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TABLE OF CONTENTS 7 FREQUENCY DISTRIBUTION TABLES AND GRAPHS 1 7.1 BASIC MEASURES OF CENTRAL TENDENCY 1 7.2 ORGANISATION OF GROUPED DATA 6 7.3 GRAPHICAL REPRESENTATION OF DATA 10 9 AREA OF PLANE FIGURES 15 9.1 AREA OF TRAPEZIUM 15 9.2 AREA OF CIRCLE 22 10 DIRECT AND INVERSE PROPORTIONS 25 10.1 DIRECT PROPORTION 25 10.2 INVERSE PROPORTION 29 10.3 COMPOUND PROPORTION 33 11 ALGEBRAIC EXPRESSIONS 35 11.1 SIMPLE OPERATIONS OF ALGEBRAIC EXPRESSIONS 35 11.2 MULTIPLYING A BINOMIAL OR TRINOMIAL BY A MONOMIAL 39 11.3 MULTIPLYING A BINOMIAL BY A BINOMIAL OR TRINOMIAL 41 11.4 WHAT IS AN IDENTITY 43 11.5 GEOMETRICAL VERIFICATION OF THE IDENTITIES 45 12 FACTORISATION 47 12.1 FACTORS OF ALGEBRAIC EXPRESSIONS 47 12.2 FACTORISATION USING IDENTITIES 50 12.3 DIVISION OF ALGEBRAIC EXPRESSIONS 53 12.4 ERROR ANALYSIS 56 14 SURFACE AREA AND VOLUME (CUBE - CUBOID) 59 14.1 SURFACE AREA OF CUBE AND CUBOID 59 14.2 VOLUME OF CUBE AND CUBOID 64 PROJECT BASED QUESTIONS 68 ADDITIONAL AS BASED PRACTICE QUESTIONS 69
CHAPTER 7 FREQUENCY DISTRIBUTION TABLES AND GRAPHS EXERCISE 7.1 BASIC MEASURES OF CENTRAL TENDENCY 7.1.1 Key Concepts i. Information in the form of numerical figures are called observations. ii. Observations gathered initially are called raw data. iii. The difference between the highest and the lowest values of the observations in a given data is called its range. iv. The number of times a particular observation occurs is called its frequency. v. When the number of observations is large, the data is usually organized into groups called class intervals. vi. A table showing the frequencies of various class intervals is called a frequency distribution table. vii. The lower value of the class interval is called lower limit and upper value is called upper limit. viii. The difference between the upper limit and lower limit is called class size. ix. Arithmetic mean is simply the average. It is given by: x¯ = xi = S um o f observations N Number o f observations x. Arithmetic mean by deviation method: Arithmetic Mean = Estimated mean + Average of deviations = Estimated Mean + S um o f deviations Number o f observations x¯ = A + (xi−A) N xi. Median is simply the middle term of the distribution when it is arranged in either ascending or descending orders. th xii. When ‘n’ is odd, n+1 observation is the median. When ‘n’ is even, then the arithmetic mean 2 n th th of two middle observations i.e., 2 and n + 1 observations is the median of the data. 2 xiii. Mode is the most frequently occurring value in the data. EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 1
7.1.2 Additional Questions Objective Questions 1. [AS3] The most evidently used measure of central tendency is . (A) Mean (B) Median (C) Mode (D) None 2. [AS1] The mean of the data 20, 19, 13, 23, 17, 21, 24, 14, 18 and 22 is . (A) 191 (B) 1.91 (C) 19.1 (D) 0.191 3. [AS1] The median of the data 8, 9, 1, 6, 13, 10, 5 and 11 is . (A) 8 (B) 8.5 (C) 9 (D) 8.9 4. [AS1] The mode of the data 21.5, 34.5, 28, 31, 21.5, 30, 34.5, 33.5, 21.5, 34.5, 34.5, 28and 30 is . (A) 28 (B) 30 (C) 21.5 (D) 34.5 5. [AS3] The measure of central tendency which uses the mid values of classes in its calculation is . (A) Mean (B) Median (C) Mode (D) None 6. [AS4] The mean height of 8 students of a class of heights 142 cm, 145 cm, 150 cm, 148 cm, 152 cm, 138 cm, 141 cm and 144 cm is . (A) 140 cm (B) 145 cm (C)146 cm (D)148 cm 7. [AS4] The median wages of a person whose wages in a week from Sunday to Saturday are Rs. 8500, Rs. 8900, Rs. 8800, Rs. 8600, Rs. 9000, Rs. 9200 and Rs. 9100 is . (A) Rs. 8871 (B) Rs. 62100 (C)Rs. 8600 (D)Rs. 8900 EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 2
8. [AS1] The mean of 'n' observations is 3. Each observation is multiplied by 9 and one is added to it. . The new mean is (A) 28 (B) 27 (C) 13 (D)28 n 9. [AS1] The mean of 11 observations is 12. If one observation 17 is deleted then the . new meanis (A) 14.9 (B) 149 (C) 11.5 (D) 13.5 (A) 29 (B) 87 (D)None of these (C) 9 2 3 Very Short Answer Type Questions 11 [AS1] Answer the following questions in one sentence. (i) Find the arithmetic mean of the first five odd natural numbers. (ii) The sum of 25 observations in a data is 745. Find the mean of the data. (iii) Calculate the median of the data 38, 23, 51, 67, 46, 32, 59 and 21. (iv) The median of x , x, x , x , x , x , x is 17 . Find the value of x. 7 43652 2 (v) The mean of 24, 72, 48, 56, 68, 39, 48, x is 49. Find the value of x. 12 [AS1] Answer the following questions in one sentence. (i) If the median of the terms x − 3, x − 2, x + 4, x + 7 and x + 9 is 23, find the value of x. (ii) The mean of 12 observations is 18. If one observation 30 is deleted, find the new mean. Short Answer Type Questions 13 [AS1] Find the arithmetic mean of 5 observations 20, 21, 13, 8 and 22 taking the assumed mean as 7. 14(i) [AS1] The number of goals were scored by a team in a series of 10 matches are 2, 3, 4, 5, 0, 1, 3, 3, 4 and 3. Find the median number of goals scored by the team. EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 3
(ii) [AS1] The following observations are in ascending order. If the median of the data is 63, find the value of x. 20, 32, 48, 50, x, x + 2, 72, 78, 84, 95 15 [AS3] When will the estimated mean become the actual arithmetic mean? 16 [AS4] The number of runs scored by 10 batsmen in a one day cricket match are as given. 23, 54, 08, 60, 18, 29, 44, 05, 86. Find the average runs scored. 17 [AS4] The following are the marks obtained (out of 100 marks) by 30 students of Class VIII of a school. 10 20 36 92 95 40 50 56 60 70 92 88 80 70 72 70 36 40 36 40 92 40 50 50 56 60 70 60 60 88 Find the mean of the marks obtained by the 30 students. 18(i) [AS4] Find the mean from the following frequency distribution of marks in a competitive exam. Marks 5 10 15 20 25 30 35 40 45 50 Number of 10 40 70 71 75 30 18 72 8 6 students (ii) [AS4] The attendance in a school for five days is as follows: 400, 430, 425, 408, 410. Calculate the arithmetic mean by assuming a mean. Long Answer Type Questions 19 [AS1] If the median of x, x , x and x when x > 0, is 10, find the mean. 5 2 3 20 [AS4] Rajesh saw around 680 animals in a zoo that he visited in his holidays. After coming home he tabulated the number of all the different animals hehas seen in the zoo as following. EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 4
Beast Land Birds Water Reptiles Animals Animals 180 Animals 75 80 200 145 Answer the questions using the given table. (i) Which animals are the maximum in number? (ii) Which animals are the least in number in the zoo? (iii) Which animals are 10 less than half of the number of birds? (iv) How many water animals are there in the zoo? (v) How many reptiles has Rajesh seen in the zoo? 21 [AS4] The ages of 40 students in a sports club are given in the table. Age (in years) 11 12 13 14 15 Number of students 8 4 10 12 6 Find their mean age. 22 [AS4] The following table shows the weights of 50 persons in a group. Weight (in kg) 40 –44 44 –48 48 –52 52 –56 56 –60 Number of Persons 12 16 9 8 5 Find their mean weight. 23 [AS4] The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the value of the missing frequency ‘f’. Daily pocket 11 – 13 13–15 15–17 17–19 19–21 21–23 23–25 allowance (in Rs.) Number of children 7 69 13 f 54 EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 5
EXERCISE 7.2 ORGANISATION OF GROUPED DATA 7.2.1 Key Concepts i. Representation of classified distinct observations of the data with frequencies is called ‘Frequency Distribution’ or ‘Distribution Table’. ii. The difference between the upper and lower boundaries of a class is called the length of the class or class length denoted by ‘C’. iii. In a class, the initial value and end value of each class are called the lower limit and upper limit of that class respectively. iv. The average of upper limit of a class and the lower limit of the successive class is called the upper boundary of that class. v. The average of the lower limit of a class and the upper limit of the preceding class is called the lower boundary of that class. vi. The progressive total of frequencies from the last class of the table to the lower boundary of a particular class is called Greater than Cumulative Frequency (G.C.F.). vii. The progressive total of frequencies from the first class to the upper boundary of a particular class is called Less than Cumulative Frequency (L.C.F.). 7.2.2 Additional Questions Objective Questions 1. [AS3] The classes in which the upper limit of one class and the lower limit of the consecutive class are not equal are called . (A) Boundaries (B) True class limits (C)Inclusive classes (D)Exclusive classes 2. [AS3] The classes which are continuous i.e., the upper limit of one class is the same as the lower limit of immediate next class are called . (A) Boundaries (B) Class limits (C)Inclusive classes (D)Exclusive classes EXERCISE 7.2. ORGANISATION OF GROUPED DATA 6
3. [AS3] The total frequency from the beginning to the upper boundary of a particular class is called the cumulative frequency. (A) Greater than (B) Less than (C) Greater than or equal to (D) Less than or equal to 4. [AS3] If the Greater than Cumulative Frequency (G.C.F) of class 70 – 80 is 12 and that of 60 – 70 is 20 then the frequency of the class 60 – 70 is . (A) 5 (B) 240 3 (C) 32 (D) 8 5. [AS3] The difference between the upper and the lower boundaries of a class is called the . (A) Class length (B) Boundaries (C) Range (D)None of these 6. [AS1] In a given data, if the minimum value and range are 12 and 27 then the maximum value is . (A) 39 (B) 15 (C) 2.25 (D)None of these 7. [AS5] The graphs of the cumulative frequencies are called . (A) Ogive curves (B) Pictographs (C)Bar graphs (D) Histograms Very Short Answer Type Questions 8 [AS1] Answer the following questions in one sentence. (i) If the minimum value of a data is 27 and the range is 15, find the maximum value of the data. (ii) The minimum and maximum values of a data are 25 and 85 respectively. If there are 10 classes in the data, find the class length of each class. 9 [AS4] Answer the following questions in one sentence. (i) The following is a list of the names and heights of five boys in an eighth grade basketball team. Devendra –167 cm, Narendra –160 cm, Sanjay –165 cm, Piyush –175 cm and Girish –155 cm. Organise the data in the ascending order of heights. EXERCISE 7.2. ORGANISATION OF GROUPED DATA 7
(ii) The marks scored by 10 students in test – 1 are Santosh – 77, Nagu – 40, Aman – 45, Amit – 81,Rakesh – 69, Sunil – 81, Vinod – 94, Anjali – 51, Aditi – 87 and Renu – 50. Arrange the data in alphabetical order. (iii) The marks scored by 20 students in a unit test out of 25 marks are as given. 12, 10, 08, 12, 04, 15, 18, 23, 18, 16, 16, 12, 23, 18, 12, 05, 16, 16, 12, 20. Find the number of students who scored less than 19. Short Answer Type Questions 10(i) [AS4] The heights (in cm) of 25 children are as given. 174, 168, 110, 142, 156, 199, 110, 101, 190, 102, 190, 111, 172, 140, 136, 174, 128, 124, 136,147, 168, 192, 101, 129, 114. Prepare a frequency distribution table taking the size of the class – interval as 20. (ii) [AS4] Consider the following marks (out of 50) scored in mathematics by 50 students of Class 8. 41, 31, 33, 32, 28, 31, 21, 10, 30, 22, 33, 37, 12, 05, 08, 15, 39, 26, 41, 46, 34, 22, 09, 11, 16, 22, 25, 29, 31, 39, 23, 31, 21, 45, 30, 22, 17, 36, 18, 20, 22, 44, 16, 24, 10, 28, 39, 28, 47, 17. Prepare a frequency distribution table. 11 [AS5] Rewrite the given table with true class limits. Classes 11 –15 16 –20 21 –25 26 –30 31 –35 36 –40 Frequency 4 37 5 7 2 EXERCISE 7.2. ORGANISATION OF GROUPED DATA 8
12 [AS5] Given are the marks scored by 40 students in an examination. 38 32 24 26 28 25 20 18 36 30 32 38 26 24 20 16 36 16 18 22 24 26 28 10 34 30 20 24 26 22 32 28 38 20 24 22 28 30 28 10 Take class intervals 0 – 10, 11 – 20, 21 – 30 and construct a frequency distribution table for the given data. 13 [AS5] Create a cumulative frequency distribution for the frequency table given. Length 11 – 16 – 21 – 26 – 31 – 36 – 41 – (mm) 15 20 25 30 35 40 45 Frequency 2 4 8 14 6 4 2 EXERCISE 7.2. ORGANISATION OF GROUPED DATA 9
EXERCISE 7.3 GRAPHICAL REPRESENTATION OF DATA 7.3.1 Key Concepts i. Histogram is a graphical representation of frequency distribution of exclusive class intervals. ii. When the class intervals in a grouped frequency distribution are varying, we need to construct rectangles in histogram on the basis of frequency density. iii. Frequency density = Frequency of class. iv. Frequency polygon is a graphical representation of a frequency distribution (discrete/ continuous). v. In frequency polygon or frequency curve, class marks or mid values of the classes are taken on X–axis and the corresponding frequencies on the Y–axis. vi. Area of frequency polygon and histogram drawn for the same data are equal. vii. A graph representing the cumulative frequencies of a grouped frequency distribution against the corresponding lower/ upper boundaries of respective class intervals is called Cumulative Frequency Curve or “Ogive Curve”. 7.3.2 Additional Questions Objective Questions 1. [AS3] The lengths of the rectangles in a histogram are proportional to the . (A) Class interval (B) Frequency (C)Upper limits (D) Lower limits 2. [AS3] If the midpoints of the widths of rectangles on the top of the bars in a histogram are joined bystraight line segments then it is called a/ an . (A) Frequency curve (B) Ogive curve (C)Frequency polygon (D)None of these 3. [AS3] The point of intersection of less than and greater than ogive curves gives . (A) Mean (B) Median (C) Mode (D) Range EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 10
4. [AS3] To construct a histogram, the classes in the given data must be . (A) Raw data (B) Grouped data (C) Discrete (D) Continuous 5. [AS3] The curve constructed by taking the upper boundaries of classes on X–axis and the corresponding cumulative frequencies on Y–axis is known a curve. (A) Less than ogive (B) Greater than ogive (C) Frequency (D)None of these 6. [AS5] If a rectangle of height 7 cm represents 140 units then another rectangle of height 4.5 cm represents units. (A) 24.5 (B) 90 (C) 15.5 (D)None of these 7. [AS3] The mid points of the top ends of the bars in a histogram are joined by means of straight line segments to form a . (A) Frequency polygon (B) Frequency curve (C)Less than ogive curve (D)Greater than ogive curve 8. [AS3] The measure of central tendency which the common point of both the ogive curves gives is the . (A) Mean (B) Mode (C) Median (D)None of these 9. [AS5] In a pictograph 10 car pictures represent 5000 cars in real. In the same pictograph, 7 and a half car pictures represent cars in real. (A) 7000 (B) 375 (C) 37500 (D) 3750 10. [AS5] In a histogram every two consecutive rectangles have . (A)One common side (B) One common height (C)An equal area (D)None of these EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 11
Very Short Answer Type Questions 11 [AS5] Answer the following questions in one sentence. Read and understand the following vertical bar graph and answer the questions. (i) What was the production of wheat in the year 1990 – 91? (ii) In which year was the production minimum and in which year was it maximum? (iii) How many more lakhs of tonnes of wheat was produced in 1980 – 81 than in 1960 – 61? (iv) What is the total production of wheat in two years 1970 – 71 and 1980 – 81? (v) What is the increase in production of wheat in the year 1950 – 51 to the year 2000 – 2001? 12 [AS5] Answer the following questions in one sentence. In a bar graph, a rectangle of height 22 cm represents 27500000 people. Find the height of the rectangle which represents 43750000 people. Short Answer Type Questions 13(i) [AS5] The following table shows the expenditure per month of a family. Items Food Clothing Rent Education Miscellaneous 2000 1500 2500 1000 Expenditure 3500 (in Rs.) Draw a bar graph to represent the given data. (ii) [AS5] Draw a histogram for the given data. Salary (in 15 –20 20 –25 25 –30 30 –35 35 –40 thousand rupees) 35 30 45 40 10 No. of employees EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 12
14 [AS5] The following frequency polygon shows the marks scored by some students in a class test. Answer these questions based on the given frequency polygon. (i) What is the class size? (ii) How many students scored less than 20? (iii) How many students scored 25 or more? 15 [AS5] The table given shows the number of employees drawing different monthly salaries. Monthly 10000 – 15000 – 20000 – 25000 – 30000 – salary (in 15000 20000 25000 30000 35000 Rs.) 5 10 20 8 7 Number of employees Construct a histogram for the data. 16 [AS5] The table gives the age distribution of a group of students. Draw the cumulative frequency curve of less than type and hence obtain the median value. Age (in 4–5 5–6 6–7 7–8 8–9 9–10 10–11 11–12 12–13 13–14 14–15 15–16 years) Frequency 36 42 52 60 68 84 96 82 66 48 50 16 EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 13
17 [AS5] For the following frequency distribution, draw a cumulative frequency curve of more than type and hence obtain the median value. Class 0 –10 10–20 20–30 30–40 40–50 50–60 60–70 interval 5 15 20 Frequency 23 17 11 9 18 [AS5] During the medical checkup of 35 students of a class, their weights were recorded as follows: Weight 38–40 40–42 42–44 44–46 46–48 48–50 50–52 (in kg) 3245 14 4 3 No. of students Draw a less than type and a more than type ogive from the given data. Hence obtain the median weight from the graph. EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 14
CHAPTER 9 AREA OF PLANE FIGURES EXERCISE 9.1 AREA OF TRAPEZIUM 9.1.1 Key Concepts i. Area of a trapezium = 1 (Sum of lengths of parallel sides) × (Distance between 2 1 them) = 2 h (a + b) ii. Area of a quadrilateral = 1 × Length of diagonal × Sum of the lengths of the perpendiculars 2 1 from the remaining two vertices on to the diagonal = 2 × d × (h1 + h2) iii. Area of a rhombus = Half of the product of diagonals = 1 × d1 × d2 2 iv. Area of a parallelogram = base × height = bh v. Area of a triangle = 1 × base × height = 1 × b × h 2 2 √ vi. Area of an equilateral triangle = 3 a2 4 9.1.2 Additional Questions Objective Questions 1. [AS1] The parallel sides of a trapezium are 9.7 cm and 6.3 cm, and the distance between them is 6.5 cm. The area of the trapezium is sq. cm. (A) 104 (B) 78 (C) 52 (D) 65 EXERCISE 9.1. AREA OF TRAPEZIUM 15
2. [AS1] ABCD is a trapezium in which AB = 40 m, BC = 15 m, CD = 28 m, AD = 9 m and CE ⊥AB. Then the area of trapezium ABCD is sq. m. (A) 306 (B) 316 (C) 296 (D) 284 3. [AS1] The length of a rectangular field is 12 m and the length of its diagonal is 15 m. The area of the field is sq. m. (A) 108 √ √ (B) 30 3 (D)None of these (C)12 15 4. [AS1] The area of a square field is 6050 sq. m. The length of its diagonal is m. (A) 110 (B) 112 (C) 120 (D) 135 5. [AS1] The area of the parallelogram one of whose sides measures 48 cm and the corresponding height measures 18.5 cm is sq. cm. (A) 444 (B) 888 (C) 1776 (D)None of these EXERCISE 9.1. AREA OF TRAPEZIUM 16
6. [AS1] The area of the trapezium given is . (A) 56 cm2 (B) 89 cm2 (C)410 cm2 (D)140 cm2 7. [AS3] A quadrilateral with two pairs of parallel opposite sides is called a . (A) Rhombus (B) Trapezium (C) Parallelogram (D) Kite 8. [AS3] The area of a rhombus is . d2 2 (A) 1 d1 d2 (B) 2 (C) bh (D) lb 9. [AS3] A quadrilateral with one pair of parallel opposite sides is called a . (A) Square (B) Rectangle (C)Trapezium (D) Rhombus 10. [AS4] A garden is in the form of a trapezium whose parallel sides are 40 m and 22 m. The perpendicular distance between them is 12 m. The area of the garden is . (A) 372 m2 (B) 462 m2 (C)732 m2 (D)550 m2 EXERCISE 9.1. AREA OF TRAPEZIUM 17
11. [AS4] The area of a hexagonal table top with each side 2 m is . (A) 11.392 m2 (B) 10.392 m2 (C)9.372 m2 (D)9.672 m2 Very Short Answer Type Questions 12 [AS1] Answer the following questions in one sentence. (i) The area of a rhombus is 50 cm2 and one of its diagonals is 15 cm. Find the other diagonal. (ii) Find the base of a parallelogram whose area is 128 cm2 and height 16 cm. (iii) Find the length of a rectangle, whose area is 154 cm2 and width 11 cm. (iv) Find the area of an equilateral triangle whose side is 6 cm. (v) Find the altitude of a trapezium, the sum of the lengths of whose bases is 8.5 cm and whose area is 34 cm2 . 13 [AS2] Answer the following questions in one sentence. (i) If the ratio of base of two triangles is x : y and that of their areas is a : b, then what is the ratio of their corresponding altitudes? 14 [AS3] Answer the following questions in one sentence. (i) Define a regular polygon. (ii) Write the formula to find the area of a quadrilateral. EXERCISE 9.1. AREA OF TRAPEZIUM 18
Short Answer Type Questions 15(i) [AS1] Find the area of the polygon given. CD BE AF⊥BE GD⊥BE CD = 14 cm BE = 18 cm AF = 6 cm DG = 8 cm. (ii) [AS1] The parallel sides of trapezium are 9 cm and 7 cm long and its area is 48 sq. cm. Find the distance between the parallel sides. 16(i) [AS4] Find the area of a rectangular park of length 200 m and breadth 150 m. (ii) [AS4] Find the area of the given field. 17(i) [AS4] The longer side of a rectangular hall is 24 m and the length of its diagonal is 26 m. Find the area of the hall. EXERCISE 9.1. AREA OF TRAPEZIUM 19
(ii) [AS4] In a four–sided field, the length of the longer diagonal is 128 m. The lengths of the perpendiculars from the opposite vertices upon this diagonal are 22.7 m and 17.3 m. Find the area of the field. 18 [AS1] Find the area of the following figure. 19 [AS1] Find the area of the shaded region in the following figure. 20 [AS1] Find the area of a trapezium whose parallel sides are AB = 16 cm, DC = 8 cm and non-parallel sides are BC = 10 cm and DA = 6 cm. 21 [AS1] Find the area of the field given. All the dimensions are given in metres. EXERCISE 9.1. AREA OF TRAPEZIUM 20
22 [AS1] Find the area of a quadrilateral whose sides measure 9 cm, 40 cm, 15 cm and 28 cm and the angle between the first two sides is a right angle. 23 [AS2] The length of a rectangle is increased by 30% and the width is decreased by the same percentage. What is the percentage change in its area? 24 [AS4] Find the length of a rectangular field whose breadth is 15 m and area is 180 m2. 25 [AS4] A rectangular lawn 75 m by 60 m, has two roads, each 4 m wide, running through the middleof the lawn, one parallel to length and the other parallel to breadth, as shown in the figure. Find the cost of gravelling the roads at Rs. 45 per sq. m. 26 [AS4] Find the breadth of the rectangular field given that its area is 256 cm.2 27 [AS4] A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. Its area is 3375 sq. m. Find the cost of fencing the lawn at Rs. 8.50 per metre. EXERCISE 9.1. AREA OF TRAPEZIUM 21
EXERCISE 9.2 AREA OF CIRCLE 9.2.1 Key Concepts i. Area of a circle = πr2, where r = radius. ii. Area of a ring = π R2 − r2 , where R = radius of outer circle and r = radius of inner circle. iii. Area of a sector = x × πr2, where x is the central angle and r is the radius. 360◦ 9.2.2 Additional Questions Objective Questions 1. [AS1] The area of a circle is 154 sq. m. Then its diameter is . (A) 7 m (B) 14 m (C) 3.5 m (D) 21 m 2. [AS2] The circumferences of two circles are in the ratio 2 : 3. The ratio between their areas is . (A) 2 : 3 (B) 3 : 2 (C)9 : 4 (D)4 : 9 3. [AS4] The circumference of a circular field is 242 m. Then its area is sq. m. (A) 9317 (B) 18634 (C) 4658.5 (D)None of these 4. [AS2] If the area of a circle is 49π sq. cm then its circumference is cm. (A) 14π (B) 21π (C) 28π (D) 7π 5. [AS2] On increasing the diameter of a circle by 40%, its area will be increased by . (A) 40% (B) 80% (C) 96% (D) 82% EXERCISE 9.2. AREA OF CIRCLE 22
6. [AS4] A circular garden has a circumference of 220 cm. There is a 7 cm wide border inside the garden along its boundary. The area of the border is . (A) 1368 m2 (B) 1683 m2 (C)1836 m2 (D)1386 m2 Short Answer Type Questions 7(i) [AS1] Find the length of the arc of the shaded part of the circle. (ii) [AS1] Find the length of the arc, which makes an angle 30° at the centre of the circle of radius 21cm. 8(i) [AS1] A chord of a circle of radius 14 cm makes a right angle at the centre. Find the area of the sector making the right angle. (ii) [AS1] The perimeter of a sector of a circle of radius 5.6 cm is 27.2 cm. Find its area. 9 [AS1] Find the area of the shaded region in the given figure. (Take π = 3.14 ) EXERCISE 9.2. AREA OF CIRCLE 23
10 [AS1] Two circles touch internally. The sum of their areas is 116 π and the distance between their centres is 6 cm. Find the radii of the circles. 11 [AS1] Two circles touch externally. The sum of their areas is 130π sq. cm and the distance between their centres is 14 cm. Find the radii of the circles. 12 [AS1] Find the area of the shaded region in the following figure. 13 [AS1] Find the area of the shaded region in the square ABCD. 14 [AS4] A steel wire, when bent in the form of a square, encloses an area of 121 sq. cm. The same wire is bent in the form of a circle. Find the area of the circle. 15 [AS4] The radii of the inner and outer boundaries of a circular park are 56 m and 70 m respectively. Find the area of the circular path. EXERCISE 9.2. AREA OF CIRCLE 24
CHAPTER 10 DIRECT AND INVERSE PROPORTIONS EXERCISE 10.1 DIRECT PROPORTION 10.1.1 Key Concepts i. Two quantities, x and y, are said to be in direct proportion if whenever the value of x increases or decreases, then the value of y increases or decreases in such a way that their ratio remains constant. ii. When x and y are in direct proportion, then x1 = x2 = x3 etc. (where x1, x2, x3 are the values of x y1 y2 y3 corresponding to the values y1, y2, y3 o f y. ) 10.1.2 Additional Questions Objective Questions 1. [AS3] If two quantities x and y are in direct proportion, then _______. (A) x + y = k (B) x − y = k (C) x × y = k (D) x = k y 2. [AS4] If a vehicle covers a distance of 60 m with 4l of petrol then the distance covered by it with 10l of petrol is km. (A) 150 (B) 180 (C) 74 (D) 2400 3. [AS4] The cost of 5 m of a particular cloth is Rs. 210. Then the cost of 3 m of the cloth is . (A) Rs. 14 (B) Rs. 126 (C) Rs. 350 (D) Rs.105 EXERCISE 10.1. DIRECT PROPORTION 25
4. [AS4] The distance travelled by a loaded truck in 5 h if the truck travels 14 km in 30 min is . (A) 140 km (B) 100 km (C) 490 km (D)None of these 5. [AS4] The cost of 8 kg of sugar is Rs 288. Then the cost of 18 kg of sugar is . (A) Rs. 324 (B) Rs. 972 (C) Rs. 648 (D) Rs. 1296 6. [AS1] Two quantities x and y are in direct proportion and the constant of variation is 4.5. If the value of x is 184.5, then the value of y is ______. (A) 830.25 (B) 180 (C) 189 (D) 41 7. [AS1] Two quantities 'm' and 'n' are in direct proportion and the constant of variation is 1.75. If the value of 'n' is 88, then the value of 'm' is _______. (A) 50.29 (B) 154 (C) 0.19 (D) 15.4 8. [AS4] The cost of 12 kg of tomatoes is Rs. 180. The amount paid by Raju for 15 kg of tomatoes is ______. (A) Rs. 225 (B) Rs.360 (C) Rs. 195 (D)None of these EXERCISE 10.1. DIRECT PROPORTION 26
9. [AS3] The cost of sugar (a) and its quantity (b) are in direct proportion. Representing this as an equation gives _____. (A) a = k(constant) (B) ab = k (constant) b (C)a + b = k (constant) (D)a − b = k (constant) Very Short Answer Type Questions 10 [AS1] Answer the following questions in one sentence. (i) The values 'a' and 'b' are in direct variation. The constant of variation is 1 . Find the value of 'a' if b = 21. 3 (ii) The cost of 30 l of diesel is Rs.1050. Find the cost of 20 l of diesel. 11 [AS2] Answer the following questions in one sentence. (i) When speed remains constant verify if the distance (x) travelled and time (y) taken are in direct proportion. (ii) Verify if the area of land (x) and its cost (y) are in direct proportion. (iii) Given that u = 3v, verify whether u and v are in direct proportion. 12 [AS3] Answer the following questions in one sentence. (i) Write the rule for direct variation. (ii) Write any one situation where you see direct proportion. 13 [AS4] Answer the following questions in one sentence. The cost of 5 metres of a particular quality of cloth is Rs. 210. Find the cost of 13 metres long cloth of the same quality. Short Answer Type Questions 14 [AS4] The scale of a map is given as 1 : 30000000. Two cities are 4 cm apart on the map. Find the actual distance between them. 15 [AS1] In the equation a = kb where k is constant, a = 6 when b = 14. (i) Find ‘a’ when ‘b’ is 10. (ii) Find ‘b’ when ‘a’ is 16. EXERCISE 10.1. DIRECT PROPORTION 27
16(i) [AS4] If 20 men assemble 6 cars in a day, identify the proportion in which they are related. Also find the number of men needed to assemble 15 cars a day. (ii) [AS4] If the weight of 70 tea–packets of the same size is 28 kg, identify the proportion in which they are related. Also find what is the weight of 32 such packets? 17(i) [AS4] If 20 oranges cost Rs. 70, what do 42 oranges cost? (ii) [AS4] If 40 metres of a cloth costs Rs. 1550, how many metres can be bought for Rs. 800? 18 [AS4] A car travels 18 km in 20 minutes. How far can it travel in 4 hours, if the speed remains the same? Long Answer Type Questions 19 [AS4] A train is moving at a uniform speed of 75 km/h. (i) How far will it travel in 20 minutes? (ii) Find the time required to cover a distance of 250 km. EXERCISE 10.1. DIRECT PROPORTION 28
EXERCISE 10.2 INVERSE PROPORTION 10.2.1 Key Concepts i. Two quantities x and y are said to be in inverse proportion, if there exists a relation of the type xy =k between them, k being a constant. ii. If y1 , y2are the values of y corresponding to the values x1 and x 2 of x respectively, then x1y1 = x2y2 (= k), or x1 = y2 . x2 y1 10.2.2 Additional Questions Objective Questions 1. [AS3] If two quantities x and y are in inverse proportion then the constant of variation is given by _______. (A) x × y (B) x (C) x + y y (D) x − y 2. [AS1] x and y are in inverse proportion. x = 27 when y = 13. If y = 39, then x = ______. (A) 33 (B) 14 (C) 223.25 (D) 9 3. [AS1] x and y are in inverse proportion. x = 18, when y = 36. If x = 54, then y = _________. (A) 79 (B) 12 (C) 81 (D) 169 9 4. [AS4] A car covers a distance in 5 h at a speed of 80 kmph. If the speed is reduced to 50 kmph, then the time required to cover the same distance is _____. (A) 10 h (B) 20 h (C)16 h (D)8 h EXERCISE 10.2. INVERSE PROPORTION 29
5. [AS4] If 100 apples are packed in one carton, 20 such cartons are needed. If 80 apples are packed in one carton then the number of cartons required is _______. (A) 25 (B) 50 (C) 40 (D) 100 6. [AS1] The quantities 'p' and 'q' are in inverse proportion and the constant of variation is 360. If the value of 'p' is 80, then the value of 'q' is _____. (A) 440 (B) 280 (C) 4.5 (D) 28800 7. [AS1] The quantites 'a' and 'b' are in inverse proportion and the constant of variation is 1975. If the value of 'b' is 125 then the value of 'a' is _____. (A) 15.8 (B) 2100 (C) 1850 (D) 2225 8. [AS4] 8 men can complete a work in 25 days. The number of days required to complete the same work by 20 men is ____ days. (A) 160 (B) 6.4 (C) 180 (D) 10 9. [AS3] The quantity of rice available is sufficient for x number of students for y number of days. The expression used to find the number of students who can finish the rice in 1 day is ____. (A) x + y (B) x − y (C) xy (D) x y Very Short Answer Type Questions 10 [AS1] Answer the following questions in one sentence. (i) 'a' and 'b' are in inverse proportion. The constant of variation is 72. Find the value of 'a' when b = 12. EXERCISE 10.2. INVERSE PROPORTION 30
(ii) The quantities x and y are in inverse proportion. Find the value of y when x = 128, if x = 16 when y = 4. (iii) x and y are in inverse proportion. Find the value of x when y = 28 if x = 35 when y = 52. 11 [AS2] Answer the following questions in one sentence. (i) Verify whether “the length (x) of a journey by bus and price (y) of the ticket are in inverse proportion or not. (ii) Verify if \"The population of a country and the area of land per person” are in inverse proportion or not. 12 Answer the following questions in one sentence. (i) [AS3] Write the rule for inverse variation. (ii) [AS3] 15 workers can build a wall in 48 hours. How many workers will be required to complete thesame work in 30 hours? 13 [AS4] Answer the following questions in one sentence. 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used? Short Answer Type Questions 14 [AS1] If x and y vary inversely, x = 16 when y = 4. Find y when x = 32.8 and find x when y = 0.25. Represent these values in the form of a table. 15(i) [AS4] If 42 men can complete a piece of work in 20 days, in how many days can 30 men complete it? (ii) [AS4] If 56 men can do a piece of work in 42 days, how many men can do it in 14 days? 16(i) [AS4] If 1600 persons can finish the construction of a building in 60 days, how many persons are needed to complete the construction of the building in 40 days? (ii) [AS1] If x and y vary inversely as each other and x = 10 when y = 6, find y when x = 15. 17(i) [AS4] 15 pipes are required to fill a tank in 2 hours 45 minutes. How long will it take if only 45 pipes and 25 pipes of the same type are used? (ii) [AS4] In a company, 10 team members can complete a project in 3 months. If the project has to be completed in 2 months, how many members should work on it? EXERCISE 10.2. INVERSE PROPORTION 31
18 [AS5] x and y are inversely proportional. Given that x = 25 when y = 50, find (i) y when x = 2 (ii) x when y = 125 (iii) x = 250 and represent these values in the form of a table. Long Answer Type Questions 19 [AS4] A garrison of 100 men had provisions for 48 days. A reinforcement of a few men arrived and the provisions last for 30 days. How many men had arrived? 20 [AS4] A hostel has provisions for 400 people for 100 days. After 10 days 50 more people joined the hostel. How long will the remaining provisions last at the same rate? 21 [AS4] Shalu cycles to her school at an average speed of 12 km/h. It takes her 20 minutes to reach the school. If she wants to reach her school in 15 minutes, what should be her average speed? 22 [AS4] A car can finish a certain journey in 10 h at the speed of 48 km/h. By how much should its speed be increased so that it may take only 8 h to cover the same distance? 23 [AS4] A tap fills a tank in 28 hours and an outlet can empty the full tank in 35 hours. In how many hours will the empty tank be filled, if both the tap and the outlet are opened simultaneously? 24 [AS4] 1000 soldiers in a fort had enough food for 20 days. If some of the soldiers were transferred to another fort and the food lasted for 25 days for the remaining members, then find the number of soldiers that were transferred. 25 [AS4] A hostel mess had provisions for 340 persons for 50 days. Identify the proportion in which they are related. After 8 days, 30 persons joined the mess. How long will the food last at the same rate? EXERCISE 10.2. INVERSE PROPORTION 32
EXERCISE 10.3 COMPOUND PROPORTION 10.3.1 Key Concepts i. Sometimes change in one quantity depends upon the change in two or more other quantities in the same proportion. Then we equate the ratio of the first quantity to the compound ratio of the other two quantities. 10.3.2 Additional Questions Objective Questions 1. [AS3] If a change in one quantity depends upon a change in two (or) more quantities in some proportion then it is known as a/ an proportion. (A) Direct (B) Inverse (C) Compound (D) None of these 2. [AS3] ‘a’ men can lay a road of ‘b’ km in ‘c’ days. Here 'c' is in inverse proportion with ‘a’ and in proportion with ‘b’. (A) Direct (B) Inverse (C) Compound (D) None of these 3. [AS4] 100 men can construct a wall of 1000 m in 20 days. The number of days that 200 men take to construct a wall of length 1500 m is ______. (A) 30 days (B) 15 days (C)20 days (D)10 days 4. [AS4] 15 men can lay a road 300 km long in 10 days. The number of days 25 men take to lay a road 100 km long is ____. (A) 5 days (B) 20 days (C)10 days (D)2 days EXERCISE 10.3. COMPOUND PROPORTION 33
5. [AS4] 350 men can clean a road 6300 km long in 72 days. The number of men required to clean a road of 7800 km long in 48 days is ______. (A) 325 (B) 650 (C) 975 (D)None of these 6. [AS4] 15 boys earn Rs. 900 in 5 days. The amount earned by 20 boys in 7 days is _____. (A) Rs.1800 (B) Rs.1680 (C) Rs.1400 (D)Rs. 2520 Very Short Answer Type Questions 7 [AS3] Answer the following questions in one sentence. Define compound proportion. Long Answer Type Questions 8 [AS4] 32 workers working 9 hours a day can finish a piece of work in 20 days. If each worker works 6 hours a day, find the number of workers needed to finish the same piece of work in 24 days. 9 [AS1] (i) If A : B = 3 : 4 and B : C = 5 : 6, then find A : C. (ii) Find the value of the unknown. 24 : 12 :: x : 15 10 [AS5] The mess charges for 65 students for 30 days is Rs. 24,375. How much will be the mess charges for 75 students for 45 days? Represent these values in the form of a table. EXERCISE 10.3. COMPOUND PROPORTION 34
CHAPTER 11 ALGEBRAIC EXPRESSIONS EXERCISE 11.1 SIMPLE OPERATIONS OF ALGEBRAIC EXPRESSIONS 11.1.1 Key Concepts i. Constant: A symbol having a fixed numerical value is called a constant. ii. Variable: A symbol which takes various numerical values is called a variable. iii. Algebraic expression: A combination of constants and variables connected by the signs of fundamental operations is called an algebraic expression. iv. Various parts of an algebraic expression which are separated by the signs ’+’ or ’–’ are called the terms of the expression. v. An algebraic expression is called a monomial, a binomial or a trinomial according to the number of terms it contains. vi. A term having no variables is called a constant term. vii. Like terms: The terms having the same literal factors are called similar terms or like terms. viii. In adding or subtracting algebraic expressions, we collect different groups of like terms and find their sum or difference. ix. To subtract an expression from another, we change the sign from ’+’ to ’–’ and from ’–’ to ’+’. 11.1.2 Additional Questions Objective Questions 1. [AS3] The number of terms in the algebraic expression 7x2yz – 5xy is . (A) 1 (B) 2 (C) 3 (D)None of these 2. [AS3] The algebraic expression which contains 3 unlike terms is known as a . (A) Monomial (B) Binomial (C) Trinomial (D) Multinomial EXERCISE 11.1. SIMPLE OPERATIONS OF ALGEBRAIC EXPRESSIONS 35
3. [AS3] The coefficient of x2 in the expression 7x4 – 5x3 + 6x2 – 4x + 9 is . (A) 7 (B) 5 (C) 6 (D) −5 4. [AS3] The number of terms in the expression 4a2b2cz5 − 5ax2y + 3abc − 7a2y + 9 is ____. (A) 5 (B) 4 (C) 3 (D) 2 5. [AS1] The value of 3x2 + 4x + 3 when x = 1 is ______. (A) 36 (B) 15 (C) 10 (D) 21 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. Find the product of 2x and 3x2. 7 [AS1] Answer the following questions in one sentence. If the degree of 4x2ym + 7xy2 − y3 is 7, then find the value of 'm'. 8. [AS3] Match the expressions given under column A with the number of terms in it under column B. Column A Column B i. 5x2 + 2y2 a. 1 ii. 3x + 5y + 2 b. 4 iii. 2x2 + y2 + z2 + p2 c. 2 iv. a + b + c + d + e d. 3 v. 6xyz e. 5 EXERCISE 11.1. SIMPLE OPERATIONS OF ALGEBRAIC EXPRESSIONS 36
9 [AS4] Answer the following questions in one sentence. The length of a rectangular field is 2x units and its breadth is (x − 5) units. Find its area. Short Answer Type Questions 10(i) [AS1] Find the value of 5x + 6 when x = 2, –5 and 1 . 4 (ii) [AS1] Find the value of 9a – 1 when a= − 1 and 2 . 3 9 11(i) [AS1] Add the algebraic expressions 2a2 + 5ab + b2 and 5a2 + 8ab + 3b2. (ii) [AS1] Add: xy 2 + 2x2y2 + 4x2y and 5xy2 + x2y2 + 2x2y 12(i) [AS1] Subtract x2 − 12xy − 7y2 from 2x2 + 8xy + 4y2 . (ii) [AS1] Subtract xy + 6y2 from 8xy + 4x2 – y2. 13(i) [AS1] What must be subtracted from the sum of a2 + b2 + c2 – 3abc and a2 + 2b2 + c2 + 3abc to get 2a2 – 2b2 – 3c2 + abc? (ii) [AS1] Subtract 3x − 4y − 7z from the sum of x − 3y + 2z and −4x + 9y − 11z. 14 [AS1] From the sum of a2 + 7a + 10 and 3a2 + 4a − 15 subtract 2a2 + a − 9. 15 [AS1] What must be added to 7x2 − 14x + 32 to get 9x2 − 10x − 15 ? 16(i) [AS1] a. Multiply: 8a2 by (−2a3) b. –9x2y × 5xy2 (ii) [AS1] 1 p3q2r × −1 p2qr3 2 5 17(i) [AS1] a. x5× x–3 × x2 = . b. p2q2 × q5 p = . (ii) [AS1] Simplify: x6y5z2 × x3y2z 18(i) [AS3] Identify the like terms: 5x2y, y2, x2y, 3xy2, 3x2y (ii) [AS3] Identify the unlike terms: p2qr, pq2r, pqr2, 1 p2qr, 3 pq, pq2r 2 5 19(i) [AS4] From an iron bar of length (20 − 7x) m, a piece of length (2x − 1) m was cut. Find the length of the remaining iron bar. EXERCISE 11.1. SIMPLE OPERATIONS OF ALGEBRAIC EXPRESSIONS 37
(ii) [AS4] Joseph had Rs.(3a + 75) with him. He spent Rs.(a − 35) to buy a book. Find the money left with him. Long Answer Type Questions 20 [AS1] Multiply − 4 xy3 by 6 x2 y and verify your result for x = 2 and y = 1. 3 7 21 [AS1] Find the value of 5a6 × −10ab2 × −2.1a2b3 f or a = 1 and b = 1 2. 22 [AS1] Determine the products and find the value of each expression for x = 2, y = 1.15, z = 0.01. (i) z2 (x − y) (ii) (2z − 3x) × (−4y) 23 [AS2] Find the product of −5x2y, −2 xy2z, 8 xyz2 and −1 z. Verify the result for x = 1; y = 2 3 15 4 and z = −3. 24 [AS1] (i) The breadth of a rectangle is 2x2y units and its length is xy2units. Find the area of the rectangle. (ii) If P = 8xyz, T = 2x2and R = 2y, find PT R 100 . EXERCISE 11.1. SIMPLE OPERATIONS OF ALGEBRAIC EXPRESSIONS 38
EXERCISE 11.2 MULTIPLYING A BINOMIAL OR TRINOMIAL BY A MONOMIAL 11.2.1 Key Concepts i. The product of two factors with like signs is positive and the product of 2 factors with unlike signs is negative. ii. The coefficient of the product of monomials is equal to the product of their coefficients. iii. If a, b, c and d are monomials, then a × (b + c) = ab + ac (a + b) × (c + d) = ac + ad + bc + bd 11.2.2 Additional Questions Objective Questions 1. [AS1] 6ab(a + b) = . (A) 6a2b + ab2 (C)6a2b + 6ab2 (B) 12 ab (D) None of these 2. [AS1] The degree of the product 3x2(4x3 –5x4 + 7x2 – 4x + 9) is _____. (A) 3 (B) 4 (C) 6 (D) 8 3. [AS1] 2a(3a − 4b + 5c) = . (A) 6a2 − 8ab + 10ac (C)8a2 bc (B) 8 abc (D)8a2 − bc 4. [AS1] If a = 1, b = 0 and c = 2, then the value of 3a(2b + 5c) is . (A) 36 (B) 30 (C) 0 (D) 72 EXERCISE 11.2. MULTIPLYING A BINOMIAL OR TRINOMIAL BY A MONOMIAL 39
5. [AS1] 2x(3x + 5y) = . (A) 16xy (C)6x2 + 10xy (B) 16x2y (D) None of these Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. Find the product of 4a and (7a – 5). 7 [AS1] Answer the following questions in one sentence. If the product of a monomial and (3x − 5) is 6x 2 − 10x then find the monomial. 8 [AS4] Answer the following questions in one sentence. The cost of one cricket ball is Rs.(7x + 5). Find the cost of (3x + 2) cricket balls. Short Answer Type Questions 9(i) [AS1] Find the product of 6x and (7x + 5). (ii) [AS1] (–8ab)(4a – 7b2) = 10(i) [AS1] Find the product 3x(5x + 2y – z). (ii) [AS1] Add the products: 2x(x2 + y2 + z2) and 3x(x2 – y2 – z2) 11(i) [AS1] Simplify: p(2p – 3q + r) + 2p(3p – 2q – r) – 3p(p – 2q + 2r) (ii) [AS1] Simplify: x(x – y) + y(y + z) + z(x + z) EXERCISE 11.2. MULTIPLYING A BINOMIAL OR TRINOMIAL BY A MONOMIAL 40
EXERCISE 11.3 MULTIPLYING A BINOMIAL BY A BINOMIAL OR TRINOMIAL 11.3.1 Key Concepts i. When multiplying a binomial with a binomial or a trinomial, one must remember to multiply each of the terms and simplify the like terms to get the product. 11.3.2 Additional Questions Objective Questions 1. [AS1] (x + y)(2x − y) = . (A) 2x2 + xy − y2 (C)2x2 − 3xy − y2 (B) 2x2 − xy − y2 (D)2x2 + 3xy − y2 2. [AS1] (7a + 3b)(2a + 3b) = . (A) 14a2 + ab2 (C)14a2 + 27ab + 9b2 (B) 14a2 + 30ab + 9b2 (D)None of these 3. [AS1] (3a − 7b)(2a − 6b + c) = . (A) 6a2 + 42b2 (C)6a2 − 18ab + 3ac − 7ab + 42b2 − 7bc (B) 6a2 + 42b2 + c (D)6a2 − 32ab + 3ac + 42b2 − 7bc 4. [AS1] If x = 1 and y = 0, then the value of (x + 2y)(x − 2y) is . (A) 0 (B) 1 (C) 2 (D) 4 EXERCISE 11.3. MULTIPLYING A BINOMIAL BY A BINOMIAL OR TRINOMIAL 41
5. [AS1] If a = 1, b = 2, and c = 0, then the value of (2a + 3b)(4a − 5b + 7c) is . (A) 48 (B) 0 (C) −48 (D) 8 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) Simplify (x – y)(a + b + c) + (x + y)(a – b – c). (ii) Simplify (x – y + z)(x + y) – (x + y – z)(x – y). (iii) Find the product of (2a + 5) and (2a −5). 7 [AS4] The cost of (3a + 7) kg of sugar is Rs. (6a2− a − 35). Find the cost of 1 kg of sugar. Short Answer Type Questions 8(i) [AS1] Multiply: (2x + 3y)(4x – y) (ii) [AS1] Multiply: (a – b)(2a – 3b) 9(i) [AS1] Find the product: (2x + 5y)(x – y – z) (ii) [AS1] Find the product: (a – b)(a2– 2ab + b2 ) Long Answer Type Questions 10 [AS1] Express the product (3x − 2) (x − 1) (3x + 5) as a single algebraic expression and also write its degree. EXERCISE 11.3. MULTIPLYING A BINOMIAL BY A BINOMIAL OR TRINOMIAL 42
EXERCISE 11.4 WHAT IS AN IDENTITY 11.4.1 Key Concepts i. An identity is an equality which is true for all values of the variables involved in it. ii. The following are some identities: i) (a + b)2 = a2 + 2ab + b2 ii) (a – b)2 = a2 − 2ab + b2 iii) (a + b)(a – b) = a2 – b2 iv) (x + a)(x + b) = x2+ (a + b)x + ab 11.4.2 Additional Questions Objective Questions . 1. [AS3] The identity (x + a)(x + b) = (B) x2 + 2ab + b2 (A) x2 + ab (D) None of these (C) x2 + x(a + b) + ab 2. [AS3] The identity used to simplify 102 × 98 is _____. (A) (a + b)2 (B) (a + b) (a − b) (C)(x + a) (x + b) (D)None of these 3. [AS1] If x + 1 = 2, then x2 + 1 is . x x2 (A) 2 (B) 4 (D)None of these (C) 8 EXERCISE 11.4. WHAT IS AN IDENTITY 43
4. [AS1] The middle term in the expansion of (2y – 3z)2 is . (A) 12yz (B) − 6yz (C) 6yz (D) −12yz 5. [AS1] The value of (13.1)2 – (6.9)2 is . (A) 124 (C) 1.24 (B) 12.4 (D) 1124 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. If the last term in the expansion of (3a + 4)(6a + a) is –28, find the value of 'a' . Short Answer Type Questions 7(i) [AS1] Find (302)2. (ii) [AS1] Find (996)2. 8(i) [AS1] Find the product: (6x + 7y)2(6x + 7y) (ii) [AS1] Find the product using a suitable identity. (3s – 2t)(3s + 2t) 9(i) [AS2] Verify: (3a + 2b)(3a + 2b) = 9a2+ 12ab + 4b2 (ii) [AS2] Verify: (206 × 194) = (200 + 6)(200 – 6) Long Answer Type Questions 10 [AS2] If a2 + b2 + c2 − ab − bc − ca = 0, prove that a = b = c. EXERCISE 11.4. WHAT IS AN IDENTITY 44
EXERCISE 11.5 GEOMETRICAL VERIFICATION OF THE IDENTITIES 11.5.1 Key Concepts i. Identities can be verified by substituting values and using geometric shapes. 11.5.2 Additional Questions Objective Questions 1. [AS4] Using the identity (a + b)2 = a2 + 2ab + b2, the value of (101)2 is _____. (A) 11011 (B) 10201 (C) 12321 (D) 1031 2. [AS4] Using the identity (a − 2 = a2 − 2ab + b2, the value of 1462 is _____. b) (A) 21316 (B) 20316 (C) 21326 (D) 22326 3. [AS4] Using identity (a + b)(a − b) = a2 − b2, the value of 107 × 93 is _____. (A) 10697 (B) 9407 (C) 200 (D) 9951 4. [AS1] 103 × 109 represented in the form of the identity (x + a)(x + b) = x2+ (a + b)x + ab is ______. (A) (100 + 3)(100 + 9) (B) (106 − 3)(106 + 3) (C)(1002 − 32) (D)(106 − 3)(100 + 9) 5. [AS1] 6.1 × 6.3 represented in the form of an identity is ______ . (A) (6.0 + 0.2)2 (B) (6.4 − 0.1)2 (C)(6.0 + 0.1)(6.0 + 0.3) (D)(6.12 − 6.32) EXERCISE 11.5. GEOMETRICAL VERIFICATION OF THE IDENTITIES 45
Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. If the middle term of (7a − b)2 is −42a, then find the value of 'b'. Short Answer Type Questions 7 [AS2] Verify the identity (a – b)2 = a2 – 2ab + b2. 8 [AS5] Verify the identity (x + a)(x + b) = x 2 + x(a + b) + ab geometrically. Long Answer Type Questions 9 [AS5] Verify the identity (a + b)2 = a + 2ab + b2 geometrically. 10 [AS5] Verify the identity a 2 b2= (a + b) (a − b) geometrically. − EXERCISE 11.5. GEOMETRICAL VERIFICATION OF THE IDENTITIES 46
CHAPTER 12 FACTORISATION EXERCISE 12.1 FACTORS OF ALGEBRAIC EXPRESSIONS 12.1.1 Key Concepts i. Factors: When an algebraic expression can be written as the product of two or more expressions, then each of these expressions is called a factor of the given expression. ii. Factorization: The process of finding two or more expressions whose product is the given expression is called factorization. (Factorization is the reverse process of multiplication.) 12.1.2 Additional Questions Objective Questions 1. [AS1] A factor of 9x2y + 3ax is . (A) 3ax (C) 3ay (B) 3xy + a (D)3 + a 2. [AS1] If the common factor of 6xy and ay2 is 3y, then the value of 'a' is . (A) 9 (B) 6 (C) 18 (D) 12 3. [AS1] The factors of 15a3b − 35ab3 are . (B) 5ab and (3a2 − 7b2 ) (A) 5 and (a3b − 7ab3) (D) 5ab and (3a2 − 7b) (C)5a2b and (3a2− 7b2) EXERCISE 12.1. FACTORS OF ALGEBRAIC EXPRESSIONS 47
4. [AS4] The area of a rectangular field of length 6b units is (6ab + 12b) sq. units. Its breadth is units. (A) (a + 2) (B) (a + 2b) (C)(a + 6b) (D)2a + 3 5. [AS1] The factorisation of x(x + z) − y(y + z) is . (A) (x + y)(x + y − z) (B) (x − y)(x + y + z) (C)(x − y)(x − y + z) (D)(x − y)(x + y − z) Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) Factorize 12a2b + 15ab2 (ii) Find the greatest common factor of 6x3y and 18x2y3 . 7 [AS1] Answer the following questions in one sentence. (i) If the H.C.F of 20a12b2 − 15amb4 is 5a8 b2 , then find the value of m. (ii) If one of the factors of a3x + a2(x − y) − a(y + z) − z is (a2x − ay − z), find the other factor. 8. [AS3] Match the numbers in column A with their right prime factorisations in column B. Column A Column B i. 48 a. 3 × 3 × 3 ii. 36 b. 2 × 2 × 2 × 3 iii. 24 c. 2 × 3 × 3 iv. 18 d. 2 × 2 × 2 × 2 × 3 v. 27 e. 2 × 2 × 3 × 3 9 [AS4] Answer the following questions in one sentence. The cost of 1 litre petrol is Rs.(3x 2 − 4x + 2) . Find the cost of (2x + 3) litres of petrol. EXERCISE 12.1. FACTORS OF ALGEBRAIC EXPRESSIONS 48
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