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ENG10

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Abhishek Mishra
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Mental Nathd DIRECTORATEIOREDUCATIONIGovtofiN!IChof{DELHI MENTAL MATHS CLASS X 2021-22 DIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI UDIT PRAKASH RAI, IAS Directorate of Edducatior Director, Education & Sports Gavt, of NGT Room No. 12. Near Vidhan Sabha, Delni-110054 swat aud Ph: 011-23890172 Mob. 8709803999 Ne PS/be/aea4/9 E-mail: diresudnir i Ob. 06/8/01 MESSAGE “The book of nature is written in the language of Mathematics” Very artfully said by Galileo. Mental Maths Project is one of the landmark projects of Directorate of Education which aims at developing skill of solving mathematical problems orally in our students. It helps make them prompt, confident and mentally agile. A confidence dawns on them that every problem has a right solution This skill definitely helps them in later years, when faced with real-life situations. Undoubtedly it contributes in holistic development of a child's personality. I would like to take this opportunity to congratulate subject experts and Mental Maths core committee members involved in developing quality study material for students, Their incessant and tireless efforts have worked tremendously in shaping future of so many students. Heartiest congratulations to team Mental Maths. UDIT PRAKASH RAI One, me Dr. Afshan Yasmin Additional Director of Education Project Direciar- Mental Maths: Regional Director of Education (CUNDINocth (West) Directorate of Education ‘Old Secretariat New Delhi ~ 110054 aaAG Ref, No, eB: tod | MESSAGE As it is said "catch them young", Mental Maths Project aims at making students adept in doing calculations mentally right from the young age, thereby it sharpen their intellect and make them quick-witted individuals, Mental Maths Project being rum under the acgis of Directorate of Education, NCT of Delhi, is an appreciable step in the direction of inculeating the habit of doing calculations without pen and paper in our young students. The tricks and techniques involved develop their mathematical and logical skills. Itis a matter of pride that yet another edition of Mental Maths question bank is ready to be in the hands of our students. I hope it will provide enough content for practice while clearing their doubis and arousing their curiosity to dive deep into the sea of numbers. Qur school children are fortunate to receive quality study material developed very diligently by subject experts and Mental Maths Core Committee Members. I wish it proves to be a stepping stone in the journey of our future mathematicians. Ke Dr. Afshan Yasmin ADDITIONAL DIRECTOR OF EDUCATION: PROJECT DIRECTOR MENTAL MATHS PROJECT SOURCE OF INSPIRATION & GUIDANCE SH. UDIT PRAKASH RAI DIRECTOR (EDUCATION) CHIEF ADVISOR & LODESTAR DR. AFSHAN YASMIN, ADDITIONAL DIRECTOR OF EDUCATION RDE (CENTRAL/ND, NORTH, WEST}, PROJECT DIRECTOR, MENTAL MATHS PROJECT SUBJECT EXPERTS & CONTENT DEVELOPMENT TEAM (Class-X) Dr. SUNIL AGGARWAL, LECTURER COORDINATOR, MENTAL MATHS PROJECT Govt. S. Co-ed Sr, Sec. School, Possangipur, B-1 Janak Puri (School ID - 1618003) Ms. SAMPDA GULATI, VICE PRINCIPAL CO-COORDINATOR, MENTAL MATHS PROJECT GSKV, C-Block, No.1, Janak Puri (School ID - 1618017) Mr. Kumar Gaurav, TGT Govt. Co-ed Sr. Sec. School, F-Block, Vikaspuri (School ID - 1618016) Ms. Vinti Singla, TGT GSKV, Samalka (School ID - 1821022) Mr. Deepak Sharma, TGT RPVV, Gandhi Nagar (School ID - 1003261) Mr. Narayan Dutt Masiwal, LECTURER Govt. S. Co-ed Sr, Sec. School, Possangipur, B-1 Janak Puri (School ID - 1618003) Ms. Ritu Tiwari, TGT RPVV, Surajmal Vihar (School ID - 1001104) COVER PAGE DESIGN & TECHNICAL SUPPORT Mr.Prem Kumar Sharma, PGT GBSSS, No. 1, C-Block, Janak Puri (School ID - 1618006) Mr. Naresh Kumar, TGT GSBV, No. 2, C-Block, Janak Puri (School ID - 1618005) STAE LEVEL MENTAL TH QUIZ COMPETITION RESULT 2020-2021 LEVEL-3 REGION WEST (1st POSITION) NAME OF NAME OF FATHER'S SCHOOL SCHOOL S.No. | CLASS STUDENT ID D.OB. GUIDE STUDENT NAME NAME CODE TEACHER ARVIND RPVV, A-6, 1 x | PRatyusHRa | KUMAR 2oieoo0e792 | 27012003 | FascHIm | 1617009 | SANJERY KR TIWARI VIHAR || DILIP KUMAR GEEV RAMESH | 1c) | SURENDER 2 x ROHAN DAS DAS 20160069717 11.12.2004 NAGAR 1516002 SINGH ASHUTOSH ROHIT 7 GBSSS, SHIV HOSHIY AR 3 x CHAUBEY CHAUBEY 20190314251 24.09.2005 VIHAR. 1618267 SINGH REGION EAST (1ST RUNNER UP) NAME OF a NAME OF FATHER'S SCHOOL SCHOOL s.Ne. | cLass | NAME OF Ane STUDENTID | D.OB. cir Cope | CODE ASTEEK DEO KANT i GSBV RADHEY a ROHITASH 1 x NARAYAN NARAYAN 20170206937 25.03.2006 SHY AM PARE 1003152 PAREEK. MUKESH 2 x ADITI PRIYA KUMAR 20160212671 24.03.2005 RPVV, IP EXTN 1002399 Sara SHRIVASTAVA 3 x KUNTI GUPTA HITESH GUPTA 20160272433 18.01.2006 RPVV, IP EXTN 1002399 AMIT KUMAR REGION CENTRAL (ND RUNNER UP) NAME OF NAME OF FATHER'S SCHOOL SCHOOL S.No. | CLASS STUDENT ID D.OB. GUIDE STUDENT NAME NAME CODE TEACHER PRAMOD VUAY KUMAR | ANARNT RPVV KISHAN 1 x ero Aen 20160039052 | 18.11.2005 vee 1208092 KUMAR GUPTA RON RARESH " 5 BACHAN 2 x SUNDRIYAL MOHAN 20160310910 03.01.2006 GBSS, BURARI 1207116 KUMAR BRAJ R 3 x AKSHIT BHUSHAN 20110265846 05.08 2005 eee re 1207113 ie _ TIWARI REGION NORTH (4TH POSITION) NAME OF NAME OF FATHER'S scHoo. | scHooL S.No. | CLASS STUDENT ID D.OB. GUIDE STUDENT NAME NAME CODE TEACHER MUSKAN NARESH i ag GSEV SEC-16 I x BHARDWAJ UMAR 20180120172 29.10.2004 ROHINI 1413070 YOGEETA RPVV BT 5 BLOCK 2 x MDIMRAN | MDMgHBOOB | 2060017680 | 28022006 | BLOCK | ianotz4 | RITU VERMA BAGH z =a GSBB E BLE = 3 x PIYUSH JHA BHAWAN JHA 20180210279 30.09.2005 MANGOL PURI 1412002 P.C.GOYAL REGION SOUTH GTH POSITION) NAME OF NAME OF FATHER'S scHoo. | scHooL S.No. | CLASS STUDENTID | D.0B. GUIDE STUDENT NAME NAME CODE | pe kcneR NARESH 29. 7 5 RPVV SEC-10 a 1 x PRIYANSHU KUMAR JHA. 20160022404 22.07.2005 DWARKA. 1821137 ANIU KUMAR TRIBHUV AN a 2 x SANJEET SINGH 20160170569 24.03 2006 Eni. 1719022 rae KUSHWAHA TANISH SANIIV 5 GBSSS, RAT HANERAT : * | suexnawar | suzxmawar | 70190320563 | 04082005 | wagarnext | 182003 | sHaRMa TENTATIVE SCHEDULE OF MENTAL MATHS QUIZ COMPETITIONS FOR THE YEAR 2021-22 DIRECTORATE OF EDUCATION Practice to students from Question Bank School level Quiz Competition Cluster level Quiz Competition Zonal level Quiz Competition District level Quiz Competition Regional level Quiz Competition State level Quiz Competition 01.07.2021 to 30.09.2021 21.10.2021 to 30.10.2021 09.11.2021 to 15.11.2021 25.11.2021 to 30.11.2021 06.12.2021 to 10.12.2021 18.01.2022 to 22.01.2022 07.02.2022 to 18.02.2022 INDEX | REAL NUMBERS |: | ewes Yt ms Yt | A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 15 pe _ = a = 12. STATISTICS AND PROBABILITY 91 CHAPTER -1 REAL NUMBERS POINTS TO REMEMBER « Euclid’s Division Lemma: Given two positive integers a and b(a = b), there exists unique integers q and r satisfying a= bq +r, where O <1 n or aftern places ifn > m. Let x= q bea rationalnumber, such that the prime factorization of q is not of the form 2™ x 5", where m and n are non-negative integers, then x has non terminating repeating decimal expansion. If a and b are two positive numbers, then HCF (a, b) x LCM (a, b)=a x b. QUESTIONS. . Find the digit at the unit place of the number 77°19 x 32019 | . Find the digit at the unit place of the number7??? x 357?, . Find the digit at the unit place of the number12345°7® + 678817545, . Find the digit at the unit place of the number 44*" x 66" x 99" + 1114. Page | 1 . Find the digit at the unit place of the number 41x 9% x 43 x oF x 4% x 98. x 499 x 9100, . What is the number of zeros in the usual form of the following : i) 200 + 1000 + 80000 + 12500000 ii) 200 x 5000 x 80000 x 12500000 . Find the number of zeroes in 2” x 5* x 4° x 108 x 64 x 15 . What is the remainder when 11/4 + 2227 + 33% is divided by 10 ? . What is the difference between the largest two digit prime number and the least 3 digit prime number? ). For p" = (a x 5)" to end with the digit 0 what will be the value of a? If [= 0.142857142857... then find the value of [z+ in decimal expansion. iA . What will be the smallest rational number by which + 5 should be multiplied so 116 . After how many places the decimal expansion of — that its decimal] expansion terminates after one place of decimal? Be | will terminate? |. In(v3 — V2 — V1) (V3 + V2 4 V7) = ay3 + by2 + cV7, then find the 15. 16. 17. value of (a+ b+ c). . Find the value of |(V2— V3)? + | (/2+ V3)? . Find the value of (V7 — V3)". (V7 +.V5)" . Find the value of (x + y) using factor tree. &) HEC Page | 2 18. Find the value of 2xy using factor tree. 20. If 7560 = 23 x 3" x q x 7, then what is the value of n+ q. 21. What is the smallest prime factor of 11 x 13 x 19 x 23+ 23? 22. If (S x 52 x 3B x ™) is a terminating decimal, then what are the least possible values of n and B? 23. Two equilateral triangles have sides of lengths 51 cm and 85 cm respectively. Find the greatest length of tape that can measure both of them exactly. 24. Two numbers are im the ratio 17: 13. If their HCF is 15, then what is the sum of the numbers? 25. The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is divided by 2, the quotient is 33, find the other number. 26. Find the HCF of (2125 — 4 and (215 — 1) 27. The LCM of two numbers is 1890 and their HCF is 30. If one of them is 270, then find the other number. 28. The HCF of two numbers is 11 and their LCM is 616. If one of the numbers is 88, find the other. 29. Given that HCF (2730, 4400) = 110 and LCM (2730, 4400) = 273 k. Find the value of k. Page | 3 30. In a seminar the number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required where in each room, the same number of participants are to be seated and all of them being of the same subject. . Six bells commence tolling together. They toll at the intervals of 2, 4, 6,8, 10 and 12 seconds respectively. In 30 minutes how many times do they toll together? 32. Three numbers are in the ratio of 3: 4: 5 and their LCM is 2400. Find their CF. 33. If the adjacent sides ‘a’ and ‘b’ of a rectangle are in the ratio 3 : 5 such that CF (a, b) = 11, then find the perimeter of the rectangle. 34. What is the HCF of smallest 3 digit number obtained using three different digits and greatest two digit composite number? 35. The length of a rectangle is LCM (a, b) and breadth of the rectangle is HCF (a, b), then what is its area? 36. If HCF (a, b) = LCM (a, b), then what is the relation between a and b? 37. If HCF (0, p)= 2 and LCM (20, p) = 60, then what is the value of p? 38. How much is (V180 ot vi08) greater than(V5 ts V3)? 39. A number which when divided by a divisor leaves remainder 23 .Whentwice the number is divided by the same divisor, the remainder is 11, what is the divisor? 40. If ) = 7, then what is the value of x? 734734734 73473473473 rr 41. The LCM of two numbers is 45 times their HCF. If one of the numbers is 125 and sum of HCF and LCM is 1150, then what is the other number? 42. If a is an odd number, b is not divisible by 3 and LCM of a and b is p, then what is the LCM of 3a and 2b? 43. What is the smallest number by which 4/81 should be multiplied so as to get a rational number? 44. What is the total number of factors of an even prime number? Page | 4 45. If HCF (144,180) = 13m —3, then what is the value of m? 46. If r is the remainder when (Sm +1) (Sm+3) Gm) is divided by 5, then what are the possible values of r, if r is a natural number? 47. Find the least positive integer which is divisible by first five natural numbers. 48. The HCF (a, b) = 29, where a, b > 29 and LCM= 4147. What is the value of |a — b|? 49. When a = b q+ 1, then what are the possible factors of (a —r)? 50. If (—1)" + (—1)™ = 0, then what is the least positive value of n? Page |5 - a co] Al al wo & ST) pl o ab sq.units 0,714285714285... a=b 3 6 10 3 places 5(v5 + V3) oe 35 2v2 3 4 225 15 op 900 [9 2 8 23 Page | 6 CHAPTER 2 POLYNOMIALS POINTS TO REMEMBER « Algebraic expressions in which power of the variable of each term is a whole number are called polynomials ie. 2x +3, 5t? + 7t+8 Degree of the polynomial in one variable: The highest power of the variable of any term in a polynomial is its degree. Following are the forms of various degree polynomials. Examples Name of the polynomial Degree 5 Constant polynomial 2x43 Linear polynomial 3x34 2x" +5x4+7 Cubic polynomial 0 1 5x? + 1x+8 Quadratic polynomial 2 3 4 ths Bets Tt + 4tt 5 Biquadratic polynomial oO Zero polynomial Not defined If for a polynomial p(x), p (a) =9, then a is called a zero of the polynomial P&S). A polynomial of degree ‘n’ has atmost ‘n’ zeroes. If wand B are the zeroes of the polynomial ax? + bx + ¢,a + O then cocfficientefx —b Sum of the zeroes (a + B) = — cocfficentofx? a Product of zeroes (a. B) = ~ cleat = £ If w and f are the zeroes of the quadratic polynomial p(x) then p= k[ x? — (sum of zeroes)x + product of zeroes] ie. p(x) =k[ x? — (a + B)x + af], where k is any real number. If a, B and y are the zeroes of the cubic polynomial f(x)=ax* + bx? + ex+d then b atBpty=—-> Page | 7 € ap + By + ya= d aBy = — = * Ifa, and y are the zeroes of the cubic polynomial p(x) = ax* + bx? + ex + dthen — p(x) =k[ x*— (a+ B+ y)x* + (aB + By + ya)x— ay], where k is any realnumber Geometrically zeroes of the polynomial f(x) are x — coordinates of the point where the graph y = f(x) intersects x — axis. Coordinates of vertex ‘A’ of graph of y = ax? + bx +c is (- opie (2.4 2). where D=b?— 4ac eg: if y= x? — 2x +4, then coordinates of its vertex are ae = land When a>0 The division algorithm states that given any polynomial p(x) and any non- zero polynomial g(x), deg p(x)= deg g(x) there are polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x), where r(x) =0 or degree r(x) < degree g(x). If (sta) is a factor of polynomials x? + px + q and x? + mx + m then a= Page | 8 QUESTIONS 1. What will be the number of zeroes of the polynomial whose graph is parallel to y axis? At how many points the graph of the quadratic polynomial intersect x-axis? Find the sum of the zeroes of the quadratic polynomial 3x*+ 15x+ 7. Find the product of the zeroes of the quadratic polynomial 2x?-7. Find the sum of the zeroes of the polynomial x*- 64. Find a quadratic polynomial whose sum and product of the zeroes are - 3 and 2 respectively. Find the quadratic polynomial whose zeroes are -9 and— ; Form a quadratic polynomial, if product and sum of its zeroes are -3 and 0, Find zeroes also. 9, Find a quadratic polynomial whose zeroes are (6 + 2) and (5- 12). 10. If a and are the zeroes of the quadraticpolynomial 2x? + 5x + 1, then find the value of «+ B+ of. . Find the cubic polynomial whose zeroes are O, 5 and -5. , If a and B are the zeroes of the quadratic polynomial 2x? - 5x + 8, then find the value of a? +B’. . If a and B are the zeroes of the quadratic polynomial 2x? - 5x +8, then find thevaeot® +6 B.« 14. Find the quadratic polynomial whose sum of the zeroes is 0 and one zero is 5. 15. Form a quadratic polynomial whose one of the zeroes is 2 + V5 and sum of the zeroes is 4. 16, If1 is a zero of the polynomial ax: + bx + c, then find the value af "5 17. If sum of the zeroes of (a + 1)x? + Qa+3)x+ Gat 4) is -1, then find a. 18. If the sum of the zeroes of the quadratic polynomial 3x* - kx + 6 is 3, then find k. 19, Form a quadratic polynomial whose zeroes are reciprocal of the zeroes of ax?+bx+c. Page |9 20. If o and B are the zeroes of x*-3x+ 2, form a quadratic polynomial whose zeroes are (a + B)? and (a —§)". 21. If one zero of the polynomial 2? + 132 — p is reciprocal of the other, then find the value of p. . For what value of p, (— 4) is a zero of the polynomial x* — 2x — (7p + 3). . If 1 is a zero of the polynomial ax* —3(a —1) x —1, then find the value of a | If sum and product of the zeroes of ax* — 5x + c is equal to 10 each, find a and c. If a and f are the zeroes of thepolynomial x'—3x + p and o — B =1, then what is the value of p? If x+2 is a factor of x? + ax + 2b and a+b =4, thenfind the value of a and b. . Which is the common factor in x’ +x — 12 and x? + 9x +20? If a polynomial of degree 5 is divided by a quadratic polynomial, then find the degree of the remainder polynomial. 29, Find the quotient when x’ — 9x + 20 is divided by x — 5. 30. Ifx+ a is a common factor of the polynomials x? — 3x — 10 and x*— 8x + 15, then find a. . What is the common factor in x? — 1,x* — 1 and (x—1)?? . Find the common zero of x* + 2x + 1, x*—1 and x’ +1. . Find the quotient when f(x) =16x' + 13 x* + x — 2 is divided by g&x)= @x+)'. . If x* + x* — ax + b is completely divisible by x* — x, then find the values of a and b. . For what value of x both the polynomials 2x* + 8x +8 and x? — 3x —10 becomes zero? What should be added to the polynomial x’ — 8x + 6 so that 4 becomes a zero of the polynomial? . What should be subtracted from x* — 3x? + 6x — 15 so that it is completely divisible by (x — 3)? Page | 10 38, If the sum of zeroes of 5x? + (p+q+1)x+ pqr is O, then what is the value of p> +q3 +13? 39. If one of the zeroes of the polynomial x’ - 9x + (7K + 4) is double of the other, then find the value of k. 40. If two zeroes of the polynomial x* — 4x? — 3x + 12 are V3 and —\3, then find its third zero. 41. Find the zeroes of the polynomial x’ — Sx* — 16x + 80, if its two zeroes are equal in magnitude but opposite in sign. 42. If a, B and + are the zeroes of the polynomial x’ + bx? + cx + d, then find the value of 1 a 11 By 43. If o, B and y are the zeroes of the polynomial x*— px? + qx — r, then find the value of 1 1 1 ap * py * ya 44. Find the coordinates of the vertex of the figure obtained by drawing the graph of 2x? — 4x+5. 45. Find the degree of the polynomial p(x) representing the given graph. ¥ Page | 11 46, Find the number of zeroes of the polynomial p(x) represented in the given graph. eg 47. Find the number of zeroes of the polynomial p(x) represented in the given graph. 48. Find the number of zeroes of the polynomial p(x) represented in the given graph. Page | 12 49. Find the number of zeroes of the polynomial p(x) represented in the given graph. 50. Find a cubic polynomial whose zeroes are 0, 4 and -4. Page | 13 1 a=3,b=1 Atmost 2 points x+4 —5 1 or O or not defined 7 2 x-4 0 k(x? + 3x+ 2) k(9x? + 82x + 9) k(5x? — 3) x? — 10x + 23 =2 x} — 25x +3) greater than or equal to3 Page | 14 CHAPTER 3 LINEAR EQUATIONS IN TWO VARIABLES POINTS TO REMEMBER © General form of pair of linear equations is ayx+byy +4 =0 a,x + bzy +c, = 0 Where a4, by, ay, bz, cy, 2 are real numbers such that (a,)? + (by)? # 0 and (az)? + (by)? #0 * In above equations if “+ ns then i. The pair of linear equations is consistent. ii The pair of linear equations represent intersecting lines. iii. The pair of linear equations has a unique solution. by « In above equations f2=2 = 5 then a bk i The pair of linear equations is dependent and consistent. ii The pair of linear equations represents coincident lines. iii. The pair of lear equations have infinitely many solutions. ; b, * Inabove equations if 2 = 2 + 4, then ar 2 ez i. The pair of linear equations is inconsistent. ii The pair of linear equations represents parallel lines. iii. The pair of linear equations has no sohution. * Area of atriangle= + x base x height « If area of triangle is zero, then the points are collinear and vice versa. Special case: When coefficient of x and y are interchanged in two equations ie. cod ax+ by =«, bx+ay=d, thenx+y=!*4andx—y= <4 Page | 15 QUESTIONS . For what value of k, (6, k) is a solution of the equation 3 x + y = 22. . Form a linear equation whose solution is (—2, 3). If one equation of a pair of dependent linear equation is —5 x + 7 y =2, then what may be the second equation? . Ifx=a,y=b, is a solution of the equations x + y = 8, x —y = 2, then what are the values of a and b? . Find the point of intersection of y= 2 and2x+3y=5. What are the values of x and y when x + 2y =9, x— 2y=1? . What is the least value of p for which x and y have same values in 2x + py =8? Find the point of intersection of the lines represented by 3x—2y= 6 and y axis. Ifx=4 and y = 3p — 1 is a solution of x + y= 6 then what is the value of p and y? 10. Find the area of triangle formed by the lines x = y, y = 4 and y -axis. 11. Find the area of the triangle farmed by the co-ordinate axes and the lines L xty=6 XV, i + pr 12. If 2x + 3y = 0 and 4x —3y =, then find the value of x+y. 13. Solve for x and y: V5x + y7y = 0, 3x — y2y =0 14. Solve for x and y 2*+ 5¥=33 15. Ifax — Vy = 0, Vbx — Vay = 0, then find the value of xy. 16. For what value of k the given system of equation has no solution? kx+2y-1=0,5x-3y+2=0 17. For what value of k the given system of equations have infinitely many solutions? Qx-3y=7, (+2 x-O@kH)y=30k-1) 18. For what value of m the given system of equations has unique solution? ox+3y-5=0, mx-6y=8 Page | 16 19. For what value of p the given system of equations represents coincident lines? Sx-y+8=0 6x —py =-16 . For what value of c the given system of equations represent parallel lines? 3x+ ey =2 2x+S5y+1=0 .1f2* = gt oY = 3*°* then find the value of x and y. . Solve for x and y:31x+ 29 y=89, 29x+3ly=91 If 47x + 3ly = 63 and 31x + 47y = 15, then find the value of x — y. . What is the value of x+ y for the following pair of linear equations 152x —378 y =—74 —378x + 152 y =—604 . Find x —y for the following: 217% + 131y =913 131x + 21 Ty = 827 . For what value of x and y: vx+ VV = Z\x- V¥= 1 . Solve for x and y: Byer = 2,%-Fa4. a1 . Salve for x andy: x—y=0.9, ep 3 4 6 8 . Solve for x and y: = = 0,—_— (5) y+) (a5) (y+a) 30. From the following figure find the values of x and y. 6 axe ( x+5)om Page | 17 31. In triangle ABC, 2A=x,/2B=y,4€=y + 20°. If y—x=50°, what type of triangle is ABC ? 32. Megha has only one rupee and two rupee coins with her. If the total number of coins that she has is SO and the amount of money with her is 275, then find the number of Ti and 22 coins. . The sum of digits of a two digit number is 9. If 27 is added to it, then digits of the number get reversed, then find the number. . At what point the linear equation 2x + 3y = -7 intersect x axis? At what point the linear equation 3x -7y = 5 intersect y axis? . If linear equation 3x + 2y =5 intersects x and y axis, then find the sum of intercepts on x and y axis. . Find the area of the triangle if its vertices are G, 2), 6, 2) and (7, 2). . Find the area of the triangle if its vertices are (3, 5), G,-7) and (3, 0) . Find the area of the triangle if its vertices are (0, 0), @, 2) and , 0). . Find the area of the triangle if its vertices are (2, 2), (4, 4) and (6, 2). . Sum of two numbers is 35 and their difference is 13. Find the numbers. . If one number is twice the other and their sum is 117, then find the numbers. . The sum of two numbers is 20 and their product is 75. Find the sum of their reciprocals. . The sum of two numbers is 20 and their product is 19. Find their difference. . The sum of numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes > Find the fraction. . Half the perimeter of a garden whose length is 12m more than its width is 60m. Find the length of the garden. . Cost of 3 books and 4 pens together is ¥257 and the cost of 4 books and 3 pens together is 7324. Find the total cost of two books and two pens. . A father is three times as old as his son. After 12 years his age will be twice as that of the age of his son. Find their present ages. . Two numbers are in the ratio 3. 4. If 8 is added to each of the number, the ratio becomes 4:5. Find the numbers. Page |18 50. The monthly income of A and B are in the ratio 9:7 and their monthly expenditure are in the ratio 4:3. If each of them saves T1600 per month, find the monthly income of each. 51. Meena went to the bank to withdraw £2000. She asks the cashier to give her %50 and %100 note only. She receives 25 nates in all, find how many notes of 50 and [100 did she received? 52. The angles of a triangle are x, y and 40°. The difference between the angles x and y is 30°. Find x andy. 33. Find x and y where the angles of a cyclic quadrilateral ABCD are 2A= (6xt 10)°, 2B=Gx)°,20=(xt+y)°, 2D = Gy—10)°. 54. When we draw the graph of the lines x = —2 and y =3, what will be the coordinates of the vertices the figure formed by the co-ordinate axes and above lines? 55. (a)The larger the two supplementary angles exceeds the smaller by 18°. Find the angles. (b) Find the area of the triangle formed by three lines y = x, x= a andy =b. 56. Find the value of x and y in the following figure: x+3y Page | 19 57. From the figure find the area of the triangle formed by the pair of linear equations: x—y+2=0, 4x—y—4=0 andy axis, 58. From the figure find the area of the shaded triangle. Page | 20 59, Find the ratio of the area of the triangle formed by given lines with x - axis and y- axis in the given figure. 60. In the given figure find the coordinates of points where —x + 3 y = 6 meets x axis and y axis. Page | 21 ANSWERS k=4 Right angled triangle xty=1 35,25 -10x + l4y = 4 or any other suitable answer 36 x=5,y=3 Ge 166 36 years, 12 years 2432 A= @14400, B= 11200 10 notes of %0 , 15 notes of M100 x=85", y=30 ©.0),@,3),(2,3),C2.0) a)99°, 81° b)e(a—b)* x= unit, y= 4 units 6 sq. units $ sq. units 3:2 €6,0), 0.2) Page | 22 CHAPTER — 4 QUADRATIC EQUATIONS POINTS TO REMEMBER Quadratic Equation : An equation of degree 2 is called a quadratic equation. The general form of a quadratic equation in one variable x is ax” + bx + c = O where a,band c are realnumbers anda # 0. Methods for solving quadratic equations are Factorization method ‘+ Completing the square method “ Quadratic formula Discriminant: For the quadratic equation ax? + bx +c=0,a +0 D=b? — 4ac is called discriminant. Nature of roots If D =0, Real and equal roots If D > O, Real and distinct roots If D <0, No real roots If D = 0, then real roots a, of the quadratic equation ax? + bx + c = Oare -bivd bb given by a = HAP and B= evn — (a #0) la Relationship between roots and coefficients : If a and B are two roots of ax? + bx + c = 0, then Sum of the roots = a+ B = ae Product of the roots= a = < Quadratic Equation : x? — (sum of roots)x + product of roots = 0 Page | 23 QUESTIONS 1. x Find the discriminant of the quadratic equation 3x’ + 8x+2=0. . Find the value(s) of k for which the quadratic equation 2x” — kx + k = Ohas equal roots. Find the value of k for which x = 2 is the root of the equation kx? + 2x —3 = 0. |. Find the value of x in the equation (2x — 4)? = 64 . Find the value of k for which roots of the equation 3x? — 10x + k= 0 are reciprocal of each other. . Find the value(s) of z if 2? +35 2,240. . If the value of discriminant for equation 3x? + rx + 4 = 0 is 1. Find the value(s) of r. . Find the value of x which satisfies the equation + iS .. The roots of the equation x? — 12x + p = 0 are in the ratio 1:2, find the value of p. . If sum and product of roots of equation kx? + 6x + 4k = O are equal, then find the value of k. Form a quadratic equation whose roots are S+ V3 and5 — V3. What is the coefficient of x in the equation whose roots are 5 and -1? If x = 1 is a common root of the equations ax’+ax+3=0 and x? +x+b =0, then find the value of ab. . If the sum of the roots of the equation x? — x = A(2x — 1) is zero, then find the value of A. Find the quadratic equation whose one root is 2 and sum of the roots is zero Form a quadratic equation whose one root is 2 + vi. Solve for x if x = /6+V6+V6 . Solve for x if x = 72 —y 72—v7Z—-. Page | 24 19. If a and B are roots of the equation x* — 3x + 2 = 0,then find value of 1 8 20. If one root is negative of the other then what is the coefficient of the middle @ 2+ iy 48 wate term of the quadratic equation. 21. For what value of ‘p’ the equation 9x? — 42x + p = 0 will be in the form of a perfect square? 22, Find the value of x which satisfy the equation x + : =—-4,x+ 0. 23. If the roots of 2x? + (4m + 1)x + 2(2m— 1) = Oare reciprocals of each other, find the value of m. 24. What is the ratio of the product and sum of the roots of the equation 5x? — 18x +12 = 0? 25. What is the sum of reciprocal of the roots of the equation x? — 7x + 12 = 0? 26. If roots of the equation ax? + bx + ¢ = O areS + V5, then find the value of arc. 27. Find the value of p for which the product of roots of the quadratic equation px? + 6x + 4p = 0 is equal to sum of the roots. 28. If quadratic equation x? — 5x — 6 = 0 is expressed as (x+a) (x+b) =O, then find the value of a and b. 29, Find the positive value of k for which the equations x” + kx + 64 = 0 and x? — 8x + k = 0 will both have real roots. 30. If 4 is the root of the equation x? + px — 4 = 0 and the quadratic equation x? + px+ k = 0 has equal roots, find the value of k. 31. If one root of the equation ax’ + bx + ¢ = 0 is three times the other, then find b?: ac. If one root of the equation kx’ — 14x + 8 = 0 is six times the other, then find the value of K. If a and B are roots of the equation x — 4x + 3 = 0, then find value of ofp? + apt 34. Find the values of k for which x? + 5kx — 16 = 0 has no real roots. Page | 25 35. If one root of the quadratic equation 2x? + kx + 4 = 0 is 2, find the other root. 36. Find the quadratic equation whose roots are twice the roots of equation 3x’ 7x+4=0. 66, find the 37. If the sum of first n natural numbers is given by S = =— = value of n. 38. If a and B are roots of the equation x? — 3x— 2 = 0, find a quadratic F 1 1 equation whose roots are —— and——. tatp Rpt 39. Find the value of k, if the difference of roots of quadratic equation x? — 5x4 (3k-—3) = 0is11. do. Solve for x, ifx = —*;— pre re 41. The sum of a natural number and its reciprocal ist. Find the number. 42. Divide 29 into two parts such that their product is 198. 43. The sum of two numbers is 15. If the sum of their reciprocals is = , then find ‘the number. 44. Find two consecutive even integers whose squares have the sum 340. 45. A two digit number is 4 times the sum of digits and twice the product of digits. Find the number. 46. If the sum of first n even natural numbers is 420, then find n. 47. If an integer is added to its square, the sum is 90, then find the integer. 48. What is the condition to be satisfied for which quadratic equations ax? + 2bx+ ¢= 0 and bx? — 2Vacx + b = 0 have equal roots? 49. Solve for x :12abx? — (9a7 — 8b*)x — 6ab = 0 50. What must be the value of k so as to solve the quadratic equation 9x? + ax +k= 0 by method of completing the square? Page | 26 3 90 a € —Sey<8 gok<5 1 3x — 14x416=0 11 16x? - 9x+1=0 Page | 27 CHAPTER -—5 ARITHMETIC PROGRESSION POINTS TO REMEMBER “+ General A P with n terms is a,a+d,a+ 2d,..., a+ (-1) d where a is the first term and d is the common difference. n'term or last term of an A P is a, ort, orl= a+ (n—1)d ™ term or general term of an A P ayort, =a+(r—1)d Sum of n terms of an A P= $, = >[2a+ (n—1)d] Or S,= <[a+]] z x" term from the end of an A P= (m-3r+ aye term from the beginning =at(n—r+1-1d =at(n-rd x" term of an A P from the end is T= a, — (1 — 1)d where a, is the last term. If a,b and c are in AP then 2b = a+c If sum of first three terms in A P is given then we take the first three terms as a-—da,at+d If sum of first four terms in A P is given then we take the first four terms as a—3da-dat+da+3d If sum of first five terms in A P is given then we take the first three terms as a—2d,a—d.a,a+d,a+ 2d To find a, when Sy given : ay = 8, —Sy_1 Common difference d = ay4i — a Page | 28 QUESTIONS 2 af V3, ¥1Z, 27, V48 are in A.P, then find next three terms. . What is the next term of the A-P V7,N28,V63.,... . For what value of k, the terms 2k, k+10 and 3k+2 are in AP . Ifa, = 5— 141n, then find the common difference. . Find the value of x if 8x + 9,6x — 2,2x— 7 are three consecutive terms of an AP. . Find the common difference of an AP where n™ term is 2n+5. '. If sum of first n terms of an AP is S, = an? + bn, find its common difference. 8. Ifthe sum of first n terms of an A-P is Sn? + Zn, then find its 2"? term. 9. Find the common difference of the A.P 11-p1-2p Pop’ oP 10. What is the n™ term of the AP 11+mi+2m mom’ m 14 eee are in A-P, then find the value of x M+R’ x43" N45 11L.If 12. If x, 13, y, 3 are in A.P, then find the value of x and y. 13. What is the sum of first n natural numbers? 14. What is the sum of first n odd natural numbers? 15. What is the sum of first n even natural numbers? 16. If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then find the value of k. 17. If 18, a, b, -3 are in A-P, then find a+b. 18. If 4, ay, a3, ay, 28 are in A.P, then find ay. 19. If the sum of n terms of an A.P is 3n? — m and common difference is 6, then find its first term. 20. If the numbers a, b, c, d, e form an A.P, then find the value of a-4b+6c-4d+e. Page | 29 21. If three consecutive terms of an A-P are a-d, a, at+d. Their sum is 33 and d is 5, then find the terms. 22. If a,b, care in A.P, then find the value of (a +2b —c) @b+ c-a) (cta-b). 23. If the sides of a right triangle are in A.P, then find the ratio of its sides. 24. If sum of three consecutive terms of an A.P is 24, then find its middle term. '5. If sum of five consecutive terms of an A.P is 115, then find its third term. 26. Angles of a triangle are in A.P. If smallest angle is 40°, then find the largest angle. 27. The angles of a quadrilateral are in A-P whose common difference is10°, find the angles. 28. Find the sum of n terms of the series (4-4)+(#-2)+(2-2 te DL 29. Find a, b and c such that the following numbers are in A.P: a, 7, b, 23, ¢ 30. Divide 16 into 4 parts which are in A-P such that the product of extremes is one less than the sum of means. 31. Which term of the AP. 72, 63, 54... is 0? 32. If the first three terms of an A.P are b, c and 2b, then find the ratio of b and c 33. If 7 times the 7™term of an A.P is equal to 11 times its 11™ term, then find its 18"term. 34. Find the 20™ term from the end of the AP 3,8, 13... ;255. 35. How many two digit natural numbers are there, which when divided by 3 yield 1 as reminder? 36. If sum of m terms of an A.P is same as the sum of its n terms, then find the sum of its (m + n)™ term. 37. Find the sum of 1-6 + 2—7+4+3-—8+4 to 100 terms. 38. Which term of the A.P 52,48, 44, ... will be the first negative term? 39. From your pocket money you save 81 on day 1, = 2 on day 2, %3 on day 3 and so on. How much money will you save in the month of February 2024? 40. Find the sum of all 11 terms of an A.P whose middle most term is 30. Page | 30 41. If the p™term of an A-P is q and q™term is p, then find its n™term. 42. Tf the m™ term of an AP is 2 and n‘ term of an A-P is 3 then find its mn™ term. 43. Find the sum of first 20 odd natural numbers. 44. The 9® term of an A.P is 449 and 449% term is 9. Find which term is equal to o? 45, For the A.P:—3,—7,—11.,... Findazo — ayo. 46. The first and last term of an A-P are 5 and 45 respectively. Find the number of terms if sum of all the terms is 500. 47. If 8® term of an AP is zero, then what is the relation between 28 and 18 term? 48, If 15 term of an A.P exceeds its 10 term by 20, then find the common difference. 49. If 3" and 9% term of an A.P are 4 and -8 respectively, then which term of the AP is 0? 50. A man got a job with monthly salary of % 7000 with an annual increment of & 500. What will be his salary after 10 years? Page | 31 80° 75°, 85°, 95°, 105° a 7 (7-1) a=-l,b=15,c=31 1,;3,5,.7 9 Tamm m 1 18,8 n@td) 15" term F435 330 ptq-n 1 400 458" term. -40 611,16 20 dabc R45 4 8 S® term 23 % 12,000 Page | 32 CHAPTER - 6 TRIANGLES POINTS TO REMEMBER « Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional(in the same ratio) AL. AABC~APQR = 2A = ZP,2B = 2Q,2C = ZR& a a PQ QR PR * Criteria of Similarity: (a2) AAA (b) SSS (c) SAS « If a line is drawn parallel to one side of triangle to mtersect the other two sides in two distinct points, then the other two sides are divided in same ratio. A AD AE AB AC AB_ AC DB EC DB EC AD AE ar(\ABc) AB? Bc? ac? * HAABC~APQR = Japan) ~ af ~ on? PRe Page | 33 * In AABC, 2B = 90° then AC? = AB? + BC? (Pythagoras theorem) A Cc In AABC, “B = 90° and BD | AC then AABD ~ AACB ~ ABCD. If two triangles are similar then their perimeters, medians, altitudes and angle bisectors are in the same ratio. The areas of two similar triangles are in the ratio of squares of their corresponding sides, altitudes, medians, perimeter and angle bisectors. A N $ pene ease) _ Ao _ se + Eg I AABC~APQR = PET pgm) 7 Pe 7 Pa Page | 34 QUESTIONS 1. In fig. AABC~ADEF, find 2D. A & LY) (\ 8 c or 2. In fig, AACB~AECD, find ZABC, 4. Infig. find ¢M and ZY B D 33. AABC And ADCE are right angled triangles in which ]X1¥2—Ys) + 2s Va) + X31 —Yo)] Sq-units. If points A (&4,y4), B &2,y2) and C (x;,y3)are collinear, then the Area of triangle formed by these three points is 0 and vice versa. Page | 47 QUESTIONS 1. Find the distance between the points (a cas 55°, 0) and (0, a cos 35°) 2. Find the distance between the points (cos0, sin@) and ( sin0, cos@). 3. In the given figure find y, if P (5-3) and Q @,y) are the paints of trisection of the line segment joining A (7,-2) and B (1,-5). al7,-2) P(S,-3) Q(3,y) B(1,-5) . In the given figure, ABC is a triangle and D is the midpoint of BC. Find the co-ordinates of D. Day Cuz} 5. In figure as given in question no. 4, find the length of AD. 6. Find the distance between the points (a cosb, a sinb) from the origin. . Find the coordinates of the point, which divides the line joining the points A G, -6) and B (2, 7) in the ratio 2:3. . In the given figure, find the coordinates of A. 9, Find the distance of the point (0, 2) from the midpoint of the line segment joining (4, 10) and (2, 2). 10. Find the value of y, if P(x, y) divides the line segment joining points (3, 3) and (1, -2) im the ratio 2:3. Page | 48 11. Find the value of x, if P(x, y) divides the line segment joining points A (7, -5) and B (2, -1) in the ratio 4:1. 12. Find the value of k, if point (0, 4) is equidistant from the points (10, k) and «, 8) 13. Find the value of k, if x axis divides the line joining the paints (-4, -6) and (5, 2) in the ratiok: 1. 14. In the given figure, A (-1, 0), B (2, -3) and C @, 5) are the coordinates of AABC. If D is the mid-point of BC, then find the coordinates of O which divides AD in the ratio 2:1. A(-1.0) D 15. In the given figure, find the area of rhombus. D(-2,-1) Page | 49 16. Find the area of triangle whose coordinates are A (1, 5), B (O, -2) and Cc @, 6). A(1,5) CO. 6) BO, -2) 17. Find the value of p for which the points (-1, 3), (2, p), and G, -1) are collinear. 18. Find x and y, if O (0,0), A @, 2), B(x, y) and C (3, 0) form a rectangle OABC. 19. In the parallelogram ABCD, coordinates of A and C are (3, 2) and (a, b) respectively. If AC and BD intersects at O (0, 0), then find the values of ‘a’ and ‘b’. po Ca. b) B 20. Find the area of triangle ABC with A (1,-4) and mid points of sides through A being @,-1) and (,-1). 21. Find the value of y such that points A (5, y), BG, 2), C @, 2) and D @, 5) form a square ABCD. 22. In the given figure, find the coordinates of the points A and D, if BACD isa rhombus and the base BC of an equilateral AABC lies on y axis. Page | 50 23. In the given figure, O is the Centre of circle and A and B are any paints on circle, find y. B(5,7) 24. Q is the midpoint of the line segment PR where coordinates of P, Q and R are (, -2), (1, 3) and (x, 8) respectively. Find ‘x’. 25. Find coordinates of point P, if P and Q trisect the line segment joining the points (6, -3) and (-1, 3) 26. In figure, find the value of Area of (AABC): Area of (AABD). A(4,6) B(3,-2) D(4,0) (5,2) . Find the area of the triangle formed by joining the mid points of the sides of a triangle, whose vertices are (3, 2), 6, 4), and @, 6). Find the values of p and q, if the line segment joining the points (, -4) and 1, 2) is trisected at the point (p, -2) and G. 4) z 29. The line joining the points (2, 1) and (5, -8) is trisected at points P and Q. If point P lies on the line 2x —y + k =O, then find k. 30. Find the coordinates of vertex C, if length of one of the sides of an equilateral triangle is ‘a’ and base BC lies on x-axis with B at the origin. Find the coordinates of P, if the distance of the point P from the point (G, 4) is V10 units and abscissa of P is double of its ordinate. . If the area of the triangle ABC formed by AG, y), B @, 2) and C Q@, Lis6 square units, then find the value of x + y. Page | 51 If ¢ - 4) is the midpoint of the segment joining the points P 6, 5) and R (2, 3), then find the value of a . Find the value of x, if the distance of the point (0, x) from @G, 5) is 5 units. Find the area of triangle formed by (a, b +c), (b, c +a) and (c, a+b). If points (a, 0), (0, b) and (1, 1) are collinear, then find the value of (= + i). If the centroid of the triangle formed by the points (a, b), (b, c) and (Cc, a) is at the origin, then find the value of a° + b* + c?. . If the centroid of a triangle is (1, 4) and two of its vertices are G, -3) and C9, 7), then find the area of triangle. . If the centroid of the triangle formed by (7, x), (y, -6) and @, 10) is at G, 3), then find the value of x and y. Find the value of y, if the points A 6, y), BG, 5), C 0, 5) and D (1, 2) are the vertices of rectangle. Find the area of triangle formed by joining the points (0, 0), (0, 2) and (2, 0). . Find the coordinates of point P which lies on x axis and equidistant from (2, 5) and @Q, -3). Find the value of p + q, if A (p, q) is the midpoint of the line segment joining the points G, 3) and (2, 4). Find the coordinates of point p that lies on y axis and equidistant from (3,4) and (-2, 5) . The points (0, — 1), (2, 1), @, 3) and (—2, 1) are the vertices of a square. Find the sum of the length of all sides and diagonals. . Find the ratio in which the line joining the points A (-4, 4) and B @, 8) is divided by (1, 5). Find the value of p and q, if the midpoints of the line segment joining (3p, 4) and (2, 2q) is @, 6). Page | 52 48. The base BC of an equilateral AABC with side 24cm lies along the x-axis such that the midpoint of the base is at origin. Find the coordinates of B. y 49. The three vertices of a rhombus taken in order are (-2, -1), (3, 0) and (4, 5). Find the coordinates of the fourth vertex. 50. Find the value of ‘a’ for which the points (0, 0), (1, 1) and (2, a) will be collinear. Page | 53 ANSWERS a units 21 V2 units 1 square unit joa p=Landq=0 63 8 Tunits (a, 0) or (—a, 0) a units ©, 3), 2,1) = 15 a,-3)) AG, 10) “1B 5 units x=19 y=l1 O square unit x=3 1 k= 2 Sabe k=3 91.5 square units 2 =S,y=2 a K=Sy 24 square units y=2 2 4 square units p=l x=3,y=2 or G,2) a=-3 and b =-2 square units 12 square units B(v2Z+ 1) units y=s 1:3 A = (3V3, 0), D (—3v3,0) p=2,q=4 y=-lor7 (12,0) x=-4 (-1,4) (3,-4) a=2 Page | 54 CHAPTER — 8 INTRODUCTION TO TRIGONOMETRY POINTS TO REMEMBER _ Perpendicular d) sin8 = ‘Hypotenuse Perpendicular ___Base Hypotenuse 2) cos@ _Pependialr i 3) tan® Base cote _ Hypotenuse 4 o=90°-9 4) cosecO = pndicalas | ~ nd = 5) seco = Hypetenuse = sin 0 = sin(90° — 9) = Ge = 89 Base cos® = Bee ~ Perendiailar — tand 6) cotO Trigonometric ratios of complementary angles 1) sin@0"—6) = cos8 2) cos(90" —6) = sin® 3) tan(@0°-6) =cot@ 4) cot(90" —9) = tane 5) sec(90" —8) = caosecO 6)cosec(90" —8) = secd Trigonometric Identities 1) sin?@ + cos¥@=1 2)) sec?@ — tan?@=1 3) cosec?@ — cot?@=1 Angles of elevation and angles of depression: - Horizontal line Ancient Db Elevation is the angle up from the Depression Horizontal. 2) Depression is the angle down from the Angle of Horizontal. Elevation 3) Angle of Elevation = Angle of Depression Horizontal line Page | 55 QUESTIONS 1. IfA+B=90", then find the value of tanA tanB. Ifa+ B = 90 and seca= 2, then find cosec B. If 0 =45', then find the value of 2sinOcos0. Find the value of sin30°cos60° + cos30°sin60". If Scos 8 =3, then find the value of =e-=82 If tand = then find the value of zo Find the value of (cos0 + sin@)* + (cos — sin0)* Tf sinA = then find value of 3cosA —4cos'A. Find the value of (sec?@ — 1) (1 — cosec*@). . Find the value of 3tan?26" — 3cosec*64". . If tan9 + cotO= 2, then find the value of tan’@ + cot*®. If cos — cos(90' - 8) = 0, then find the value of 6. Evaluate tan5‘tan25‘tan45‘tan65"tan85" . Find the value of acute angle @ if sin (0 + 26°) = cos0. Find the value of sin*l* + sin*5° + sin*9" +. .. + sin?89° If a= 3sec*@ -1 and b = 3tan?@ + 2, then find the value of (a —b). . If sec — tan@ =k, then what is the value of sec + tan@. . Ifx = 15", then find the value of 4sin2xcos4xsin6x. If sinx + sin?x = 1, then find the value of cos*x (1 + cos*x), If 6x = secO and ‘= tan, find the value of O(x? =o z Pa If k -2 = sec? (1 + sinA) (1 — sinA), then find the value of k Evaluate Stan*A —Ssec*A + 1 If sinA — cosA =O, then find the value of (sin?) + (costA} . Find the value of sin*10° + sin?20° + sin?30°+ ... + sin?80° tan2A+cot'a If tana + cotA = 4, then find the value of 7 sas pocmAcok Find the value of cotA — cosect + cot?A + cosectA If 7sin?A + 3costA = 4 and 0* < A < 90°, then find the value of tanA. . If cos + secO= 2, find the value of cos + sec®*0. Page | 56 2 2 . If x= acos*@, y = bsin’®, then find the value of @ + (BF. Find the value of 3(sec'@ + tan'd), if sec’ © If cos8 = 3 then find the value of 2sec*6 + tan*6 + 1. . If 1 + tan’O + 4ktan*Osec*@ = sec’ + 2tan*Osec’O, then find the value of k. 2 2 33. Ifx =a (sin@ + cos6), y = b (sin@ - cos@), then find the value of (5) + ® . \* If xsin45° = ycosec30", then find the value of @) . . If cos x + cosy = 2, then find the value of sinx + siny. . What is the value of 50, if tan20 = cot30. sin6—sin?6 cosb-cos'8 . Find the angle of elevation of the sun at an instant when the length of the . Find the value of tan@ x shadow of a pole is V3 times its height. . A ladder was placed against a wall in such a way that it makes an angle of 30° with the ground. If its top is 10m above the ground, find the distance ‘between wall and foot of the ladder. . Two posts are ‘k’ meter apart and the height of the one is double that of the other. If from the middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then find the height Gn m) of shorter post. . If a tower of 6 meter height casts a shadow of 23 meter along the ground, then what is the angle of elevation of the sun at that time? 42. In the given figure, if BE = ED then find xy Page | 57 43. In right angle AABC, 7B= 90" and AC — AB=1, then find the value of cosA + cosB + cosC A 44. In the given figure, find the height ‘h’ p 7m 45. In the given figure, find the value of ‘p’. A 46. In the given figure, find AE D Page | 58 47. In the given figure, find AC + AD 48. In the given figure, find the height of the tower AB (in m). B 6cm © 75cm 49. In the given figure, find QS. 50. In the given figure, 7B =90", find the height of the tower AB (in m) i“ { c—20m —— D Page | 59 ANSWERS Page | 60 CHAPTER - 9 CIRCLES POINTS TO REMEMBER *® A tangent to a circle is a line that touches the circle at only one point. iP The tangent at any point of a circle is perpendicular to the radius through the point of contact. OA 1 PA. ie. 2OAP = 90°. There are exactly two tangents to a circle through a point lying outside the circle. PQ and PR are exactly two tangents from the external point P. The length of tangents drawn from an external point to a circle are equal PQ=PR Q R * Inthe above figure, the sum of opposite angles of a Quadrilateral OQPR is 180° Page | 61 QUESTIONS 1. The radii of two concentric circles are Scm and 3cm. AB is a diameter of the bigger circle and BD is tangent to the smaller circle touching it at D and the bigger circle at E. Find the length of AD. E 2. Find the value of Gp + 7), where p is the distance between two parallel tangents to a circle whase radius is 12.5cm. 3. In the given figure, find BP. TA <0) |. Find the radius of the circle passing through the vertices of a right angled triangle, when lengths of perpendicular sides are 6cm and 8cm. . In the given figure, PA and PB are tangents to the circles with Centre O such that 2APB = 50°, what is the value of zOAB. <{ Page | 62 6. In the given figure, AABC is right angled at B. Find the radius of circle, if AB =S5cm and BC = 12cm. c B Sem 7. In the given figure, find the value of OQ. Q 8. In the given figure, ~RPQ = 50", and O is the center of circle, then find 2BAC. Page | 63 9. In the given figure, if O is the Centre of circle, find the value of x. 10. In the given figure, find the value of (PR + OR) =3 11. In the given figure, find the value of (¢ACB+ 2CAQ). <<) S 12. In the given figure, find the length BC Page | 64 13. In the given figure, O is the Centre of circle, find the value of ZOAB. A 14. In the given figure, O is the Centre of circle with radius r. if OP = 21, find the value of 2OST. s 15. In the given figure, AB is the diameter of circle with Centre O and AT is tangent. Find the value of 2ATQ. B Page | 65 16. In the given figure, Find the value of zAOB,if 2ACB+ CBO = 120. A, B 17. In the given figure, OPQ = 40°, find the value of 2ROQ. alt é Q R 18. In the given figure, O is the Centre of circle with radius r, find the radius of circle. 29 ce Ty 23cm, Page | 66 19. In the given figure, find the radius of circle, if area of APQR = 189 sq.cm. Zi x 21. Find the perimeter ofAPQR, where PM = a cm, RN= b cm, QL = cm. PB Q N R 22. The tangent at a point ‘C’ of a circle and a diameter AB when extended intersect at ‘P’. If

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