

−2cot230

0

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙̇˙̇˙̇˙̇˙̇˙̇˙̇˙̇˙̇˙̇

,

find if possible (i)AB , (ii) BA , (iii)A2.

154. Use ruler and compasses only for the following

question. Construct triangle BCP where CB = 5

cm , BP = 4 cm and ∠ PBC = 45°,Complete the

rectangle ABCD such that:

(i) P is equidistant from AB and BC; and

(ii) P is equidistant from C and D.

Measure and write down the length of AB.





155. A is represented on number line as given below.

B is also represented on a number line given below

.

a)Write A in the set builder form.

b) Write B in the set builder form.

c) On the given number line show A ∩ B.

d) On the given number line show A ∪ B

.

e) On the given number line show A ∩ B'.

f) On the given number line show A'∩ B.

156. Parul holds 1350; Rs 100 shares of a company

that pays 15% dividend annually Calculate her

annual dividend. If she had bought these shares at

40% premium, what percentage return does she

get on her investment? Give your answer to the

nearest integer.




Name or write equations for the lines L1 and L2.

158. A man invests Rs 2,400 for two years at

compound interest. After one year this money

amounts to Rs 2,550. Find the interest for the

second year correct to the nearest paise.

159. In the figure given alongside BC parallel to DE.

Area of triangle ABC = 150 cm2, area of

trapezium BCED = 66 cm2, DE = 42 cm.

Calculate the length of BC.

160. The value of a machinery in 2007 is Rs.29160. If

the rate of depression is 10 % p.a., find the value

of the computer in 2004.

161. A certain sum of money is invested for three years

at 10% p.a., if the compound interest for the

second year is Rs 880, calculate the amount at the

end of three years.

162. The line passing through (−4, −2) and (2, −3) is

perpendicular to the line passing through (a, 5)

and (2, −1). Find a.

163. Two isosceles triangles have equal vertical angles

and their areas are in the ratio 16 : 25. Find the

ratio of their corresponding heights.

164. A foot path of uniform width runs round the inside

of a rectangular field 16m long and 14 m wide. If

the path occupies 56 m2, Find the width of the

path.

165. If A, B and C are 2X2 matrices, is A. (B.C ) =

(A.B) C? Why?


at a certain rate is Rs 300 for the first year and the

compound interest for 2 years is Rs 612. Find the

sum and the rate of interest.

167. Calculate the co−ordinates of the centre of the

circle which passes through the points (4,2),

(2,−2) and (7,−7). Find the value of its radius.

Also, calculate the centroid of ∆ formed by joining

the three points.





168. If A and B are square matrices and if A.B = 0,

then A = 0 or B = 0? Why?

169. For what values of a and b, the expression 2x3 +

ax2 + b x + 3 is divisible by

2x2 + x −1.

170. The mean of 1,7,5,3,4 and 4 is m.The numbers

3,2,4,2,3,3 and p have mean m−1 and median q.

Find p and q.

171. If A= , B= and (A+B)2= A2 +

B2, find the value of x.

172. Find the co−ordinates of the points of trisection of

the line joining the points (−3,0) and (6,6).

173. A man invests Rs 46,875 at 4% per annum

compound interest for 3 years. Calculate : (i) The

interest for the 1st year; ( ii) The amount

standing to his credit at the end of the second year;

(iii) The interest for the 3rd year.

174. If x =

175. By investing Rs 7500 in a company paying 10%

dividend an income of Rs.500 is received. What

price is paid for each Rs 100 share?

176. Find the ratio compounded of the reciprocal ratio

of 15:28 , the sub−duplicate ratio of 36:49 and

triplicate ratio of 5 :4.

177. Solve :

178. Lines mx+3y+7=0 and 5x−ny−3=0 are

perpendicular to each other. find the relation

connecting m and n.

179. A motor boat whose speed is 15 km/hr in still

water goes 30 km downstream and comes back in

a total of 4 hour 30 minutes. Determine the speed

of the stream.

180. In what time will Rs 1500 yield Rs 496.50 as

compound interest at 20% per compounded

semi-annually?

181. Using the information in the given histogram,

calculate the mean.

182. A model of a missile is made to a scale of 1:600.

(i) The length of the model is 6 m; calculate the

length of the missile.

(ii) The area of the surface of the missile is

7,200,000 m2; find the surface area of the model.

(iii) The volume of the model is 200 litre;

calculate the volume of the missile is m3.

183. Prove that the points A(−5,4),B(−1,−2) and C(5,2)

are vertices of an isosceles right angled triangle.

Find the co−ordinates of D so that ABCD is a

square.





184. The results of an examination are tabulated below:

Marks less than 10 20 30 40 50 60 70 80 90 100

No. of candidates 0 25 42 65 95 120 128 135 148 150

Draw an ogive for the above data and find :(i) the number of candidates who got less than 45; (ii) the number of

candidates who got more than 75.

185. If [2x−1] : [5x+15] be in the duplicate ratio of 3 :

5, find x.

186. A can do a piece of work in ‘x’ days and B can do

the same work in x+8 days . If both working

together can do it in days; Calculate ‘x’.

187. The marks obtained by 19 boys are given :

27,36,22,31,25,26,33,24,37,32,29,28,36,35,27,26

,32,35,28, find the (i) median, (ii) lower quartile,

(iii) upper quartile and (iv) Interquartile range.

188. The machinery of a particular factory is valued at

Rs 18400 at the end of 1990. If it is supposed to

depreciate each year at 8% of the value at the

beginning of the year, calculate the value of the

machinery at the end of 1989 and 1991.

189. Calculate the mean from the following ogive.





190. In the given ∆ PQR, LM is parallel to QR and

PM:MR = 3:4.Calculate the value of ratio:(i)

and then . (ii) . (iii)

, (iv ) .

191. If x + a is a common factor of expressions

f(x) = x2 + px + q and g(x)=x2 +mx + n; show that

.

192. If .

193. John deposited Rs.260/- per month in a bank for 9

months under the Recurring Deposit Scheme. If

the maturity value of his deposits is Rs.2437.5,

what is the rate of interest given that the interest is

calculated at the end of every month?

194. If

195. The following data gives the frequency distribution for the no. of apples in the farm:

Weight.in grams 60−64 65−69 70−74 75−79 80−84

No. of apples 7 11 22 16 4

Calculate: (a) the mean weight of the apples in grams,(b) If the weight of each apple is increased by 5 gm, what

will be the new mean weight be? (c) the total weight of all the apples in the above table. (d) if the number of

apples in each category is doubled, what is the new mean weight (e)the medial class of the above data. (f) the

modal class of the above data.

196. A page from the pass book of Mr. Shah is given below. Find the interest for the period January to December

1998 at 4.5% per annum.

Date Particulars Debit (Rs.) Credit (Rs.) Balance (Rs.)

Jan. 1 Balance B/F 7,500 = 00

March 7 By Cheque 1,875 = 00

March 10 To Cash 625 = 00

July 17 To Self 3,250 = 00

Oct. 5 By Cheque 2,160 = 00

Dec. 19 To Cheque 1,340 = 00





197. By selling at Rs 77, some 2 ¼ % shares of face

value Rs 100 and investing the proceeds in 6%

shares of face value Rs 100 selling at Rs 110 a

person increased his income by Rs 117 a year.

Find : (i) How many shares did he sell? (ii) How

many shares did he buy? (iii)What was his original

income?


a, −b


−1 ± 2

200. Factorise the following, using Factor theorem:

x3 − 4x2 + x +6.

201. Solve : x2 = 25

202. Find the equations of the lines passing through

point (−2,0) and equally inclined to the

co−ordinate axes.

203. Given . Find x

and y.

204. One pipe can fill a cistern in 3 hours less than the

other. The two pipes together can fill the cistern in

6 hours 40 minutes. Find the time that each pipe

will take to fill the cistern.

205. Evaluate x and y if ;

206. A circus tent is cylindrical to a height of 4 m and

conical above it. If it is diameter is 105 m and its

slant height is 80 m, calculate the area of canvas

required. Also find the total cost of canvas used at

Rs.15 per metre if the width is 1.5 m.

207. Find the order of matrix M and hence, find matrix

M in each of the following

M×

208. The compound interest, calculated yearly, on a

certain sum of money for the second year is Rs

630 and for the third year is Rs 661.50. Calculate

the rate of interest and the sum.

209. Solve :x2 − 5x = 0

210. A sector of radius 35 cm is cut out of a thin

cardboard with angle 1440. It is folded into a cone

of maximum size. Find the curve surface and the

volume of the cone. Take = 22/7.

211. Find the ratio in which the line joining (3,4) and

(−2,8) is divided by the line x-=1.

212. Mr. Ashok has an account in the Central Bank of India. The following entries are from his pass book:

Date Particulars Withdrawals Deposits Balance

01.01.05 B/F 1200.00

07.01.05 By cash 500.00 1700.00

17.01.05 To cheque 400.00 1300.00

10.02.05 By cash 800.00 2100.00

25.02.05 To cheque 500.00 1600.00

20.09.05 By cash 700.00 2300.00

21.11.05 To cheque 600.00 1700.00

05.12.05 By cash 300.00 2000.00

If Mr.Ashok gets Rs.167.50 as interest at the end of the year where the interest is compounded annually,

calculate the rate of interest paid by the bank in his Savings Bank Account on 31st December, 2005.



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2011 LMR contd

213. A card is drawn from a well-shuffled pack of 52

cards. Find the probability that the card drawn is:

(i)a spade (ii)a red card (iii)a face card (iv)5 of

heart or diamond (v)Jack or Queen (vi) ace and

king (vii)a red and a king (viii)a red or a king

Read carefully and answer fully showing all steps

of working

214. An aeroplane flying horizontally 1 km above the

ground is observed at an elevation of 600. After 10

seconds, its elevation is observed to be 300; find

the uniform speed of the aeroplane in km per hour.

215. Quad OABC is a parallelogram. Calculate

∠OAB.

216. Prove : sin (90−A)cos (90−A)=tanA cos2A

217. The length of the direct common tangent to two

circles of radii 12 cm and 4 cm is 15 cm.

Calculate the distance between their centres.

218. A person standing on the bank of a river observes

that the angle of elevation of the top of a tree

standing on the opposite bank is 600. He moves 40

m away from the bank, he finds the angle of

elevation to be 300, find : i the height of the tree;

iithe width of the river.

219. A copper wire when bent in the form of a square

encloses an area of 121 cm2. If the same wire is

bent into the form of a circle, find the area of the

circle.

220. The area of a circle, inscribed in an equilateral

triangle, is 154 cm2. Find perimeter of the triangle.

Take = 1.73. Give answer correct to one

decimal place. Calculate the area of circumcircle

to the above triangle.

221. Given that two circles intersect at T and S. BA

and CD are produced to meet at P.

i Prove that PATD is a cyclic quadrilateral.

iiProve PA PB = PC PD.

iiiIf PA = 3 cm, AB = 7 cm, PD = 4 cm, Find PC.

222. Evaluate: sin 150 cos 750 + cos 150 sin 750

223. The height of a tree is times, the length of its

shadow. Find the angle of elevation of the sun.

224. In the given figure ∠XYZ = 52° and ∠YZX =

70°. Calculate ∠ABC, ∠BAC, ∠ACB.

225. Prove :

226. In a circle with centre O. Chords BA and DC are

produced to meet at a point P. If PA=2 cm,

AB=7cm and PC=3 cm. Find CD.

227. The given diagram represents the area swept by

the wiper of a car with the dimensions given in the

diagram, calculate the shaded area swept by the

wiper. Take



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228. The angles of elevation of the top of a tower from

two points on the ground at distances a and b

metres from the base of the tower and in the same

straight line with it are complementary. Prove that

the height of the tower is metre.

229. In a circle with centre O, diameter AB is produced

to point T. TC is tangent to the circle, if∠CAB =

24°,

Calculate ∠CTB.

230. In the adjoining diagram chords AB and CD of

the circle are produced to meet at O. Prove that

triangle ODB and OAC are similar. Given that

CD = 2 cm, DO = 6 cm and BO = 3 cm, Calculate

AB.

231. An equilateral triangle of side 8 cm is inscribed in

a circle. Find the radius of the circle.

232. In the given figure, AB is a side of a regular five

sided polygon and AC is a side of a six sided

polygon inscribed in the circle, centre S. Calculate

the sizes of i∠ASB; ii∠ACB; iii∠ABC.

233. Evaluate: sin420sin480−cos420cos480

234. In the diagram I is the incentre of triangle XYZ,

XI produced meets the circum circle of triangle

XYZ at W. ∠YXZ = 50°, ∠XYZ=70°. Calculate :

i∠WYZ;

ii∠IYZ;

iii∠YIW.

235. A chord AB subtends an angle of 1210 at the

centre O of the circle. If the length of the chord is

20 cm; calculate the radius of the circle.

236. At a point on level ground, the angle of vertical

tower is found to be such that its tangent is 5/12.

On walking 192 metres towards the tower, the

tangent of the angle is found to be 3/4. find the

height of the tower.

237. Calculate the circumference of a circle whose area

is equal to the sum of areas of the circles with

diameters 24 cm, 32 cm and 96 cm.

238. Prove :

239. The upper part of tree, broken over by the wind,

makes on angle of 450 with ground ; and distance

from the root to the point where the top of the tree

touches the ground, is 15m. What was the height

of the tree before it was broken?

240. Prove :

241. In the figure PT is a tangent to the circle. PA = 2

cm and AB = 16 cm. Prove that PT2 =PA. PB.

Hence, find PT.





242. From the top of a light house, it is observed that a

ship is sailing directly towards it and the angle of

depression of the ship changes from 300 to 450 in

10 minutes. Assuming that ship is traveling at

uniform speed, calculate how long will it take to

reach the light house.

243. A chord of length 24 cm is at a distance of 5 cm

from the centre of the circle. Find the length of the

chord of the same circle which is at a distance of

12 cm from the centre.

244. AB and CD are two chords in a circle that

intersect at P. If AP = 2 cm ,DP= 3 cm and CP= 5

cm. Find BP.

245. Evaluate: tan 100 tan 150 tan750 tan 800

246. Draw a line AP = 7.5 cm. Mark a point Q on AP

such that PQ = 4 cm. Using a ruler and

compasses only, construct:

(1) a circle of radius 2.5 cm to pass through A and

Q.

(2) a tangent to the above circle from P. Measure

its length.

247. Calculate the length of the radius of the

circumcircle and the incircle of a right angled

triangle of side 5 cm and 12 cm, that contain the

right angle. .

248. In the diagram, if the length of the chord is 30 cm

and CE:AB=5:6, find the radius of the circle.

249. O is the centre of the circle, PQR is an isosceles

triangle inscribed in a circle PQ = PR = 25 cm and

QR = 14 cm. Calculate the radius of the circle.

250. ABCD are the vertices of a rectangle in the given

figure, on the circumference of a circle. If the

length of the rectangle is 24 cm and the width is 7

cm. Find the area of the shaded region. Take π =

3.14

251. In the adjoining diagram AB and CD are two

chords intersecting at P. If CD = 9 cm, find CP

and PD.

252. If 00 < A < 900 , solve the following eqns: (i)sin

3A = cos 2A ,(ii) tan 5A = cot A , (iii) sin (3A−4)

= cos (5A+6) ,(iv) sin2 300 + cos2 (2A + 20)=1

253. A piece of card board is in the shape of quarter of

a circle of a radius 7 cm, bounded by

perpendicular radii OX and OY, points A and B

lie on OX and OY respectively such that OA = 3

cm and OB = 4 cm. The triangular part is

removed. Calculate the area and the perimeter of

the remaining piece.





254. Construct a regular pentagon of side 4cm.

Construct all the lines of symmetry for this

pentagon.

255. Two circles of radii 25 cm and 9 cm touch each

other externally. Find the length of the direct

common tangent.

256. If tan A + sin A = m and tan A – sin A = n ;

prove that ; m2 – n2 = 4

257. A vertical tower is 20m high. A man at some

distance from the tower knows that the cosine of

the angle of elevation of the top of the tower is

0.53. How far is he standing from the foot of the

tower.

258. From the top of a cliff, 60 metres high the angle of

depression of the top and bottom of a tower are

observed to be 300 and 600. Find the height of the

tower.

259. The figure given below shows a circle with centre

O in which diameter CD bisects the chord AB at

point E. If AE = EB = 8 cm and ED = 4 cm, find

the radius of the circle.

260. Construct triangle ABC, ∠B=60°, ∠A = 45° and

the circumradius = 4 cm. Hence, complete cyclic

quadrilateral ABCD such that D is equidistant

from A and C.

261. AC, BQ and BP are the tangents to the circle.

Prove that BQ= semi−perimeter of ∆ABC.

262. In the figure alongside, PR is a diameter of the

circle, PQ = 7 cm, QR = 6 cm and RS = 2cm.

Calculate the perimeter of the cyclic quadrilateral.

263. Two parallel chords are drawn in a circle of

diameter 30 cm. The length of one chord is 24 cm

and the distance between the two chords is 21 cm;

find the length of another chord.

264. Prove : sec 700 sin 200 + cosec 700cos200 =2.

265. Prove :

266. A boy of height 1.7 m is standing 20m away from

a flag staff on the same level ground. He observes

that the angle of elevation of the top of the

flagstaff is 240. Calculate the height of the

flagstaff.

267. In the area enclosed between the concentric circles,

is 770 cm2. Given that the radius of the outer

circle is 21 cm. Calculate the radius of the inner

circle. ( )





268. From a boat, 300 m away from a vertical cliff, the

angles of elevation of the top and the foot of a

vertical concrete pillar at the edge of the cliff are

55040′ and 54020′ respectively. find the height of

the pillar correct to the nearest metre.

269. A kite is flying at a height of 175 metres from the

level ground, attached to a string inclined at 600 to

the horizontal. Find the length of the string to the

nearest metre.

270. Prove :

271. In a circle with centre O, given that AB is parallel

to diameter CD. If ∠ABC = 25°. Calculate

∠AEB.

272. The angle of elevation of a stationary cloud from a

point 25m above a lake is 150 and the angle of

depression of its reflection in the lake is 450. What

is the height of the cloud above the lake−level?

273. The length of common chord of two intersecting

circles is 30 cm. If the diameters of two circles be

50 cm and 34 cm, calculate the distance between

their centres.

274. The radii of two concentric circles are 17 cm and

10 cm ; a line PQRS cuts the larger circle at P and

S and the smaller circle at Q and R. If QR = 12

cm. Calculate PQ.

275. In the figure the tangent PT = 8 cm. PA = 5 cm,

find the length of the chord AB.

276. PA = 10 cm, find AB. If PT is a tangent as shown,

find its length.

277. Prove : cos4 A – sin4 A = 2 cos2 A – 1

278. Given that O is the centre of a circle. Chord QP is

produced to any point R such that ∠TRQ= 26°.

Also given that ∠OQP = 48°. Calculate ∠PTQ

and ∠TPQ.

279. MN is a diameter of a semicircle with centre O. P,

Q, and R are points on the circumference such that

PM = PQ. If ∠QRN = 136°. Calculate ∠MPQ

and ∠PNQ. Prove that OP is parallel to QN.

280. In the given diagram chord AB and CD of a circle

intersect at E. (i)Prove that triangle ADE and

CBE are similar. ( ii)Given DC = 12 cm, DE = 4

cm and AE = 16 cm, Calculate the length of BE.





281. Given two circles that intersect at points B and E.

HEC and AEG are straight lines.

Prove ∠AHE + ∠EGC = 180°.

282. The perimeter of a semi−circle is 90 cm. Calculate

its area.

283. Draw a circle of radius 3 cm. Mark P, any point

on the circle. Construct a tangent to the circle at

point P. Mark point A on the tangent such that AP

= 4.5 cm, construct another tangent to the circle

from point A.

284. The angle of elevation of a cloud from a point

20m above lake level is xo, where tan xo= and

the angle of depression of its reflection in the lake

water is yo where tan yo= . Calculate the height

of the cloud above the lake level, correct to the

nearest m.

285. Two parallel chords of lengths 16 cm and 12 cm

are drawn in a circle of diameter 20 cm. Find the

distance between the chords, if both the chords are

ion the same side of the centre, (ii) on the opposite

sides of the centre.

286. In the figure AB is a diameter of the circle, centre

O ,CD // AB. If angle CAB = x , find the value of

i ∠COB ii∠DOC ,iii∠DAC ,iv∠ADC.

287. Prove :

288. Three spheres of diameters x, 8 cm and 10 cm are

melted and recasted into a single sphere of

diameter 12 cm. Calculate the value of x.

289. O is the centre of the circle. TP bisects ∠OPQ.

i Prove TO // PQ

ii if ∠POQ = 64°,

calculate ∠TQR.

290. Prove :sec2 A cosec2 A = tan2 A + cot2 A + 2

291. A box contains 150 bulbs out of which 15 are

defective. It is not possible to just look at a bulb

and tell whether or not it is defective. One bulb is

taken out at random from this box. Calculate the

probability that the bulb taken out is :

(i)a good one

(ii)a defective one

292. In the given figure, if

∠ACE = 430

and ∠CAF = 620 find the values of

a, b and c.

293. Find the values of x which satisfy the inequation:

Graph the solution set on the real number line.

294. Prove that :sin2A + cos

2A = 1

295. The mid point of the line segment joining (2a , 4)

and (-2 , 2b) is (1 , 2a + 1). Find the values of a

and b.





296. An article is bought at a certain price. It is marked

up by 30% on the purchased price. A discount of

10% is given on the marked price. A sales tax of

10% is marked on the discounted price and the

article is sold for Rs 3861.Find:

a) the cost price of the article;

b) the marked price on the article.

297. Let a, b, c and d be integers such that a < b,

b < c and c = d.

The mode of these four numbers is 11.

The range of these four numbers is 8.

The mean of these four numbers is 8.

Calculate the value of each of the integers a,

b, c, d.

298. An inverted cone of height 10 cm and base radius

6.4cm contains water to a depth of 5 cm,

measured from the vertex. Calculate the volume of

water in the cone.

299. An archery target has three concentric regions.

The diameters of the regions are in the ratio 1:2:3.

Find the ratio of their areas.

300. (i)If A and B are two complementary events then

what is the relation between P(A) and P(B)?

(ii)If the probability of happening of an event A is

0.46. What will be the probability of not

happening of the event A?

301. Using a ruler, construct a triangle ABC with BC =

6.4 cm, CA = 5.8 cm and ∠ABC = 600. Draw its

incircle. Measure and record the radius of the

incircle.

302. Two dice are rolled together. Find the probability

of getting:

(i)a total of at least 10

(ii)a multiple of 2 on one die and an odd number

on hte other die

303. In a single throw of a die, find the probability of

getting :

(i)5 (ii)8 (iii)a number less than 8 , (iv) a prime

number.

304. If P(E) = 0.39; find P (not E).

305. A bag contains 3 white, 5 black and 2 red balls, all

of the same shape and size. A ball is drawn from

the bag without looking into it, find the probability

that the ball drawn is:

(i)a black ball ,

(ii)a red ball ,

(iii) awhite ball,

(iv)not a red ball,

(v)not a black ball

306. In a single throw of two dice, find the probability

of :

(i)a doublet

(ii)a number less than 3 on each die

(iii)an odd number as a sum

(iv)a total of at most 10

(v)an odd number on one die and a number less

than or equal to 4 on the other die.

307. An inverted cone of height 15cm and base radius 4

cm contains water to a depth of 10 cm. Calculate

the volume of water in the cone.

308. Draw a circle of radius 4 cm. Draw two tangents

to this circle so that the angle between the tangents

is 45°. Measure distance of centre from exterior

point.

309. A card is drawn from a pack of 100 cards

numbered 1 to 100. Find the probability of

drawing a number which is a square.

310. Two dice are thrown simultaneously. Find the

probability that :

(i)both the dice show the same number,

(ii)the first die shows 6,

(iii)the total (sum) of the numbers on the dice is 9,

(iv)the product of the numbers on the dice is 8,

(v)the total of the numbers on the dice is greater

than 9.





311. A straight line passes through the points

P(-1,4) and Q (5,-2). It intersects the

co-ordinate axes at points A and B

respectively. If M is the midpoint of the

segment AB. Find:L

i) the equation of the line,

ii) The co-ordinates of A and B,

iii) The co-oridnates of point M.

312. A shopkeeper buys a camera at a discount of 20%

from the wholesaler, the printed price of the

camera being Rs.1600 and the rate of sales tax is

6%. The shopkeeper sells it to the buyer at the

printed price and charges tax at the same rate.

Find :

(i)The price at which the camera can be bought.

(ii)The VAT (Value Added Tax) paid by the

shopkeeper.

313. A pair of dice is thrown. Find the probability of

getting a sum of 10 or more, if 5 appears on the

first die.

314. Use a ruler and compass only in this question.

(i) Draw a circle, centre O and radius 4 cm.

(ii) Mark a point P such that OP = 7cm.

Construct the two tangents to the circle from P.

Measure and record the length of the tangents.

315. In the given figure, AB is a diameter. The tangent

at C meets AB produced at Q.

If find: (i) (ii)

316. An inverted cone of height 12cm and base radius

6cm contains 20 cm3 of water. Calculate the depth

of water in the cone, measured from the vertex.

317. In the given figure, ABC is a triangle. DE is

parallel to BC andAD

DB=

3

2.

(i) Determine the ratios AD

AB,DE

BC

(ii) Prove that DEF is similar to CBF.

Hence, find EF

FB.

(iii) What is the ratio of the areas of

DFE and BFC ?

318. A die is thrown once. Find the probability of

getting: (i)an even number , (ii) a number between

3 and 8 , (iii)an even number or a multiple of 3.

319. From the top of a hill, the angle of depression of

two consecutive kilometer stones, due east are

found to be 300

and 450 respectively. Find the

distance of the two stones from the foot of the bill.

320. Using a ruler and compasses only, construct a

triangle ABC in which

BC = 6 cm, ∠ACB = 45° and perpendicular from

A on BC is 4 cm. Draw an inscribed circle and

measure its radius.

321. In the given figure, AB is the diameter of a circle

with centre O and OA = 7 cm.

Find the area of the shaded region.





322. In the figure, and

. Find: (i) (ii) Hence

show that AC is a diameter.

323. The median of the following observations 11, 12,

14, 18, (x + 4), 30, 32, 35, 41 arranged in

ascending order is 24. Find x.

324. A box contains some black balls and 30 white

balls. If the probability of drawing a black ball is

two-fifths of a white ball; find the number of black

balls in the box.

325. A circular paper of radius 20 cm is cut in half and

each half is made into a hollow cone by joining the

straight edges. Find the slant height and base

radius of each cone.

326. Mukesh deposited Rs.400/- per month in a bank

for 39 months under the Recurring Deposit

Scheme. What will be the maturity value of his

deposits, if the rate of interest is 12 % per annum

and interest is calculated at the end of every

month?

327. Three identical coins are tossed together. What is

the probability of obtaining :

(i)all heads (ii)exactly two heads (iii)exactly one

head (iv)at least one head (v)at least two heads

(vi)all tails

328. A single letter is selected at random from the word

‘Probability ‘. Find the probability that it is a

vowel.

329. The price of a TV set including sales tax at

8% is Rs 16,200. Find its marked price. When

the rate of sales tax was increased, the

customer had to pay Rs. 300 more. Find the

new rate of sales tax.



ID: A


Last Minute Revision 2011

Answer Section

1. Rs4,783.25

2. show that

3. (i)12cm, (ii)1:9,1:8

4. 12 cm; 95.47 cm2

5. S.S= {1,2,3,4,5}

6. Rs 30, 000

7. Rs 1960

8.

(i)200 , (ii)300 , (iii)640 , (iv)7.11%

9. 16 litres

10. Rs 6000,10%

11. {4}

12. (16/7,27/7)

13. R = 5%

14. (i) 1cm on map represents 210000 cm on the ground = 210000

100000=2.1 km on the ground.

(ii)1 cm2 on map represents ( 2.1)2 km2 on the ground = 4.41 km2 on the ground.

(iii) the area on the map that represents the plot = 83.79

4.41=19 cm2

15. Rs.7,200; 400 shares

16. 100

17. s. s. = x: − 3 < x ≤ 3 ;x ∈ R{ }

18. 5:1;



ID: A


19. square

20. 39.5,38,40,2:

21. Rs.1,815.

22. (i) For L1 as the y-coordinate is the same, the equation of L1 is y = –4.

For L2 as the x coordinate is the same, the equation of L2 is x= –6 .

(ii) P(3,4)y = − 4

→ P’(3,y2) where using midpoint formula we get, y2= –12

(iii)Q(−5,−2)x = −6

→ Q’(x2,-2) where using midpoint formula we get, x2= –7

23. Rs. 50,000; Rs.10,000, Rs 15000; Rs 12,500

24. a = −7, b = 10

25. a=-7, b= -6; (x+2)(x-3)(x+1)

26. 38

27. x2 − 12x + 36 = 0

28. 100

29. S.S= {1,2,3….}

30. y=x+3

31.

s.s. = {1 , 2 , 3}

32. (i) A(−2,4),B(2,6),E(6,0),( ii)− ½ ,( iii) 2y+x=14,( iv) C(0,7, D(14,0(v1:6

33.

Third proportion = xy

34. 11cm , 3 cm

35. Rs.3,410

36. Rs 1872. Rs 1989



ID: A


37. 14

3

38. −1 ≤ y < 7

39.

40.

41. Loss =Rs 48

42. 603.43 cm3 ; 603.4 cm2

43. x2−6x + 7=0

44. x=

45. (i) 6,938 (ii) Rs.60,310.

46. 2x+3y=12; P=(6,0); Q=(0,4); 12 unit2.

47.

48. 9000, 10%

49. 400

50. irrational and unequal

51. 22.5 cm

52. A=(−3,0); 3x+4y+9=0

53. M=

54. 2y = 3x + 12

55. (i) Son;s age 5 years ago is 5 years

(ii) Present age of women = x2 + 5 = 52 + 5 = 30 years.



ID: A


56. ±3

57. ..

58. AC is 2y=3x+7; BD is 2x+3y=4

59. Rs.179,138

60. ..

61. M = [1 2]

62. 15o

63. 150

64. a) 10%, b) Rs. 1756.92, c) Rs. 1320.

65. rational and unequal

66. 12

67. a = −2, b = −5

68. Rs. 40.8 less

69. Rs. 45,000

70.

71. i AB and BC ; ii B and D.

72. {−.60,−8.4}



ID: A


73.

74. 40 %

75. P′′ ( 2 , 4) , Q′′ (7,8)

76. ix=2;y=5; ii (2,11); iii(7,5)

77. 30 km/hr

78. Rs.45

79. 4x2−3=0

80. 30

81. 7 , 8 , 9

82. Rs.19,200; 400; Rs3,000

83. 8cm;15cm

84. ..

85.

first investment is better

86. Rs.883.05.

87. prove

88. 12.58 years

89. ±2

90. Rs. 750

91. False , AB≠BA



ID: A


92. 1, −4 and 450

93. 1:2 and (10/3,0)

94. Rs 20,000

95. (4,1) or

96. ab= −6

97. a=8;b=4

98. −

99. Rs.2,640.

100. 3x−4y+10=0; x−2y−8=0; 3x+5y+8=0

101. x−3y−14=0

102. p=3 and q=4 or p=4 and q =3

103. 81 kg

104. 24 and −6

105. 24 units

106. (i) M(3,0);( ii) D(2,−6), rhombus; (iii)Yes,M'(3,−4)

107. i y=7 , iix+5=0

108.

109. Rs 7200

110. (x + 3) ( x− 7) (x + 4 )

111. 2,

112. Rs.64,000.

113. prove

114. DIAGRAM



ID: A


115. l

116. 67 , 64.18

117. (i) Mirror line is the perpendicular bisector of the segment joining the two points, hence the mirror line is y= –3

(ii) The equation of the line in which these points are invariant is the line that passes through both these points,

i.e x = –1

118. 6 and24

119. Rs 238

120. M=

121. (i)47.5,(ii)35.5, (iii)58.5,(iv)33%, (v)6

122. {3 , -2}

123. m=3,c=5

124. (i) 10 cm, ( ii) 4 cm2

125. (3 , 0)

126. x = 0 , 2

127. l2 & 30

128. -D is line L1, C is line L2, A is line L3, and B is L4

129. y = x - 3

130. 3

131. Rs 2600

132.



ID: A


133. p = - 1

134. mean= 4.7, median=5, mode=6

135. P′ (3 , −4) , Q′(−5 , 2)

136. [

137. prove

138. i parallelogram; ii No, lines of symmetry; iii P , 8,8 and Q , 8,1, pentagon.

139.

24 Rs. 5665

140. 6.4 %; Rs.192

141. ..

142. {9.4,.64}

143. Rs 800; Rs 810;

144. real and equal

145. Rs 1260, 7 ½ %

146. iA(0,6);B(−3,0);C(0,−2);D(4,0); iiS−A−S test; iiiYes ABCD is cyclic

147. prove

148. 8

149. prove

150. ..

151. 4

152. DIAGRAM



ID: A


153. i

154. l

155. a) A = { x: − 4 < x ≤ 3,x ∈ R}

b) B ={ x: − 3 ≤ x <5,x ∈ R}

c)A ∩ B

d)A ∪ B

e)A ∩ B'

f) A'∩ B

156. Rs 20,250, 11%

157. L1 is x-axis or y = 0; L2 is y-axis or x = 0

158. Rs 159.38

159. In In ABC and ADE

∠BAC=∠DAE ( common angle )

∠ABC = ∠ADE ( corresponding angles)

∴ ABC ∼ ADE (A-A postulate)

ADE= ABC+ trap.BCED = 150 + 66 =216 cm2

ABC

ADE=

BC

DE

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

2

(the ratio of areas of similar triangles equals square of ratio of its sides)

150

216=

BC

42

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

2

⇒ 25 / 36=BC

42⇒

5

6× 42 = BC ∴ BC = 35cm

160. P = Rs.40000

161. Rs.10,648

162. a=3



ID: A


163. 4:5

164. 1 m

165. True, multiplication of matrices is associative

166. Rs 7500,4%

167. (7,−2); 5 units

168. False

169. a = −5 , b = - 4

170. 4,3

171. 1

172. (0,2) and (3,4)

173. Rs 1875; Rs 50,700; Rs 2,028

174. show that

175. Rs 150

176. 25

8

177. x=[

178. 5 = m , 3 = n

179. 5 km/hr

180. 1.5 years



ID: A


181. mean=38

182. (i)k =length of the model

length of the missile

1

600=

6

length of the missile⇒ ∴ length of the missile = 600 × 6 = 3600m.

(ii) k2

=area of the model

area of the missile

1

6002

=area of the model

7,200,000⇒ ∴

1

6002

× 7,200,000 = area of the model

∴ area of the model= 20m2.

(iii) k3

=volume of the model

volume of the missile

1

6003

=350

volume of the missile⇒ ∴ volume of the missile = 600

3× 350 = 75,600,000,000 litres.

To

convert litres to m3 we divide by 1000

therefore the volume in m3 is 75,600,000,000

1000=75,600,000 m3

183. (1 , 8)

184. 80,19

185. x = 32

186. 8

187. 29,26,35,9

188. Rs 20,000; Rs16928

189. 38.85 yrs

190. (i)3:7, 3:7 ,(ii)3:7 ,(iii)10:7 , (iv)9:49

191. ..

192. 13 : 1



ID: A


193.

r = 10 %

194. prove

195. (i) 71.92g,( ii)76.92g (iii)4315g, (iv)71.92 g, (v)69.5−74.5 (vi) 69.5−74.5

196. Rs.377.40

197. 60; 42; Rs 135

198. a. x2−:[a−b]x−ab=0

199. x2+ 2x−1=0

200. (x−2)(x−3)(x+1)

201. {5 , -5}

202. y=x+2 and x+y+2=0

203. x=3 and y=2

204. 12 hrs ; 15 hrs.

205. x=3, y=2

206. 14520 sq m and Rs.145200

207. M = [−4 −5]

208. Rs 12000, 5%

209. {0 , 5}



ID: A


210. 1540 cm2; 6587.09 cm3

211. 2:3

212. R = 10% p.a.



ID: A


2011 LMR contd

Answer Section

213. (i)1

4 (ii)

1

2(iii)

3

13 (iv)

1

26(v)

2

13 (vi)0 (vii)

1

26 (viii)

7

3

214. 415.67 km/hr

215. 60 cm

216. prove

217. 17 cm

218. 20 m,20m

219. 154 cm2

220. 72.7 cm

221. 7.5 cm

222. ..

223. 600

224. 76°,64°,40°

225. prove

226. 3 cm

227. 77 cm2

228. prove

229. 42°

230. 13 cm

231.

232. 72°,36°,30°

233. ..

234. 25°,35°,60°



ID: A


235. 11.49 cm

236. 180 m

237.

238. prove

239. 36.21m

240. prove

241. 6 cm

242. 13.66 minutes

243. 10 cm

244. 7.5 cm

245. ..

246. ..

247. 2cm, 6.5 cm

248. 17 cm

249.

250. 322.625 cm2

251. CP 1.15 cm, PD = 7.85 cm

252. (i)180 (,ii) 150 ,(iii) 11o ,(iv) 50

253. 38.5 cm2, 23 cm

254. ..

255. 30 cm



ID: A


256. prove

257. CB=12.5m

258. 40 m

259. 10 cm

260. ..

261. ..

262. 24 cm

263. 18cm

264. prove

265. prove

266. 10.6 m

267. 14 cm

268. 21 m

269. 202 m

270. prove

271. 40°

272. 43.3 m

273. 28cm

274. 9 cm

275. 7.8 cm

276. 4 cm ; 7.75 cm

277. prove

278. 42°,37°

279. 134°,23°

280. 2 cm



ID: A


281. ..

282. 481.2 sq. cm.

283. ..

284. 24.44m

285. i 2 cm, ii 14 cm

286. 2x°; 180−4x; 90−2x; 90+x

287. prove

288. 6 m

289. 61°,

290. prove

291. (i)9

10 (ii)

1

10

292. In

= 1800 – (62 + 43) = 75 (sum of angles of a triangle)

a = 180 – 75 = 105 (opp. angles of cyclic quadrilateral)

In

62 + 105 + b = 180 (sum of angles of a triangle)

b = 13

= 43 + 62 = 105 (exterior angle property)

13 + 105 + c = 180

c = 62

293.



ID: A


294. Using the definition of sin A and cos A we get;

sinA =opp

hyp=

BC

AC; cosA =

adj

hyp=

BA

AC

L. H. S:

sin2A + cos

2A

=BC

AC

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

2

+BA

AC

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

2

=BC

2+ BA

2

AC2

=AC

2

AC2

(By Pythagoras Theorem; BC2

+ BA2

= AC2)

= 1 Hence Proved.

295.

2 = 2a – 2 2(2) + 1 =

2a = 4 10 = 4 + 2b

a = 2 b = 3

296. a) Rs 3000/- ; b) Rs. 3900/-

297. d = 11; c = 11

d – a = 8 therefore

therefore b = 7

298. 53.6 cm

299. 1:3:5

300. (i) 1 , (ii)0.054



ID: A


301. radius of incircle 1.1 cm

302. (i)1

6, (ii)

1

2

303. (i)1

6 (ii)0 (iii)1 (iv)

1

2

304. 0.61

305. (i)1

2 (ii)

1

5 (iii)

3

10 (iv)

4

5(v)

1

2

306. (i)1

6, (ii)

1

9, (iii)

1

2,(iv)

11

12 ,(v)

4

13

307. 74.5 cm3



ID: A


308. 10.5 cm

309. Total number of all possible outcomes = 100

Since required (favourable) outcomes are : 1 , 4, 9 , 16 ,25 , 36 , 49 , 64 , 81 or 100

The number of favourable outcomes = 10

Required probability = Number of favourable outcomes

Totalnumber of outcomes=

10

100=

1

10



ID: A


310. When the two dice are thrown simultaneously, all the possible outcomes are as shown alongside.

Clearly, the total number of all possible outcomes = 62

= 6 × 6 = 36

(i)When both the dice show the same number:

The favourable outcomes are : (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) , (6 , 6)


Required probability = 6

36=

1

6

(ii)When the first die shows 6 :

The favourable outcomes are : (6 , 1) , (6 , 2) , (6 , 3) , ( 6 , 4) , (6 , 5) and (6 , 6)



36=

1

6

(iii)When the total of the numbers on the dice is 9:

The favourable outcomes are : (3 , 6) , (4 , 5) , (5 , 4) and (6 , 3)



36=

1

9

(iv)When the product of the numbers on the dice is 8:

The favourable outcomes are : (2 , 4) and (4 , 2)



36=

1

18

(v)When the total of the numbers on the dice is greater than 9:

The favourable outcomes are : (4 , 6) , (5 , 5) , (5 , 6) , (6 , 4) , (6 , 5) and (6 , 6).



36=

1

6



ID: A


311.



ID: A


312.

313. 1

18



ID: A


314. 5.9 cm

315. (i) ∠BCA = 900 (angle inscribed in a semicircle is 90)

90 + 34 + ∠CBA = 180 (Sum of angles of a triangle)

∴ ∠CBA = 560

(ii) ∠BCQ = 340 (angles in the alternate seg. are equal)

∴ ∠ CBA = ∠BCQ + ∠CQB (ext. angle property of a triangle)

56 = 34 + ∠CQA

∴ ∠CQA = 220

316. 4.24 cm



ID: A


317. (i)

(ii) (vertically opp. angles)

(iii)

318. (i)1

2 (ii)

2

3 (iii)

2

3

319. Let BC= x,

therefore, the distance of the stones from the foot of the hill is 1.366 km and 2.366 km respectively.



ID: A


320.

321. area of semicircle CBD = 1

2πr

2

=1

2×

22

7× 7 × 7 = 77 cm

2

area of BCD =1

2b × h =

1

2× 14 × 7 = 49 cm

2

area of semicircle - BCD = 77 - 49 = 28 cm2

radius of circle (smaller) = 3,5 cm

area of small circle = πr2

=22

7× 3.5 × 3.5 = 38.5 cm

2

area of shaded part = 28 + 38.5 = 66.5 cm2

322. (i) ∠BAD + ∠BCD = 1800 (Opp. angles of a cyclic quad. are supplemenrtary)

∴ ∠ BCD = 1150

(ii) In ABD ; 650

+ ∠ADB + 700

= 1800

(Sum of angles of a triangle)

∴ ∠ADB = 180 - 135

= 45

In ADC ; ∠ADC = 45 + 45 = 90

AC is the diameter of the circle. (Converse of angle inscribed in a semicircle is 90)

323. 11 , 12, 14, 18, (x + 4) , 30 , 32 , 35 , 41

N = 9

Median rank = N + 1

2 = 5th observation

x + 4 = 24 = x = 20

324. x = 12

325. 20cm, 10 cm.

326.

monthly deposit= x = Rs.400 ; n = 39 months ; r = 12 %

Interest =n(n + 1)x

2×

r

100×

1

12=

39(39 + 1) × 12

2400= 3120

maturity value = nx + I = 39×400 + 3120 = Rs. 18720 (Ans)



ID: A


327. When three coins are tossed together (or , a single coin is tossed three times): the possible outcomes are : HHH ,

HHT , HTH , THH , HTT , THT , TTH , and TTT

i.e., the total number of possible outcomes = 8

For the favourable outcomes, we can form a table as shown below:

Favourable outcomes No. of favourable outcomes

(i)All heads : HHH 1

(ii)Exactly two heads : HHT , HTH , THH 3

(iii)Exactly one head : HTT , THT , TTH 3

(iv)At least one head : HTT , THT , TTH ,

HHT , HTH , THH , HHH

7

(v)At least two heads : HHT , HTH , THH ,

HHH

4

(vi)All tails : TTT 1

(i) P (all heads) = 1

8

(ii)P (exactly two heads) = 3

8

(iii)P(exactly one head) = 3

8

(iv)P (at least one head) = 7

8

(v)P (at least two heads) = 4

8=

1

2

(vi)P (all tails) = 1

8

328. (i) 5

31 (ii)

5

31 (iii)

8

31

329.



PC LMR 2011

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