202110723-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G10-FY_Optimized

11. Constructions ThepointCdividesthelineABintheratioof3:2(m:n): Step 1: Draw the DABC. C Step 2: Draw a line AX making an acute angle with AB the side AB of DABC. A1 QuaAd2 ratiAc3 EqAu4ations Step 3: Locate 5 points o(n53th, eglrienaetAerX number in the fraction), A1 , A2 , A3 , A4 , A5 such that CHAPTER: X =AA1 A=1 A2 A=2 A3 A=3 A4 A4 A5 . Method 2 Step 4: Join BA5 Y B2 B1 Step 5: Draw a line through the 3rd point ( 53pa, rsamllealllteor C number in the fraction) from A i.e., A3 and BA5 such that it intersects the line AB at D A B Step 6: Draw a line through D and parallel to BC so that A1 A2 A3 X it intersects the side AC at E. Step 7: DADE is the required triangle whose sides are 3 times the length of the sides of DABC. 5 Step 1: Locate the line AB. C Step 2: Draw a line AX from A so that ∠XAB is an E acute angle. Step 3: Draw a line BY from B such that BY || AX . A DB A1 A2 A3 A4 A5 X Step 4: Locate points A1, A2 , A3 (m = 3) points on AX and B1,B2 (n = 2) points on line BY such that Procedure to construct tangents to a circle from a point outside it A=A1 A1=A2 A2=A3 B=B1 B1B2 . Step 5: Join A3B2 so that it intersects line AB at C. From a point P outside the circle, two tangents can be AC : BC = 3 : 2 constructed. Let O be the center of a circle to which the tangents from P are to be constructed. The point C divides the line AB in the ratio 3:2 (m:n) Scale factor means the ratio of the lengths. A scale Step 1: Join OP. Step 2: Draw a perpendicular bisector of OP and it factor less than 1 indicates the length has reduced and a scale factor more than 1 indicates that the intersects OP at Q. Here Q will the midpoint of OP. length has increased. Step 3: With Q as center and QO as radius draw a circle Example: and it intersects the initial circle at R and S. Step 4: Join PR and PS. PR and PS are the two tangents i) A line of 4 cm is extended till 6 cm, then the 6 to the circle from point P. scale factor will be 4 = 1.5 R ii) Two similar triangles have one set of corresponding sides of length 4 cm and 8 cm. 4 The scale factor will be 8 = 0.5 Procedure to construct a triangle similar to a given PQ triangle as per a given scale factor. To construct a similar triangle with sides 3 times the O 5 length of sides of DABC. The construction can be started over any side of the triangle, say AB of D ABC. S 87

11. Constructions Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 • Basic Proportionality Theorem PS – 2 PS - 3 Pre-requisites • Construction of triangle PS-4 • Tangents to Circle Self Evaluation Sheet Quadratic Equation • Division of a Line segment • Construction of triangle similar to another triangle • Construction of Tangents to a Circle Worksheet for “Constructions” Evaluation with Self Check ---- or Peer Check* 88

PRACTICE SHEET - 1 (PS-1) 1. Draw a triangle of sides 6 cm, 4 cm and 5 cm and explain the steps. 2. In D ABC shown in the figure, PQ || BC. If AP = 4.5 cm, BP = 2.5 cm, AC = 6 cm and BC = 8 cm, then find the length of PQ, AQ and AC. A P BC 3. Draw a circle of radius 4 cm and draw a tangent to the circle at any point of the circle. Explain the construction steps. 4. Explain the construction steps to draw a line CD parallel to a given line AB and passing through a given point C. 89

PRACTICE SHEET - 2 (PS-2) 1. Draw a line of 7.3 cm and divide it in the ratio of 4:3. Justify the construction and determine the lengths of the two segments using a scale. 2) Draw a line of 6.5 cm and divide it in the ratio of 4:1. Justify the construction and determine the lengths of the two segments using a scale. 3) Extend a line segment of length 7.3 cm by a factor of 7/6. Give the measure of the extended line. 4) A length of 8.6 cm is to be divided in the ratio of 2:3:4. Justify the construction and measure the length of the segments. 5) Draw a ∆ABC of sides 7.5 cm, 6.5 cm and 4.5 cm and then construct a triangle whose sides are 2/3 the corresponding sides of ∆ABC. Justify the construction and find the lengths of the smaller triangle. 6) Construct a similar triangle whose sides are 7/5 times the corresponding sides of the triangle with base length 6 cm and the with two base angles 90° and 30° Give the length of the size of both triangles. Explain and justify the construction. 9 7 7) An equilateral triangle of side 5.2 cm is enlarged by a scale factor of . Construct the enlarged triangle. Give the measure of the triangle. 90

PRACTICE SHEET - 3 (PS-3) 1. Draw tangent to the circle of diameter 6 cm from a point which is 8 cm from the center of the circle. Measure the lengths of the tangents. 2. OP is a horizontal line passing through the center of the circle of diameter 6 cm. Line OQ is is 5.6 cm long and is at an angle of 45° to OP. Draw tangents to the circle from point Q if O is the center of the circle. Measure the length of the tangents. 3. From point P, draw a pair of tangents to a circle of diameter 6.5 cm at the end of the chord of 4 cm length. Determine the distance of the point P from the center of the circle. 4. Draw a circle such that its center is at a distance of 7.2 cm from point P and the length of tangents from P are 5.3 cm. Determine the radius of the circle. 5. Draw two tangents to a circle of diameter 7.6 cm from point P which are at an angle of 120° to one another. Determine the distance of the point P from the center of the circle. 91

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. The first step to divide a line segment in a given ratio m : n is to _______. (A) find the sum of the ratio (B) decide the point which divides the line segment (C) draw the acute angle (D) draw the line segment 2. The number of equal parts the ray making an acute angle with the line segment is divided into __________. (A) is the difference of the terms of the ratio (B) depends on the given ratio (C) depends on the length of the line segment (D) All of these 3. If XP : XY is 3 : 8, in what ratio does P divide XY? (A) 3 : 5 (B) 5 : 8 (C) 8 : 11 (D) 8 : 5 4. M divides AB in the ratio 4 : 7. AM is _______. (A) 3AB (B) 4 AB (C) 4AB (D) 4 AB 11 7 3 11 5. The sides of a triangle are smaller than the sides of another triangle by a scale factor of __________. (A) 13 2 (C) 13 13 7 (B) 5 4 (D) 3 6. The scale factor may be ________. (A) equal to 1 (B) less than 1 (C) greater than 1 (D) Both (B) and (C) II. Very short answer questions. 1. To divide a line PQ in a given ratio how is the ray PX drawn? 2. Which theorem or criterion is used to justify that the division of line segment in a given ratio is correct? III. Short answer questions. 1. From the figure, justify that the line segment AB is divided in the ratio 5 : 8. 2. Justify the construction of a tangent to a circle from an external point. IV. Long answer questions. 1. Explain the steps involved in constructing a tangent to a circle from an external point. 2. Draw a pair of tangents to a circle of radius 6 cm, such that the angle between them is 70º. 3. Draw a ∆ABC of sides 4 cm, 5 cm and 6 cm. Then draw ∆AB’C similar to ∆ABC whose sides are 5 times the corresponding sides of ∆ABC. 3 92

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Draw a line 7.9 cm length and divide it in the ratio of 2:4. Justify the construction and give the length of the segments. (4 marks) 2. A triangle of sides 6 cm, 6 cm and 7 cm is to be enlarged so that the size 7 cm will become 9 cm. Explain the construction steps. Give the measure of the sides of the triangles. (4 marks) 3. Two sides of a triangle are 7 cm and 7 cm and the included angle is 45°. Draw a triangle whose sides are 2/3 times the sides of the given triangle. Give the measure of the sizes of the triangles. (4 marks) 4. AB is a chord of 5 cm length in a circle of diameter 6.4 cm. Draw a tangents from a point P which is at a distance of 5 cm from A and exists on extension of line AB. Measure the length of the tangents. (3 marks) 93

12. Areas Related to Circles Learning Outcome By the end of this chapter, a student will be able to: • Calculate areas of regions formed by combination • Calculate areas of sectors and segments of circles. of plane figures. Concept Map Key Points • Circumference of a circ=le 2=πr πd , where r is the radius of the circle and d is the diameter of the circle. Area of a circ=le π=r 2 πd 2 4 • π is taken as 22 or 3.14 approximately. • The piece of circle between two points is called 7 an arc. When two points or a chord divide a circle into two parts, the longer arc is called major arc • A line joining two points on the circle is called and the shorter one is called minor arc. Major arc chord. A chord which passes through the center of is represented using the two end points and an the circle is called diameter of the circle. Diameter intermediate point and minor arc is represented is the longest chord and all diameters have the using the two end points of the arc. same length which is equal to two times the radius. If the end points of the arc form the diameter, then both arc are equal and each is called a semi circle. The region between a chord and either of its arcs is called segment of circular region or simply a segment of the circle. The region between chord and minor arc is called minor segment and the region between major arc and segment is called major segment. Example: P, Q are any two points on the circle dividing it into two arcs. R is a point on the circle as shown. 94

12. Areas Related to Circles • The region between arc and two radii joining the Area of the sector OPQ=R α ×π r 2 center to the end points of the arc is called a sector. 360° The region between the major arc and two radii is called major sector and is represented using three Area of the segment PQR = Area of the sector OPQR – Area of ∆OPR points (Two end points and one point in between). The region between minor arc and two radii joining the end points is called minor sector. Example: P, Q are any two points on the circle dividing it into two arcs. R is a point on the circle as shown. Let O be the center of the circle. • When the two arcs are equal, then each is known as semicircle, then both segments and both sectors become the same and each is known as semicircular region. • For a sector of a circle of radius r and angle θ Area of a sec=tor θ ×π r 2 • Trigonometric Ratios for Definite Angles 360° Length of arc of a sect=or θ × 2πr Trigonometric Ratio Angle (θ ) 360° 30° 45° 60° Area of minor segment = Area of the sector – Area of triangle formed by radii & chord. sinθ 11 3 Area of the major segment = Area of the circle – Area 2 22 of the minor segment cosθ 31 1 Example: A circle of radius r has an arc PRQ which subtends 2 22 an angle α at the center of the circle. PR is the tanθ 1 1 3 chord of the circle. 3 • To calculate the area of the combination of plane figures, divide the given area into simpler regions. Add or subtract area of individual regions to get the final area. Example: In the given figure, the shaded region is the area of a combination of three plane figures. 95

12. Areas Related to Circles Area of the shaded region = Area of region A + Area of region B – Area of region C A, C are sectors of circle and B is a rectangle. By calculating the areas of the individual regions, the area of the shaded portion can be calculated. • Area and Perimeter of simple plane figures Geometry, Size Area & Perimeter Rectangle, l ×b Area = lb Perimeter = 2(l +b ) Square, a Area = a2 Triangle (a,b,c ) Perimeter = 4a Area of right triangle ( Area b,h ) = (s (s −a )(s −b )(s −c ) Where s =a +b +c 2 Perimeter = a +b +c Area = 1 bh 2 Parallelogram ( b,h ) Area = bh 96

12. Areas Related to Circles Work Plan Concept Coverage Coverage Details Practice Sheet PS – 1 Pre-requisites • Perimeter and Area of a Circle – A review PS - 2 PS - 3 Areas Related to Circles • Areas of Sector and Segment of a Circle PS - 4 • Areas of Combinations of Plane figures Self Evaluation Sheet Worksheet for “Areas Related to Cirles” Evaluation with Self Check ---- or Peer Check* 97

PRACTICE SHEET - 1 (PS-1) 1) A racing track as shown is made using three different grades of tar. In the middle of the track, a large lawn is made. i) What is the distance covered by the driver if he drives along track 1, track 2 and track 3 if the distance between the tracks is 3 m? ii) What is the area of each track? iii) What is the area of the lawn in the middle of the track? (Use π = 3.14) 2) An electric cable is cut and it was found that there was one copper wire in the middle and two layers of plastic insulation one over the other on the copper wire. If the diameter of the copper wire is 5 mm and the thickness of the two insulation layers is 1 mm each, what is the outer diameter of the wire? What is the area of the plastic insulation? (Use π = 3.14 ) 3) Determine the area and perimeter of the following geometries: i) A right angled triangle with legs 8 cm and 6 cm ii) A semi circle of radius 7 cm radius. iii) An isosceles triangle with the unequal side as 40 mm and the angle between the equal sides as 60°. (Use 3 = 1.73 ) 98

PRACTICE SHEET - 2 (PS-2) 1) An arc of a circle of radius 35 m subtends an angle of 90° at the center of the circle. Find the area of the corresponding segment of the circle. Determine the length of the arc. 2) From a 30 cm diameter pizza and portion of 30° is cut. What is the area of the pizza that is cut? 3) In a circular ground, at one end a stage is built such that the end points of the stage lie on the circumference of the ground. The length of the stage is 20 m and the stage subtends an angle of 60° at the center of the ground as shown. Determine the area of the stage and the remaining area left in the ground. (Use 3 = 1.73 ) 4) An old farmer had a circular field. He wanted to divide it in the radio of 1:6 so that he could give the smaller portion to charity and the larger portion to his son. If the old man wants to divide it in the form of the sectors, what is the angle of the sectors? 5) An arc of length 44 cm of a circle subtends 120° at the center the of circle. Determine the area of sector formed by the arc. (Use 3 = 1.73 ) 6) An arc of length 44 cm subtends 60° at the center of the circle. Determine the area of sector formed by the arc and the area of the segment formed by the chord. 7) In a building each floor was in the shape of a circle and had three equal partitions out of which one portion was not used for making houses. In each floor a garden was created whose area is equal to the segment of the arc formed in one sector. If the radius of the building is 21 m, determine the area of the garden in each floor of the building. (Use 3 = 1.73 ) 99

PRACTICE SHEET - 3 (PS-3) 1) Determine the area of the shaded region in the 5) An arc ABC is drawn with the O as the center. square of 50 cm. Point O is the point of intersection of the diagonals of the rectangle, where AC = 70 mm amd CD = 60 mm as shown. Determine the area of the shaded region. 2) Determine the area of the shaded region if the area of the quadrilateral is 3250 mm2 and each sector is drawn at the corners of the quadrilateral is of radius 17.5 mm. 3) Determine the area of the shaded region if the 6) Raghu makes paper plates by cutting 5 circular area of the quadrilateral is 2346 mm2 and the portions of 20 cm diameter from a sheet of size diameter of circle passing through the vertices 50 cm × 50 cm as shown in the figure. Determine of the quadrilateral is 70 mm. the percentage of paper wasted. Suggest alternate size of sheet to make the same size of paper plates but with less paper wastage. 4) A work around a pillar needed a granite slab 7) A tailor stitches a bag using pieces of cloth as of 60 cm × 40 cm size with two of its opposite shown in the figure. The bag is of square shape corners shaped as shown in the figure. If the cost and the strap of the bag is 5 cm wide. For every of finished granite slab is Rs. 300 for 1000 cm2 of bag, the tailor needs one set of strap pieces and finished area, then find the cost of finishing the two sets of pieces of the bag portion. Determine required granite slab. the cloth needed to make one bag. 100

PRACTICE SHEET - 3 (PS-3) golden letters is Rs. 2. Determine the cost of 22 8) In the figure given below, ∆ABC is an equilateral printing 20 logos. (Use πð = 7 , 3 = 1.73 ) triangle and a semicircle is drawn over the side AC with center O. Determine the area of the shaded region. (The dimensions are in cm and useðπ= 3.14 , 3 = 1.73 ) 9) A circle of 60 mm diameter is drawn through 12) ∆CAB is a right angled triangle with right angle the four vertices of a square ABCD as shown in at A as shown with sides AB, AC of length 7 cm the figure. Determine the area of the segments and 5 cm respectively. An arc is drawn passing formed by the four sides of the square and the through the three vertices with midpoint of circle. hypotenuse as its center. Another arc is drawn with vertex A as center and tangential to the hypotenuse. Determine the area of the shaded portion. 10) A circle is drawn through the three vertices 13) Determine the area of the shaded portion, of a rhombus whose side is 60 mm as shown. if the O is the midpoint of the diagonal and Determine the area of the shaded region. an arc is drawn with O as center and passing through three vertices of the quadrilateral. The (Use 3 = 1.73 and all dimensions are in mm) dimensions provided are in mm. 11) The crescent shaped logo of a company was made by drawing a chord AB to a circle of diameter 60 mm such that it subtends an angle of 120° at the centre. A semicircle is then drawn over the chord of as shown. The shaded region is the logo and the cost of printing 1 mm2 in 101

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. The circumference of a circle with diameter ‘d’ units is _______. (A) πr2 units (B) πr units (C) πd sq. units (D) πd units 2. Area of a circle of radius ‘r’ units is given by _______. (A) πd sq. units (B) πr2 sq. units (C) πr2 units (D) πr sq. units 3. The length of a sector of a circle with radius ‘r’ and angle with degree measure θ is _____________. (A) 2π rθ (B) πr2 (C) 2πθ (D) π r2θ 360 360θ 360r 360 4. The circumference of a circle with diameter 56 cm is ______. (A) 176 cm (B) 176 cm2 (C) 176 sq. m (D) All of these 5. The area of a sector of a circle of radius ‘r’ is given by _______. (A) 2π rθ (B) πr2 (C) 2πθ (D) π r2θ 360 360θ 360r 360 6. Choose the area of a circle of radius 4 cm. (D) 64 cm2 (A) 64 cm (B) 16 cm (C) 16 cm2 7. Which of these gives the area of a segment of a circle? (A) The difference of the areas of corresponding circle and triangle. (B) The difference of the areas of corresponding sector and triangle. (C) The sum of the areas of corresponding sector and triangle. (D) The product of the areas of corresponding sector and triangle. 8. The diagonal of a square of side 20 cm is _____. (A) 20 2 cm (B) 20 2 sq.cm (C) 2 20 cm (D) 25 cm 9. The perimeter of a semicircular plastic sheet is 36 cm. Find the diameter of the sheet. (A) 12 cm (B) 15 cm (C) 14 cm (D) 22 cm 10. The area of a sector of a circle of radius 7 cm making an angle of 60º at the centre is ___________. (A) 25.6 cm2 (B) 25.65 cm2 (C) 25.6 cm (D) 205.6 cm2 II. Short answer questions. 1. What is the area of the quadrant of a circle of radius 42 cm? 2. Find the area of the shaded region in the figure given that OABCD is a quadrant of a circle with radius 4 cm and OA = 3 cm. II. Long answer questions. 1. A square of side 16 cm is inscribed in a quadrant of a circle as shown. Find the area of the shaded region correct to 2 decimal places. 2. From the four corners of a square piece of decorative paper, of side 8 cm, four quadrants of radius 2 cm are cut. Also, a circle of diameter 4 cm is cut from the centre of the square to get a decorative shape, shown by the shaded region in the figure. Find the area of the decorative shape. 3. A rectangular table cloth of sides 160 cm and 130 cm is put on a circular table of radius 80 cm. Find the area of the table cloth that falls out of the table. 102

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1) A sector of a circle of radius 28 cm has an area of 616 cm2. Determine the area of the segment. (3marks) 2) In order to make some paper designs, Rahul cut color papers from 60 cm square sheet as shown. Determine percentage of paper left cutting it. (Shaded region). (Use π = 3.14) (3 marks) 3) A right angled triangle PQR has a right angle at Q, If ∠QPR = 56°, PQ = 40 cm, QR = 60 cm as shown in the figure, then determine the area of the shaded region. (3 marks) 103

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 4) In ∆ABC as shown in the figure, the corners of the triangle are cut such the portions cut are in the shape of sectors of radius 15 cm. Determine the area of the shaded region. (Dimensions are in cm and use π = 3.14)                    (3 marks) 5) The shaded portion of a square floor of size 60 m is to be painted. The cost of painting is Rs. 10 per every sq. meter. Determine the cost of painting if all dimensions are in meters. (Use π = 3.14)(3 marks) 104

13. Surface Areas and Numbers Learning Outcome By the end of this chapter, a student will be able to: • Determine the surface area of volume of Combination of Solids • Determine the size of a solid as it is converted from one shape to another. • Determine the volume and surface area of a frustum of a cone. Concept Map Key Points Ver tex or apex • A cuboid has length (l ) , breadth (b ) and height (h ) Total surface area of a cuboid = 2(lb +bh +hl ) Slant Total length of all edges of a cuboid = 4(l +b +h ) h edge Volum=e lb=h base area ×height Circular r base Total surface area of the co=ne πr (l +r ) 1 Volume of a cone is 3 of the volume of the cylinder lh with same base circle and height. b • In a right cylinder, with radius of the base r and Volume of the cone = 1 π r 2h height of cylinder h , 3 Curved surface area = 2πrh • A sphere has only curved surface and no flat Total surface area of a cylin=der 2πr (r +h ) Volume of cylinder = πr 2h surfaces. 22 Surface area of the sphere = 4πr 2 where r is the 7 radius of the sphere Here π is approximately taken as or 3.14 4 3 Volume of the sphere = π r 3 rr h r r r r • Curved surface of the hemisphere = 2πr 2 • In a right circular cone, the height of the cone, Total surface area of hemisphere = 3πr 2 radius and slant height (length of slant edge) of the cone are denoted by h , r and l . l =2 r 2 + h 2 Curved surface area of the cone = πrl 105

13. Surface Areas and Numbers Volume of hemisphere = 2 πr 3 Solid A 3 (Hemisphere) rr Overlapping area rr Solid B • Unit conversions (Cuboid) 1000 cm3 = 1 litre 1000 litres = 1 m3 • When some material changes its shape the volume • Mass of a solid object = Volume of the object × of the material remains the same but its surface area can change depending on the shape which the Density of the object. material takes. • Volume of liquid flowing in a unit time is equal to Unless some loss of material is provided, the initial the velocity multiplied by the area of the flow. volume is equal to the final volume. • Surface area of a Combination of Solids Step 1: Divide the give solid into simpler solids Example: i) A wax candle when melted and then made into whose surface area can be calculated. some other shape will have the same volume. Step 2: Determine the overlapping or common ii) A block of ice when it melts can take any shape but its volume remains the same. surface areas. iii) Metals can be squeezed and converted into dif- Step 3: Combine the surface areas of the individual ferent shapes. surfaces solid portions and then subtract the • When a right circular cone is cut by a plane which common surface areas. is parallel to the base of the cone, the portion with Example: In the given figure, hemisphere (A) rests the base circle of the original cone is called the on a cuboid (B). The base area of the hemisphere is frustum of the cone. The remaining portion of the common to both the solid portions. cone will also be a right circular cone but will have Total surface area = curved surface area of the reduced height and reduced base circle. hemisphere + total surface area of the cuboid – base area of the hemisphere (shaded area) Solid A (Hemisphere) Overlapping area Solid B (Cuboid) Right Circular Cone Frustum of a cone • Volume of a combination of solids Let h be the height, l be the slant height and r1 and The volume of solid formed by joining two basic r2 be the radii of the ends of the frustum ( r1 > r2 ) of a cone, then we have solids will be the sum of volumes of the constituents. If a volume is removed from the solid, then the ( )Volume of the frustum of co=ne1 difference of the volumes of the constituents is to 3 π h r12 + r22 + r1r2 be considered. Example: In the figure, volume of a combination of Slant height of the frustum of cone l = h 2 + (r1 −r2 )2 solids will be sum of the hemisphere (A) rests on solid B, hence Curved surface area of the frustum of cone Total volume = volume of hemisphere (A) + Volume of the cuboid (B) = π (r1 +r2 )l Total surface area of the frustum of cone = πl (r1 +r2 ) +πr12 +πr22 106

13. Surface Areas and Numbers Work Plan Concept Coverage Details Practice Sheet Coverage PS – 1 Pre-requisites • Surface area and volume of cuboid, cone, cylinde- and sphere PS – 2 Surface Areas and Vol- • Surface area of a Combination of Solids PS - 3 umes • Volume of Combination of Solids PS - 4 PS - 5 • Conversion of Solid from One shape to another Self Evaluation Sheet • Frustum of a Cone Worsksheet for “Surface Areas and Numbers” Evaluation with Self ---- Check or Peer Check* 107

PRACTICE SHEET - 1 (PS-1) 1. A cuboid of surface area 400 m2 is needed with a length of 8 m and breadth 10 meters. Determine the height of the cuboid and its volume. 2. Determine the volume and total surface area of a hollow cylinder whose outer diameter is 24 m and inner diameter is 10 m and has a length of 10 m. 3. An ice cream of diameter 7 cm and length 14 cm is newly released in market. Find its weight if the one 1 cm3 of ice cream weighs 1.1 gm. Also find its total surface area. 4. Determine the volume and surface of a sphere and hemisphere, each having 42 cm diameter. 108

PRACTICE SHEET - 2 (PS-2) 1. In a solid block of wood of size 30 cm × 20 cm × 20 4. A circus tent which has a cylindrical base of cm, a hole is made from a surface with size 70 m diameter and a height of 3 m. Its roof is 20 cm × 20 cm for a depth of 20 cm. The diameter conical in shape with a slant height of 42 m. of the hole is 14 cm. Determine the total surface of Determine the total amount of cloth required wood before and after making the hole in it. If the to make the tent and the volume inside the hole is made from one end to another, find the total circus tent. surface area and its volume. Slant height 42m 20 cm Height 3 m Diameter 70m 30 cm 5. In a cylindrical block of diameter 20 cm and height 14 cm, a conical cavity is made of diameter 10 cm diameter and 12 cm depth starting from its top flat surface. Determine the 14 cm 20 cm total surface of the block and the volume of the block with the cavity. 6. A wedding cake is made of three layers. At 20 cm bottom is a cardboard on which a 56 cm diameter and 14 cm height cake is placed. On 2. A brick of length 20 cm × 8 cm × 6 cm are placed top of this another cake of diameter 49 cm next to one another such that the face of size 20 cm diameter and 14 cm height cake is placed and × 6 cm touch one another along the longest side. on top of this, a 42 cm diameter and 14 cm What will be total surface area of the arrangement height cake piece is placed. If the baker wanted of 4 bricks and the volume of the 4 bricks to apply cream over this stack of cakes, then combined? find out the area over which the cream is to 3. Raju brings an ice cream pack from a shop in which be applied? Determine the total volume of the the ice cream was in the shape of cuboid of size 15 cakes. cm × 6 cm × 6 cm. He takes out the ice cream and places it on a plate on its 15 cm × 6 cm side and then takes out 4 scoops (hemispheres) of diameter 3 cm as shown in the figure and eats them. The ice cream takes 1 min to melt for every 50 cm2 of exposed C surface to air. Determine the total time required for the ice cream pack to melt. Determine the amount of ice cream left on the plate. (Use π = 3.14 ). B Scoops of A 20 cm diameter 3 cm Plate Ice cream 15 cm 9 cm Cardboard 6 cm 7. A medicine capsule is having hemispherical ends and a total length of 21 mm. If the diameter of the capsule is 5 mm, determine the total surface area of the capsule. Find the volume of the medicine that can be filled in the capsule. 109

PRACTICE SHEET - 2 (PS-2) 8. A vertical concrete pillar has a square base of size 20 cm × 21 cm and a height 28 cm over which a cylindrical portion of diameter 15 cm and length 280 cm is made. At end of a cylindrical portion a hemispherical shape with diameter is 35 cm is created. Determine the total surface area of the concrete pillar neglecting the area on which the pillar is standing. Determine the volume of concrete needed to make the pillar. 9. A musical drum of diameter 35 cm diameter and height 63 cm is packed in a box of size 40 cm × 40 cm × 80 cm. To provide protection to the drum, the gap between the box and the drum is filled with thermocole pieces. Determine the volume of thermocole in the box. 110

PRACTICE SHEET - 3 (PS-3) 1. 4 ice cubes of size 25 mm are added to a glass of 42 mm diameter and 70 mm height. If the glass had 27720 mm3 of juice already in it determine the level to juice in the glass before adding the ice cubes and the level of juice after the ice cubes have melted fully. 2. Milk is packed in tetra packs by a company. The inner dimensions of the tetra pack are 4 cm × 5 cm × 8 cm. Milk is boiled in a large cylindrical vessel of diameter 35 cm and height of milk in the vessel is 20 cm. How many tetra packs can be fully filled using the milk from the container, if 4% of the milk gets wasted during filling process? 3. A steel bar of 4.2 cm diameter and 2 m length is melted to make balls for a bearing of 4.2 cm diameter. How many steel balls are made? 4. 1000 metal cones of diameter 4.2 cm and 4.2 cm height are needed for a special application. A bar of metal is to be melted whose diameter is 14 cm. Determine the length of the metal to be chosen for melting. 5. Soil from a hill (conical shape) of base diameter 70 m and height 70 m is excavated to fill up a dried up lake which is hemispherical in shape. Determine the size of the lake. 6. A company gets an order to supply 84 ice creams in plastic balls of diameter 6 cm. The ice cream mix is prepared in a large cylindrical container of diameter 28 cm. Determine the height to which the container need to be filled with ice cream mix in order to fill required number of ice cream balls. 7. During the digging a well of 14 m diameter and 42 m deep the soil was put aside in a vacant plot. At the end of the digging of the well a huge pile of soil (conical shape) was formed. If the height of the heap of soil is 14 m, then find the radius of the heap of soil. 8. Water is pumped out to fields from well of diameter 14 m filled with water to a height of 7 m. The diameter of the pipe carrying water is 7 cm and water is flowing through the pipe at a velocity of 35 m/s. It takes 30 minutes to fill the field with a water to a height of 30 cm. If the field is square in shape then determine the size of the field. Also find the final height of water in the well. 9. A cylindrical pillar of diameter 1.4 m and height of 14 m is to be constructed using cement, sand, gravel and water which are used in the volume ratio of 1:4:4:1 respectively. The size of the cement bag is 60 cm × 40 cm × 15 cm. Determine the number of cement bags needed in the construction of the pillar. 111

PRACTICE SHEET - 4 (PS-4) 1. A frustum of a cone has a height of 10 cm and the diameters at the two ends are 14 cm and 21 cm. Determine the slant height, curved surface area and its volume. 2. Aluminum storage container has a height of 83 cm with lid of diameter of 70 cm. The base diameter of the container is 56 cm. Determine the capacity of the container and the area of the aluminum sheet required to make it. 3. The circumference of a flower pot at the bottom is 44 cm and at the top is 88 cm. If the height of the flower pot is 30 cm, determine the amount of soil needed to fill the pot. 4. A metal bucket has a capacity of 25 liters. It base diameter of 30 cm and the diameter at the top end is 45 cm. Determine the height of the bucket. 5. A cup is filled with ice cream from a cylindrical vessel. The size of the cup is 5 cm diameter at the top, 4 cm diameter at the bottom and 3 cm height. The cylindrical vessel is having a diameter of 20 cm. 1% of the contents of the cylindrical vessel cannot be used as it will be sticking to the walls of the container. 100 cups are filled with the mix in the cylindrical continer. Determine the level to which the container had the ice cream mix. 112

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. The slant height ‘ ’ of a cone of height ‘h’ and base radius ‘r’ is given by _____________. (A)  = r2 − h2 (B)  = r2 + h2 (C)  = r2 + h2 (D)  = r2 − h2 2. _____________ is the Curved Surface Area of a cone of base radius ‘r’ and slant height ‘ ’. ≠r A) π r (B) π r2 (C) π r2 (D)  3. Volume of hemisphere of radius ‘r is given by _________. (A) 4≠ r3 (B) 2≠ r3 (C) 3≠ r3 (D) 2≠ r3 3 5 4 3 4. What is the height of a cone with slant height 6.5 cm and radius 2.5 cm? (A) 7 cm (B) 6 cm (C) 8 cm (D) 13 cm 5. The volume of a sphere of radius 3 cm is _______. (A) 122 cm3 (B) 116 cm3 (C) 114 cm3 (D) 113 cm3 6. The quantity of water that a glass of inner radius 2.5 cm and height 10 cm can hold is _________. (A) 491 cm3 (B) 490 cm3 (C) 422 cm3 (D) 320 cm3 7. The volume of a cuboidal box of length 15 m, width 8 m and height 7 m is ______. (A) 820 m3 (B) 840 m3 (C) 410 m3 (D) 225 m3 8. The curved surface area of the frustum of cone of radii r1 and r2 and slant height ‘ ’ is _______. (A) (B) (C) (D) 9. A cube has an edge 6 cm and a cuboid has length 5 cm, width 4 cm and height 2 cm. Which of these is correct? (A) The cuboid has a greater surface area than the cube. (B) The cube has a lesser volume than the cuboid. (C) The cube has a greater volume than the cuboid. (D) All of these 10. The height ‘h’ of a cylinder of base radius ‘r’ and curved surface area ‘C’ is _________. (A) c (B) c (C) 2≠ r (D) ≠ r ≠r 2≠ r II. Short answer questions. 1. Find the curved surface area of a cylindrical vessel of radius 7 cm and height 21 cm. 2. A cuboid is formed by joining two cubes. If its surface area is 160 cm2 what is the volume of each cube? III. Long answer questions. 1. A wooden paper weight is in the shape of a cone placed on a cylinder. The circular parts of the cone and the cylinder are of the same radius equal to 2 cm. The height of the cylindrical part is 2.1 cm and the slant height of the conical part is 2.8 cm. Find the volume of wood used to make it. 2. A cement sit out in the shape of a frustum of cone as shown in the figure, is constructed in front of a house. The vertical height of the sit out is 0.60 m and the diameters of the two circular faces are 6 m and 3 m. What is the surface area of the sit out? 3. A copper rod of diameter 5 cm and length 16 cm is drawn into a wire of length 24 m of uniform thickness. Find the thickness of the wire. 113

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. A rocket has a cylindrical body of length 20 m and diameter of 6 m. At the top end of the rocket is a cone of length 4 m. The rocket is to be painted on all sides at a cost of Rs. 10 per sq m, determine the cost of painting. (2 Marks) 2. A carpenter makes hemispherical cavity at one end and a conical cavity at the other end of a cylindrical block of wood to create a musical instrument. The diameter of the block of wood is 28 cm and its length is 56 cm. If the diameter of the hemisphere is 21 cm and diameter of the cone is 21 cm and its slant height is 15 cm, determine the total surface area of the wooden block before and after the work by the carpenter. Determine the volume of the wood left out. (4 Marks) 3. A swimming pool of size 50 m × 25 m × 7 m is to be filled with water using water brought from the tankers. The tankers have a hemispherical ends connected with a cylindrical portion in the middle and diameter of the tanker is 7 m and the total length is 28 m. Determine the number of tankers of water needed to fill the swimming pool. (3 Marks) 4. A cylinder of 10 cm diameter is filled with water to a height of 7 cm height. 21 Glass marbles are dropped into the cylinder until the water level increases by 3.5 cm. Determine the size of the marbles. (Use 3 25 = 2.92 ) (2 Marks) 5. Ravi makes paper cups whose ends are circular with diameters 5.6 cm and 4.2 cm. If the height of the cup is 5.6 cm then determine the capacity of the cup and amount of paper required to make each cup. (4 Marks) 114

14. Statistics Learning Outcome By the end of this chapter, a student will be able to: • Create a graph of cumulative frequency and • Calculate mean, median and mode of the given determine the median from the graphs. data. Concept Map Key Points can be chosen to represent the observations falling in the class. • An ungrouped set of data is a collection of numbers. Midpoint of a class = Example: Marks of students: 22, 25, 23, 16, 18, 17, Upper limit of aclass +Lower limit of aclass etc. 2 Student age, student height, etc. Example: • Frequency means the number of times a particular For the class interval 10 − 25 the midpoint is 25 + 10 = data value occurs. For2 the 17.5 interval 0−5 . the midpoint is Example: Marks of students: 22, 25, 23, 16, 18, 17, 5 0 class 23, etc. + = 2.5 . Marks 10-20 occurs 3 times, 16, 18, 17 2 Marks 20-30 occurs 4 times 22, 25, 23, 23 For the class interval 0 −10 the midpoint is Mark 23 occurs 2 times while all other occur once. 10 + 0 • Grouped data is the data that is combined together 2 = 5 into categories. • When the given data is not continuous, then the Example: upper and lower limits of two consecutive class Marks No. of Students intervals are averaged and the class intervals are (Frequency) modified. The size of the class intervals changes 0-10 0 according to the amount of discontinuity in the 10-20 3 class intervals. All calculations are made using the 20-30 4 new class intervals. • By convention, a class interval contans values Example: Given class intervals are values from (including ) upto (does not include q). 1-10 11-20 21-30 31-40 Converting Ungrouped Data into Grouped Data Averaging the upper and lower limits of two • If the data provided is an ungrouped data, then consecutive limits we get ⇒ 10 +2=11 10.5, 20 +2=21 20.5 convert it into grouped data using appropriate The new class intervals will be class intervals. Then the frequency of each class interval is assumed to be centered on its midpoint of the class interval. So the midpoint of each class 0.5-10.5 10.5-20.5 20.5-30.5 30.5-40.5 115

14. Statistics Mean of Grouped Data o Step-deviation Method • The mean or average of observations is the sum of Step 1: Convert the given data into grouped values of all the observations divided by the total data. number of observations. Step 2: Select one among the xi as the assumed Let x1 , x 2 , … , xn are observations with f1 , f2 , … , mean and denote it by a. Usually the middle fn frequencies respectively, then value among the values of xi from the table is Sum of the values of all observations selected as a . The value of the mean does not = f1x1 +f2x 2 +…+fn xn depend on the value of a . dh=i xi −a Number of observations= f1 +f2 +…+fn Step 3: Find Step deviation, u=i h , where f x +f2x 2 +…+f x Mean x = 1 1 1 +f2 +…+fn n n h is the class interval or any suitable value. f Step 4: Find the product of frequency (fi ) and The greet capital letter S means summation and step deviation (ui ) using in the formula of mean we can write, nf Step 5: Calculate ∑fi ui and ∑fi ∑∑ Mean x = i x i ∑fi ui i =1 Step 6: Mean Step deviation, u ∑fi nf = i =1 i Or Mean x = ∑fi xi Step 7: Mean, x =a +hu =a +h ∑fi ui ∑fi ∑fi In the formula of mean for the grouped data, the This method will be convenient to apply if all the deviations (di ) have a common factor. Even value of xi is the midpoint of the class interval. if the class sizes are unequal, this method can be employed by taking the value of h which is a o Direct Method common divisor of all the value of di Step 1: Convert the given data into grouped Mode of Grouped Data data. • Mode is that value among the observations which Step 2: Find the product of frequency (fi ) and occurs most often, that is the value of observation having maximum frequency. It is possible that midpoint of the class ( xi ) more than one value may have the same maximum Step 3: Calculate ∑fi xi and ∑fi Step 4: Mean i sxu=se∑d∑fifixif i values of xi and fi are frequency and such data is said to be multimodal. This method To calculate the mode of a grouped data the the following steps are followed: sufficiently small. Step 1: Convert the given data into grouped data. o Assumed Mean Method Step 2: Determine the class having the maximum Step 1: Convert the given data into grouped frequency. This class is called modal class. data. Step 3: Mode =l +  2f1f1−−f0f0−f 2 ×h Step 2: Select one among the xi as the assumed  Here l = Lower limit of the modal class mean and denote it by a. Usually the middle h = Size of the class interval (Assuming all class value among the values of xi from the table is sizes to be equal) selected as a . The value of the mean does not f1 = Frequency of modal class depend on the value of a . Step 3: Find the difference di between each a f0 = Frequency of class preceding the modal class and x i , d=i xi −a Step 4: Find the product of frequency (fi ) and f2 = Frequency of the class succeeding the modal deviation (di ) class Step 5: Calculate ∑fi di and ∑fi Note: i) The data provided should not be multimodal in Step 6: Mean of deviations  d ∑fi di nature. = ∑fi ii) The size of the class interval should be the same in Step 7: Mean,   x =a +d =a + ∑fi d i the data provided to calculate the mode. ∑fi iii) If the modal class is the first class of the table, then fo can be assumed to be zero and if the modal class This method is used if xi and fi are large is the last class of the table, then f2 can be assumed numbers. 116

14. Statistics to be zero. the largest and the smallest observations of the entire data. It also enables for the comparison Median of Grouped Data of two or more distributions. Extreme values in • Median gives the value of the middle-most the data affect the value of mean. o If the requirement was to find the typical observation in the data. observation, the median is more appropriate To calculate Median of the grouped data, the such as average salary levels in a country, etc. The extreme values do not affect the value of following steps are followed: median. Step 1: Convert the given data into grouped data. o When the requirement is to determine the most Step 2: Calculate the cumulative frequency. frequent value, calculating the value of mode for the given data is the best choice. Cumulative frequency can be calculated for less • The relation between the three measures of central tendency is given by 3 Median = Mode + 2Mean than or more than type for the limit of the class intervals. For less than type, the frequencies are added and for more than type, the frequencies are subtracted from the total frequency cumulative (n). For less than type, the upper limits of class interval are used and for more than type the lower limits of class interval are used. Example: Given data Class Interval 0-10 10-20 20-30 30-40 Frequency 5 6 4 3 Less than type More than type Class range Class Range Cumulative frequency More than or equal to 0 Cumulative frequency 5 + 6 + 4 + 3 =18 Less than 10 5 More than or equal to 18 − 5 =13 10 Less than 20 5 + 6 =11 More than or equal to 13 − 6 =7 20 Less than 30 11 + 4 =15 More than or equal to 7 − 4 =3 30 Less than 40 15 + 3 =18 Any of the two types of cumulative frequencies can be calculated to determine the median. Step 3: Calculate the middle of the value of the cumulative frequency  n  and the class whose  2  (and n 2 frequency is greater than nearest to) is called the median class. Step 4: Median =l +  n2 f −cf ×h Here l = lower limit of median class n = number of observations or total of all frequencies cf = Cumulative frequency of class preceding the median class f = frequency of median class h = class size (assuming class size to be equal). • Mean, Mode and Median are three measures of central tendency. o Mean is most frequently used measure of central tendency because it takes into account all the observations and lies between the extremes i.e., 117

14. Statistics Graphical Representation of Cumulative Distribution • The following steps are followed to graphically represent cumulative distribution. Step 1: The limits of the class intervals are marked on the horizontal axis -(x-axis). Step 2: The corresponding cumulative frequencies are marked on the vertical axis (y-axis). The scale for x-axis and y-axis need not be same. Step 3: The points are joined with a free hand smooth curve and the curve is called cumulative frequency curve or Ogive. If the lower limits of class intervals form the x-coordinate and more than type cumulative frequency is used as y-coordinate to plot the curve, then it is called More than type Ogive. If the upper limits of class intervals form the x-coordinates and less than type cumulative frequency is used as y-coordinate to plot the curve, then it is called Less than type Ogive. • The value of median can be calculated from the Ogives: o Lxo-ccoaoterdinn2ateoonftthheeyp-oaixnist and draw a line parallel to x-axis from this point. The abscissa or the of intersection is the median of the given data. This method is applicable both Ogives. (Less than type and More than type) for o Draw less than type ogive and more than type ogive for the given data and the abscissa or the x-coordinate of the point of intersection of the two O gives is the median value for the given data. Work Plan Concept Coverage • Coverage Details Practice Sheet Pre-requisites • Grouped Data and Ungrouped Data PS – 1 Mean of Grouped Data Statistics • o Direct Method PS – 2 • o Assumed mean Method • o Step Deviation Method PS - 3 Mode of Grouped Data PS - 4 Worksheet for “Statistics” Median of Grouped Data PS-5 Evaluation with Self Graphical Representation of Cumulative Frequency distri Self Evaluation Check or Peer Check* bution Sheet ---- 118

PRACTICE SHEET - 1 (PS-1) 1. Which of the following values can be contained in the class interval 20 – 25 ? i) 20.5 ii) 22.5 iii) 20 iv) 25 v) 25.5 vi) 24.5 vii) 19.5 2. The marks obtained by the 30 students in a Maths test are as follows. 22, 16, 11, 18, 06, 12, 20, 15, 16, 03, 18, 21, 19, 17, 13, 14, 15, 12, 10, 05, 09, 23, 06, 02, 25, 08, 15, 17, 25, 07. Convert the given data into grouped data with a class interval of 5, 10. Can the data be grouped into class interval of 50? 119

PRACTICE SHEET - 2 (PS-2) 1. The height of 30 students is measured and tabulated as shown in the table. Find the average height of the students. i) Using the ungrouped data ii) Grouping the data into class intervals of 10 starting from 75. Student height (cm) 79 82 89 92 94 99 100 101 103 104 110 No of Students 532163 2 1 2 1 4 2. The census of a village was undertaken to determine the age of the population and the distribution is as follows: Age 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of villagers 100 50 60 35 40 25 15 10 5 2 Calculate mean age of the villagers using the three methods and compare the results. 3. A farmer has mango trees of different ages in this orchid and each tree was giving him different amount of mangoes. Determine the average age of the trees the farmer was having using three methods if the table below gives the details of the age and the number of trees. Age of mango tree 0-3 3-6 6-9 9-12 12-15 15-18 18-21 21-24 No. of trees 5 15 25 15 20 17 18 20 4. A bank wanted to analyse the deposited amounts in the various accounts and tabulates the data available with them as given in the table below. Determine the average balance in the account using the three methods. Deposited Amount 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 (in thousands) (Rs.) No. of accounts 400 220 190 300 50 200 250 80 5. The Municipal Corporation of a city decided to check the water quality in a number of wells and recorded the data as given in the table. Find the mean level of impurities in the city ground water using step deviation method. (Here impurities are measured in ppm = parts per million) Amount of 0.000- 0.015- 0.030- 0.045- 0.060- 0.075- 0.090- impurities (ppm) 0.015 0.030 0.045 0.060 0.075 0.090 0.105 No. of wells 10 6 7 2 2 2 1 6. If the average salary of employees of a certain company is Rs. 17500, then find the value of m . (Use Step deviation method) Salary (Rs.) 0-5000 5000-10000 10000-25000 25000-30000 30000-50000 50000- 100000 No. of 15 m 10 5 1 Employees 4 7. During the inspection of certain jewellery made, a particular design was found with varying defects. If the average variation is 0.00255 grams, then find the value of n. (Use Assumed mean method) Variation in weight of jewellery (mg) 0-1 1-2 2-3 3-4 4-5 5-6 No. of pieces detected 25 10 25 n 15 5 8. The average marks scored by the students in a class test were 10. Find the value of m. (Use direct method to calculate mean) 120

PRACTICE SHEET - 3 (PS-3) 1. The census of cows was undertaken to see if there was sufficient capacity for milk production in the various dairy farms in various parts of the state. Calculate the mean and mode and discuss the results. Number of cows 0-100 100-200 200-300 300-400 400-500 500-600 600-700 Number of dairy farms 200 200 800 900 400 400 200 2. A printing press prints books as given in the table. Find the mean and mode of the data given in the table. No. of books printed (thou- 0-2 2-4 4-6 6-8 8-10 10-12 sands) No. of days 312576 3. Find the mode of the given data which represents the forest cover and the number of villages surveyed. Forest cover (%) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of villages 100 150 100 150 140 30 10 4. A lot of complaints were received by the BCCI regarding the injuries to cricketers so it decided to look at the data. The data has the number of years a cricketer has played at the national and international levels before he retires. What is mean and mode of the data? No. of years a Cricketer played 0-4 4-8 8-12 12-16 16-20 No. of cricketers 280 300 390 200 30 5. A shopkeeper sells mobile phones and wants to see which price phones are mostly purchased by the customers and the collected data for a month as given below. Find the mean and mode of the data and explain the results. Cost of Mobile 1000-4000 4000-6000 6000-7000 7000- 10000- 13000- 16000- 19000- Phone (Rs.) 10000 13000 16000 19000 22000 No. of pieces 100 150 50 300 200 400 500 600 sold 6. The mode of the given data 54. Find the value of m Class interval 0-10 10-20 20-30 30-40 40-50 50-60 60-70 35 40 m 35 Frequency 5 15 20 121

PRACTICE SHEET - 4 (PS-4) 1. The education department decided to check the number of schools in the district and tabulated the data as follows. What is mean, median and mode of the following data and explain the results. Total Students 0-100 100-200 200-300 300-400 400-500 500-600 No. of schools 10 15 30 25 20 25 2. In a cross country practice race 50 runners have participated and after some distance the runners quit due to a variety of injuries. What is median of the distance covered by the runners? Running distance (km) 0-20 20-40 40-60 60-80 80-100 No. of runners who quit 1 3 0 15 20 3. The median age of a family is 32.5 years. Find the value of m. Age (years) 0-10 10-20 20-30 30-40 40-50 50-60 60-70 8 6 No. of persons 5 9 m 12 5 4. The temperature of a city was measured and recorded for 52 weeks and is as provided in the table. Determine the median temperature of the city and verify the same by drawing less than and more than ogives for the same. Temperature °C 20-22 22-24 24-26 26-28 28-30 30-32 32-34 34-36 No. of days 2 7 12 10 8 8 3 2 5. The students of Class 10 planned to donate money and bring money according to their choice. Find the median amount brought by the students. Draw the less than and more than ogives for the given data and verify the value of the median from graph. Amount (Rs.) No. of Students More than 10 280 More than 20 240 More than 30 215 More than 40 175 More than 50 95 More than 60 45 More than 70 15 122

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. The class mark of a class interval is _______. (A) Upper limit + Lower limit (B) Upper limit - Lower limit 2 (C) Upper limit – Lower limit (D) Upper limit + Lower limit 2 2. The mean of observations x1, x2, x3,…,xⁿ is _______. (A) x1 + x2 + x3 + ... + xn (B) x1 + x2 + x3 + ... + xn nn n (C) x1 − x2 − x3 − ... − xn (D) x1 − x2 − x3 − ... − xn nn n 3. ________ is calculated using the cumulative frequency graph. (A) Mean (B) Mode (C) Median (D) All of these 4. ________ is the most widely used measure of central tendency. (A) mean (B) median (C) mode (D) All of these 5. If ∑fi xi = 200 and number of observations 10, the mean is ____. (A) 220 (B) 17 (C) 190 (D) 20 6. The frequency obtained by adding the frequencies of all the classes preceding the given class gives __________. (A) mean (B) median (C) cumulative frequency (D) mode 7. In calculating ________, we use the mid values of the classes. (A) median (B) mean (C) mode (D) All of these 8. The mean of 4, 7, x, 6, 9, y, 13 is 8 then ________. (A) x + y = 17 (B) x – y = 17 (C) x + y = 15 (D) x – y = 10 9. Identify the correct one. (A) Mean = 3 Median – 2 Mode (B) Median = 3 Mean – 2 Mode (C) Mode = 3 Median – 2 Mean (D) All of these 10. When do we use the median? (A) When we wish to find a typical observation. (B) When we wish to compare two or more distributions. (C) When we wish to find the most frequent value. (D) All of these II. Short answer questions. 1. Explain the terms in the formula for computing the median of a grouped frequency distribution. 2. Arun noted the following information from a grouped frequency distribution to calculate its mode. The lower limit of the modal class is 50, the class size is 12, the frequency of the modal class is 14, the frequency of the class preceding the modal class is 9, and that succeeding the modal class is 13. What is the value that Arun arrived at? III. Long answer questions. 1. The heights of students in class10 of a school are tabulated as follows: Height 130 – 136 136 – 142 142 – 146 146 – 152 152 – 156 156 – 162 (in cm) 10 3 Number of 9 12 18 23 students Using the direct method, calculate the mean height of the students. 123

PRACTICE SHEET - 5 (PS-5) 2. The following table shows the number of males in a locality in the age group of 10 years to 58 years. Age 10 – 16 16 – 22 22 – 28 28 – 34 34 – 40 40 – 46 46 – 52 (in years) Number of 175 325 100 150 250 400 525 males What is their median age? 3. Using the step deviation method, calculate the mean of the given frequency distribution. Class Inter- 8 – 13 13 – 18 18 – 23 23 – 28 28 – 33 33 – 38 38 – 43 val Frequency 320 780 160 540 260 100 80 124

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. The mean and median of a frequency distribution are 5, 6 respectively, determine the mode. (1 Marks) 2. Determine the less than type and more than type cumulative distribution for the following data: (2 Marks) Class Interval 0-10 10-20 20-30 30-40 40-50 Frequency 24 8 3 1 3. A school has decided to increase the games hour for the students in the coming year if their average marks in the final exams of the current year is more than 65. The marks of the students are as given in the table. Find out if the students get an additional games hour. (Use Assumed Mean Method) (3 Marks) Student Marks 0-30 30-40 40-50 50-60 60-70 70-80 No. of students 5 20 25 30 35 10 4. The price of tomatoes was recorded for 200 days in a year by the agriculture department and the average price of tomatoes was found to be Rs. 22.50. Find the value of a in the given table. (3 Marks) Price of Tomatoes in a day 5-10 10-15 15-a a-30 30-40 40-50 50-55 55-60 No. of days 50 40 30 20 30 10 10 10 125

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 5. A cola company wants to know the age of their biggest customers to improve their marketing strategy. They tabulated data of the age of the customers and the number of customers who consume the particular cola. Find the mean and mode of the following data and explain the results. (3 Marks) Age group 5-15 15-25 25-35 35-45 45-55 55-65 No. of customers (lakhs) 1.1 1.5 0.9 1.1 1.3 1.1 6. Find the median number of months before which a mango tree starts giving fruits based on the given data. (3 Marks) No. of Months 10-13 14-17 18-21 22-25 26-29 30-33 No. of plants 20 23 12 8 13 4 126

15. Probability Learning Outcome By the end of this chapter, a student will be able to: • Explain the concepts of Experimental probability and Theoretical probability • Explain the different types of events in an experiment. • Determine the probability of an event in a given experiment or trial. Concept Map Key Points • Possible outcomes of simple Experiments Coin: Heads, Tails • A trial is an action which results in one or several Dice: 1, 2, 3, 4, 5, 6. outcomes. Deck of Cards: 52 cards in 4 Suites (spades (black Examples of outcomes: color), hearts (red color), diamond (red color), Getting head, tail in coin tossing clubs (black color)) of 13 cards each. The cards in Getting a number of dice in the game of dice each suit are ace, king, queen, jack, 10, 9, 8 , 7, 6, 5, 4, 3, 2. King, queen and jack are called Face Cards. Examples of Trial: • An event having only one outcome of the Coin Tossing experiment is called elementary event. The sum Rolling a dice of the probabilities of all the elementary events of Pulling a card or cards from a deck of cards an experiment is 1. • In probability, the set of outcomes from an Example of elementary events: - In coin tossing experiment, getting head, experiment or trial is known as an Event. • Experimental Probability or Empirical Probability getting tail are both elementary events. - In rolling dice, getting 1 is an elementary event. is based on the results of actual experiments and adequate recordings of the happening of the Also obtaining any of the other numbers is an events. elementary event. Experimental Probability, - In rolling of a dice, getting a number more than 4 is not an elementary event because there are ental Probability, P (E) = No.of Trails in which the event happened two possible outcomes(5,6 numbers on the Total number of trials dice) • In an experiment equally likely events have same • Theoretical Probability or Classical Probability of an event, E, is based on certain assumptions to probability of occurrence. avoid experiments (failure of satellite launches, - Examples of Equally likely events building destruction due to earth quake, etc) as - Getting 3 on a toss of die, Getting 5 on a toss of the assumptions help in directly calculating the exact or theoretical probability. The assumption of die, etc are equally likely events. equally likely outcomes is one such assumption. - Getting a red card, getting a black card while Theoretical Probability, pulling a card from a deck of cards are equally al Probability, P(E) = Number of outcomes favourable to event,E likely events. Number of all possible outcomes • For an event (E), there will be an event ( E ) of the experiment representing ‘not E’ called complement of the event E. The events E and E are called Complementary Note: From the definition of P(E), the numerator is events. always less than or equal to the denominator. 0 ≤ P(E) ≤ 1. P(E) +P(E) =1 (or) P(E ) =1−P(E) Example complementary events: 127

15. Probability - Getting head or tail in coin tossing experiment are complementary events. - Getting Odd number and getting even numbers are complementary events. - Also getting a number ‘not greater than 4’ is complementary of getting a number “greater than 4’. Getting a number ‘not greater than 4’ is same as getting a number less than or equal to 4. - Getting a black card and getting a red card in a deck of cards are complementary events. • The probability of an event which is impossible to occur is 0. Such an event is called impossible event. Examples of Impossible event: - Getting a number greater than 7 in a game of dice. - Getting neither head nor tail is an impossible event in coin tossing. • Sure event or Certain event is one which is certain to occur during the experiment. Examples of Sure event: - Getting a number less than 7 in a game of dice. - Getting either black or red card from a deck of cards. - Getting either heads or tails. • Experiments in which the outcome is any number between two given numbers or in which outcome is every point within a circle or rectangle, etc have infinitely many possible outcomes. In such cases, the definition of theoretical probability cannot be applied directly. The problem has to be reduced in a way that there are finite numbers of solutions using some inequality or other mathematical notations. 128

15. Probability Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Pre-requisites • Probability • Experimental Probability PS - 2 Probability – • Numerical problems on Experimental Probability PS-3 A theoretical Approach • Different types of events Self Evaluation Sheet Worksheet “Probability” • Experimental Probability definition Evaluation with Self Check • Calculating the probability of different types of events. or Peer Check* ---- 129

PRACTICE SHEET - 1 (PS-1) 1. Ramesh and Suresh tossed a coin alternately 50 times each and recorded the results. When the results were counted, they get 49 heads and 51 tails. Determine the experimental probability of getting i) head ii) tail 2. Two friends play a game of rolling dice. They roll the dice simultaneously for 20 times and 6 six times both of them get the same number. Only in two cases do the friends get an even number. Find the probability of getting same number on the dice and getting an even number on the dice based on the experiment. 3. The date of birth of the students of a class is noted and sregated by months as follows: Months Jan.- Apr.- Jul.– Oct.– Mar. Jun. Sep. Dec. Frequen- 5 15 12 8 cy Determine the probability of student being born in the month of i) August ii) January. (The numbers can be used or made into an activity) 4. A hotel manager wanted to see who is eating more food and wanted to know the behavior of his customers. For the next one week, he makes entries of the age of the customers who made large bills as follows: Determine the probability that a customer chosen at random i) Makes a bill amount less than Rs. 100 ii) Makes a bill amount more than Rs. 200 iii) A teenager (age group 15-19) making bill less than Rs. 200. 130

PRACTICE SHEET - 2 (PS-2) 1. Two students in a class are born in a leap year. 12. From a bunch of old calendars, one calendar What is the probability that their date of births is picked at random and it was found to be a is the same? leap year. Find the probability of the following cases. 2. What is the probability of getting a number greater than or equal to 5 when a dice is i) 53 Fridays thrown? Mention the complementary event. ii) There are only 52 Sundays. iii) 53 Mondays and 53 Saturdays. Name the 3. A manager of a company gives a gift to his employee if the birthday falls on Sunday. What type of is the probability that an employee gets a gift? event. 4. In a class of 40 students there are 10 boys and iv) 52 Sundays, 53 Mondays and 53 Tuesdays 30 girls. What is the probability that a student 13. Fatima sees a chess board and puts a camel chosen at random is a girl? piece on the board. What is the probability that 5. A football ground has a length 40 m and the camel piece is placed on breadth 30 m. In the middle of the ground i) White square there is a circle marked with a radius of 3m. ii) Black Square What is the probability that a student going iii) Any of the corners into the ground stops inside the circle. iv) Top row or Bottom row of the board 14. Anil and Sunil play a game of dice and record 6. In a leap year selected at random, what is their numbers. A player will win the game if the probability of having Republic Day on a the number on the dice thrown by him is more Sunday. than the number on the dice thrown by the other player. 7. A bag contains 10 red balls and a ball is taken i) What is the probability of Anil winning the out from the bag at random. What is the game? probability of getting a red ball? What is the ii) What is the probability of Sunil winning the type of event? game? iii) What is the probability of nobody winning 8. A card is drawn at random from a deck. Find the game? the probability that the card is a red king. Find the probability of getting a black jack card. Whether the two events are equally likely events? 9. A bag contains x white balls, y black balls. 30 red balls are added to the bag and one ball is picked at random. The probability of selecting a white ball is twice the probability of selecting black ball. If the probability of picking a red ball is 0.2 find the number of white and black balls in the bag. 10. A bag contains red balls numbered from 1 to 40. If a ball is picked at random, what is the probability that the number on the ball is: i) Prime number ii) Multiple of 9 iii) Greater than 38 11. All cards of spade suite from a deck of cards are spread on a table. Find the probability of the following events: i) Picking a numbered card ii) The number on the card is less than 5 iii) Picking a black card. What is the type of event? 131

PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 1. Which of these is an equally likely event? (A) The outcomes of head or tail when a coin is tossed. (B) The outcomes of 1, 2, 3, 4, 5 or 6 when a die is thrown. (C) The outcomes of picking a king of spades in a well – shuffled pack of cards. (D) Both (A) and (B) 2. Which of these is an impossible event? (A) Getting a 7 when a die is thrown. (B) Getting a 2 when a die is thrown. (C) Picking a marble from a bag containing 10 marbles. (D) All of these 3. The probability of an event E is such that it satisfies ________. (A) 0 < P(E) ≤ 1 (B) 0 ≤ P(E) ≤ 1 (C) 0 ≤ P(E) < 1 (D) 0 < P(E) < 1 4. In an experiment, the probability of occurrence of an event E is given by the ratio of _______. (A) The number of outcomes favourable to E to the total number of possible outcomes in the experiment. (B) The number of outcomes not favourable to E to the total number of possible outcomes in the experiment. (C) The number of outcomes favourable to E to the number of outcomes not favourable to E. (D) The number of outcomes not favourable to E to the number of outcomes favourable to E. 5. For any event E, E is called its _____________. (A) complementary event (B) supplementary event (C) impossible event (D) sure event 6. Identify the correct one. (A) P(E) – P( E ) = 1 (B) P(E) + P( E ) = -1 (C) P( E ) – P(E) = 1 (D) P(E) + P( E ) = 1 7. The probability of getting a 0 when a die is thrown is ____. (A) 0 (B) 1 (C) 0.5 (D) 0.25 8. The probability of an event E is 0.45. What is the probability of ‘not E’? (A) 0 (B) 1 (C) 0.55 (D) 0.65 9. The probability of picking a red marble from a bag containing 10 red and 4 blue marbles is ________. (A) 4 (B) 6 10 1 14 14 (C) 14 (D) 14 10. Which of these cannot be the probability of an event? (A) 0.025 (B) 12 % (C) - 1 (D) All of these 14 II. Short answer questions. 1. From a bag containing 5 red pens, 6 green pens and 11 blue pens, a pen is taken out at random. What is the probability that the pen taken out is green? 2. Find the probability of getting a composite number when a die is thrown. III. Long answer questions. 1. The table gives the number of school going children in each family of a particular locality. Number of school going 0 1 23 children in a family Number of families 40 55 75 60 If a family is selected at random, what is the probability that (i) it has 3 school going children? (ii) no school going children? (iii) less than 2 school going children? (iv) at least 1 school going child? 132

PRACTICE SHEET - 3 (PS-3) 2. Heads appeared 285 times and tails appeared 255 times when a coin is tossed 540 times. If the coin is tossed at random, what is the probability of getting (i) a head and (ii) a tail? 3. In a survey, the number of employees in a company are distributed as follows: Age (in years) 20 - 29 30 - 39 40 - 49 50 - 59 Above 60 years 6 Number of 60 92 68 74 employees If an employee is chosen at random, find the probability that the employee is (i) under 30 years (ii) aged between 30 and 50 (iii) above 50 years (iv) over 39 years but under 60 133

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. What is probability of drawing a king from a pack of cards? Write the complementary event of the given event. (1 Mark) 2. Sachin tosses a coin thrice. What is the probability that Sachin gets tails in the first two tosses? (2 Marks) 3. A news reporter mentions that on 21 January 2019, the probability of raining is 0.2. Ramu will go out only if the probability of not raining is more than 0.85. Determine whether Ramu will go out or stays at home.    (2 Marks) 4. The face cards are removed from a deck of cards and then placed on a table face down. i) If card is drawn at random, what is the probability that the card is a king of spades. ii) The first card is noted as jack of hearts. This card is put aside and a second card is drawn from the cards on the table, what is the probability that the card is a black Jack. (3 Marks) 134

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 5. A box contains 5 red balls, 8 black balls, 7 yellow balls and x green balls. If probability of getting a black ball is 0.2, then find the number of green balls in the box. Also find the probability of not getting a yellow ball when one ball is taken out from the box. (3 Marks) 6. A disc has numbers 0-5 and when it stops rotating an arrow points to a number. The disc is rotated twice and the numbers are added up. What is the probability that the total is i) 0 ii) 11 iii) Even number iv) Odd number (4 Marks) 135