16 [AS2] In the adjoining figure, O is the centre of a circle and PO bisects ∠APD. Prove that AB = CD. EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 48
EXERCISE 12.5 CYCLIC QUADRILATERALS 12.5.1 Key Concepts i. If a circle passes through all the four vertices of a quadrilateral, then the quadrilateral is called a cyclic quadrilateral. ii. If a line segment joining two points subtends the same angles at two other points lying on the same side of the line segment, then the four points lie on a circle. iii. The sum of a pair of opposite angles of a cyclic quadrilateral is supplementary. iv. If a parallelogram is cyclic, then it becomes a rectangle. v. If a rhombus is cyclic, then it becomes a square. 12.5.2 Additional Questions Objective Questions 1. [AS1] In the given figure, AOB is a diameter and ABCD is a cyclic quadrilateral. If ∠ADC =120◦ then ∠BAC = . (A) 60◦ (B) 30◦ (C) 20◦ (D) 45◦ EXERCISE 12.5. CYCLIC QUADRILATERALS 49
2. [AS1] In the given figure, if ABCD is a cyclic quadrilateral in which AB DC and ∠BAD = 100◦, then ∠ABC = . (A) 80◦ (B) 100◦ (C) 50◦ (D) 40◦ 3. [AS1] In the given figure, if ABCD is a cyclic quadrilateral in which BC = CD and ∠CBD = 35◦, then ∠BAD = . (A) 65◦ (B) 70◦ (C) 110◦ (D) 90◦ EXERCISE 12.5. CYCLIC QUADRILATERALS 50
4. [AS1] In the given figure, if equilateral triangle △ ABC is inscribed in a circle and ABCD is a quadrilateral as shown, then ∠BDC = . (A) 90◦ (B) 60◦ (C) 120◦ (D) 150◦ 5. [AS1] In the given figure, sides AB and AD of quadrilateral ABCD are produced to E and F respectively. If ∠CBE = 100 ◦, then ∠CDF = . (A) 100◦ (B) 80◦ (C) 130◦ (D) 90◦ Very Short Answer Type Questions 6 [AS3] State true or false. (i) If all the vertices of a quadrilateral lie on a circle, it is called a cyclic quadrilateral. ] [ ] (ii) An isosceles trapezium is a cyclic quadrilateral. [ EXERCISE 12.5. CYCLIC QUADRILATERALS 51
Fill in the blanks. [AS3] . (iii) The opposite angles of a cyclic quadrilateral are (iv) If two opposite sides of a cyclic quadrilateral are equal, then the other two sides are . [AS3] Choose the correct answer. . (v) In a cyclic quadrilateral the sum of any pair of opposite angles is (A) 360◦ (B) 90◦ (C) 180◦ (D) 45◦ 7 [AS1] Choose the correct answer. (i) In the given figure, AB and CD are two intersecting chords of a circle. If ∠CAB = 40◦ and ∠BCD = 80◦ , then ∠CBD = . (A) 80◦ (B) 60◦ (C) 50◦ (D) 70◦ (ii) In the given figure, AOB is a diameter of a circle and CD AB. If ∠BAD = 30◦, then ∠CAD = . (A) 30◦ (B) 60◦ (C) 45◦ (D) 50◦ EXERCISE 12.5. CYCLIC QUADRILATERALS 52
(iii) In the given figure, O is the centre of a circle and ∠OAB = 50◦. Then, ∠BOD = . (A) 130◦ (B) 50◦ (C) 100◦ (D) 80◦ (iv) In the given figure, ABCD and ABEF are two cyclic quadrilaterals. If ∠BCD = 110◦, then ∠BEF = . (A) 55◦ (B) 70◦ (C) 90◦ (D) 110◦ (v) In the given figure, ABCD is a cyclic quadrilateral in which DC is produced to E and CF is drawn parallel to AB such that ∠ADC = 95◦ and ∠ECF = 20◦ . Then, ∠BAD = . (A) 95◦ (B) 85◦ (C) 105◦ (D) 75◦ EXERCISE 12.5. CYCLIC QUADRILATERALS 53
8 [AS1] Choose the correct answer. (i) In the given figure, O is the centre of a circle. If ∠OAB = 40 ◦ and C is a point on the circle, then ∠ACB = . (A) 40◦ (B) 50◦ (C) 80◦ (D) 100◦ (ii) AB and CD are two equal chords of a circle with centre O such that ∠AOB = 80◦ . Then ∠COD = . (A) 100◦ (B) 80◦ (C) 120◦ (D) 40◦ EXERCISE 12.5. CYCLIC QUADRILATERALS 54
(iii) In the given figure, AB is a chord of a circle with centre O and AB is produced to C such that BC = OB. Also, CO is joined and produced to meet the circle in D. If ∠ACD = 25 ◦, then ∠AOD = . (A) 50◦ (B) 75◦ (C) 90◦ (D) 100◦ (iv) In the given figure, BOC is a diameter of a circle with centre O. If ∠BCA = 30 ◦, then ∠CDA = . (A) 30◦ (B) 45◦ (C) 60◦ (D) 50◦ EXERCISE 12.5. CYCLIC QUADRILATERALS 55
CHAPTER 13 GEOMETRICAL CONSTRUCTIONS EXERCISE 13.1 BASIC CONSTRUCTIONS 13.1.1 Key Concepts i. A geometrical construction is the process of drawing geometrical figures using only two instruments –a graduated ruler and a compass. ii. The following are some basic constructions. a) Perpendicular bisector of a given line segment. b) Bisector of a given angle. c) Construction of 60°angle at the initial point of a given ray. 13.1.2 Additional Questions Objective Questions 1. [AS1] The angle that can be constructed using a ruler and compass only is _______. (A) 25◦ (B) 50◦ (C) 22.5◦ (D) 42.5◦ 2. [AS1] The angle that cannot be constructed using a ruler and compass only is ______. (A) 135◦ (B) 37.5◦ (C) 120◦ (D) 40◦ 3. [A(SA1)]22T12h◦e angle that cannot be constructed using a ruler and compass only is ____. (B) 15◦ (C) 52 1 ◦ (D) 32 1 ◦ 2 2 EXERCISE 13.1. BASIC CONSTRUCTIONS 56
4. [AS2] Is it possible to construct an angle of 35◦ using a ruler and compass only? (A) Yes (B) No (C)Cannot be said (D)None of these 5. [AS2] Is it possible to construct an angle of 67.5 ◦ using a ruler and compass only? (A) No (B) Yes (C)Cannot be said (D)None of these Very Short Answer Type Questions 6 [AS3] Fill in the blanks. (i) To construct geometrical figures, such as a line segment, an angle, a triangle, a quadrilateral etc., some basic are needed. (ii) A geometrical construction is the process of drawing a geometrical figure using only two instruments - an and a . (iii) In a triangle if two of the angles are equal then the triangle is known as . (iv) In a triangle, if all sides are equal then it is known as an triangle. (v) It is to construct an angle of 135o , using only a ruler and a compass. Short Answer Type Questions 7(i) [AS5] Construct the perpendicular bisector of a given line segment. (ii) [AS5] Construct the bisector of a particular angle. 8(i) [AS5] Draw an equilateral triangle of side 10 cm. (ii) [AS5] Construct an angle of 45◦ at the initial point of a given ray. Long Answer Type Questions 9 [AS5] Draw the perpendicular bisector of a given line segment AB of length 6 cm and write justification. EXERCISE 13.1. BASIC CONSTRUCTIONS 57
EXERCISE 13.2 CONSTRUCTION OF TRIANGLES (SPECIAL CASES) 13.2.1 Key Concepts The following are a few special cases of constructions of triangle i. To construct a triangle, given its base, a base angle and the sum of the other two sides. ii. To construct a triangle given its base, a base angle and the difference of the other two sides. iii. To construct a triangle, given its perimeter and its two base angles. iv. To construct a circle segment given a chord and an angle. 13.2.2 Additional Questions Objective Questions 1. [AS2] The construction of a △ABC in which AB = 6 cm, and ∠A = 45◦ is possible when (BC + AC) is . (A) 7 cm (B) 5.8 cm (C) 5 cm (D) 4.9 cm 2. [AS2] The construction of a ABC in which AB = 7 cm and∠A = 75◦ is possible when (BC − AC) is equal to . (A) 7.5 cm (B) 7 cm (C)8 cm (D)6. 5 cm 3. [AS2] Is it possible to construct a ABC in which BC = 5 cm, ∠B = 120◦ and ∠C = 60◦? (A) Yes (B) No (C)Cannot be said (D)None of these 4. [AS2] Is it possible to construct a ABC in which ∠A = 60◦, ∠B = 70◦ and ∠C = 60◦ ? (A) Yes (B) No (C)Cannot be said (D)None of these EXERCISE 13.2. CONSTRUCTION OF TRIANGLES (SPECIAL CASES) 58
5. [AS2] Is it possible to construct a triangle whose sides measure 7 cm, 5 cm and 12 cm? (A) Yes (B) No (C)Cannot be said (D)None of these Long Answer Type Questions 6 [AS5] Construct a XYZ given YZ = 5 cm, ∠Y = 60◦ and XY + XZ = 9 cm. 7 [AS5] Construct a △ ABC in which BC = 6 cm, ∠B = 30◦ and AB − AC = 2.4 cm. 8 [AS5] Construct △ ABC in which BC = 4 cm, ∠B = 75◦ and ∠C = 60◦. 9 [AS5] Construct a segment of a circle with a chord of length 5 cm and containing an angle of 60◦. EXERCISE 13.2. CONSTRUCTION OF TRIANGLES (SPECIAL CASES) 59
CHAPTER 14 PROBABILITY EXERCISE 14.1 PROBABILITY/USES OF PROBABILITY IN REAL LIFE 14.1.1 Key Concepts i. The words most likely, equal chance, no chance are used to make judgements about the chance of a particular occurrence. ii. Probability is nothing but common sense reduced to calculation. iii. In many situations we make some statements and use our past experience and logic to take decision. iv. We take decisions by guessing the future happening, i.e. whether an event occurs or not. v. We measure the chance of occurrence or non–occurrence of some events by measuring numerically. This kind of measurement helps us to take decision in a more systematic manner. vi. We study probability to figure out the chance of something happening. vii. In a random experiment, all outcomes have equal chance of occurring. viii. As the number of trials increase, the probability of equally likely outcomes comes very close to each other. ix. The probability of an event P (A) = No. o f f avourable outcomes . No. o f total possible outcomes x. The probability of event always lies between 0 and 1 (0 and 1 inclusive). 14.1.2 Additional Questions Objective Questions 1. [AS1] In a cricket match a batsman hits a boundary 4 times out of the 32 balls he plays. In a given ball, the probability that he does not hit a boundary is _______. (A) 1 (B) 7 8 8 (C) 1 (D) 6 7 7 EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 60
2. [AS1] There are 600 electric bulbs in a box out of which 20 bulbs are defective. If one bulb is chosen at random from the box, the probability that the chosen bulb is defective is _____. (A) 1 (B) 1 19 20 (C) 1 (D) 29 30 30 3. [AS1] A coin is tossed 100 times with the following outcomes: head 43 times and tail 57 times. The probability of getting a head is _______. (A) 43 (B) 57 57 43 (C) 43 (D) 7 100 50 4. [AS1] In 50 tosses of a coin, tail appears 32 times. If a coin is tossed at random, the probability of getting a head is _______. (A) 1 (B) 1 32 18 (C) 16 (D) 9 25 25 5. [AS1] In 50 throws of a die, the outcomes were noted as given in the following table. Outcomes 1234 5 6 Number of times 8 9 6 7 12 8 A die is thrown at random. The probability of getting an even number is ______. (A) 12 (B) 3 25 50 (C) 1 (D) 1 8 2 Very Short Answer Type Questions 6 [AS4] Choose the correct answer. (i) The probability that an ordinary year has 53 Mondays is ______. (A) 2 (B) 1 7 7 (C) 7 (D) 7 52 53 EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 61
(ii) One card is drawn at random from a well–shuffled deck of 52 cards. The probability of getting a 6 is _____. (A) 3 (B) 1 26 52 (C) 1 (D)None of these 13 (iii) One of the following that cannot be the probability of an event is ____. (A) 1 (B) 0.3 3 (C) 33% (D) 7 6 (iv) The probability that two friends have different birthdays is ____. (A) 1 (B) 2 365 365 (C) 364 (D) 363 365 365 (v) In a lottery, there are 8 prizes and 16 blanks. The probability of getting a prize is ____. (A) 1 (B) 1 2 3 (C) 2 (D)None of these 3 Short Answer Type Questions 7(i) [AS1] If P(A) = 1 , P(B) = 1 , P(C) = 3 and P(S um) = 5 then find the value of P(D). 6 6 6 6 (ii) [AS4] The following table gives the number of students of different age groups. Age group Boys Girls Under 4 36 20 4 –7 12 36 7 –10 24 30 10 –12 60 36 12 –14 70 40 EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 62
a) Find the probability of number of students of age above 12 years. b) Find the probability of number of students of age below 7 years. c) Find the probability of number of boys above 10 years. 8(i) [AS4] In a cricket match a batsman hits boundary 6 times out of the 30 balls he plays. Find the probability that he does not hit a boundary. (ii) [AS4] The table given shows the ages of 75 teachers in a school. Age (in years) 20 –29 30 –39 40 –49 50 –59 27 37 8 Number of teachers 3 A teacher from this school is chosen at random. What is the probability that the chosen teacher is a) 40 or more than 40 years old? b) of an age lying between 30 – 39 years (including both)? c) 20 years or more old? d) above 60 years of age? Hint: Here 20 – 29 means equal to or greater than 20 and less than or equal to 29. 9(i) [AS4] 12 packets of salt, each marked 2 kg, actually contained the following weights (in kg) of salt: 1.950, 2.020, 2.060, 1.980, 2.030, 1.970, 2.040, 1.990, 1.985, 2.025, 2.000, 1.980 Out of these packets, one packet is chosen at random. What is the probability that the chosen packet contains more than 2 kg of salt? EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 63
(ii) [AS4] 1500 families with 2 children each were selected randomly and the following data were recorded. Number of girls in a family 210 Number of families 102 675 723 Out of these families, one family is selected at random. What is the probability that the selected family has a) 2 girls? b) 1 girl? c) no girl? 10(i) [AS4] A glass jar contains 6 red, 5 green, 3 yellow and 8 blue marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a a) red marble? b) green marble? c) blue marble? d) yellow marble? (ii) [AS5] In a particular section of Class IX, 40 students were asked about the months of their birth and the following graph was prepared for the data so obtained. Find the probability that a student of the class was born in August. 11(i) [AS1] A coin is tossed 200 times and it is found that head comes up 114 times and tail 86 times. If a coin is tossed at random, what is the probability of getting a) a head? b) a tail? EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 64
(ii) [AS1] Three coins are tossed simultaneously 150 times and it is found that 3 tails appeared 24times, 2 tails appeared 45 times, 1 tail appeared 72 times and no tail appeared 9 times. If the three coins are tossed simultaneously at random find the probability of getting a) 3 tails. b) 2 tails. c) 1 tail. d) 0 tail. 12(i) [AS5] Which kind of marble is most likely to be pulled out of this bag? Find the probability of picking that marble. (ii) [AS5] This eight–sided spinner is spun; calculate the probability that it lands on the following shapes. a) A circle b) A star c) A shape with less than 12 sides EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 65
13(i) [AS5] If a shape is picked from the container at random what is the probability of the following shapes being picked? a) Any shape apart from triangle. b) Any shape with more than one side. (ii) [AS5] Sameer has six tins of soup. The labels have fallen off. Here are the labels and tins. Sameer chooses a tin. a) What is the probability that it is a tin of Pea Soup? b) What is the probability that it is a tin of Chicken Soup? EXERCISE 14.1. PROBABILITY/USES OF PROBABILITY IN REAL LIFE 66
CHAPTER 15 PROOFS IN MATHEMATICS EXERCISE 15.1 MATHEMATICAL STATEMENTS 15.1.1 Key Concepts i. The sentences that can be judged on some criteria, regardless of the process, whether they are true or false are called statements. ii. Mathematical statements are of a distinct nature from general statements. They cannot be proved or justified by getting evidence but they can be disproved by finding a counter example. iii. Making mathematical statements through observing patterns and thinking of the rules that may define such patterns. iv. A hypothesis is a statement of idea which gives an explanation to a sense of observation. v. A process which can establish the truth of a mathematical statement based purely on logical arguments is called a mathematical proof. 15.1.2 Additional Questions Objective Questions 1. [AS2] The statement “There are 8 days in a week” is . (A) Always true (B) Sometimes true (C) Never true (D)None of these 2. [AS2] The statement “dogs can fly” is . (A) Never true (B) Sometimes true (C)Always true (D)None of these EXERCISE 15.1. MATHEMATICAL STATEMENTS 67
3. [AS2] “ For any real number x, x2 0 ” is . (A) Sometimes true (B) Always true (C)Never true (D)None of these 4. [AS2] Let a and b be real numbers such that ab 0. The true statement among the following is____. (A) Both a and b must be zero. (B) Either a or b must be non–zero. (C)Both a and b must be non–zero. (D)None of these 5. [AS2] “The product of two even integers is an even integer” is . (A) Always false (B) Sometimes true (C)Always true (D)None of these Very Short Answer Type Questions 6 [AS3] State true or false. (i) A process which cannot establish the truth of a mathematical statement based purely on logical arguments is called a mathematical proof. [] [AS3] Fill in the blanks. . (ii) The statements which are assumed to be true without proof are called (iii) A mathematical statement, the truth of which has been established or proved is called a . EXERCISE 15.1. MATHEMATICAL STATEMENTS 68
[AS2] Choose the correct answer. (iv) “Human beings are mortal–Ajay is a human being”. Based on these two statements, we can conclude that Ajay is . (A) Animal (B)Mortal (C) Immortal (D)None of these (v) “The sum of an even number and an odd number is an even number” is . (A) Always true (B) Sometimes true (C)Never true (D)None of these EXERCISE 15.1. MATHEMATICAL STATEMENTS 69
EXERCISE 15.2 REASONING IN MATHEMATICS 15.2.1 Key Concepts i. The prime logical method in proving a mathematical statement is deductive reasoning. ii. A proof is made up of a successive sequence of mathematical statements. iii. Beginning with what is given (hypothesis of the theorem) and arriving at the conclusion by means of a chain of logical steps is mostly followed to prove theorems. iv. The proof in which, we start with the assumption contrary to the conclusion and arrive at a contradiction to the hypothesis is another way that we establish that the original conclusion is true. This is another type of deductive reasoning. v. The logical tool used in the establishment of the truth of an un–ambiguous statement is called deductive reasoning. vi. The reasoning which is based on examining a variety of cases or sets of data and discovering pattern and forming conclusions is called inductive reasoning. 15.2.2 Additional Questions Objective Questions 1. [AS2] “Martians have red tongues–Gulag is a Martian”, based on these statements, we can conclude that Gulag has a tongue. (A) Black (B) Red (C)Blue (D)Green 2. [AS2] If it rains for more than four hours on a particular day, the gutters will have to be cleaned on the next day. It has rained for 6 hours today, then the gutters are to be tomorrow. (A) Changed (B) Removed (C)Cleaned (D)Kept as it is EXERCISE 15.2. REASONING IN MATHEMATICS 70
3. [AS2] Humans are mammals. All mammals are vertebrates. Based on these two statements, we can conclude that humans are . (A) Animals (B) Birds (C)Insects (D)Vertebrates 4. [AS1] Given that y = − 6x + 5 and x = 3, the value of y is ______. (A) 13 (B) –13 (C) 23 (D)– 23 5. [AS1] Given that √√ is . P is irrational for all primes ‘P’, and that 19423 is a prime number then 19423 (B) Complex (A) Rational (C) Irrational (D)None of these Short Answer Type Questions 6(i) [AS2] All men are mortal. Socrates is a man. Write the conclusion from these statements. (ii) [AS2] Everyone who eats carrots is a quarter back. John eats carrots. Write the conclusion from these statements. EXERCISE 15.2. REASONING IN MATHEMATICS 71
EXERCISE 15.3 THEOREMS, CONJECTURES AND AXIOMS 15.3.1 Key Concepts i. Axioms are statements which are assumed to be true without proof. ii. A conjecture is a statement we believe is true based on our mathematical intuition, but which we have yet to prove. iii. A mathematical statement, the truth of which has been established or proved, is called a theorem. 15.3.2 Additional Questions Objective Questions 1. [AS1] Every integer greater than 4 can be expressed as a sum of two odd primes. (A) Odd (B) Even (C) Prime (D)None of these 2. [AS1] For the triangular numbers T1, T2, T3, . . . . . if T1 + T2 = 4 ; T2 + T3 = 9 and T3 + T4 = 16 then T4 + T5 = . (A) 25 (B) 20 (C) 36 (D) 18 3. [AS1] Every prime number greater than 3 is of the form where K is some integer. (A)6K + 1 (B) 6K + 5 (C)6K + 3 (D)Either (A) or (B) 4. [AS1] A line parallel to side BC of a triangle ABC, intersects AB and AC at D and E respectively. Then AD = . DE (A) AB (B) EC AC AE (C) AE (D)None of these EC EXERCISE 15.3. THEOREMS, CONJECTURES AND AXIOMS 72
5. [AS3] Every even number greater than 4 can be expressed as sum of two primes. This conjecture is stated by _________ . (A) Ramanujan (B) Goldbach (C) Euler (D) Euclid Long Answer Type Questions 6 [AS2] In a polynomial p(x) = x 2 + x+41 for different values of x, find p(x). Can you conclude that p(x) is prime for all values? For x = 41, find p(x). Is it prime or composite? EXERCISE 15.3. THEOREMS, CONJECTURES AND AXIOMS 73
EXERCISE 15.4 WHAT IS A MATHEMATICAL PROOF? 15.4.1 Key Concepts i. The logical tool used in the establishment of the truth of an un–ambiguous statement is called deductive reasoning. ii. Beginning with the given hypothesis of the theorem and arriving at the conclusion by means of a chain of logical steps is mostly followed to prove theorems. iii. A proof is made up of a successive sequence of mathematical statements. 15.4.2 Additional Questions Objective Questions 1. [AS3] The sum of the interior angles of a triangle is . (A) 1 right angle (B) 1 right angle 2 (C) 2 right angles (D)None of these 2. [AS3] The product of two odd numbers is . (A) An odd number (B) An even number (C)A prime number (D)None of these 3. [AS3] The smallest of all the line segments drawn onto a line segment from a point which is not on it, is to the given line. (A) Intersecting (B) Parallel (C) Perpendicular (D)None of these EXERCISE 15.4. WHAT IS A MATHEMATICAL PROOF? 74
4. [AS4] A counter example for “a quadrilateral with all angles equal is a square” is . (A) Rectangle (B) Rhombus (C) Parallelogram (D)None of these 5. [AS4] A counter example for “2n2+ 11 is a prime for all whole numbers n” is . (A) 5 (B) 8 (C) 9 (D) 11 Long Answer Type Questions 6 [AS2] (a) Prove that the sum of two odd numbers is even. (b) Prove that the product of any two consecutive even natural numbers is divisible by 4. EXERCISE 15.4. WHAT IS A MATHEMATICAL PROOF? 75
—— Project Based Questions —— (i) Construct a triangle △ ABC in which AB = 8 cm ; BC = 6 cm and CA = 10 cm and measure the angles of ∆ ABC. What do you observe? Can you construct another triangle with hypotenuse 10 cm and one of the other two sides of the triangle as 8 cm. How many such triangles can be constructed? (ii) Construct a parallelogram gm ABCD. Mark two points P and Q on one of the diagonals BD such that P and Q are equidistant from the vertices D and B respectively. Join these two points with the other two vertices of the parallelogram to form a quadrilateral APCQ and four triangles △ APD ; △ AQB ; △ CQB and △ CPD. What can you say about these triangles? Are they congru- ent? If so which triangles are congruent ? What can you conclude about the quadrilateral APC Q? (iii) Consider a paragraph from an English daily news paper which consists of atleast 100 words. Write that paragraph in your note book and note down the number of letters in each word. Now prepare a frequency table for the above data and then draw a bar graph showing this data. Also find the mean of the data. (iv) Selvi’s house has an overhead tank which is in the shape of a cylinder. This overhead tank has to be filled by pumping up the water from a sump (an underground tank), which is in the shape of a cuboid. The sump has dimensions 1.57 m ×1.44 m×0.95 m. The overhead tank has a diameter 60 cm and height 95 cm. Can you help her in finding the height of the water left in the sump after the overhead tank has been completely filled with water from the sump which had been full. Compare the capacity of the tank with that of the sump. Also help her in finding the time taken for the overhead tank to be filled if the water is flowing through a circular tube of diameter 5 cm at the rate of 20 cm/ sec . PROJECT BASED QUESTIONS 76
(v) In countries like USA and CANADA, temperature is measured in Fahrenheit, where as in coun- tries like India, it is measured in Celsius. Here is a relation between Fahrenheit (F) and Celsius (C) by which we can convert the given temperature from one unit to another. F= 9 C + 32 5 Prepare a graph of the above relation by considering it as a linear equation in two variables using x – axis for Celsius and y – axis for Fahrenheit. From the graph identify: a) The temperature in Fahrenheit if the temperature in Celsius is 30◦C. b) The temperature in Celsius if the temperature in Fahrenheit is 95◦ F. c) If the temperature is 0◦ C , what is the temperature in Fahrenheit and if the temperature is 0◦ F , what is the temperature in Celsius? d) Is there a temperature which is numerically equal in both Fahrenheit and Celsius? If yes, find it. PROJECT BASED QUESTIONS 77
Additional AS Based Practice Questions Q1 [AS4] If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 + c2. Q2 [AS5] In the given figure, ABCD and AEFG are two parallelograms. If∠C = 55º, determine ∠ F. Q3 [AS4] The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is: A) A rhombus B) A rectangle C) A square D) Any parallelogram Q4 [AS5] Refer to the figure in Q16, and draw a rough figure for the following. A sphere of diameter 7 cm is dropped in a right circular cylinder vessel partly filled with water. The diameter of the cylindrical vessel is 14 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel? Q5 [AS3] Geometrical construction is the process of drawing a geometrical figure using only two instruments . Q19 [AS3] In a triangle, if all sides are equal, then it is known as an _________________________ triangle. Q6 [AS4] If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form A) A square B) a rhombus C) a rectangle D) any other parallelogram ADDITIONAL AS BASED PRACTICE QUESTIONS 78
Q7 [AS5] The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If ∠ DAC = 32º and ∠ AOB is 70º, then ∠ DBC is equal to A) 24º B) 86º C) 38º D) 32º Q8 [AS5] A right circular cylinder just enclosed a sphere of radius r as shown in given figure. Find the surface area of the sphere, curved surface area of the cylinder and also their ratio. ADDITIONAL AS BASED PRACTICE QUESTIONS 79
Q9 [AS3] The probability of an event P(A) = ________________________________. Q10 [AS3] The probability of an event always lies between ________ and ________. Q11 [AS2] Can the experimental probability of an event be greater than 1? Justify your answer. Q12 [AS2] Bulbs are packed in cartons each containing 40 bulbs. Seven hundred cartons were examined for defective bulbs and the results are given in the following table: No. of defective bulbs 0 1 2 3 4 5 6 More than 6 Frequency 400 180 48 41 18 8 3 2 One carton was selected at random. What is the probability that it has i) No defective bulb? ii) Defective bulbs from 2 to 6? iii) Defective bulbs less than 4? Q13 [AS4] Restate the following statements with appropriate conditions, so that they become true statements. i) If you divide a number by two, you will always get half of that number. ii) If a quadrilateral has all its sides equal, then it is a square. Q14 [AS4] Take any three consecutive odd numbers and find their product. Write the conjecture for the pattern observed. ADDITIONAL AS BASED PRACTICE QUESTIONS 80
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