MATHEMATICS
S T A N D A R D
MINIMUM LEVEL
LEARNING MATERIAL
for
CLASS – X
2022 – 23
Prepared by
M. S. KUMARSWAMY, TGT(MATHS)
M. Sc. Gold Medallist (Elect.), B. Ed.
Kendriya Vidyalaya gachibowli
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MATHEMATICS
S T A N D A R D
MINIMUM LEVEL
LEARNING MATERIAL
for
CLASS – X
Prepared by
M. S. KUMARSWAMY, TGT(MATHS)
M. Sc. Gold Medallist (Elect.), B. Ed.
Kendriya Vidyalaya gachibowli
DEDICATED TO MY FATHER LATE SHRI. M. S. MALLAYYA
INDEX OF MINIMUM LEVEL LEARNING STUDY MATERIAL
CLASS X : MATHEMATICS (STANDARD)
- 1 Real Numbers – Concepts with Important Questions 1 – S. NO. CHAPTER/CONTENT PAGE NO.
- 2 Triangles Theorem – Proof and Exercise 6.2 4 –
- 3 Circles – Concepts with Important Questions 6 –
- 4 Probability – Concepts with Important Questions 14 –
- 5 Statistics – Concepts with Important Questions 20 –
- 6 Coordinate Geometry– Concepts with Important Questions 26 –
- 7 Polynomials – Concepts with Important Questions 31 –
- 8 Quadratic Equations – Important Questions 35 –
- 9 Arithmetic Progression – Concepts with Important Questions 40 –
- 10 Linear Equation in two variables – Important Questions 45 –
- 11 Triangles – 1 mark Important Questions 50 –
- 12 Trigonometry Chapter 08 & 09 – Important Questions 54 –
- 13 Areas related to Circles – Important Questions 62 –
- 14 Surface Areas and Volumes – Important Questions 70 –
CHAPTER – 1
REAL NUMBERS
The Fundamental Theorem of Arithmetic Every composite number can be expressed ( factorised ) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. The prime factorisation of a natural number is unique, except for the order of its factors. Property of HCF and LCM of two positive integers ‘a’ and ‘b’: HCF a b ( , ) LCM a b ( , ) a b ( , ) ( , ) a b LCM a b HCF a b
( ,^ )^
a b HCF a b LCM a b
PRIME FACTORISATION METHOD TO FIND HCF AND LCM
HCF(a, b) = Product of the smallest power of each common prime factor in the numbers. LCM(a, b) = Product of the greatest power of each prime factor, involved in the numbers. IMPORTANT QUESTIONS Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two numbers Solution: 510 = 2 x 3 x 5 x 17 92 = 2 x 2 x 23 = 2^2 x 23 HCF = 2 LCM = 2^2 x 3 x 5 x 17 x 23 = 23460 Product of two numbers = 510 x 92 = 46920 HCF x LCM = 2 x 23460 = 46920 Hence, product of two numbers = HCF × LCM Questions for practice
1. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method. 2. Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM. 3. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers: (i) 26 and 91 (ii) 336 and 54 4. Find the LCM and HCF of the following integers by applying the prime factorisation method: (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25 5. Explain why 3 × 5 × 7 + 7 is a composite number. 6. Can the number 6n, n being a natural number, end with the digit 5? Give reasons. 7. Can the number 4n, n being a natural number, end with the digit 0? Give reasons. 8. Given that HCF (306, 657) = 9, find LCM (306, 657). 9. If two positive integers a and b are written as a = x^3 y^2 and b = xy^3 ; x, y are prime numbers, then find the HCF (a, b). 10. If two positive integers p and q can be expressed as p = ab^2 and q = a^3 b; a, b being prime numbers, then find the LCM (p, q). 11. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case. 12. Find the largest number which divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
MCQ QUESTIONS (1 mark)
1. If HCF and LCM of two numbers are 4 and 9696, then the product of the two numbers is: (a) 9696 (b) 24242 (c) 38784 (d) 4848 2. (^) 2 3 (^5) is : (a) a rational number (b) a natural number (c) a integer number (d) an irrational number 3. If 3 2 6 9
x
^
, the value of x is: (a) 12 (b) 9 (c) 8 (d) 6
4. If (m)n^ =32 where m and n are positive integers, then the value of (n)mn^ is: (a) 32 (b) 25 (c) 5^10 (d) 5^25 5. The number (^) 0.57 in the p q form q 0 is (a)
(b)
(c)
(d)
6. The number 0.57 in the p q form q 0 is (a)
(b)
(c)
(d)
7. If p is a prime number and p divides k^2 , then p divides: (a) 2k^2 (b) k (c) 3k (d) none of these 8. The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is (a) 13 (b) 65 (c) 875 (d) 17 50 9. If two positive integers a and b are written as a = x^3 y^2 and b = xy^3 ; x , y are prime numbers, then HCF ( a , b ) is (a) xy (b) xy^2 (c) x^3 y^3 (d) x^2 y^2 10. If two positive integers p and q can be expressed as p = ab^2 and q = a^3 b ; a , b being prime numbers, then LCM ( p , q ) is (a) ab (b) a^2 b^2 (c) a^3 b^2 (d) a^3 b^3 11. The product of a non-zero rational and an irrational number is (a) always irrational (b) always rational (c) rational or irrational (d) one 12. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is (a) 10 (b) 100 (c) 504 (d) 2520
CHAPTER – 6
TRIANGLES
IMPORTANT THEOREMS
BASIC PROPORTIONALITY THEOREM OR THALES THEOREM
If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
CHAPTER – 10
CIRCLES
THEOREMS
- The tangent to a circle is perpendicular to the radius through the point of contact.
- The lengths of tangents drawn from an external point to a circle are equal. IMPORTANT QUESTIONS
1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle 2. In the below figure, if TP and TQ are the two tangents to a circle with centre O so that POQ = 110°, then find PTQ. 3. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then find POA 4. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle. 5. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 6. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC 7. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre. 8. Prove that the parallelogram circumscribing a circle is a rhombus. 9. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
10. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. 11. XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that AOB = 90°. 12. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC. 13. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ = 2 OPQ. 14. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. 15. Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral. 16. If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that DBC = 120°, prove that BC + BD = BO, i.e., BO = 2BC. 17. Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord. 18. Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A. 19. From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the the perimeter of the triangle PCD. 20. In a right triangle ABC in which B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC. 21. If d 1 , d 2 (d 2 > d 1 ) be the diameters of two concentric circles and c be the length of a chord of a circle which is tangent to the other circle, prove that d 2^2 c^2 d 12 22. If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by 2 a b c r
23. Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.
31. If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that AQ =
(BC + CA + AB)
32. If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + FA. 33. Let s denote the semi-perimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, prove that BD = s – b. 34. From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD. 35. If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in below figure. Prove that BAT = ACB 36. In below figure, tangents PQ and PR are drawn to a circle such that RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the RQS. 37. AB is a diameter and AC is a chord of a circle with centre O such that BAC = 30°. The tangent at C intersects extended AB at a point D. Prove that BC = BD. 38. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc. 39. A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ. 40. In below figure, the common tangent, AB and CD to two circles with centres O and O' intersect at E. Prove that the points O, E, O' are collinear. 41. The tangent at a point C of a circle and a diameter AB when extended intersect at P. If PCA =110º , find CBA
42. In below figure. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB. 43. Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord. 44. Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A. 45. If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle. 46. Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ. 47. In a right triangle ABC in which B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC. 48. A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ΔABC. MCQ QUESTIONS (1 mark) 1. If angle between two radii of a circle is 130º, the angle between the tangents at the ends of the radii is : (a) 90º (b) 50º (c) 70º (d) 40º 2. If radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is (a) 3 cm (b) 6 cm (c) 9 cm (d) 1 cm 3. In the below figure, the pair of tangents AP and AQ drawn from an external point A to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. Then the radius of the circle is (a) 10 cm (b) 7.5 cm (c) 5 cm (d) 2.5 cm
7. From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is (a) 60 cm^2 (b) 65 cm^2 (c) 30 cm^2 (d) 32.5 cm^2 8. At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is (a) 4 cm (b) 5 cm (c) 6 cm (d) 8 cm 9. In below figure, AT is a tangent to the circle with centre O such that OT = 4 cm and OTA = 30°. Then AT is equal to (a) 4 cm (b) 2 cm (c) 2 3 cm (d) 4 3 cm 10. In below figure, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then POQ is equal to (a) 100° (b) 80° (c) 90° (d) 75° 11. In below figure, if PA and PB are tangents to the circle with centre O such that APB = 50°, then OAB is equal to (a) 25° (b) 30° (c) 40° (d) 50°
12. If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to (a)
cm (b) 6 cm (c) 3 cm (d) 3 3 cm
13. In below figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and BQR = 70°, then AQB is equal to (a) 20° (b) 40° (c) 35° (d) 45°
IMPORTANT QUESTIONS
Two dice are thrown together. Find the probability that the sum of the numbers on the top of the dice is (i) 9 (ii) 10 Solution: Here, total number of outcomes, n(s) = 36 (i) Let A be the event of getting the sum of the numbers on the top of the dice is 9 then we have n(A) = 4 i.e. (3, 6), (4, 5), (5, 4), (6, 3) Therefore, Probability of getting the sum of the numbers on the top of the dice is 9,
n A P A n S
P A
(ii) Let B be the event of getting the sum of the numbers on the top of the dice is 10 then we have n(B) = 3 i.e. (4, 6), (5, 5), (6, 4) Therefore, Probability of getting the sum of the numbers on the top of the dice is 10,
n B P B n S
P B
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) red colour ace card (ii) a face card or a spade card (iii) a black face card Solution: Here, total number of outcomes, n(s) = 52 (i) Let A be the event of getting red colour ace card and we know that the number of red ace card is 2 then we have, n(A) = 2 Therefore, Probability of getting red colour ace card,
n A P A n S
P A
(ii) Let B be the event of getting a face card or a spade card and we know that there are 12 face cards, 13 spade cards and 3 face cards are spade then we have, n(B) = 12 + 13 – 3 = 22 Therefore, Probability of getting a face card or a spade card,
n B P B n S
P B
(ii) Let B be the event of getting a black face card and we know that there are 6 face cards are black then we have, n(C) = 6 Therefore, Probability of getting a black face card,
n C P C n S
P C
Questions for Practice
1. Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is (i) 6 (ii) 12 (iii) 7 2. A die is thrown twice. What is the probability that (i) 5 will not come up either time? (ii) 5 will come up at least once? 3. A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that (i) She will buy it? (ii) She will not buy it?
4. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face card (iii) a red face card (iv) the jack of hearts (v) a spade (vi) the queen of diamonds 5. Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. (i) What is the probability that the card is the queen? (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen? 6. 12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one. 7. A piggy bank contains hundred 50p coins, fifty Re 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a 50 p coin? (ii) will not be a Rs 5 coin? 8. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red? (ii) white? (iii) not green? 9. (i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective? (ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective? 10. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5. 11. A carton consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Jimmy, a trader, will only accept the shirts which are good, but Sujatha, another trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton. What is the probability that (i) it is acceptable to Jimmy? (ii) it is acceptable to Sujatha? 12. Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days? 13. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag. 14. A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find x. 15. A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is
. Find the number of blue marbles in the jar. MCQ QUESTIONS (1 mark) 1. Which of the following can be the probability of an event? (a) – 0.04 (b) 1.004 (c)
(d)
2. A card is selected at random from a well shuffled deck of 52 playing cards. The probability of its being a face card is (a)
(b)
(c)
(d)