PC Last Minute Revision for ICSE Math by Prashant Gohil

2011 Last Minute Revision L.M.R.

prashantgohil@gmail.com PRASHANT CLASSES 19/01/11

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.

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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.

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

•

His earlier assignments include:

•

IB Mathematics Teacher at BD Somani International School,

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

ICSE Mathematics Teacher at Bombay International School

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.)

IB Mathematics Higher Level and Standard Level ( Dhirubhai Ambani International School, Ecole Mondiale, B.D Somani, Jamnabai Narsee, Fazlani, D.Y. Patil etc)

•

HSC Mathematics(Std 11 and 12),

ICSE Mathematics and Physics (Std 9 and 10)

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

SAT

SAT Subject Tests- Mathematics and Physics

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.

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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.

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.

x + 2y − z y + 2z − x z + 2x − y = = 2. If 3a − b 3b − c 3c − a x+y+z show that each fraction = . a +b +c

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.

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

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.

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.

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.

5. Solve the following inequation and represent the solution set on a number line. 30 − 4(2x−1) > −8 and x ∈ { positive integers}

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.

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13. At what rate per cent per annum C.I. will Rs 2,000 amount to Rs 2,315.25 in 3 years?

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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.

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.

. 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.

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.

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

No. of boys

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.

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?

24. Find the values of a and b so that the polynomial x3 −4x2 + ax +b is exactly divisible by x2 + x − 2.

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.

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.

(ii)Write down the images of points P (3 ,4 ) on reflection in L 1. Name the images as P′. (iii)Write down the images of Q(-5,-2) on reflection in L2. Name the images as Q′.

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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.

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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?

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.

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.

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).

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.

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

Graph the solution set on the real number line. 39. If A =

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.

. 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 :

Neighbour:

5 eggs

4 pies

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.

33. Find the third proportional to

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.

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.

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43. Framing quadratic equations from the given roots 3+

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50. Without solving, comment upon the nature of roots of each of the following equations: 6x 2 − 13x + 4 =0

,3−

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.

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)

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.

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.

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.

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.

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 ,

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.

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

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.

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. 64. The compound interest on a certain sum of money 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.

where Mt is transpose of M.

61.

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62. The figure alongside

65. Without solving, comment upon the nature of roots of each of the following equations: 2 7x − 9x + 2 =0

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

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]

No. of plants

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.

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.

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 %.

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 + BP 2 where A and B are two fixed points.

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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. 79. Framing quadratic equations from the given roots

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?

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

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.

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

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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 1 2 3 4 5 6

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(2 ,4) (-6 , 8) (3 , -2) (2 , 3) (-3 , 7) (2 , 5)

Equation of the mirror line

(2 , 8) (4 , 2) (3 , 0) (6 , 3) (-4 , 0) (- 1 , 8)

85. Which is a better investment : 12 % at Rs 120 or 8% at Rs 90 ?

91. If A and B are square matrices, is ( A+ B)2 = A2 + 2 AB + B2? Explain.

86. Calculate the amount and compound interest on Rs.14500 at 6 % p.a. for 12 months compounded half yearly.

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.

87. If q is the mean proportional between p are r, prove that:p2−q2 + r2 = q4

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.

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 x 2−4kx + 12 = 0 are k and 3k. Find the value of k.

95. The abscissa of a point is four times its ordinate. Find the co−ordinates of the points if its distance units. from (6,2) is

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.

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

The mean daily expenditure per student is Rs.40.20 p. Find the values of a and b. 103. Two solid spheres have surface areas of 5 cm 2 and 45 cm 2 respectively and the mass of the smaller sphere is 3 kg. Find the mass of the larger sphere.

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.

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.

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.

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 .

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 ÈÍ ˘˙ Í ˙ ÍÈÍ p q ˙˘˙ ÍÍÍÍ p ˙˙˙˙ = [25] ÍÍÎ ˙˙˚ ÍÍÍ ˙˙˙ ÍÍ q ˙˙ Î ˚

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. down A and B in set builder notation. A ∩ B′ on two different number lines.

(i) Write (ii) Represent A ∩ B,

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.

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.

110. Factorise: x3 − 37x − 84.

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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?

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. a + 3b +

a − 3b

a + 3b − 2ax + 3b = 0.

a − 3b

113. If x =

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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.

, prove that 3bx -

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

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.

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.

118. Find two numbers such that the mean proportional between them is 12 and the third proportional to them is 96.

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

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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.

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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 cm 2 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.

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

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134. Calculate the mean, median and mode:

135. Points (5 , 0) and (−2, 0) are invariants points under reflection in the line L1. Points (0, −6) and (0 ,3) are invariant points on reflection in line L2. 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.

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?

137. If

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′

1 2 3 4

(2 , 0) (0 , 3) (3 , 1) (2 , 7)

(- 4 , 0) (0 , - 6) (9 , 1) (2 , -1)

Mirror line

Line in which points are invariant

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

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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.

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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.

144. Without solving, comment upon the nature of roots of each of the following equations: 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.

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:

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150.

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Fill in the blanks: Sr.no.

point

Reflect in

(2 , 6)

x-axis

(3 , −4)

y-axis

(−2 , 4)

y= 0

(6 , −2)

x=0

(−2 , −4)

origin

(−3 , 2)

(−3, −2)

(−3 , 7)

(3 , 7)

(3 , 4)

x = −1

(−2 , 3)

y= 5

(−2 , 8)

(−2 , 4)

(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. ÈÍ ˘˙ ÍÍ ˙˙ ÍÍ 0 4 ˙˙ 6 ˙˙ and 153. A = ÍÍÍ ˙˙ ÍÍ ÍÎ 3 0 −1 ˙˙˚ ÍÈÍ ˙˘ 0 ÍÍ sin00 tan45 ˙˙˙˙ ÍÍ ÍÍ ˙˙ Í 0 0 ˙ ˙ B = ÍÍÍ −cot 45 cos e c 30 ˙˙˙ , ÍÍ ˙˙ ÍÍ ˙ 2 0 ˙ ÍÍ −5 ˙˙ −2cot 30 ÍÎÍ ˙˚˙

Image

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.

find if possible (i)AB , (ii) BA , (iii)A 2 .

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

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159. In the figure given alongside BC parallel to DE. Area of triangle ABC = 150 cm 2, area of trapezium BCED = 66 cm2, DE = 42 cm.

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 Calculate the length of BC. . e) On the given number line show A ∩ B'.

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.

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

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.

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. 157. Points (5 , 0) and (−2, 0) are invariants points under reflection in the line L1. Points (0, −6) and (0 ,3) are invariant points on reflection in line L2. Name or write equations for the lines L1 and L 2. 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.

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?

166. The compound interest on a certain sum of money 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.

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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 2x 2 + 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.

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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.

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 :

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.

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184. The results of an examination are tabulated below: Marks less than

100

No. of candidates

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.

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190. In the given ∆ PQR, LM is parallel to QR and PM:MR = 3:4.Calculate the value of ratio:(i) and then

. (ii)

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191. If x + a is a common factor of expressions f(x) = x2 + px + q and g(x)=x2 +mx + n; show that .

. (iii)

, (iv )

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

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

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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? 198. Framing quadratic equations from the given roots a, −b 199. Framing quadratic equations from the given roots −1 ±

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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

2 M×

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. 212.

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.

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

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.

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.

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.

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.

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

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

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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 metre. the height of the tower is 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.

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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. 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.

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.

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.

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.

233. Evaluate: sin420sin480−cos420cos480

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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.

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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.

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.

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 Ď&#x20AC; = 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 (3Aâ&#x2C6;&#x2019;4) = cos (5A+6) ,(iv) sin 2 300 + 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.

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254. Construct a regular pentagon of side 4cm. Construct all the lines of symmetry for this pentagon.

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261. AC, BQ and BP are the tangents to the circle. Prove that BQ= semi−perimeter of ∆ABC.

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 ; m 2 – 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.

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.

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. 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 :

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.

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 cm 2. Given that the radius of the outer circle is 21 cm. Calculate the radius of the inner circle. ( )

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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 55 040′ 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.

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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. 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.

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.

In the figure the tangent PT = 8 cm. PA = 5 cm, find the length of the chord AB.

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Call 2380 00 92.

281. Given two circles that intersect at points B and E. HEC and AEG are straight lines. Prove ∠AHE + ∠EGC = 180°.

289. O is the centre of the circle. TP bisects ∠OPQ. i Prove TO // PQ ii if ∠POQ = 64°, calculate ∠TQR.

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

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

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

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.

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

Graph the solution set on the real number line. 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.

294. Prove that :sin2 A + cos 2 A = 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.

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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.

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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.

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 â&#x2C6; 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

308. Draw a circle of radius 4 cm. Draw two tangents to this circle so that the angle between the tangents is 45Â°. 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.

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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. find: (i) (ii) If

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317. In the given figure, ABC is a triangle. DE is AD 3 = . parallel to BC and DB 2 AD DE (i) Determine the ratios , AB BC (ii) Prove that DEF is similar to CBF. EF Hence, find . 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 45 0 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, â&#x2C6; ACB = 45Â° 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.

316. An inverted cone of height 12cm and base radius 6cm contains 20 cm 3 of water. Calculate the depth of water in the cone, measured from the vertex.

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322. In the figure, . Find: (i) show that AC is a diameter.

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and (ii)

Hence

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 â&#x20AC;&#x2DC;Probability â&#x20AC;&#x2DC;. 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. Shop No. 12, Mahavir Apartments, Opp Swati Snacks, Tardeo Road, Mumbai 7.

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Last Minute Revision 2011 Answer Section 1. Rs4,783.25

2. show that

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

12 cm; 95.47 cm2

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% 210000 =2.1 km on the ground. 100000 (ii)1 cm2 on map represents ( 2.1)2 km2 on the ground = 4.41 km2 on the ground. 83.79 (iii) the area on the map that represents the plot = =19 cm2 4.41

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

15. Rs.7,200; 400 shares

16. 100 17. s. s. = {x: − 3 < x ≤ 3 ;x ∈ R} 18. 5:1; Shop No. 12, Mahavir Apartments, Opp Swati Snacks, Tardeo Road, Mumbai 7.

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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 . y=−4

 →P ’(3,y2) where using midpoint formula we get, y2= –12 (ii) P(3,4)  x = −6

 →Q’(x2,-2) where using midpoint formula we get, x2= –7 (iii)Q(−5,−2)  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

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37.

14 3

38. −1 ≤ y < 7

39.

40.

41. Loss =Rs 48 42. 603.43 cm3 ; 603.4 cm 2

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 unit 2. 47. 48. 9000, 10% 49. 400 50. irrational and unequal

51. 22.5 cm

52. A=(−3,0); 3x+4y+9=0

53.

54. 2y = 3x + 12 55. (i) Son;s age 5 years ago is 5 years (ii) Present age of women = x 2 + 5 = 52 + 5 = 30 years. Shop No. 12, Mahavir Apartments, Opp Swati Snacks, Tardeo Road, Mumbai 7.

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56. ±3

57. ..

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

59. Rs.179,138

60. ..

61. M = [1 2]

62. 15 o 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}

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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. 4x 2 −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

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

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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.

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 cm 2

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.

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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

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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 cm

2 ABC ÊÁÁÁ BC ˆ˜˜˜ = (the ratio of areas of similar triangles equals square of ratio of its sides) ADE ÁÁË DE ˜˜¯ 2 150 ÁÊÁÁ BC ˜ˆ˜˜ =ÁÁ ˜ 216 Ë 42 ˜¯

160.

⇒

25 / 36=

BC 42

⇒

5 × 42 = BC 6

∴ BC = 35cm

P = Rs.40000

161. Rs.10,648 162. a=3

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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

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181. mean=38 182. (i)k =

length of the model length of the missile

1 6 = 600 length of the missile (ii) k 2 =

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

area of the model area of the missile

1 area of the model = 2 7,200,000 600

⇒ ∴

1 × 7,200,000 = area of the model 600 2

∴ area of the model = 20m2 . volume of the model (iii) k 3 = volume of the missile

1 350 = ⇒ ∴ volume of the missile = 6003 × 350 = 75,600,000,000 litres. 3 volume of the missile 600 convert litres to m3 we divide by 1000 75,600,000,000 =75,600,000 m3 therefore the volume in m3 is 1000 183. (1 , 8) 184. 80,19

185. x = 32

186.

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

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

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210. 1540 cm2; 6587.09 cm 3

211. 2:3 212.

R = 10% p.a.

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2011 LMR contd Answer Section 213. (i)

1 1 3 1 2 1 7 (ii) (iii) (iv) (v) (vi)0 (vii) (viii) 4 2 13 26 13 26 3

214. 415.67 km/hr 215. 60 cm 216. prove

217. 17 cm

218. 20

m,20m

219. 154 cm 2 220. 72.7 cm 221. 7.5 cm 222. ..

223. 60 0 224. 76°,64°,40° 225. prove 226. 3 cm 227. 77 cm 2

228. prove 229. 42° 230. 13 cm 231.

232. 72°,36°,30° 233. .. 234. 25°,35°,60°

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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 cm 2 251. CP 1.15 cm, PD = 7.85 cm 252. (i)18 0 (,ii) 150 ,(iii) 11o ,(iv) 50 253. 38.5 cm 2, 23 cm 254. .. 255. 30 cm

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

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

289. 61°, 290. prove 291. (i)

9 1 (ii) 10 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 b = 13 = 43 + 62 = 105

(sum of angles of a triangle) (exterior angle property)

13 + 105 + c = 180 c = 62

293.

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294.

Using the definition of sin A and cos A we get; adj BA cos A = = hyp AC

opp BC sinA = = ; hyp AC L. H. S: 2

sin A + cos A 2 2 ÁÊ BC ˜ˆ˜ ÁÊ ˜ˆ ˜˜ + ÁÁÁ BA ˜˜˜ = ÁÁÁÁ ˜ Á AC ˜ Ë AC ¯ Ë ¯

BC2 + BA 2 AC2

AC AC 2

(By Pythagoras Theorem; BC 2 + BA 2 = AC 2 ) Hence Proved.

295.

2 = 2a – 2 2a = 4 a=2

2(2) + 1 = 10 = 4 + 2b 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

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301. radius of incircle 1.1 cm

1 1 302. (i) , (ii) 6 2 1 1 303. (i) (ii)0 (iii)1 (iv) 6 2 304. 0.61 1 1 3 4 1 (iv) (v) 305. (i) (ii) (iii) 2 5 10 5 2 1 1 1 11 4 306. (i) , (ii) , (iii) ,(iv) ,(v) 6 9 2 12 13 307. 74.5 cm 3

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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 Number of favourable outcomes 10 1 Required probability = = = Total number of outcomes 100 10

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PRASHANT CLASSES --TUITIONS FOR IB, IGCSE- MATHS & PHYSICS. Experienced IB teacher and IB examiner. 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 Ă&#x2014; 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) The number of favourable outcomes = 6 6 1 Required probability = = 36 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) The number of favourable outcomes = 6 6 1 Required probability = = 36 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) The number of favourable outcomes = 4 4 1 = Required probability = 36 9 (iv)When the product of the numbers on the dice is 8: The favourable outcomes are : (2 , 4) and (4 , 2) The number of favourable outcomes = 2 2 1 = Required probability = 36 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). The number of favourable outcomes = 6 6 1 Required probability = = 36 6

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PRASHANT CLASSES --TUITIONS FOR IB, IGCSE- MATHS & PHYSICS. Experienced IB teacher and IB examiner. ID: A 311.

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312. 313.

1 18

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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 0 (ii) ∠BCQ = 34 (angles in the alternate seg. are equal) ∴ ∠ CBA = ∠BCQ + ∠CQB (ext. angle property of a triangle) 56 = 34 + ∠CQA ∴ ∠ CQA = 22 0 316. 4.24 cm

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PRASHANT CLASSES --TUITIONS FOR IB, IGCSE- MATHS & PHYSICS. Experienced IB teacher and IB examiner. ID: A 317. (i)

(ii)

(vertically opp. angles)

(iii)

318. (i)

1 2 2 (ii) (iii) 2 3 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.

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320. 1 2 πr 2 1 22 = × × 7 × 7 = 77 cm2 2 7 1 1 area of BCD = b × h = × 14 × 7 = 49 cm2 2 2 2 area of semicircle - BCD = 77 - 49 = 28 cm

321. area of semicircle CBD =

radius of circle (smaller) = 3,5 cm area of small circle = π r

22 × 3.5 × 3.5 = 38.5 cm2 7 2

area of shaded part = 28 + 38.5 = 66.5 cm

322. (i) ∠BAD + ∠BCD = 180 0 (Opp. angles of a cyclic quad. are supplemenrtary) ∴ ∠ BCD = 115 0 ABD ; 65 0 + ∠ADB + 700 = 180 0 (Sum of angles of a triangle) (ii) In ∴ ∠ADB = 180 - 135 = 45 ADC ; ∠ADC = 45 + 45 = 90 In 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 = 5th observation 2

x + 4 = 24 = x = 20

324. x = 12 325. 20cm, 10 cm. 326. monthly deposit= x = Rs.400 ; n = 39 months ; r = 12 % n(n + 1)x 39(39 + 1) × 12 r 1 Interest = × × = = 3120 2 100 12 2400 maturity value = nx + I = 39×400 + 3120 = Rs. 18720

(Ans)

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PRASHANT CLASSES --TUITIONS FOR IB, IGCSE- MATHS & PHYSICS. Experienced IB teacher and IB examiner. 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 , 7 HHT , HTH , THH , HHH (v)At least two heads : HHT , HTH , THH , 4 HHH (vi)All tails : TTT 1 1 (i) P (all heads) = 8 3 (ii)P (exactly two heads) = 8 3 (iii)P(exactly one head) = 8 7 (iv)P (at least one head) = 8 4 1 (v)P (at least two heads) = = 8 2 1 (vi)P (all tails) = 8 5 5 8 328. (i) (ii) (iii) 31 31 31

329.

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