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2 Way to Success SPECIAL GUIDE MATHS Production Team Mr. K. Nagarajan M.A., M.A., M.Phil., B.Ed Mrs. A. Leema Rose M.Sc., B.Ed. Mr. K. Dinesh M.Sc., M.Phil., P.G.D.C.A.,(Ph.D.,) For subject related clarifications Mail us: & Call us : Visit us: You can download free study materials from our website 1

3 Government Question Pattern Section Qu. Given To be Question Type No Questions written Marks Section A 1 15 Choose the best answer (1 mark questions) Section B 16 9 Mark questions Mark Compulsory question 1 Section C Mark questions Mark Compulsory question 1 5 Section D 46 Geometry Graph Blue Print Chap. No Chapter One marks Two marks Five marks 1 Sets and Functions 1 Sequences and series of Real Numbers 1 3 Algebra 3 4 Matrices Coordinate Geometry 6 Geometry Trigonometry 1 8 Mensuration 1 11 Statistics Probability Total Chap. No Chapter Ten marks 9 Practical Geometry 1 10 Graphs 1 For Single Copy by VPP : Send your address through SMS to For Bulk Orders: Please contact your district Co-ordinators from the list given in the last page of this book or Contact following phone Numbers , , Note : At most care is taken to fulfill your requirements without mistakes. If you find any mistakes, kindly inform us by to ways100@gmail.com. Corrections will be updated then and there in our website and that will be carried out in our further edition.

4 Dear students Greetings You have Way to Success Maths special guide now in your hands. This guide is not written like other usual guides. There is a huge difference between this guide and other guides. This guide will help the students understand the concepts clearly and make them to score very high marks. This guide gives you the exact answers for textbook questions, previous years Govt. Examination questions and additional questions. Tips to slow learners: Make notes while you study the lessons and revise the notes very often. Get rid of any unnecessary 'time wasting mechanisms' while studying. This includes TV, computer, mobile phones etc. Do not study till the last minute of your examination. Go through the prepared notes before your exams. Read the question paper very carefully and select the questions which you are going to answer. First of all you should attend Practical geometry & Graph, then followed by other sections. While writing the answers for mark questions select the questions which get you full marks. Don t forget to put the question number correctly. Do not write extra questions. Believe in yourself that you can get good marks if you do practice and hard work. Study the following chapters should well practiced. One mark 05 questions with answers for all chapters Practical Geometry Tangents 8 sums, Triangle 8 sums Special graph 9 sums should be practice regularly. Then, 1. Sets and functions,. Sequences and series, 3. Algebra, 4. Matrix, 5. Coordinate geometry 6. Geometry, 8. Mensuration, 1. Probability Try to follow the above tips and score more in MATHS. Wish you all the best. Wish you all the best to get 100 out of 100 marks in the Public Examinations 3 Way to Success team

5 CONTENT Chap. No Chapter Page Number One mark Qns. Two mark Qns. 1 Sets and Functions 5 18 Sequences and series of Real Numbers Algebra Matrices Coordinate Geometry Geometry Trigonometry Mensuration Statistics Probability Practical Geometry Graphs 86 Points to Remember 91 SYMBOLS equal to not equal to < less than less than or equal to > greater than greater than or equal to equivalent union intersection belongs to does not belong to proper subset of subset of or is contained in not a proper subset of not a subset of or is not contained in A or A c complement of A or empty set or null set or void set P A power set of A n A number of elements in the set A ly similarly P A probability of the event A symmetric difference N natural numbers R real numbers Z integers triangle angle perpendicular parallel to implies therefore absolute value congruent identically equal to π pi ± plus or minus end of the proof (or) : such that 4

6 10 th Maths One Mark Questions Way to Success ONE MARK QUESTIONS Choose the correct answer 1. Sets and Functions 1. For two sets A and B, A B A if and only if A) B A B) A B C) A B D) A B. If A B, then A B is A) B B) A\B C) A D) B\A 3. For any two sets P and Q, P Q is A) {x: x P or x Q } B) {x:x P and x Q} C) {x:x P and x Q} D) {x:x P and x Q} 4. If A {p, q, r, s}, B {r, s, t, u}, then A\B is: A) {p, q} B) {t, u} C) {r, s} D) {p, q, r, s} 5. If n p A 64, then n(a) is: A) 6 B) 8 C) 4 D) 5 6. For any three sets A, B and C, A (B C) is A) (A B) (B C) B) (A B) (A C) C) A (B C) D) (A B) (B C) 7. For any two sets A and B, {(A\B) (B\A)} (A B) is A) B) A B C) A B D A B 8. Which one of the following is not true? A) A\B A B B) A\B A B C) A\B (A B) B D) A\B (A B)\B 9. For any three sets A, B and C, B\(A C) is: A) (A\B) (A\C) B) (B\A) (B\C) C) (B\A) (A\C) D) (A\B) (B\C) 10. If n(a) 0, n(b) 30 and n(a B) 40, then n(a B) is equal to A) 50 B) 10 C) 40 D) If {(x,), (4,y)} represents an identity function, then (x,y) is A) (,4) B) (4,) C) (,) D) (4,4) 1. If { 7,11, 5,a } represents a constant function, then the value of a is A) 7 B) 11 C) 5 D) Given f(x) ( 1) x is a function from N to Z. Then the range of f is A) {1} B) N C) {1, 1} D) Z 14. If f {(6,3), (8,9), (5,3), (-1,6)}, then the pre-images of 3 are A) 5 and 1 B) 6 and 8 C) 8 and 1 D) 6 and Let A {1,3,4,7,11}, B{ 1,1,,5,7,9} and f:a B be given by f{(1, 1),(3,),(4,1),(7,5), (11,9)} Then f is A) one - one B) onto C) bijective D) not a function 16. The given diagram represents A) an onto function B) a constant function C) an one-one function D) not a function 17. If A {5,6,7}, B {1,,3,4,5} and f: A B is defined by f x x, then the range of f is A) {1,4,5} B) {1,,3,4,5} C) {,3,4} D) {3,4,5} ways100@gmail.com

7 10 th Maths One Mark Questions Way to Success 18. If f(x) x +5, then f( 4) A) 6 B) 1 C) 0 D) If the range of a function is a singleton set, then it is A) a constant function B) an identity function C) a bijective function D) an one-one function 0. If f:a B is a bijective function and if n(a) 5, then n(b) is equal to A) 10 B) 4 C) 5 D) 5. Sequences and Series of Real Numbers 1. Which one of the following is not true? A) A sequence is a real valued function defined on N. B) Every function represents a sequence. C) A sequence may have infinitely many terms. D) A sequence may have a finite number of terms.. The 8 th term of the sequence 1, 1,, 3, 5, 8,.. is A) 5 B) 4 C) 3 D) 1 3. The next term of 1 0 in the sequence 1, 1 6, 1 1, 1 0, is A) 1 4 B) If x, x+, 3x+3 are in G.P, then 5x, 10x+10, 15x + 15 form A) an A.P. B) a G.P. C) a constant sequence D) neither A.P. nor G.P. ways100@gmail.com C) 1 30 D) If a, b, c, l, m are in A.P, then the value of a 4b+6c 4 l + m is A) 1 B) C) 3 D) 0 5. If a, b, c are in A.P. then a b A) a b a c B) b c is equal to 6. If the n th term of a sequence is 100 n +10, then the sequence is A) an A.P. B) a G.P. C) a constant sequence D) neither A.P. nor G.P. C) a c D) 1 7. If a 1, a, a 3,. are in A.P. such that a 4 3 then the a 7 13th term of the A.P. is A) 3 B) 0 C) 1a 1 D) 14a 1 8. If the sequence a 1, a, a 3,. is in A.P., then the sequence a 5, a 10, a 15,... is A) a G.P B) an A.P. C) neither A.P nor G.P. D) a constant sequence 9. If k +, 4k 6, 3k are the three consecutive terms of an A.P. then the value of k is A) B) 3 C) 4 D) If a, b, c, l, m, n are in A.P., then 3a + 7, 3b + 7, 3c + 7, 3l + 7, 3m + 7, 3n + 7 form A) a G.P. B) an A.P. C) a constant sequence D) neither A.P. nor G.P. 11. If the third term of a G.P. is, then the product of first 5 term is A) 5 B) 5 C) 10 D) If a, b, c are in G.P, then a b A) a b b c B) b a is equal to C) a c D) c b

8 10 th Maths One Mark Questions Way to Success 14. The sequence 3, 3, 3, is A) an A.P. only B) a G.P. only C) neither A.P. nor G.P. D) both A.P. and G.P. 15. If the product of the first four consecutive terms of a G.P. is 56 and if the common ratio is 4 and the first term is positive, then its 3 rd term is A) 8 B) 1 16 C) 1 3 D) In a G.P. t 3 and t 3 1. Then the common ratio is 5 5 C)1 D) 5 A) 1 5 B) If x 0, then 1 + sec x+ sec x + sec 3 x + sec 4 x + sec 5 x is equal to A) (1+sec x) (sec x+ sec 3 x +sec 4 x) B) (1+sec x) (1+sec x+ sec 4 x) C) (1 sec x) (sec x+sec 3 x+sec 5 x) D) (1+sec x)(1+sec 3 x+sec 4 x) 18. If the n th term of an A.P. is t n 3 5n, then the sum of the first n term is A) n [1 5n] B) n(1 5n) C) n (1+5n) d) n (1+n) 19. The common ratio of the G.P. a m-n, a m, a m+n is A) a m B) a -m C) a n D) a -n 0. If n k then n 3 is equal to A) k B) k 3 C) k(k+1) 3. Algebra D) (k+1) 3 1. If the system 6x y 3, kx y has a unique solution, then A) k 3 B) k 3 C) k 4 D) k 4. A system of two linear equations in two variables is inconsistent, if their graphs A) coincide B) intersect only at a point C) do not intersect at any point D) cut the x-axis 3. The system of equations x 4y 8, 3x 1y 4 A) has infinitely many solutions B) has no solution C) has a unique solution D) may or may not have a solution 4. If one zero of the polynomial p(x) (k + 4) x + 13x + 3k is reciprocal of the other, then k is equal to A) B) 3 C) 4 D) 5 5. The sum of two zeros of the polynomial f x x + p + 3 x + 5 is zero, then the value of p is A) 3 B) 4 C) -3 D) The remainder when x x + 7 is divided by x + 4 is A) 8 B) 9 C) 30 D) The quotient when x 3 5x + 7x 4 is divided by x-1 is A) x + 4x + 3 B) x 4x + 3 C) x 4x 3 D) x + 4x 3 8. The GCD of (x 3 + 1) and x 4 1is A) x 3 1 B) x C) x + 1 D) x 1 9. The GCD of x xy + y and x 4 y 4 is A) 1 B) x + y C) x y D) x y ways100@gmail.com

9 10 th Maths One Mark Questions Way to Success 10. The LCM of x 3 a 3 and (x a) is A) (x 3 a 3 ) (x + a) B) (x 3 a 3 ) (x a) C) (x a) (x + ax + a ) D) (x + a) (x + ax + a ) 11. The LCM of a k, a k+3, a k+5 where k N is A) a k+9 B) a k C)a k+6 D) a k+5 1. The lowest form of the rational expression x +5x+6 A) x 3 x+3 B) x+3 x 3 x x 6 C) x+ x If a+b and a 3 b 3 a b a 3 +b 3 are the two rational expressions, then their product is 4. If α and β are the roots of ax + bx + c 0, then one of the quadratic equations whose roots are 1 α and 1 is β A) ax + bx + c 0 B) bx + ax + c 0 C) cx + bx + a 0 D) cx + ax + b 0 5. Let ba+c. Then the equation ax +bx +c 0 has equal roots, if A) a c B) a c C) a c D) a c ways100@gmail.com is D) x 3 x+ A) a +ab+b B) a ab +b C) a ab b D) a +ab +b a ab+b a +ab +b a +ab +b a ab b 14. On dividing x If by x+5 x+3 x 9 is equal to A) (x 5) (x 3) B) (x 5) (x + 3) C) (x + 5) (x 3) D) (x + 5) (x + 3) a 3 a b is added with, b3 b a, then the new expression is A) a + ab + b B) a ab + b C) a 3 + b 3 D) a 3 b The square root of 49 (x xy + y ) is A) 7 x y B) 7(x + y) (x y) C) 7(x + y) D) 7(x y) 17. The square root of x + y + z xy + yz zx A) x + y z B) x y + z C) x + y + z D) x y z 18. The square root of 11 x 4 y 8 z 6 (l m) is A) 11x y 4 z 4 l m B) 11x 4 y 4 z 3 (l m) C) 11x y 4 z 6 l m D) 11x y 4 z 3 (l m) 19. If ax + bx + c 0 has equal roots, then c is equal A) b a B) b 4a 0. If x +5k x has no real roots, then A) k > 8 5 B) k > 8 5 C) b a C) 8 5 < k < 8 5 D) b 4a D) 0 < k < A quadratic equation whose one root is 3 is A) x 6x 5 0 B) x + 6x 5 0 C) x 5x 6 0 D) x 5x The common root of the equations x bx + c 0 and x + bx a 0 is A) c+a b B) c a b C) c+b a 3. If α, βare the roots of ax + bx + c 0 a 0, then the wrong statement is A) α + β b ac B) αβ c a a C) α + β b a D) a+b c D) 1 α + 1 β b c

10 10 th Maths One Mark Questions Way to Success 4. Martices 1. Which one of the following statements is not true? A) A scalar matrix is a square matrix B) A diagonal matrix is a square matrix C) A scalar matrix is a diagonal matrix D) A diagonal matrix is a scalar matrix. Matrix A [a ij ] m x n is a square matrix if A) m < n B) m > n C) m 1 D) m n 3. If 3x y + 1 3x 1 y 8 8 then the values of x and y respectively are A), 7 B) 1, 7 3 C) 1, 3 3 D), If A (1 3) and B 3 then A+B 0 A) (0 0 0) B) 0 0 C) ( 14) D) not defined 5. If a matrix is of order 3, then the number of elements in the matrix is A) 5 B) 6 C) D) If x then the value of x is 1 A) 1 B) C) If A is of order 3 x 4 and B is of order 4 x 3, then the order of BA is A) 3 x 3 B) 4 x 4 C) 4 x 3 D) not defined D) If A (1 ), then the order of A is 0 A) x 1 B) x C) 1 x D) 3 x 9. If A and B are square matrices such that AB I and BA I, then B is A) Unit matrix B) Null matrix C) Multiplicative inverse matrix of A D) A 1 x 10. If 1 y, then the values of x and y respectively, are 4 A), 0 B) 0, C) 0, D) 1, If A and A+BO, then B is A) B) C) D) If A 6 3, then A is A) B) C) D) A is of order m n and B is of order p q, addition of A and B is possible only if A) m p B) n q C) n p D) m p, n q 14. If a , then the value of a is 0 A) 8 B) 4 C) D) 11 ways100@gmail.com

11 10 th Maths One Mark Questions Way to Success α β 15. If A γ α is such that A I, then A) 1+ α + βγ 0 B) 1 α + βγ 0 C) 1 α βγ 0 D) 1+ α βγ If A [a ij ] x and a ij i + j, then A A) B) C) D) a b c d 1 0, then the values of a, b, c and d respectively are 0 1 A) 1, 0, 0, 1 B) 1, 0, 0, 1 C) 1, 0, 1, 0 D) 1, 0, 0, If A and A+B 1 0, then the matrix B 4 A) B) C) If 5 x 1 1 (0), then the value of x is 3 A) 7 B) 7 C) 1 7 D) D) Which one of the following is true for any two square matrices A and B of same order? A) (AB) T A T B T B) (A T B) T A T B T C) (AB) T BA D) (AB) T B T A T 5. COORDINATE GEOMETRY 1. The midpoint of the line joining (a, b) and (3a, 5b) is A) ( a, b) B) (a, 4b) C) (a, b) D) ( a, 3b). The point P which divides the line segment joining the points A (1, 3) and B ( 3, 9) internally in the ratio 1:3 is A) (, 1) B) (0, 0) C) ( 5, ) D) (1, ) 3 3. If the line segment joining the points A (3, 4) and B (14, 3) meets the x-axis at P, then the ratio in which P divides the segment AB is A) 4:3 B) 3:4 C) :3 D) 4:1 4. The centroid of the triangle with vertices at (, 5), (, 1) and (10, 1)is A) (6, 6) B) (4, 4) C) (3, 3) D) (, ) 5. If (1, ), (4, 6) (x, 6) and (3, ) are the vertices of a parallelogram taken in order, then the value of x is A) 6 B) C) 1 D) 3 6. Area of the triangle formed by the points (0, 0), (, 0) and (0, ) is A) 1sq.units B) sq.units C) 4sq.units D) 8sq.units 7. Area of the quadrilateral formed by the points (1, 1), (0, 1), (0, 0) and (1, 0) is A) 3sq.units B) sq.units C) 4sq.units D) 1sq.units 8. The angle of inclination of a straight line parallel to x-axis is equal to A) 0 0 B) 60 0 C) 45 0 D) Slope of the line joining the points (3, ) and ( 1, a) is 3, then the value of a is equal to A) 1 B) C) 3 D) 4 ways100@gmail.com

12 10 th Maths One Mark Questions Way to Success 10. Slope of the straight line which is perpendicular to the straight line joining the points (, 6) and (4, 8) is equal to A) 1 3 B) 3 C) 3 D) The point of intersection of the straight lines 9x y 0 and x+y 9 0 is A) ( 1, 7) B) (7, 1) C) (1, 7) D) ( 1, 7) 1. The straight line 4 x+3y 1 0 intersects the y-axis at A) (3, 0) B) (0, 4) C) (3, 4) D) (0, 4) 13. The slope of the straight line 7y x11 is equal to A) 7 B) 7. In ABC, DE is to BC, meeting AB and AC at D and E. If AD 3cm, DB cm and AE.7 cm, then AC is equal to A) 6.5 cm B) 4.5 cm C) 3.5 cm D) 5.5 cm ways100@gmail.com C) 7 D) The equation of a straight line passing through the point (, 7) and parallel to x-axis is A) x B) x 7 C) y 7 D) y 15. The x and y-intercepts of the line x 3y+6 0, respectively are A), 3 B) 3, C) 3, D) 3, 16. The centre of a circle is ( 6, 4). If one end of the diameter of the circle is at ( 1, 8), then the other end is at A) ( 18, 1) B) ( 9, 6) C) ( 3, ) D) (0, 0) 17. The equation of the straight line passing through the origin and perpendicular to the straight line x+3y 7 0 is A) x+3y0 B) 3x y0 C) y+50 D) y The equation of a straight line parallel to y-axis and passing through the point (, 5) is A) x 0 B) x+0 C) y+50 D) y If the points (, 5), (4, 6) and (a, a) are collinear, then the value of a is equal to A) 8 B) 4 C) 4 D) 8 0. If a straight line yx+k passes through the point (1, ), then the value of k is equal to A) 0 B) 4 C) 5 D) 3 1. The equation of a straight line having slope 3 and y-intercept 4 is A) 3x y 40 B) 3x+y 40 C) 3x y+40 D) 3x+y+40. The point of intersection of the straight lines y0 and x 4 is A) (0, 4) B) ( 4, 0) C) (0, 4) D) (4, 0) 3. The value of k if the straight lines 3x +6y +7 0 and x+ky 5 are perpendicular is A) 1 B) 1 C) D) 1 6. GEOMETRY 1. If a straight line intersects the sides AB and AC of a ABC at D and E respectively and is parallel to BC, then AE A) AD DB AC B) AD AB C) DE BC D) AD EC

13 10 th Maths One Mark Questions Way to Success 3. In PQR, RS is the bisector of R. If PQ 6 cm, QR 8 cm, RP 4cm then PS is equal to A) cm B) 4 cm C) 3 cm D) 6 cm 4. In figure, if AB AC BD, DC B400, and C60 0, then BAD A) 30 0 B) 50 0 C) 80 0 D) In the figure, the value x is equal to A) 4. B) 3. C) 0.8 D) In triangles ABC and DEF, B E, C F, then A) AB CA DE EF B) BC AB EF FD 7. From the given figure, identify the wrong statement. A) ADB~ ABC B) ABD~ ABC C) BDC~ ABC D) ADB~ BDC C) AB BC DE EF D) CA AB FD EF 8. If a vertical stick 1 m long casts a shadow 8m long on the ground and at the same time a tower casts a shadow 40m long on the ground, then the height of the tower is A) 40 m B) 50m C) 75m D) 60m 9. The sides of two similar triangles are in the ratio :3, then their areas are in the ratio A) 9:4 B) 4:9 C) :3 D) 3: 10. Triangles ABC and DEF are similar. If their areas are 100cm and 49cm respectively and BC is 8: cm then EF A) 5.47 cm B) 5.74cm C) 6.47 cm D) 6.74 cm 11. The perimeters of two similar triangles are 4 cm and 18 cm respectively. If one side of the first triangle is 8 cm, then the corresponding side of the other triangle is A) 4 cm B) 3 cm C) 9cm D) 6 cm 1. AB and CD are two chords of a circle which when produced to meet at a point P such that AB 5 cm, AP 8cm, and CD cm then PD A) 1 cm B) 5 cm C) 6 cm D) 4 cm 13. In the adjoining figure, chords AB and CD intersect at P. If AB 16cm, PD 8cm, PC 6 and AP>PB, then AP A) 8 cm B) 4 Cm C) 1 cm D) 6 cm 14. A point P is 6 cm away from the centre O of a circle and PT is the tangent drawn from P to the circle is 10 cm, then OT is equal to A) 36 cm B) 0 cm C) 18 cm D) 4 cm 15. In the figure, if PAB 10 0 then BPT A) 10 0 B) 30 0 C) 40 0 D) If the tangents PA and PB from an external point P to circle with centre O are inclined to each other at an angle of 40 0 then POA A) 70 0 B) 80 0 C) 50 0 D) 60 0 ways100@gmail.com

14 10 th Maths One Mark Questions Way to Success 17. In the figure, PA and PB are tangents to the circle drawn from an external point P. Also CD is a tangent to the circle at Q. If PA 8 cm and CQ 3 cm, then PC is equal to A) 11 cm B) 5 cm C) 4 cm D) 38 cm 18. ABC is a right angled triangle where B 90 0 and BD AC. If BD 8cm, AD 4cm, then CD is A) 4 cm B) 16 cm C) 3 cm D) 8 cm 19. The areas of two similar triangles are 16cm and 36cm respectively. If the altitude of the first triangle is 3 cm, then the corresponding altitude of the other triangle is A) 6.5 cm B) 6 cm C) 4 cm D) 4.5 cm 0. The perimeter of two similar triangles ABC and DEF are 36 cm and 4 cm respectively. If DE 10cm, then AB is A) 1 cm B) 0 cm C) 15 cm D) 18 cm 7. Trigonometry 1. (1 sin θ )sec θ A) 0 B) 1 C) tan θ D) cos θ. (1+tan θ) sin θ A) sin θ B) cos θ C) tan θ D) cot θ 3. (1 cos θ) (1+cot θ) A) sin θ B) 0 C) 1 D) tan θ 4. sin (90 0 θ) cos θ + cos(90 0 θ) sin θ A) 1 B) 0 C) D) sin θ 1+cosθ A) cos θ B) tan θ C) cot θ D) cosec θ 6. cos 4 x sin 4 x A) sin x 1 B) cos x 1 C) 1+sin x D) 1 cos x. 7. If tan θ a x, then the value of x a +x A) cos θ B) sin θ C) cosec θ D) sec θ 8. If x a secθ, yb tanθ, then the value of x y a b A) 1 B) 1 C) tan θ D) cosec θ secθ cotθ +tanθ A) cot θ B) tan θ C) sin θ D) cot θ sin 90 0 θ sinθ + cos 900 θ cosθ tanθ cotθ A) tan θ B) 1 C) 1 D) sin θ 11. In the adjoining figure, AC A) 5 m B) 5 3m C) 5 m D) 5 m 3 1. In the adjoining figure ABC A) 45 0 B) 30 0 C) 60 0 D) 50 0 ways100@gmail.com

15 10 th Maths One Mark Questions Way to Success 13. A man is 8.5 m away from a tower. His eye level above the ground is 1.5 m. The angle of elevation of the tower from his eyes is Then the height of the tower is A) 30 m B) 7.5 m C) 8.5 m D) 7 m 14. In the adjoining figure, sin θ 15. Then BC 7 A) 85 m B) 65m C) 95m D) 75m 15. (1+tan θ) (1 sin θ) (1+sin θ) A) cos θ sin θ B) sin θ cos θ C) sin θ +cos θ D) (1+cot θ) (1 cos θ) (1+cos θ) A) tan θ sec θ B) sin θ cos θ C) sec θ tan θ D) cos θ sin θ 17. (cos θ 1) (cot θ +1) +1 A) 1 B) 1 C) D) ta n θ 1+co t θ 19. sin θ + A) cos θ B) tan θ C) sin θ D) cot θ 1 1+ta n θ A) cosec θ + cot θ B) cosec θ cot θ C) cot θ cosec θ D) sin θ cos θ 0. 9tan θ 9sec θ A) 1 B) 0 C) 9 D) 9 8. Mensuration 1. The curved surface area of a right circular cylinder of radius 1 cm and height 1 cm is equal to A) πcm B) π cm C) 3πcm 3 D) cm. The total surface area of a solid right circular cylinder whose radius is half of its height h is equal to A) 3 πh sq.untis B) 3 πh sq.untis C) 3 πh sq.untis D) πh sq.untis 3 3. Base area of a right circular cylinder is 80 cm. If its height is 5 cm, then the volume is equal to A) 400 cm 3 B) 16 cm 3 C) 00 cm 3 D) cm3 4. If the total surface area a solid right circular cylinder is 00 π cm and its radius is 5 cm, then the sum of its height and radius is A) 0 cm B) 5 cm C) 30 cm D) 15 cm 5. The curved surface area of a right circular cylinder whose radius is a units and height is b units, is equal to A) πa b sq.cm B) πab sq.cm C) πsq.cm D) sq.cm 6. Radius and height of a right circular cone and that of a right circular cylinder are respectively, equal. If the volume of the cylinder is 10 cm 3, then the volume of the cone is equal to A) 100 cm 3 B) 360 cm 3 C) 40 cm 3 D) 90 cm 3 7. If the diameter and height of a right circular cone are 1 cm and 8 cm respectively, then the slant height is A) 10 cm B) 0 cm C) 30 cm D) 96 cm 8. If the circumference at the base of a right circular cone and the slant height are 10 πcm and 10cm respectively, then the curved surface area of the cone is equal to A) 100π cm B) 600π cm C) 300π cm D) 600 cm ways100@gmail.com

16 10 th Maths One Mark Questions Way to Success 9. If the volume and the base area of a right circular cone are 48π cm 3 and 1π cm respectively, then the height of the cone is equal to A) 6 cm B) 8 cm C) 10 cm D) 1 cm 10. If the height and the base area of a right circular cone are 5 cm and 48 sq. cm respectively, then the volume of the cone is equal to A) 40cm 3 B) 10 cm 3 C) 80 cm 3 D) 480 cm The ratios of the respective heights and the respective radii of two cylinders are 1: and :1 respectively. Then their respective volumes are in the ratio A) 4:1 B) 1:4 C) :1 D) 1: 1. If the radius of a sphere is cm, then the curved surface area of the sphere is equal to A) 8π cm B) 16 cm C) 1π cm D) 16π cm 13. The total surface area of a solid hemisphere of diameter cm is equal to A) 1 cm B) 1 π cm C) 4π cm D) 3π cm 14. If the volume of a sphere is 9 πcu.cm, then its radius is 16 A) 4 3 cm B) 3 4 cm C) 3 cm D) 3 cm 15. The surface areas of two spheres are in the ratio of 9:5. Then their volumes are in the ratio A) 81:65 B) 79:1565 C) 7:75 D) 7: The total surface area of a solid hemisphere whose radius is a units, is equal to A) π a sq.untis B) 3π a sq.units C) 3π a sq.untis D) 3a sq.untis 17. If the surface area of a sphere is 100π cm, then its radius is equal to A) 5 cm B) 100cm C) 5 cm D) 10 cm 18. If the surface area of a sphere is 36π cm, then the volume of the sphere is equal to A) 1π cm 3 B) 36π cm 3 C) 7π cm 3 D) 108π cm If the total surface area of a solid hemisphere is 1 π cm then its curved surface area is equal to A) 6π cm B) 4π cm C) 36π cm D) 8π cm 0. If the radius of a sphere is half of the radius of another sphere, then respective volumes are in the ratio A) 1:8 B) :1 C) 1: D) 8:1 1. Curved surface area of solid sphere is 4 cm. If the sphere is divided into two hemispheres, then their the total surface area of one of the hemispheres is A) 1 cm B) 8 cm C) 16 cm D) 18 cm. Two right circular cones have equal radii. If their slant heights are in the ratio 4:3, then their respective curved surface areas are in the ratio A) 16:9 B) :3 C) 4:3 D) 3:4 11. Statistics 1. The range of the first 10 prime numbers, 3, 5, 7, 11, 13, 17, 19, 3, 9 is A) 8 B) 6 C) 9 D) 7. The least value in a collection of data is If the range of the collection is 8.4, then the greatest value of the collection is A) 4.5 B) 43.5 C) 4.4 D) The greatest value of a collection of data is 7 and the least value is 8. Then the coefficient of range is A) 44 B) 0.7 C) 0.44 D) 0.8 ways100@gmail.com

17 10 th Maths One Mark Questions Way to Success 4. For a collection of 11 items, x13, then the arithmetic mean is A) 11 B) 1 C) 14 D) For any collection of n items, Σ(x x ) A) Σ x B) x C) nx D) 0 6. For any collection of n items, (Σx) x A) nx B) (n ) x C) (n 1) x D) 0 7. If t is the standard deviation of x, y, z, then the standard deviation of x+5, y +5, z +5 is A) t 3 B) t +5 C) t D) x y z 8. If the standard deviation of a set of data is 1.6, then the variance is A) 0.4 B).56 C) 1.96 D) If the variance of a data is 1.5, then the S.D is A) 3.5 B) 3 C).5 D) Variance of the first 11 natural numbers is A) 5 B) 10 C) 5 D) The variance of 10, 10, 10, 10, 10 is A) 10 B) 10 C) 5 D) 0 1. If the variance of 14, 18,, 6, 30 is 3, then the variance of 8, 36, 44, 5, 60 is A) 64 B) 18 C) 3 D) Standard deviation of a collection of data is. If each value is multiplied by 3, then the standard deviation of the new data is A) 1 B) 4 C) 6 D) Given (x x ) 48, x 0 and n 1. The coefficient of variation is A) 5 B) 0 C) 30 D) Mean and standard deviation of a data are 48 and 1 respectively. The coefficient of variation is A) 4 B) 5 C) 8 D) If is an impossible event, then P( ) A) 1 B) Probability. If S is the sample space of a random experiment, then P(S) A) 0 B) 1 8 C) 0 D) 1 3. If p is the probability of an event A, then p satisfies A) 0 <p <1 B) 0 p 1 C) 0 p < 1 D) 0 < p 1 4. Let A and B be any two events and S be the corresponding sample space. Then P (A B) A) P(B) P(A B) B) P(A B) P(B) C) P(S) D) P[(A B) ] 5. The probability that a student will score centum in mathematics is 4. The probability that he will 5 not score centum is A) 1 5 B) 5 C) 1 C) If A and B are two events such that P(A) 0.5, P(B) 0.05 and P(A B) 0.14, then P(A B) A) 0.61 B) 0.16 C) 0.14 D) 0.6 ways100@gmail.com D) 1 D) 4 5

18 10 th Maths One Mark Questions Way to Success 7. There are 6 defective items in a sample of 0 items. One item is drawn at random. The probability that it is a non-defective item is A) 7 B) 0 C) 3 D) If A and B are mutually exclusive events and S is the sample space such that P(A) 1 P(B) and 3 S A B, then P(A) A) 1 4 B) 1 9. The probabilities of three mutually exclusive events A, B and C are given by 1 3, 1 4, and 5 1. Then P(A B C) is A) 19 1 B) 11 1 ways100@gmail.com C) 3 4 C) 7 1 D) 3 8 D) If P(A) 0.5, P(B) 0.50, P(A B) 0.14 then P (neither A nor B) A) 0.39 B) 0.5 C) 0.11 D) A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected at random, the probability that it is not red is A) 5 1 B) 4 1 C) Two dice are thrown simultaneously. The probability of getting a doublet is A) 1 36 B) A fair die is thrown once. The probability of getting a prime or composite number is C) 1 6 A) 1 B) 0 C) Probability of getting 3 heads or 3 tails in tossing a coin 3 times is A) 1 8 B) A card is drawn from a pack of 5 cards at random. The probability of getting neither an ace nor a king card is A) 13 B) C) 3 8 C) The probability that a leap year will have 53 Fridays or 53 Saturdays is A) 7 B) 1 7 C) 4 7 D) 3 4 D) 3 D) 1 6 D) 1 D) The probability that a non-leap year will have 53 Sundays and 53 Mondays is A) 1 7 B) The probability of selecting a queen of hearts when a card is drawn from a pack of 5 playing cards is A) 1 B) C) 3 7 C) 1 13 D) 3 7 D) 0 D) Probability of sure event is A) 1 B) 0 C) 100 D) The outcome of a random experiment results in either success or failure. If the probability of success is twice the probability of failure, then the probability of success is A) 1 3 B) 3 C) 1 D) 0

19 TWO MARK QUESTIONS 1. Sets and Functions 1. If A B, then show that A B B (use Venn diagram). A B A B A B B. If A B, then find A B and A \ B (Use Venn diagram). (Oct-014) A B A B A A\B 3. Let P {a, b, c}, Q {g, h, x, y} and R {a, e, f, s}. Find the following: (i) P \R (ii) Q R (iii) R \ (P Q) (Jun-015) (i) P \R {a,b,c}\{a,e,f,s}{b, c} (ii) Q R {g, h,x,y} {a,e,f,s} { } (iii) R \ (P Q) (P Q) {a,b,c} g, h, x, y { } R \ (P Q) a, e, f, s \{ } {a, e, f, s} 4. If A {4, 6, 7, 8, 9}, B {, 4, 6}, C {1,, 3, 4, 5, 6}, then find (i) A (B C) (Apr-014) (ii) A (B C) (Jun-01, Oct-01) (iii) A\(C\B) (i) A (B C) (B C), 4, 6 1,, 3, 4, 5, 6 {,4,6} A (B C) 4, 6, 7, 8, 9, 4, 6 {,4,6,7,8,9} (ii) A (B C) (B C), 4, 6 1,, 3, 4, 5, 6 {1,, 3, 4, 5, 6} A (B C) 4, 6, 7, 8, 9 1,, 3, 4, 5, 6 {4, 6} (iii) A\ (C\B) (C\B) 1,, 3, 4, 5, 6 \, 4, 6 {1, 3, 5} A\ (C\B) 4, 6, 7, 8, 9 \{1, 3, 5}{4, 6, 7, 8, 9} 5. Draw Venn diagram of three sets A,B and C. If A and B are disjoint but both are subsets of C. ways100@gmail.com

20 6. A (B\C) Draw Venn diagram: (B\C) 7. Draw Venn diagram for (B C) \A A (B\C) (B C) (B C) \A 8. Draw Venn diagram for A (B C) (Apr-015) (B C) 9. C (B\A) Draw Venn diagram: (B\A) 10. C (B A) Draw Venn diagram: A (B C) C (B\A) (B A) C (B A) ways100@gmail.com

21 11. Let A {1,, 3, 4} and B { 1,, 3, 4, 5, 6, 7, 9, 10, 11, 1}. Let R {(1, 3), (, 6), (3, 10), (4, 9)} A X B be a relation. Show that R is a function and find its domain, co-domain and the range of R. (Jun-01, Oct-013) All the elements of A has only one image in B R is a function Domain {1,, 3, 4} Co-domain {-1,, 3, 4, 5, 6, 7, 9, 10, 11, 1} Range {3, 6, 9, 10} 1. Write the pre-images of and 3 in the function f {(1, ), (13, 3), (15, 3), (14, ), (17, 17)}. Pre images of is 1, 14 Pre images of 3 is 13, If R {(a, ), ( 5, b), (8, c), (d, 1)} represents the identity function, find the values of a, b, c and d. (Apr-013) a, b 5, c 8, d A {, 1, 1, } and f x, 1 : x A. Write down the range of f. Is f a function from A to A? x Range { 1, 1,1, 1 }, f, 1, 1, 1, 1, 1,, However, 1 A. Hence, it is not a function from A to A., Let A {1,, 3, 4, 5}, B N and f : A B be defined by f(x) x. Find the range of f. Identify the type of function. (Apr-015, Jun-015) f (x) x f (1) 1 1 f () f (3) 3 9 f (4) 4 16 f ( 5) 5 5 (i) Range of f {1, 4, 9, 16, 5} (ii) Since, distinct elements are mapped into distinct images. It is a one-one function 16. Represent the function f {( 1, ), ( 3, 1), ( 5, 6), ( 4, 3) as (i) a table (ii) an arrow diagram (i) Table (ii) Arrow diagram x f(x) ways100@gmail.com

22 x if x 0, 17. Let x where x R Does the relation. (Jun-013) x if x < 0 {(x, y) y x, x R } define a function? Find its range. (i) For every value of x, there exists a unique value y x. Therefore, the given relation defines a function. (ii) The range will be the set on non-negative real numbers. 18. For the given function F {(1, 3), (, 5) (4, 7) (5, 9), (3, 1)} write the domain and range. (i) Domain {1,, 3, 4, 5} (ii) Range {3, 5, 7, 9, 1} 19. Let A {10, 15, 0, 5, 30, 35, 40, 45, 50} B {1, 5, 10, 15, 0, 30} and C {7, 8, 15, 0, 35, 45, 48}. Find A\ (B C). (Apr- 01) (B C) {1, 5, 10, 15, 0, 30} {7, 8, 15, 0, 35, 45, 48} {15, 0} A\ B C 10, 15, 0, 5, 30, 35, 40, 45, 50 \ 15, 0 {10, 5, 30, 35, 40, 45, 50} 0. For A {5, 10, 15, 0}; B {6, 10, 1, 18, 4} and C {7, 10, 1, 14, 1,8}, verify whether A\(B\C) (A\B) \C. (Jun-013) LHS A\(B\C) (B\C) 6, 10, 1, 18, 4 \ 7, 10, 1, 14, 1,8 {6, 18, 4} A\(B\C) 5, 10, 15, 0 \ 6, 18, 4 {5, 10, 15, 0.. (1) RHS (A\B) \C (A\B) 5, 10, 15, 0 \ 6, 10, 1, 18, 4 {5, 15, 0} (A\B) \C 5, 15, 0 \ 7, 10, 1, 14, 1,8 {5, 15, 0}. () LHS RHS (1) () From (1) and () A\(B\C) (A\B)\C 1. X {1,, 3, 4, 5}, Y {1, 3, 5, 7, 9} determine which of the following relations from X to Y are functions? Give reason for your answer. If it is a function, state its type. {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)} (Apr-1) R{ (1, 1) (1, 3) (3, 5), (3, 7),(5,7)} The element 1 X has two images 1,3 in Y. So, it is not a function. Verify the commutative property of set intersection for A {l, m, n, o,, 3, 4, 7} and B {, 5, 3, -, m, n, o, p}. (Apr-013) A B {l, m, n, o,, 3, 4, 7} {, 5, 3,, m, n, o, p} {m, n, o,, 3} 1 B A {, 5, 3,, m, n, o, p} {l, m, n, o,, 3, 4, 7} {m, n, o,, 3} From (1) and (), we get A B B A. That is, the intersection of a set is commutative. 3. Let U {4, 8, 1, 16, 0, 4, 8}, A {8, 16, 4} and B {4, 16, 0, 8}. Find (A B and A B. (Jun-14) A B 8, 16, 4 4, 16, 0, 8 {4, 8, 16, 0, 4, 8} A B U\ A B {1} A B 8, 16, 4 4, 16, 0, 8 {16} A B U\ A B 4, 8, 1, 16, 0, 4, 8 \ 16 {4, 8, 1, 0, 4, 8} 4. Which of the following relations are functions from A {1, 4, 9, 16} to B { 1,, 3, 4, 5, 6}? f { 1, 1, 4,, 9, 3, (16, 4)} In case of a function, write down its range. (Oct-01) Each element in A is associated with a unique element in B. It is a function Range {-1,, -3, -4} ways100@gmail.com

23 5. Let A {5, 6, 7, 8}; B { 11, 4, 7, 10, 7, 9, 13} and f {(x, y): y 3 x, x A, y B }. (i) What is the range? (Oct-014) (ii) Write down the elements of f (Mar-016) (i) y 3 x x 5, then y 3 (5) x 6, then y 3 (6) x 7, then y 3 (7) x 8, then y 3 (8) Range of f { 7, 9, 11, 13} (ii) f { 5, 7, 6, 9, 7, 11, (8, 13)} 6. Let X {1,, 3, 4}. f {(, 3), (1, 4), (, 1), (3, ), (4, 4)} Examine whether each of the relations given below is a function from X to X or not. Explain. (Apr-014) f {(, 3), (1, 4), (, 1), (3, ), (4, 4)} f is not a function, because is associated with two different elements 3 and If A and B are two sets and U is the universal set such that n (U) 700, n (A) 00, n (B) 300 and n (A B) 100, find n (A B. n (A B) n(a) + n(b) n(a B) n A B n(a B n(u) n(a B) n A B Given n (A) 85, n (B) 195, n (U) 500, n (A B) 410, find n A B. n(a B) n(a) + n(b) n(a B) n A UB n(a B n(u) n(a B) n A B For any three sets A, B and C if n (A) 17, n (B) 17, n (C) 17, n (A B) 7 n (B C) 6, n (A C) 5 and n (A B C), find n (A B C). n (A B C) n(a) + n(b) + n(c) n(a B)- n(b C) n(a C) +n(a B C) n (A B C) The following table represents a function from A {5, 6, 8, 10} to B {19, 15, 9, 11} where f(x) x 1. Find the value of a and b. x f(x) a 11 b 19 f(x) x 1 f(5) (5) f(8) (8) a 9, b 15 ways100@gmail.com - -

24 31. Draw Venn diagram (A B) (Oct-013) 3. The function f: [ 3, 7) R defined as follows (Jun- 014) 4x 1; 3 x < f x 3x ; x 4 x 3; 4 < x < 7 Find f 5 + f 6 f [ if 4 < x < 7, f x x 3] f x {5, 6} f 5 + f Given A a, x, y, r, s, B {1, 3, 5, 7, 10} verify the commutative property of set union. (Mar-16) Let us verify that A B B A A B a, x, y, r, s 1,3,5,7, 10 {a, x, y, r, s, 1,3,5,7, 10} 1 B A 1,3,5,7, 10 a, x, y, r, s {a, x, y, r, s, 1,3,5,7, 10}. From (1) and (), we get A B B A That is, the union of a set is commutative.. Sequences and Series of Real Numbers 1. Find the 1 th term of the A.P., 3, 5, (Mar-16) a, d 3, n1 t n a+(n 1)d t 1 + (1 1) x + 3 t 1 3. Find the 17 th term of the A.P. 4, 9, 14,.. (Apr-014) a 4, d 9 4 5, n 17 t n a+(n 1)d t (17 1) x x t ways100@gmail.com

25 3. How many two digit numbers are divisible by 13? (Apr-01) a 13, d 13, l 91 n l a d n n 7 Hence, there are 7 two digit numbers divisible by Find the sum of the arithmetic series (Apr-013, Jun-014) a 5, d , l 95 n l a +1 d n S n n a + l 16 x [5 + 95] 8 x Find the sum series (Oct-014) n 3 n(n+1) (0+1) 0x1 6. If n 10, find n 3. n 10 Aliter n 3 n n n 3 n(n+1) ways100@gmail.com n n If n , then find n. (Jun-013) n 3 ( n) Aliter n n 3 n(n+1) n n 8. If p 171, then find p 3. n 171 Aliter n 3 ( k) n n 3 9. If k 3 881, then find k. n 3 ( n) Aliter n(n+1) n(n+1) n n 3 n(n+1) n n n(n+1) n(n+1)

26 10. If 5 times the 5 th term of an A.P is equal to 7 times its 7 th term, then show that the 1 th term of the A.P. is zero (Oct-01) t n a+ (n 1) d 7t 7 5 t 5 7(a + 6d) 5(a + 4d) 7a + 4d 5a + 0d 7a -5a + 4d 0d +0 a + d 0 a + 11d 0 t In an A.P S n 175, first term a 3, common difference d 4, then find value of n (Jun-1) S n n [a + n 1 d] S n n [ 3 + n 1 4] 175 n[6 + (4n 4)] 175 x 6n + 4n 4n 550 4n +n (Divide by ) n + n (n + 51) (n 5) 0 n 5 1. Three numbers are in the ratio : 5:7. If the first number, the resulting number on the substraction of 7 from the second number and the third number form an arithmetic sequence, then find the numbers. Given ratio :5:7 Let the numbers be x, 5x and 7x for some unknown x, (x 0) By the given information, we have that x, 5x 7, 7x are in A.P. d (5x 7) x 7x (5x 7) [d t - t 1 ] 3x 7 x + 7 x x, 5x, 7x (14), 5(14), 7 (14) 8, 70, 98 n + n The first term of an A.P. is 6 and the common difference is 5. Find the A.P. and its general term. The general form of an A.P is a, a + d, a + d, a + 3d,. Given that a 6, d 5 Thus, the A.P is 6, 6 + 5, 6 + 5, ,. That is, the required A.P is 6, 11, 16, 1,. The general term t n a + (n 1)d 6 + (n 1)(5) 6 + 5n 5 5n +1 t n 5n Find the first term and common difference of the A.P. 1, 5, 7, 3,, (Apr-015) First term a 1 Common difference d ways100@gmail.com

27 15. If a clock strikes once at 1 O clock, twice at O clock and so on, how many times will it strike in a day? Number of times the clock strikes each hour form an A.P. Then, the first 1 hours, the arithmetic series S n n [a + l] S 1 1 [1+1] 6 x Hence, the clock strikes in a day (in 4 hours) x times 16. A man has saved Rs.640 during the first month, Rs.70 in the second month and Rs.800 in the third month. If he continues his savings in this sequence, what will be his savings in the 5 th month? The amount saved by the man during first month, second month, third month 640, 70, 800,. This is an A.P with a 640, d a 640 d t n a + (n 1)d t (5 1) ( 80) ( 80) Hence, his saving in the 5 th month is Rs Which term of the geometric sequence. 1,, 4, 8, is 104? 1,, 4, 8, a 1, r t t 1 1 t n 104 t n arn 1 ar n x n-1 10 n-1 10 n 1 10 n ,, 4, 8, Then, the 11 th term of the given geometric sequence is Find the common difference and 15 th term of the A.P. 15, 10, 115, 110,.. 15, 10, 115, 110, First term of the A.P is a 15 Common difference d t (15 1) x ( 5) t n a + (n -1)d x ( 5) Find the sum of the first 10 terms of (Jun-015) first 10 terms (1 4) + (9 16) terms ( 3) + ( terms ( 3) + ( terms a 3, d ( 7) ( 3) S n n [a + n 1 d] ways100@gmail.com

28 S 5 5 [ ] x Find the sum of the first 5 terms of the geometric series a 16, r S n a(1 rn ), r 1 1 r S 5 16[1 ( 3)5 ] 1 ( 3) 4 ( ) 16[1+35 ] 4 1. Find the 18 th and 5 th terms of the sequence defined by n (n + 3), if n N and n is even a n n if n N and is odd. n + 1, If n is even, a n n (n + 3) a (18 + 3) 18 x n If n is odd a n n +1 a 5 x Give an example for a function which is not a sequence. (Oct-013) A function is not necessarily a sequence. For example, the function f: R R given by f x x + 1, x R is not a sequence since the required listing is not possible. Also, note that the domain of f is not N or a subset 1, n of N. 3. Find the sum of all numbers between 50 and 00 which are divisible by 10. The sum of all numbers between 50 and 00 which are divisible by a 60, d 10, l 190 n l a d n 14 S n n [a + l] 14 S n 1750 [ ] 7 x Write the first three terms of the sequence whose n th term is given by a n n(n ) a n n(n ) 3 a 1 1(1 ) a ( ) a 3 3(3 ) a n 3 n n 3 The first three terms are 1, 0, 1. 3 ways100@gmail.com (Oct-015)

29 3. Algebra 1. Using cross multiplication method, solve 3x+5y 5, 7x+6y30 (Jun-01) x y x y x x 0 y x 0 0 y Hence, the solution is {0,5} y Solve each of the following system of equations by elimination method. 3x+y 8, 5x+y 10 3x+y x+y ( ) () (1) x x 1 Substitute x 1 in (1) 3(1) +y y 8 y y 5 Thus the solution is {1, 5} 3. Find a quadratic polynomial with zeros at x 1 and x 1. 4 (Jun-013) Quadratic polynomial x (α + β) x+ αβ Here α 1, β -1 4 Quadratic polynomial x ( 1-1) x+ (1 x 1) x x + 3 x x Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively. 3, Sum of the zeros α + β 3 Product of the zeros αβ Quadratic polynomial p (x) x (α + β) x + αβ x 3 x + 5. Find a quadratic polynomial if the sum and product of zeros of it are -4 and 3 respectively. Sum of the zeros α + β 4 Product of the zerosαβ 3 p (x) x (α + β) x + αβ x ( 4) x + 3 x +4 x Find the Greatest Common Divisor of 5bc 4 d 3, 35b c 5, 45c 3 d. 5bc 4 d bc 4 d 3 35 b c b c 3 45c 3 d 9 5 c 3 d G.C.D 5c 3 ways100@gmail.com

30 7. Find the GCD of the following. m 3m 18, m + 5m +6. (Apr-014) m 3m 18 (m 6) (m + 3) m + 5m +6 (m + ) (m + 3) G.C.D (m + 3) 8. Find the LCM of the following. 3(a 1), (a 1), a 1. (Jun-015) 3(a 1) 3(a 1) (a 1) (a 1) (a 1) a 1 (a 1) (a+1) L.C.M 3 (a 1) (a 1) (a + 1) 6(a 1) (a + 1) 9. Find the LCM of the following. x y + xy, x + xy (Oct-013) x y + xy xy (x + y) x + xy x (x+ y) LCM xy (x+ y) 10. Simplify into lowest form: x 4 x a 3 a a 1 x 3 +x x 4 x a 3 a x x a(a 1) a 1 x 3 +x a 1 x (x+) 11. Simplify into lowest form: 5x+0 5x+0 5(x+4) 5 7x+8 7 x+4 7 x+ (x ) x a(a 1) a(x ) a 1 x (x+) x 7x+8 1. Simplify into lowest form: x3 7 x 9 x 3 7 x x 3 (x +3x+9) (x +3x+9) x 9 x 3 x+3 x 3 (x+3) 13. Simplify into lowest form: x 81 x 4 x x +6x+8 x 5x 36 x 81 x x +6x+8 x 9 x x+ (x+4) x 4 x 5x 36 x x+4 (x 9) 14. Simplify into lowest form: x 9 x+9 x x+ (x+4) x x+ x+4 (x 9) x+9 x x x x+1 x 1 x x x x x x x+1 (x 1) x 1 x+1 x 1 x+1 x+1 (x 1) x+1 x x 15. What rational expression should be added to x3 1 to get x3 x +3? x + x + Let p(x) be the required rational expression p(x) x 3 x +3 (Apr-015) x + x 3 1 x + x 3 x +3 x 3 +1 x 3 x +4 x + x + ways100@gmail.com

31 16. Find the quotient and remainder using synthetic division ( 3x 3 + 4x 10x + 6) (3x-) (Oct-014) 3 3x 0 3x x Quotient 1 3 [3x + 6x 6] x + x, Remainder 17. Find the square root of (x+11) 44x (x + 11) 44x x + x x x x Find the square root of x x4 +. x x 4 x + 1 x + 1 (x 11) (x 11 x 19. Find the square root of 11x 8 y 6 81 x 4 y 8 11x 8 y 6 81 x 4 y 8 x 11x 8 y 6 11 x 4 81 x 4 y 8 9 y 11 9 x y 0. Solve : 3x 8 (Oct-013) x Solution: 3x 8 3x 8 x Solution 4 3, x 3x 8 x 3x x 8 0 (x ) (3x + 4) 0 x or x The product of two consecutive positive even numbers is 4. Then find the numbers. (Oct-1) Let us consider the two consecutive positive even numbers are x, x + Given x (x+) 4 x +x 4 0 (x- 6) (x + 8) 0 x 6 or 8 x, x + are positive. So, the numbers are 6, 8 ways100@gmail.com

32 . Solve x + 1 x 6 Solution: x + 1 x x +1 6 x 5 5x + 5 6x 5x 6x (x 5) (5x 1) 0 x 5 or x 1 5 Solution 5, Determine the nature of roots of the following quadratic equations x 11 x ax + bx + c 0 x 11x 10 0 a 1, b 11, c 10 b 4ac ( 11) 4 x 1 ( 10) > 0 The roots are real but unequal 4. Determine the nature of roots of the following quadratic equations x +5 x ax + bx + c 0 x +5x+5 0 a, b 5,c 5 b 4ac (5) 4 x x < 0 No real roots or imaginary roots. 5. Determine the nature of the roots of the equations. x 8 x+1 0. ax + bx + c 0 x 8x+1 0 a 1, b 8, c 1 b 4ac ( 8) 4 ( 1 )( 1) > 0 The roots are real but unequal 6. Determine the nature of the roots of the equations. x 3x+4 0. ax + bx + c 0 x 3x+4 0 a, b 3, c 4 b 4ac ( 3) 4 ()( 4) < 0 The equation has no real roots 7. Determine the nature of roots of the following quadratic equations 4x 8 x ax + bx + c 0 4x 8x a 4, b 8, c 49 b 4ac ( 8) (4 x 4 x 49) The roots are real and equal. (Apr-013, Jun-013) 8. Determine the nature of the roots of the equations. (x a) (x b) 4ab. (Jun-014) (x a) (x b) 4ab x x (a+b) +4ab 4ab x x (a + b) 0 ax + bx + c 0 a 1, b (a + b) c 0 b 4ac [ (a + b)] (4 x 1 x 0) 4(a + b) > 0 Thus, the roots are real but unequal ways100@gmail.com

33 9. Find the values of k for which the roots are real and equal in each of the following equations. 1x + 4kx (Jun-015) Given : 1x + 4kx The roots are real and equal b 4ac 0, a 1, b 4k, c 3 (4k) 4 ( 1 )( 3) 0 16k k 144 k 144 k 9 k ± Form the quadratic equation whose roots are and 7 3. (Apr-014, Oct-014) Sum of the roots Product of the roots (7 + 3 ) x (7 3 ) 7 ( 3) The required equation is x sum of the roots x + product of the roots 0 x 14x Form the quadratic equation whose roots are and 3 7. Sum of the roots Product of the roots (3+ 7 ) x (3 7 ) 3 ( 7) 9 7 The required equation is x sum of the roots x + product of the roots 0 x 6x Form the quadratic equation whose roots are 3, 4. (Jun-01, Oct-01) The required equation is x sum of the roots x + product of the roots 0 x (3 + 4) x + (3 x 4) 0 x 7x If the sum and product of the roots of the quadratic equation ax - 5x + c 0 are equal to 10, then find the values of a and c. (Apr-013) The given equation is ax 5x + c 0 Sum of the roots b 10 ( 5) a a a Product of the roots C a 10 c 10a 10 x 1 5 c 5 a If α and β are the roots of the equation 3x 6x + 4 0, find the value of α + β. α and β are the roots of the equation 3x - 6x (Apr-01)(Mar-016) α + β b 6 a 3 αβ c a 4 3 α + β (α + β) αβ () x 4 3 α + β ways100@gmail.com

34 35. If α and β are the roots of the equation 3x 5x + 0, then find the values of α β + β α 3x -5x + 0, here a3, b-5, c α + β b ( 5) a αβ c a 3 α + β α +β β α αβ (α+β ) αβ αβ Solve 3x 5y 16, x + 5y 31 (Oct-015) Solution: 3x 5y x + 5y 31 (1) + () 5x 15 x 3 x 3 sub in (1) 3x 5y y 16 5y y 5 y 5 Solution {3, 5} 37. Simplify x3 + 8 x x x x 3 3 x x x x x 3 3 x x x +x+4 x x + x Martices If A 4 5, then verify that (A T ) T A A A T A T T 4 5 A Hence verified (Mar-016) a 3 b 3 a b (a + ab + b ) (Apr-015) ways100@gmail.com

35 8 5. If A 1 3 4, then find AT and A T T. (Jun-01, Apr-013, Jun-014, Apr-015) A A T (A T ) T If A Transpose of A A T, then find the transpose of A If A , (i) find the order of the matrix (ii) write down the elements a 4 and a 3 (iii) in which row and column does the element 7 occur? i)the order of the matrix 3 x 4 ii) a 4 4, a 3 0 iii) The element 7 occurs a Let A [a ij ] 6 5. Find (i) the order of the matrix (ii) the elements a 13 and a 4 (iii) the position of the element. i) Since the matrix A has 4 rows and 3 columns, A is of order 4 x 3 ii) The element a 13 is in the first row and third column. / a Similarly, a 4, the element in 4 th row and nd column. iii) The element occurs in nd row and nd column / a. 6. If A and B 8 find 6A - 3B. (Apr-01) A 3B Let A and B. Find the matrix C if C A+B. (Jun-014) C A + B If A, then find the additive inverse of A. (Oct-1, 13, Apr- 015, Jun-015) A The additive inverse of A 16 6 ways100@gmail.com

36 Let A and B. Find A + B if it exists Order of A x 3 Order of B x Order of A Order of B Addition of matrices A and B is not possible If A and B , then find A+B. 8 3 A+B Find the values of x, y and z if x z 5 y 1. x z 5 y 1 x 3, y 9, z 4 1. A matrix consists of 30 elements. What are the possible orders it can have? (Jun-013) The possible orders of the matrices having 30 elements are 1 30, 15, 3 10, 5 6, 6 5, 10 3, 15, A matrix has 8 elements. What are the possible orders it can have? The possible orders of the matrices having 8 elements are 1 8, 4, 4, Construct a x matrix A [a ij ] whose elements are given by a ij ij A a 11 a 1 a 1 a ij A a , a 1 1, a 1 1, a Construct a x matrix A [a ij ] whose elements are given by a ij i j a 11 a 1 A a 1 a i j 1 4 a , a a , a A Construct a x matrix A [a ij ] whose elements are given by a ij i j A a a 1 1 A a 11 a 1 a 1 a i j i+j 0, a 1 1 3, a Find the product of the matrices, if exists, ( 1) ( 1) ways100@gmail.com i+j.

37 18. Construct a x3 matrix A [a ij ] whose elements are given by a ij i 3j. (Apr-014) A a 11 a 1 a 1 a a 13 a 3 i 3j a (1) a 1 1 3() a 13 (1) 3(3) a 1 3(1) a 3() 4 6 a 3 3(3) A Find a and b if a 1 + b a 1 + b Adding (1) & () 5a 15 a 3a + b 10 b 5 a 15/5 a 3 a b 3a + b 10 5 Substituting a 3 in () a b ( 3) + b b 5 3a + b 5.. b a 15 b If A 4 and B, then find AB and BA if they exist AB 4 BA If A 4 0 and B 3 0 then find AB and BA. Are they equal? AB BA AB BA. 5 Prove that A 7 3 and B are inverses to each other under matrix multiplication. AB BA I AB I 1 ways100@gmail.com

38 3 5 BA I. From (1) and (), we have AB BA I Thus, the given matrices are inverses to each other under matrix multiplication Prove that 1 and are multiplicative inverses to each other. AB BA I AB I BA I The given matrices are inverses to each other under matrix multiplication A, then verify AI IA A, where I is the unit matrix of order. (Apr-014) AI A IA A Hence AI IA A (Oct-013) 5. Find the product of the matrices, if exists, (Apr-01, Jun-01) x 4 + ( x ) 3 x 1 + ( x 7) 5 x x 5 x 1 + (1 x 7) 1 + ( 4) 3 + ( 14) (7) 1 6. Find the product of the matrices, if exists, (Oct-014,Mar-016) Find the product of the matrices, if exists, ( 7) ( 7) Solve for x and y if x + y x 3y (Oct-014) x + y x 3y 5 13 x + y 5 1 x 3y 13. () + (1) () x 3y 13 (1) 3 6x + 3y 15 7x 8 substituting x 4 in (1) ( 4) + y 5 y Solution {4, 3} ways100@gmail.com

39 9. Determine whether each matrix product is defined or not. If the product is defined, state the dimension of the product matrix. A x 5 and B 5 x 4. (Jun-015) Now, the number of columns in A and the number of rows in B are equal. So, the product AB is defined. Also, the product matrix AB is of order x Determine whether each matrix product is defined or not. If the product is defined, state the dimension of the product matrix. A 1 x 3 and B 4 x 3. Now, the number of columns in A and the number of rows in B are not equal. So, the matrix product AB is not defined. 31. For the matrices A and B, the product AB exists but BA does not exist. What can you say about the order of A and B? (Apr-013) The number of columns in A and the number of rows in B are not equal. So, order of A and order of B are not equal. 3. A 1 1, B 1 4, then find (AB) T (Jun-013) A (1 1) B 1 4 AB 1 1 (AB) T (0) 3 x 33. Solve 4 5 y 8 13 Solution: 3 x 4 5 y x + y 4x + 5y ( + 4) (0) Cross multiplication method 8 3 3x + y 8 3x + y x + 5y 13 4x + 5y 13 0 x y x y 1 x 14 and y x & y Find the values of x, y and z, if x + y y + z 7 9 x + y 7 z 5 0 y + z 9 In (3) z 5 0 z 5 z 5 0 substitute z 5 in () x + y 7 1 y y y + z 9 substitute y 4 in (1) x x z The values of x 3, y 4, z 5 (Oct-015) (Mar-016) ways100@gmail.com

40 5. COORDINATE GEOMETRY 1. Find the midpoint of the line segment joining the points (3, 0) and ( 1, 4). x 1, y 1 x, y (3, 0), (-1, 4) Mid point x 1 + x, y 1+ y 3 1, 0+4, 4 (1,). Find the midpoint of the line segment joining the points (1, 1) and ( 5, 3). x 1, y 1 x, y (1, 1), ( 5, 3) Mid point x 1 + x, y 1+ y 1 5, 1+3 4, (, 1) 3. Find the midpoint of the line segment joining the points (0, 0) and (0, 4). x 1, y 1 x, y (0, 0), (0, 4) Mid point x 1 + x, y 1+ y 0+0, 0+4 0, 4 (0,) 4. The centre of a circle is at ( 6, 4). If one end of a diameter of the circle is at the origin, then find the other end. (Jun-01, 014, Apr-015) x, y x 1, y 1, (0, 0) Centre of the circle is the midpoint of the diameter. x 1 + x, y 1+ y x 1 + 0, y 1+ 0 x 1 ( 6, 4) ( 6, 4) 6 x 1 1 y 1 4 y 1 8 Other end is ( 1, 8) 5. Find the centroid of the triangle whose vertices are (1, 3), (, 7) and (1, 16). x 1, y 1 x, y x 3, y 3 (1, 3), (, 7), (1, 16) Centroid G x 1 + x +x 3, y 1+ y +y , , (5, ) 6. Find the centroid of the triangle whose vertices are (3, 5), ( 7, 4) and (10, ). x 1, y 1 x, y x 3, y 3 (3, 5), ( 7, 4), (10, ) Centroid x 1 + x +x 3, y 1+ y +y , 5+4 6, (, 1) ways100@gmail.com

41 7. Find the centroid of the triangle whose vertices are A (4, 6), B (3, ) and C (5, ). (Oct-01) x 1, y 1 x, y x 3, y 3 (4, -6), (3, ), (5, ) Centroid x 1 + x +x 3, y 1+ y +y , 6 + 1, (4, ) 8. If the centroid of a triangle is at (1, 3) and two of its vertices are ( 7, 6) and (8, 5) then find the third vertex of the triangle. (Apr-01)(Mar-016) x 1, y 1 x, y x 3, y 3 ( 7, 6), (8, 5) (x 3, y 3 ) Centroid x 1 + x +x 3, y 1+ y +y x 3, 6+5+y x 3, 11+y x (1, 3) (1, 3) (1,3) y ways100@gmail.com x y x y x 3 y 3 Third vertex (, ) 9. Find the area of the triangle formed by the points (0, 0), (3, 0) and (0, ). A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } A {6 0} sq. units A Find the area of the triangle formed by the points (5, ), (3, 5) and ( 5, 1). A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } A A 1 { 38 6} A 1 { 64} A 3 Sq. units 11. Find the area of the triangle formed by the points ( 4, 5), (4, 5) and ( 1, 6). A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } A [ } A 1 { } A 1 { 38} A 19 Sq. units 1. Find the area of the triangle whose vertices are (1, ) ( 3, 4) and ( 5, 6). A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } A [ } 4 6

42 A 1 {1 + 3} A 1 44 Sq. units 13. Show that the points A (, 3), B (4,0) and C (6, 3) are collinear. A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } 4 6 A [ } A 1 {6 6} A The given points are collinear 14. Determine if the following set of points are collinear or not. A (4, 3), B(1, ) and C (, 1). A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } A [ } A 1 {3 3} A The given points are collinear 15. Determine if the following set of points are collinear or not. A (, ), B ( 6, ) and C(,) A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } 6 A [ } A 1 { 4 1} A 1 { 16} A 8 A 0 Hence the given three points are not collinear. 16. Determine if the following set of points are collinear or not. ( 3, 3), (6, ) and ( 3, 4) A 1 { x 3 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } 6 3 A [ } A 1 {18 18} A Hence, the given three points are collinear. 17. If the three points a, 1, 1,, (0, b + 1) lie on a straight line, then show that (Apr-013) a b a b+1 0 a 1 a + b ab + a 0 ways100@gmail.com

43 a + b ab a 0 a + b ab 0 a + b ab (Divided by ab on both sides) a+b ab ab ab a + b ab 1 a + 1 b ab 1 1 Hence proved. 18. If the area of the ABC is 68 sq.units and the vertices are A 6, 7, B 4, 1 and C(a, 9) taken in order, then find the value of a. (Jun-013, Jun-015) A 68 A 1 { x 1y + x y 3 + x 3 y 1 x y 1 + x 3 y + x 1 y 3 } a a [ 8 + a 54 } 68 x a [ 8 + a } a + 8 a a 136 6a 1 a 1/6 a 19. Find the equations of the straight lines parallel to the coordinate axes and passing through the point ( 5, ). (Jun-013) Equation of the straight line passing Equation of the straight line passing through ( 5, ) and parallel to x axis through ( 5, ) and parallel to y axis y k x k y x 5 y + 0 x Show that the straight lines x + y and 3x + 6y + 0 are parallel. Slope of x + y is m 1 x 1 Slope of 3x + 6y + 0 is m x m 1 m Hence, the two straight lines are parallel. y y Show that the straight lines 3x + y 1 0 and 6x + 4y are parallel. Slope of 3x + y -1 0 is m 1 Slope of 6x + 4y is m coefficient of x coefficient of y coefficient of x coefficient of y m 1 m Hence, the two straight lines are parallel Prove that the straight lines x + y and x y are perpendicular to each other. Slope of x + y is m 1 coefficient of x 1 coefficient of y coefficient ofx coefficient of y Slope of x -y is m 1 m 1 m 1 The two straight lines are perpendicular. ways100@gmail.com

44 3. Show that the straight lines 3x 5y and 15x+9y are perpendicular. (Apr-013) Slope of 3x 5y is m 1 Slope of 15x+9y is m m 1 m 3 x The two straight lines are perpendicular. coefficient of x 3 3 coefficient of y 5 5 coefficient of x 15 5 coefficient of y Find the value of a if the straight lines 5x y 9 0 and ay + x 11 0 are perpendicular to each other. (Mar-016) Slope of 5x y 9 0is m 1 Slope of x+ay 11 0 is m m 1 x m 1 5 x 1 a a 5 a 5 coefficient of x coefficient of y coefficient of x coefficient of y a 3 5. Find the equation of straight line whose angle of inclination is 45 0 and y-intercept is 5 (Oct-013) Slope m tan θ tan y intercept c 5 The equation of the straight line is y mx + c y 1x+ 5 5y 5x + 5x 5y Find the equation of the straight line parallel to the line 3x y+70 and passing through the point (1, ). Equation of the straight line parallel to 3x y is 3x y + k 0 (Jun-015) It passes through (1, ), we have x 1, y 3(1) ( ) + k k 0 5+k 0 k 5 Thus, the equation of the required straight line is 3 x y Find the slope and y-intercept of the line whose equation is 4 x y (Apr-015) 4 x y Aliter Slope y- intercept coefficient of x 4 coefficient of y constant 1 1 coefficient of y y mx + c y 4x 1 y x + m, c 1 Slope, y- intercept 1 8. Find the x and y intercepts of the straight line x y x y constant x intercept coefficient of x constant y-intercept coefficient of y 1 x intercept 8, y intercept 16 ways100@gmail.com

45 9. Find the equation of the straight line passing through the points (, 5) and (3, 6). Equation of the straight line in two-points form is y y 1 y 5 x y 5 x y 5 x + 5y 5 x + x 5y Thus, the required equation is x 5y x x 1 y y 1 x x Find the equation of a straight line whose slope is -3 and y-intercept is 4. Slope m 3 y -intercept c 4 y mx + c y 3x + 4 3x y Thus, the required equation is 3x y If 7, 3, 6, 1, 8, and (p, 4) are the vertices of a parallelogram taken in order, then find the value of p (Jun-01) Let the vertices of the parallelogram ABCD be A 7,3, B 6,1, C 8, and D(p, 4) We know that the diagonals of a parallelogram bisect each other Mid points of the diagonal AC Midpoint of the diagonal BD Mid point 7+8, 3+ Equating the x- coordinates 6+P P 15 x 1 +x, y 1+y 6+P, 1+4 P P 9 3. Find the equation of the straight line which passes through the midpoint of the line segment joining (4,) and (3,1) whose angle of inclination is 30 (Oct-01) Given that the angle of inclination θ 30 0 Thus, the slope m tan θ tan The midpoint of the straight line joining (4, ), (3, 1) x 1 +x, y 1+y 4+3, +1 7, 3 The equation of the straight line in slope point form is y y 1 m( x x 1 ) y x 7 Note: m 1 3 y x 7 3(y 3) x 7 (x 1, y 1 ) ways100@gmail.com , 3

46 3y 3 3 x x 3 y + (3 3 7) The coordinates of the midpoint of the line segment joining the points a +, 3, (4, b + 1) are (a, b). find the values of a, b. (Apr-014) The midpoint of the line segment joining the points (a +, 3), 4, b + 1 (a, b) x 1 +x, y 1+y a++4, 3+b+1 a+6 a (a, b) (a, b) b+4 b a + 6 4a b + 4 4b a 6 b 4 a 3 b a 3, b 34. Find the equations of the straight lines parallel to the coordinate axes and passing through the point (3, 4) (Apr-014) Let L and L be straight lines passing through the point (3, 4) and parallel to x-axis and y-axis respectively The y-coordinate of every point on the line L is 4, The equation of the line L is y 4 The x-coordinate of every point on the line L is 3, The equation of the line L is x Find the equation of the straight line whose slope is 1 3 (x 1, y 1 ) (,3) Slope m 1 3 Equation of straight line y y 1 m x x 1 y x + 3y 9 x + x 3y and passing through (, 3) (Jun- 014) 36. Find the coordinates of the point which divides the line segment joining 3, 5, (4, 9) in the ratio 1 :6 internally (Oct-014) The point P divide & internally lx +mx ( 3) (5), , , 1 7 7, ly +my 1 l+m l+m x 1 3, x 4, y 1 5, y 9 l 1, m 6 P (,3) ways100@gmail.com

47 37. P and Q trisect the line segment joining the points (,1) and (5, 8). If the point P lies on x y + k 0, then find the value of k. (Oct-014) P and Q trisect the line segment AB. P lx +mx 1, ly +my 1 l+m l+m 1 5 +() (1), , , 6 3 P (3, ) P lies on x y + k k k 0 k 8 [x 3, y ] 38. The side AB of a square ABCD is parallel to x-axis. Find the slope of the diagonal A ( Oct-015) The side AB of a square ABCD is parallel to x-axis. Now, the diagonal AC bisects the angle DAB Thus BAC 45 0 θ 45 0 Slope of diagonal AC- m tanθ tan GEOMETRY 1. In ABC, AE is the external bisector of A, meeting BC produced at E. If AB 10 cm, AC 6 cm and BC 1 cm, them find CE. In ABC, AE is the external bisector of A, meeting BC produced at E. Let CE x By, angle bisector theorem BE AB CE AC 1+x x x x 5 3 3(1 + x) 5x x 5x 36 5 x 3x x 36 x 18 EC 18 cm. In the figure, find CD (Apr-015) EA x EB EC x ED 8 x 5 (x + 4) x 4 8 x 5 4 x + 4 x x CD 6 cm ways100@gmail.com

48 3. In ABC, DE BC and AD DB 3 By Thales theorem, we have AD EC EC 3.7 x cm. If AE 3.7 cm, find EC. AE DB EC (Jun- 01, Jun-014) 4. In a ABC, D and E are points on the sides AB and AC respectively such that DE BC. If AD 6cm, DB 9 cm, and AE 8 cm, then find AC. By Thales theorem, we have AD EC EC 8 x 9 AC cm 6 AE DB EC 1cm 5. In a ABC, D and E are points on the sides AB and AC respectively such that DE BC. If AD8 cm. AB1cm and AE1cm, then find CE. By Thales theorem, we have AD AE DB EC EC EC 1 x cm 6 8 CE 6 cm 6. In ABC, the internal bisector AD of A meets the side BC at D. If BD.5 cm, AB 5 cm and AC 4. cm, then find DC. (Apr-01, Oct-01, 013, June-015) By angle bisector theorem, BD.5 5 DC DC DC.1 cm 5.1 AB DC AC 7. In PQR, give that S is a point on PQ such that ST QR and PS SQ 3 5 If PR 5. 6cm, then find PT. In PQR we have PT x / TR PR PT 5.6 x ST QR by Thales theorem PS PS PT TR PT SQ TR x (5.6 x) 5x x 5x + 3x 16.8 x SQ 8 / PT.1 cm ways100@gmail.com

49 8. In a ABC, AD is the internal bisector of A, Meeting BC at D. If BD cm, AB 5cm, DC 3 cm find AC. By angle bisector theorem, BD 3 5 AC AC 5 3 AC 7.5 cm AB DC AC 9. In a ABC, AD is the internal bisector of A, Meeting BC at D. If AB 5.6 cm, AC 6cm and DC 3 cm find BC. By angle bisector theorem. BD BD BD 5.6 x 3.8 BD.8 cm BC cm 10. In the figure, find x PA x PB PC x PD 4 x x 8 x 3 x 8 x 3 x AB DC AC 11. AB and CD are two chords of a circle which intersect each other internally at P. If CP4cm, AP 8cm, PBcm then find PD (Apr- 014) PA x PB PC x PD 8 4 x x x 4 x 4 cm PD 4 cm 1. AB and CD are two chords of a circle which intersect E ach other internally at P. If AP1cm, AB 15cm, CPPD then find CD PA PB PC PD 1 3 x x 36 x x 6 CD 6+6 1cm 13. Find the value of x in each of the following diagrams. PA PB PC PD 9 4 (x + ) (x+) (x+) x+ 18 x x 16 cm ways100@gmail.com

50 14. AB and CD are two chords of a circle which interest each other externally at P If AB 4 cm BP 5 cm and PD 3 cm, then find CD. PA PB PC PD 9 5 (x + 3) x+3 x+3 3 x x CD 1cm 15. AB and CD are two chords of a circle which interest each other externally at P If BP 3 cm, CP 6cm and CD cm, then find AB. PA PB PC PD (x+ 3) x x+3 8 x AB 5 cm 16. In the figure TP is a tangent to a circle. A and B are two points on the circle. If BTP 7 0 and ATB 43 0 find ABT. (Apr-013,Mar-016) BAT BTP 7 0 [Angles in alternate segment] ABT, we have ATB + BAT + ABT ABT ABT Let PQ be a tangent to a circle at A and AB be a chord. Let C be a point on the circle such that BAC 54 0 and BAQ 6 0. Find ABC. Since PQ is a tangent at A and AB is a chord, we have BAQ ACB 6 0 (tangent-chord theorem) BAC + ACB + ABC (Sum of all angles in a triangle is ) ABC ( BAC + ACB) ( ) 64 0 Hence, ABC Pythagoras theorem (Baudhayan theorem). In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. BC AB + AC 19. Converse of Pythagoras theorem In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. BC AB + AC, Then BAC 90 0 ways100@gmail.com

51 0. Tangent Chord theorem If from the point of contact of tangent (of a circle), a chord is drawn, then the angles which the chord makes with the tangent line are equal respectively to the angles formed by the chord in the corresponding alternate segments. BAT BPA & BAS AQB 1. Converse of Tangent Chord theorem. If in a circle, through one end of a chord, a straight line is drawn making an angle equal to the angle in the alternate segment, then the straight line is a tangent to the circle.. In MNO, MP is the external bisector of M, meeting NO produced at P. If MN 10cm, MO 6cm, NO 1cm then find OP (Jun-013) In MNO, MP is the external bisector of M, meeting NO produced at P MNO, by the angle bisector theorem, we have NP MN OP MO 1+OP MN OP MO 1+OP 10 OP x OP 10x OP 10 x OP 6 x OP 7 4 x OP 7 OP cm 7. Trigonometry 1. A ladder leaning against a vertical wall, makes an angle of 60 with the ground. The foot of the ladder is 3.5m away from the wall. Find the length of the ladder. (Oct-1, Apr-13, Jun-14, Jun-15) In the right CAB cos AB AC 3.5 AC AC 3.5 x 7 The length of ladder 7m. Find the angular elevation (angle of elevation from the ground level) of the Sun when the length of the shadow of a 30 m long pole is 10 3m. In the right CAB tanθ BC AB (Apr-01, Apr-014) 3 tan / The angular elevation of the sun from the ground level θ 60 0 ways100@gmail.com

52 3. A ramp for unloading a moving truck, has an angle of elevation of If the top of the ramp is 0.9m above the ground level, then find the length of the ramp. (Oct-014, Apr-015) In the right ABC sin AB AC 0.9 AC AC 0.9 x 1.8 The length of the ramp 1.8 m 4. A girl of height 150 cm stands in front of a lamp-post and casts a shadow of length cm on the ground. Find the angle of elevation of the top of the lamp-post. (Jun-01) In the right ABC tan θ AB BC tan /The angle of elevation θ Prove the following identities 1 sinθ 1+sinθ Hence proved. 1 sinθ 1+sinθ 1 sinθ 1 sinθ (1 sinθ ) 1+sinθ 1 sinθ 1 sin θ (1 sinθ ) co s θ secθ tanθ. (1 sin θ cos θ) 1 sin θ (Oct-01, Jun-014) ways100@gmail.com cosθ 1 1 sin θ cosθ 1 sinθ cosθ cosθ secθ tanθ 6. Prove the following identities sec θ + cosec θ tan θ + cot θ. (Apr-014, 015) sec θ + cosec θ 7. Prove the identity. sinθ + cosθ cosecθ sinθ + cosθ sinθ 1 + cosθ 1 cosecθ secθ sinθ cosθ 1 + tan θ cot θ tan θ + cot θ + tan θ + cot θ + tanθ. cotθ (tanθ + cotθ) tan θ + cot θ Hence proved secθ 1. sinθ. sinθ + cosθ. cosθ sin θ + cos θ 1 Hence proved cosθ 8. Prove the following identities. 1+sinθ. secθ tanθ cosθ secθ tanθ Hence proved cosθ secθ tanθ cosθ (secθ +tanθ ) sec θ tan θ 1+ sinθ secθ +tanθ secθ +tanθ cosθ x 1 cosθ + cosθx sinθ cocθ (Jun- 01, Oct-014) (sec θ tan θ 1) (Jun-013)

53 1 1 sin θ sin θ 1 cos θ 1 1 sin θ sin θ 9. Prove that 1. (Apr-013) 1 cos θ cosec θ cos θ sin θ cosec θ cot θ 1 Hence proved (cosec θ cot θ 1) 10. Prove the identity. (sin 6 θ + cos 6 θ) 1 3sin θ cos θ. (Apr-01) sin 6 θ + cos 6 θ (sin θ) 3 + (cos θ) 3 (a 3 +b 3 (a+b) 3 3ab(a+b)) (sin θ + cos θ) 3 3 sin θ cos θ (sin θ + cos θ) 1 3sin θ cos θ (sin θ +cos θ 1) Hence proved 11. Prove that 1+secθ 1+secθ secθ secθ 1+ 1 cosθ 1 cosθ 1+cos θ sin θ. 1 cosθ cosθ +1 x cosθ cosθ 1 (1+cos θ) x (1 cosθ ) (1 cosθ ) (1+cos θ) x (1 cosθ ) 1 cos θ 1 cosθ sin θ 1 cosθ Hence proved (1 cosθ ) (Oct-013)(Mar-016) 1. A kite is flying with a string of length 00m. If the thread makes an angle 30 with the ground, find the distance of the kite from the ground level. (Here, assume that the string is along a straight line) Let h denote the distance of the kite from the ground level BC h AC 00m In right CAB sin30 0 BC 1 h 00 h 00 AC 100 BC h 100m Hence, the distance of the kite from the ground level is 100m. 13. Prove that identity. 1 cosθ 1+cosθ 1 cosθ 1+cosθ cosecθ cotθ 1 cosθ x 1 cosθ 1+cosθ 1 cosθ (1 cosθ ) sin θ (1-cos θ sin θ ) 1 cos θ sinθ 1 1 cos θ sinθ 1 cosθ sinθ sinθ cosecθ cotθ ways100@gmail.com

54 14. If ABC is right angled at C, then find the values of cos(a + B) and sin (A + B) (Mar-016) Sum of all angles in a triangle is C 90 0, So A + B 90 0 cos(a + B) cos sin (A + B) sin A boy of height 5 feet away from the pillar of distance 100 feet saw the top of the pillar with angle of elevation Find the height of the pillar. (Jun-013) AB A Boy 5 feet BC Distance between boy and pillar Height of the pillar CE CD + DE In right angle ADE tan 45 0 DE / DE 100 AD DE 5 + DE (CD AB 5feet ) Height of the pillar CE feet 16. If sinθ cos θ, θ is an acute angle show that tan θ cos θ 0 (Oct-013) θ is an acute angle, and sinθ cos θ then ; θ 45 LHS tan θ cos θ tan 45 cos ways100@gmail.com RHS 8. Mensuration 1. A solid right circular cylinder has radius 7cm and height 0 cm. Find its (i) curved surface area and (ii) total surface area. Right circular cylinder: r 7cm h 0cm (i)curved Surface Area πrh x x 7 x sq.cm 7 (ii)total Surface Area πr(h+r) x 7 x 7 (0+7) 1188 sq.cm. A solid right circular cylinder has radius 14cm and height 8 cm. Find its curved surface area and total surface area. (Apr-014) r 14cm h 8cm Curved Surface Area πrh x x 14 x sq.cm 7 Total Surface Area πr(h+r) x 7 x 14 (8+14) 1936 sq.cm

55 3. The radii of two right circular cylinders are in the ratio of 3: and their heights are in the ratio 5:3. Find the ratio of their curved surface areas. (June-015) cylinder : r 1 : r 3 :, h 1 : h 5 : 3 The ratio of the curved surface areas πr 1 h 1 : πr h 3 x 5 : x 3 5 : 4. Find the volume of a solid cylinder whose radius is 14 cm and height 30 cm. (Oct-01) cylinder : r 14cm h30cm Volume πr h 7 x 14 x 14 x cu.cm 5. The radii of two right circular cylinders are in the ratio :3. Find the ratio of their volumes if their heights are in the ratio 5:3. cylinder : r 1 : r : 3, h 1 : h 5 : 3 The ratio of the volumes πr 1 h 1 : πr h ( 5) : (3 3 3) 0 : 7 6. Radius and slant height of a solid right circular cone are 35 cm and 37 cm respectively. Find the curved surface area and total surface area of the cone. Cone : r 35 cm l 37cm Curved surface Area πr l 7 x 35 x sq.cm Total surface Area πr (l + r) 7 7 x 35 (37+35) x 35 x7 790 sq.cm 7. If the circumference of the base of a solid right circular cone is 36 cm and its slant height is 1cm, find its curved surface area. (Apr-013, Jun-013) Cone : l 1cm Circumference πr 36cm πr Curved surface area π rl 118 x sq.cm 8. The radius and height of a right circular solid cone are 7 cm and 4 cm respectively. Find its curved surface area and total surface area. (Jun-01) Right circular solid cone : r7 cm h 4 cm l r + h cm ways100@gmail.com

56 (i)curved Surface Area πrl x 7 x sq.cm (ii)total Surface Area πr(l+r) 7 x 7 x (5+7) 704 sq.cm 9. The circumference of the base of a 1 cm high wooden solid cone is 44 cm. Find the volume. (Oct-013) Solid cone: h1cm Base circumference of the wooden solid πr 44cm x 7 x r 44 r 44 x 1 x 7 r7cm Volume 1 3 πr h 1 3 x 7 x 7 x 7 x cm Radius and slant height of a right circular cone are 0 cm and 9 cm respectively. Find its volume. Right circular cone: (Apr-01) r 0cm l9cm Volume 1 3 πr h h l r h 1cm 1 x x 0 x 0 x cm The volume of a solid hemisphere is 115 π cu.cm. Find its curved surface area. Volume of a solid hemisphere 115 π π 3 r3 115 π r x 3 r r 1cm CSA π r x π x 1 x 1 88 π sq.cm 1. The total surface area of a solid right circular cylinder is 660 sq.cm. If its diameter of the base is 14 cm, find the height and curved surface area of the cylinder. (Oct-01) Right circular cylinder: Diameter 14cm Radius 7 cm ways100@gmail.com

57 Total surface Area 660 sq.cm πr (h + r) (h+7) 660 (h + 7) h h cm h8 cm Curved surface area πrh x 7 x 7x 8 35 sq.cm 13. The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of volumes of the balloon in the two cases. Sphere: r 1 7cm.. r 14 cm The ratio of volumes 4 3 πr 1 3 : 4 3 πr 3 (7 x 7 x 7) :( 14 x 14 x 14 ) 1: Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 14 cm. Diameter of the cone 14 cm Radius(r) 7 cm Height(h) 14 cm Volume 1 3 πr h 1 x x 7x 7x cm Find the volume of a sphere shaped metallic shot-put having diameter of 8.4 cm. (Apr-13, Jun-15) Diameter of the sphere 8.4 cm Radius(r) 4. cm 4 Volume 4 3 πr3 4 3 x 7 x x 4 x cm If the curved surface area of a solid hemisphere is 77 sq.cm, then find its total surface. CSA of a solid hemisphere πr 77 sq.cm πr 77 πr 1386 TSA of a solid hemisphere 3πr 3 x sq.cm 17. Radii of two solid hemispheres are in the ratio 3:5. Find the ratio of their curved surface areas and the ratio of their total surface areas. Hemisphere : r 1 : r 3 : 5 Ratio of the curved surface areas πr 1 : πr (3 x 3) : (5 x 5) 9 : 5 Ratio of the total surface area 3πr 1 : 3πr (3 x 3) : (5 x 5) 9 : 5 ways100@gmail.com

58 18. If the curved surface area of solid a sphere is 98.56cm, then find the radius of the sphere. (Oct-013) Curved surface area of a solid sphere cm 4πr x 7 r r x 1 x 7 4 r 7.84 Radius r.8 cm 19. Total surface area of a solid hemisphere is 675 sq. cm. Find the curved surface area of the solid hemisphere. (Jun-014) TSA of a solid hemisphere 3πr 675π πr 675π 5π 3 CSA πr 5π x 450π sq.cm 0. The internal and external radii of a hollow cylinder are 1 cm and 18 cm respectively. If its height is 14cm, then find its curved surface area and total surface area. (Apr-01) HOLLOW CYLINDER: r 1 cm R 18cm h14 cm CSA πh (R+r) TSA π(r+r) (R-r+h) x 7 x 14 x (18 + 1) x 7 x 7 x30x0 x x x sq.cm sq.cm x (18+)( ) 1. The radius of a circular well is 5m. Find the cost of cementing of surface to the depth of 14m at the rate of Rs. per square metre. (Oct-014) Right circular cylinder: r 5 m h14m Inner surface of the well (CSA) πrh x 7 x 5 x 14 x x 5 x 440 sq.m The cost of cementing, the rate of RS. per square metre Rs. The cost of cementing 440sq.metre Rs. 440 x Rs.880. The radii of the spheres are 3cm, 4cm respectively. Find the ratio of their volume. (Jun-01) Sphere: R 4cm r 3cm Ratio of the spheres volume 4 3 πr3 : 4 3 πr3 R 3 :r :3 3 64:7 3. Volume of a solid cylinder is 6.37 cu. cm. Find the radius if its height is 4.5 cm. ( Jun-013) Height of the cylinder h4.5 cm Volume V 6.37 r? V πr h x r x r 6.37 x 7 x 4.5 r cm ways100@gmail.com

59 4. The thickness of a hemispherical bowl is 0.5cm. The inner radius of the bowl is 5cm. Find the outer curved surface area of the bowl (Take π (Apr-015) Hemisphere: Inner radius r 5cm Thickness w 0.5 cm Outer radius R cm Outer curved Surface Area πr x 7 ways100@gmail.com ) x 5.5 x sq.cm 5. The outer and the inner radii of a hollow sphere are 1 cm and 10 cm. Find its volume. Hollow sphere : Outer radius R 1cm Inner radius r 10cm Volume of the hollow sphere V 4 3 π (R3 r 3 ) 4 3 x 7 ( ) ( ) x cm3 6. The volume of a solid right circular cone is 498 cu.cm. If its height is 4 cm, then find the radius of the cone. Volume of the right circular cone 498cm 3, height h 4cm 1 3 πr h x x 3 7 r x r 498 x 3x x 4 r cm 7. The volume of a cone with circular base is 16 πcu. cm. If the base radius is 9 cm, then find the height of the cone. Cone: Radius r 9cm Volume 16 π Volume of the cone 1 3 π r h 16 π 1 3 x π x 9 x h 16 π h 16 x 3 9x9 8 cm 8. A hollow sphere in which a circus motorcyclist performs his stunts, has an inner diameter of 7 m. Find the area available to the motorcyclist for riding. Inner diameter of the hollow sphere r 7m Available area to the motorcyclist for riding Inner surface area of the sphere 4 πr π(r) 7 x7 7 x 7 x m 9. A cone, a hemisphere and cylinder have equal bases. If the heights of the cone and a cylinder are equal and are same as the common radius, then find the ratio of their respective volumes. Let r be the common radius of the cone, hemisphere, and cylinder. Let h be the common height of the cone and cylinder The volumes of the cone, hemisphere and cylinder respectively. 1 π 3 r h: π 3 r3 : π r h 1 : :1 [rh] 3 3 Hence, the required ratio 1::3

60 30. How many litres of water will a hemispherical tank hold whose diameter is 4.m? (Apr-014) Hemisphere: Diameter d 4. m Radius.1 m Volume of the hemisphere shaped water tank 3 πr3 x x.1 x.1 x m litre [1m lit] 31. The largest sphere (with maximum volume) is carved out o f a cube of sides 14cm. Find the volume of the sphere. (Jun- 014) diameter of the sphere d 14cm Radius of the sphere 7 cm Volume of the sphere 4 3 πr cm3 3. A sector containing an angle of 10 is cut off from a circle of radius 1 cm and folded into a cone. Find the curved surface area of the cone. (Oct-014) Radius of the sector, R 1 cm Angle of the sector, θ 10 0 θ Length of the arc l x πr When the sector is folded into a right circular cone, we have circumference of the base of the cone Length of the arc πr π 1 r 7cm 360 Slant height of the cone l 1 cm CSA of the cone πrl ways100@gmail.com Or 3 CSA of the cone Area of two sector x 7 x cm θ0 x πr x 7 x 1 x 1 46cm 33. A cone of height 4cm has a curved surface area of 550 cm cm. Find the volume of the cone (Oct-15) Cone: h 4 cm πr 550cm The volume of the cone 1 3 πr h 1 x 3 πr xh 1 x 550x cm The external diameter of a cylindrical shaped iron pipe is 5cm and its length is 0cm. If the thickness of the pipe is 1 cm, find the total surface area of the pipe (Oct-015) Cylindrical pipe: Let R, r and h be the external, internal radii and length of the pipe respectively, D 5 cm, R 1.5 cm thickness w 1cm Internal radius r R w r 11.5 cm, h 0 cm

61 Total surface area π R + r (R r + h) π ( ) x x 4 x cm 35. Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq.cm and 110 cm respectively. Find its height and diameter. (Mar-016) CSA of the solid right circular cylinder4400sq.cm Circumference of the base of the cylinder πr 110 cm r Diameter r cm CSA πrh 110 h 4400 Thus, the height of the cylinder h cm The central angle and radius of a sector of a circular disc are 180 and 1 cm respectively. If the edges of the sector are joined together to make a hollow cone, then find the radius of the cone.(mar-16) Given that the central angle of the sector θ 180 and the radius of the sector r1cm. By joining the edges of the sector, a hollow cone is formed. Let R be the radius of the cone. Circumference of the base of the cone Arc length of the sector πr θ πr 360 R cm Statistics 1. Find the range and the coefficient of range of 43, 4, 38, 56,, 39, 45. (Apr-01, 015) L 56, S Range L S Coefficient of Range L S 56 L+S The weight (in kg) of 13 students in a class are 4.5, 47.5, 48.6, 50.5, 49, 46., 49.8, 45.8, 43.,48, 44.7, 46.9, 4.4. Find the range and coefficient of range. L 50.5, S 4.4 Range L S Coefficient of Range L S L+S Find the range and coefficient of range of the following data. 59, 46, 30, 3, 7, 40, 5, 35, 9. (Oct-14) L 59, S 3 Range L S Coefficient of Range L S L+S ways100@gmail.com

62 4. Find the range and coefficient of range of the following data. 41., 33.7, 9.1, 34.5, 5.7, 4.8, 56.5, 1.5. L 56.5, S1.5 Range L S Coefficient of Range L S L+S The smallest value of a collection of data is 1 and the range is 59. Find the largest value of the collection of data. Range 59, S 1 Range L S 59 L L 71 L L The largest of 50 measurements is 3.84 kg. If the range is 0.46 kg, find the smallest measurement. (Apr-013) L 3.84, Range 0.46 Range L S S S S 3.38 kg 7. The largest value in a collection of data is If the range is.6, then find the smallest value in the collection. (Oct-01) L 7.44, Range.6 Range L S S S S The standard deviation of 0 observations is 5. If each observation is multiplied by, find the standard deviation and variance of the resulting observations. standard deviation 5 SD of new data x 5 5 Variance of new data ( 5) 4 x Calculate the standard deviation of the first 13 natural numbers. (Oct-013, Jun-015) n13 standard deviation n ways100@gmail.com

63 10. Find the standard deviation of the first 10 natural numbers. (Apr-014, Jun-014) n10 standard deviation n If the coefficient of variation of a collection of data is 57 and its S.D. is 6.84, then find the mean. (Jun-01, Jun-013,Mar-016) C.V. 57, σ 6.84 C.V. 57 x x x 1 σ x 6.84 x x 100 x 100 x A group of 100 candidates have their average height cm with coefficient of variation 3.. What is the standard deviation of their heights? x 163.8, C.V. 3., σ? C.V. 3. x σ x σ x x x σ The mean of 30 items is 18 and their standard deviation is 3. Find the sum of all the items and also the sum of the squares of all the items. n 30, x 18, σ 3 x n x 30 x σ x n x x n n 18 9 x x x x 540, x 9990 ways100@gmail.com

64 14. If n 10, x 1 and x 1530, then calculate the coefficient of variation. n 10, x 1, x 1530 S.D x x n n C.V. σ x x x C.V. 5 x Probability 15. Mean of 100 items is 48 and their standard deviation is 10. Find the sum of all the items and the sum of the squares of all the items. The mean of 100 items x 48 The sum of 100 items x 48 x Given that standard deviation σ 10 Variance σ x P(A) 4 5 ways100@gmail.com x 4800 n 100 x n x n x n n 100 x n (48) x 100 x 404,40, The probability that it will rain on a particular day is What is the probability that it will not rain on that day? P(A) 0.76 P(A ) 1 P (A) P( A ) 0.4 The probability that it will not rain is 0.4. An integer is chosen from the first twenty natural numbers. What is the probability that it is a prime number? (Apr-01) S 1,,3,..0 n(s) 0, A be the event of choosing a prime number A {,3,5,7,11,13,17,19} n(a) 8 P(A) n(a) 8 n(s) 0 5 P(A) 5 3. There are 7 defective items in a sample of 35 items. Find the probability that an item chosen at random is non-defective. (Jun-01, Apr-013) n(s) 35 A be the event of choosing a non-defective item n(a) 8 P(A) n(a) 8 4 n(s) 35 5

65 4. Find the probability that a leap year selected at random will have 53 Fridays. S {(Sun,Mon),(Mon,Tue),(Tue,Wed),(Wed,Thur),(Thur,Fri),(Fri,Sat),(Sat,Sun)} n(s) 7 A be the event of getting one Friday in the remaining two days A{(Thur,Fri),(Fri,Sat)} n(a) P(A) n(a) P(A) 7 n(s) 7 5. Find the probability that a leap year selected at random will have only 5 Fridays. S {(Sun,Mon),(Mon,Tue),(Tue,Wed),(Wed,Thur),(Thur,Fri),(Fri,Sat),(Sat,Sun)} n(s) 7 A be the event of not getting a Friday in the remaining two days A {(Sun,Mon),(Mon,Tue),(Tue,Wed),(Wed,Thur),(Sat,Sun)} n(a) 5 P(A) n(a) P(A) 5 7 n(s) Find the probability that a non-leap year selected at random will have 53 Fridays. S {Sun,Mon,Tue,Wed,Thur,Fri,Sat} n(s) 7 A be the event of getting a Friday in the remaining one day n(a) 1 P(A) n(a) P(A) 1 7 n(s) A die is thrown twice. Find the probability of getting a total of 9. (Oct-01, 014) S { 1,1, 1,, 1,3, 1,4, 1,5, 1,6, n(s) 36,1,,,,3,,4,,5,,6, A {(3, 6), (4, 5), (5, 4), (6, 3)} 3,1, 3,, 3,3, 3,4, 3,5, 3,6, n(a) 4 4,1, 4,, 4,3, 4,4, 4,5, 4,6, 5,1, 5,, 5,3, 5,4, 5,5, 5,6, P(A) n(a) 4 1 n(s) ,1, 6,, 6,3, 6,4, 6,5, (6,6)} P(A) Three rotten eggs are mixed with 1 good ones. One egg is chosen at random. What is the probability of choosing a rotten egg? n(s) 15 n(a) 3 P(A) n(a) n(s) 15 5 P(A) 1 5 ways100@gmail.com

66 9. Two coins are tossed together. What is the probability of getting at most one head. (Jun-015) S {HH, HT, TH, TT} n(s) 4 A {TT, HT, TH} n(a) 3 P(A) n(a) P(A) 3 4 n(s) For a sightseeing trip, a tourist selects a country randomly from Argentina, Bangladesh, China, Angola, Russia and Algeria. What is the probability that the name of the selected country will begin with A? n(s) 6 n(a) 3 P(A) n(a) 3 1 n(s) 6 P(A) From a well shuffled pack of 5 playing cards, one card is drawn at random. Find the probability of getting i) a black king ii) a spade card. (Oct-013) n(s) 5 n(s) 5 (i) A black king n(a) (ii) A spade card n(b) 13 P(A) n(a) 1 n(s) 5 6 P(B) n(a) 13 1 n(s) From a well shuffled pack of 5 playing cards, one card is drawn at random. Find the probability of getting i) a king ii) a diamond 10. n(s) 5 A king n(a) 4 P(A) n(a) 4 1 n(s) 5 13 n(s) 5 A diamond 10 n(b) 1 P(B) n(a) 1 n(s) Three dice are thrown simultaneously. Find the probability of getting the same number on all the three dice. (Apr-014) S 1, 1, 1, 1, 1,,.. 6, 6, 6 n(s) 16 A {(1, 1, 1), (,, ), (3, 3, 3), (4, 4, 4), (5, 5, 5), (6, 6, 6)} n(a) 6 P(A) n(a) P(A) n(s) Three coins are tossed simultaneously. Find the probability of getting (i) at least one head (ii)exactly two tails (iii) at least two heads S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} n(s) 8 ways100@gmail.com

67 (i) at least one head A {HHH, HHT, HTH, HTT, THH, THT, TTH} n(a) 7 P(A) n(a) n(s) 7 8 (ii)exactly two tails B {HTT, TTH, THT} n(b) 3 P(B) n(b) n(s) 3 8 (iii) at least two heads C {HHH, HTH, HHT, THH} n(c) 4 P(C) n(c) 4 1 n(s) 8 P(A) 7 8, P(B) 3 8, P(C) There are 0 boys and 15 girls in a class of 35 students. A student is chosen at random. Find the probability that the chosen student is i) boy ii) girl. n(s) 35 Probability of choosing a boy Probability of choosing girl n(a) 7 n(b) 15 P(A) n(a) P(B) n(b) 0 4 n(s) n(s) A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of drawing a red ball, then find the number of blue balls in the bag. Let the number of blue balls be x Total number of balls n(s) 5+ x B be the event of drawing a blue ball and R be the event of drawing a red ball P(B) 3P(R) n(b) n R 3 n(s) n(s) x 5+x x x The number of blue balls x If A is an event of a random experiment such that P(A) : P(A ) 7:1, then find P(A). P(A) : P(A ) 7:1 P(A) 1 x P(A) 7x P(A ) P(A) 7 1 1P(A) 7[1 P(A) ] 7 7P(A) 19P(A) 7 P (A) If A and B are mutually exclusive events such that P(A) 3 and P(B) 1, then find P(A B). 5 5 If A and B are mutually exclusive events, then P(A B) 0 P(A B) P(A) + P(B) P(A B) ways100@gmail.com

68 19. If A and B are two events such that P(A) 1, P(B) and P(A B) 1,then find P(A B). 4 5 P(A B) P(A) + P(B) P(A B) P(A B) P(A) +P(B) P (A B) A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random. Find the probability that the ball drawn is white or red. Total number of balls, n(s) 6 Number of white balls n(w) 10, Number of red balls n(r) 6, Number of blue balls n(b) 10 (W and R are mutually exclusive events ) P(W R) P(W) + P(R) [P(W R) 0] A number is selected at random from integers 1 to 100. Find the probability that it is a perfect square. Sample space S 1,, 3,..100 n(s) 100 A be the event of getting a perfect square A {1, 4, 9, 16, 5, 36, 49, 64, 81, 100} n(a) 10 P(A) n(a) 10 1 n(s) A number is selected at random from integers 1 to 100. Find the probability that it is not a perfect cube. Sample space S 1,, 3,.100 n (S) 100 A be the event of getting a perfect square A {1, 8, 7, 64} n(a) 4 P(A) n(a) 4 1 n(s) Hence, the probability that the selected number is not a cube is P A 1 P(A) cards are numbered from 1 to 0. One card is drawn at random. What is the probability that the number on the card is (i) a multiple of 4. (Mar-016) (ii) not a multiple of 6 (Mar-016) Sample Space S 1,, 3,.0 n(s) 0 (i) A be the event of drawing a card such that a number on it, is a multiple of 4 A {4, 8, 1, 16, 0} n(a) 5 P(A) n(a) 5 1 n(s) 0 4 (ii) Let B be the event of drawing a card such that the number on the card is a multiple of 6 B{6,1,18} ways100@gmail.com

69 n B 3, P B n(b) n(s) 3 0 P B 1 P B A ticket is drawn from a bag containing 100 tickets. The tickets are numbered from one to hundred. What is the probability of getting a ticket with a number divisible by 10? Sample space S 1,, 3,.100 n(s) 100 Let A be the event of getting a ticket with a number divisible by 10 A 10, 0, 30,.100 n(a) 10 P(A) n(a) 10 1 n(s) Two dice are thrown together. Find the probability that the two digit number formed with the two numbers turning up is divisible by 3. S 1,1, 1,, 1,3,.. 6,6 n(s) 36 Let A be the event of the two digit number formed with the two numbers turning up is divisible by 3 A {1, 15, 1, 4, 33, 36, 4, 45, 51, 54, 63, 66} n(a) 1 P(A) n(a) 1 1 n(s) A point is chosen at random inside a circle of radius cm. What is the probability that this point is nearer to the centre than to the circumference? (Oct-015) S be the sample space n S πr π x 4π n A πr π 1 π Probability P A n(a) π 1 n(s) 4π 4 7. A card is drawn at random from a well shuffled deck of 5 cards. Find the probability that it will be a spade or a king (Jun-013) n S 5 Let A be the event of getting a spade, So n A 13 P A n(a) 13 n(s) 5 Let B be the event of getting a king, So n B 4 P B n(b) n (S) 4 5 Number of spade king n A B 1 P A B n(a B) n(s) 1 5 Required probability P A B P A + P B P(A B) ways100@gmail.com

70 10 th Maths Practical Geometry Way to Success 9. PRACTICAL GEOMETRY Blue Print 10 Mark Total Marks 1 10 In Practical geometry, we shall learn how to draw tangents to circles, triangles with the help of given actual measurements. Construction of a tangent to a circle: Definition: Tangent The straight line PQ and the circle have exactly one common point. Equivalently the straight line touches the circle at only one point. The straight line PQ is called the tangent to the circle at A. O as the centre, OA as the radius, A point of contact, PAQ tangent Basic hints for tangents and circles: 1. In a circle, the radius drawn at the point of contact is perpendicular to the tangent at that point.. Only one tangent can be drawn on a circle. 3. Two tangents can be drawn to a circle from an external point. 4. Two tangents are equal if they are drawn from an external point of a circle to same circle. 5. Diameters subtend 90 0 on the circumference of a circle. Construction of tangents to a circle: Now let us learn below, how to draw a tangent to a circle. (i) Using centre (ii) Using tangent chord theorem (iii) Construction of tangents to a circle from an external point ways100@gmail.com

71 10 th Maths Practical Geometry Way to Success I. Construction of a tangent to a circle using the centre: Example 9.1, Ex:9.1- (1) 1. Example 9.1 Draw a circle of radius 3. cm. Take a point P on this circle and draw a tangent at P. (using the centre) Given: Radius of the circle 3. cm Fair diagram: Construction: 1. With O as the centre draw a circle of radius 3. cm.. Take a point P on the circle and join OP. 4. Mark M and N on the arc such that PM PN 5. Draw the bisector PT of the line segment MN Rough diagram 6. Produce TP to T to get the required tangent T PT. 3. Draw an arc of a circle with centre at P cutting OP at L. ways100@gmail.com

72 10 th Maths Practical Geometry Way to Success. Ex.9.1 (1) Draw a circle of radius 4. cm, and take any point on the circle. Draw the tangent at that point using the centre. Fair diagram: Construction: 1. With O as the centre draw a circle of radius 4. cm.. Take a point P on the circle and join OP. 3. Draw an arc of a circle with centre at P cutting OP at L. 4. Mark M and N on the arc such that PM PN 5. Draw the bisector PT of the line segment MN Rough diagram 6. Produce TP to T to get the required tangent T PT. ways100@gmail.com

73 10 th Maths Practical Geometry Way to Success II. The tangent chord theorem The angle between a chord of a circle and the tangent at one end of the chord is equal to the angle subtended by chord on the alternate segment of the circle. Example 9., Ex.9.1 () 3. Example 9. Draw a circle of radius 3. cm AT a point P on it, draw a tangent to the circle using the tangent chord theorem. Given: The radius of the circle 3. cm. Fair diagram Construction: Rough diagram 1. With O as the centre, draw a circle of radius 3. cm. 7. Produce TP to T to get. Take a point P on the 5. Join PR and QR. the required tangent circle. 6. At P, construct line T PT. 3. Through P, draw any QPT PRQ. chord PQ. 4. Mark a point R distinct from P and Q on the circle so that P, Q and R are in counter clockwise direction. ways100@gmail.com

74 10 th Maths Practical Geometry Way to Success 4. Ex.9.1 () Draw a circle of radius 4.8 cm. Take a point on the circle. Draw the tangent at that point using the tangent chord theorem Rough diagram Construction: 1. With O as the centre, draw a circle of radius 4.8cm.. Take a point P on the circle. 3. Through P, draw any chord PQ. 4. Mark a point R distinct from P and Q on the circle so that P, Q and R are in counter clockwise direction. 5. Join PR and QR. 6. At P, construct QPT PRQ. 7. Produce TP to T to get the required tangent line T PT. ways100@gmail.com

75 10 th Maths Practical Geometry Way to Success III. Construction of pair of tangents to a circle from an external point Example 9.3 Ex. 9.1, (3), (4), (5) 5. Example 9.3 Draw a circle of radius 3 cm. From an external point 7 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. (March-014) Given: Radius of the circle 3 cm. OP 7 cm. Fair diagram Verification: In the right angled OPT > PT OP OT PT 6.3 cm (approximately) Construction: 1. With O as the centre draw a circle of radius 3 cm.. Mark a point P at a distance of 7 cm from O and join OP. 3. Draw the perpendicular bisector of OP. Let it meet OP at M. 4. With M as centre and MO as radius, draw another circle. 5. Let the two circles intersect at T and T. Rough diagram 6. Join PT and PT. They are the required tangents. ways100@gmail.com

76 10 th Maths Practical Geometry Way to Success 6. Ex. 9.1 (3) Draw a circle of diameter 10 cm. From a point P, 13cm away from its centre, draw the two tangents PA and PB to the circle, and measure their lengths.. (June-013, March-015,June-015) Rough diagram Fair diagram Verification: PA cm Construction: 1. With O as the centre draw a circle of radius 5 cm.. Mark a point P at a distance of 13 cm from O and join OP. 3. Draw the perpendicular bisector of OP. Let it meet OP at M. 4. With M as centre and MO as radius, draw another circle. 5. Let the two circles intersect at T and T. 6. Join PT and PT. They are the required tangents. ways100@gmail.com

77 10 th Maths Practical Geometry Way to Success 7. EX.9.1 (4) Draw the two tangents from a point which is 10cm away from the centre of a circle of radius 6cm. Also, measure the lengths of the tangents. (Oct-014) Fair diagram: Verification: PA cm Construction: 1. With O as the centre draw a circle of radius 6 cm.. Mark a point P at a distance of 10 cm from O and join OP. 3. Draw the perpendicular bisector of OP. Let it meet OP at M. 4. With M as centre and MO as radius, draw another circle. 5. Let the two circles intersect at T and T. Rough diagram 6. Join PT and PT. They are the required tangents. ways100@gmail.com

78 10 th Maths Practical Geometry Way to Success 8. Ex.9.1 (5) Take a point which is 9cm away from the centre of a circle of radius 3 cm, and draw the two tangents to the circle from that point. (Oct-01, June-013, Oct-013, Oct-015, March-016) Fair diagram Verification: PA cm Construction: 1. With O as the centre draw a circle of radius 3 cm.. Mark a point P at a distance of 9 cm from O and join OP. 3. Draw the perpendicular bisector of OP. Let it meet OP at M. 4. With M as centre and MO as radius, draw another circle. 5. Let the two circles intersect at T and T. Rough diagram 6. Join PT and PT. They are the required tangents. ways100@gmail.com

79 10 th Maths Practical Geometry Way to Success Construction of triangles: We have already learnt how to construct triangles when sides and angles are given. In this section, let us construct a triangle when, i) The base, vertical angle and the altitude from the vertex to the base are given ii) The base, vertical angle and the median from the vertex to the base are given First, let us describe the way of constructing a segment of a circle on a given line segment containing a given angle. 9. Ex.9. (1) Construct a segment of a circle on a given line segment AB 5.cm containing an angle 48 0 Construction 1. Draw a line segment AB 5. cm. At A, make BAX 4 0, BA Draw AY AX 4. Draw the perpendicular bisector of AB which meets AX at O. 5. With O as cenre and OA as radius draw a circle BAX 4 6. Take any point C on the circle. By the tangent-chord theorem, the major arc ACB is the required segment of the circle containing the angle 48 0 ways100@gmail.com

80 10 th Maths Practical Geometry Way to Success 10. Example 9.4 Construct a ABC such that AB 6 cm, C 40 0 and the altitude from C to AB is of length 4.cm. (March-013, June-013, June-014) Given: In ABC, AB 6 cm, C 40 0 The length of the altitude from C to AB is 4. cm. Rough diagram Fair diagram Construction 1. Draw a line segment AB 6 cm.. Draw AX such that BAX Draw AY AX. 4. Draw the perpendicular bisector of AB intersecting AX at O and AB at M. 5. With O as centre and OA as radius, draw the circle. 6. The segment AKB contains the vertical angle On the perpendicular bisector MO, mark a point H such that MH 4.cm BAX Draw CHC parallel to AB meeting the circle at C and at C. 9. Complete the ABC, which is one of the required triangles. ways100@gmail.com

81 10 th Maths Practical Geometry Way to Success 11. Ex. 9. () Construct a PQR in which the base PQ 6cm, R 60 0 and the altitude from R to PQ is 4 cm Fair diagram Rough Diagram Construction 1. Draw a line segment PQ 6 cm.. Draw PX such that QPX Draw PY PX. 4. Draw the perpendicular bisector of PQ intersecting PY at O and PQ at M. 5. With O as centre and OP as radius, draw the circle. 6. The segment PKQ contains the vertical angle QPY On the perpendicular bisector MO, mark a point H such that MH 4 cm. 8. Draw RHR parallel to PQ meeting the circle at R and at R. 9. Complete the PQR, which is one of the required triangles. ways100@gmail.com

82 10 th Maths Practical Geometry Way to Success 1. Ex. 9. (3) Construct a PQR such that PQ 4cm, R 5 0 and the altitude from R to PQ is 4.5 cm Fair diagram Construction: 1. Draw a line segment PQ 4 cm.. Draw PX such that QPX Draw PY PX. 4. Draw the perpendicular bisector of PQ intersecting PY at O and PQ at M. 5. With O as centre and OP as radius, draw the circle. 6. The segment PKQ contains the vertical angle Rough Diagram QPY On the perpendicular bisector MO, mark a point H such that MH 4.5 cm. 8. Draw RHR parallel to PQ meeting the circle at R and at R. 9. Complete the PQR, which is one of the required triangles. ways100@gmail.com

83 10 th Maths Practical Geometry Way to Success 13. Example 9.5 Construct a ABC in which BC5.5 cm, A 60 0 and the median AM from the vertex A is 4.5 cm. (March-01) Given: In ABC, BC5.5 cm, A 60 0, Median AM 4.5 cm. Fair diagram: Rough diagram Construction: 1. Draw a line segment BC 5.5 cm. ;.. Through B draw BX such that CBX Draw BY BX. 4. Draw the perpendicular bisector of BC intersecting BY at O and BC at M. 5.With O as centre and OB as radius, draw the circle. 6. The major arc BKC of the circle, contains the vertical angle CBY With M as centre, draw an arc of radius 4.5 cm meeting the circle at A and A. 8. ABC or A BC is the required triangle. ways100@gmail.com

84 10 th Maths Practical Geometry Way to Success 14. Ex-9. (4) Construct a ABC such that BC 5 cm, A 45 0 and the median from A to BC is 4 cm (June-01, June-015, Oct-015) Construction: 1. Draw a line segment BC 5 cm. Fair diagram. Through B draw BX such that CBX Draw BY BX. 4. Draw the perpendicular bisector of BC intersecting BY at O and BC at M. 5.With O as centre and OB as radius, draw the circle. 6. The major arc BKC of the circle, contains the vertical angle Rough Diagram CBX With M as centre, draw an arc of radius 4 cm meeting the circle at A and A. 8. ABC or A BC is the required triangle. ways100@gmail.com

85 10 th Maths Practical Geometry Way to Success 15. Example 9.6 Construct a ABC in which BC 4.5 cm, A 40 0 and the median AM from A to BC is 4.7cm. Find the length of the altitude from A to BC Rough diagram Construction: 1. Draw a line segment BC5 cm.. Through B draw BX such that CBX Draw BY BX. 4. Draw the perpendicular bisector of BC intersecting BY at O and BC at M. 5. With O as centre and OB as radius, draw the circle. 6. The major arc BKC of the circle, contains the vertical angle With M as centre, draw an arc of radius cm meeting the circle at A and A. 8. ABC or A BC is the required triangle. 9. Produce CB to CZ CBX Draw AE CZ. ways100@gmail.com

86 10 th Maths Practical Geometry Way to Success 16. Ex.9. (5) Construct a ABC in which the base BC5 cm, BAC 40 0 and the median from A to BC is 6 cm. Also, measure the length of the altitude from A. (March-015) Given: BC5 cm, BAC 40 0, median from A to B 6cm. Construction: 1. Draw a line segment BC5 cm.. Through B draw BX such that CBX Draw BY BX. 4. Draw the perpendicular bisector of BC intersecting BY at O and BC at M. ; 5. With O as centre and OB as radius, draw the circle. 6. The major arc BKC of the circle, contains the vertical angle ; 7.With M as centre, draw an arc of radius cm meeting the circle at A and A. 8. ABC or A BC is the required triangle. 9. Produce CB to CZ. Rough diagram CBY Draw AE CZ. ways100@gmail.com

87 10 th Maths Graphs Way to Success 10. GRAPHS Blue Print 10 Mark Total Marks 1 10 Graphs are diagram that show information. In this chapter we have two types. (i) Quadratic graphs (ii) Some special graphs Clear notes and graphs for second type, given below. Well practice in this type we get full marks in GRAPHS section Relation between the variables of polynomial are (i) Direct Variation y x k (ii) Indirect Variation xy k We are going to learn the methods of special graph drawings Graph of Direct variation is a straight line Graph of Indirect variation is a smooth curve known as a Rectangular Hyperbola Tabular columns be given for the following Example 10.7, 10.8, Ex 10. (), (3), (6) We prepared tabular column for the following Example 10.9, Ex 10. (1), (4), (5) Direct Variation Graphs: Example 10.7,10.9, Ex.10. (1),(),(3),(4) Indirect Variation Graphs: Example 10.8, Ex 10. (5), (6) 1. Example 10.7 Draw a graph for the following table and identify the variation x y Hence, find the value of y when x 4. Solution: Points (, 8), (3, 1), (5, 0), (8, 3), (10, 40) Thus, the variation is a direct variation. / y k x k (i) The constant of proportionality 4 (ii) When, x 4, y 4x (4, 16) ways100@gmail.com

88 10 th Maths Graphs Way to Success. Example 10.8 A cyclist travels from a place A to a place B along the same route at a uniform speed on different days. The following table gives the speed of his travel and corresponding time he took to cover the distance. Speed in km /hr x Time in hrs y Draw the speed-time graph and use it to find (i) the number of hours he will take if he travels at a speed of 5 km/hr (ii) the speed with which he should travel if he has to cover the distance in 40 hrs. (Oct-014) Solution: Points: (,60) (4,30) (6,0) (10,1) (1,10) Thus, the variation is called indirect variation. xy k k (1 10) k 10 i) When x 5, 5 y 10 y 10 4 hours 5 ii) When y 40,; x x 10 3 km / hour Ex.10. () The following table gives the cost and number of notebooks bought No.of note books x Cost Rs. y Draw the graph and hence (i) find the cost of seven note books (ii) How many note books can be bought for Rs. 165 Solution: Points :; (,30) (4,60) (6,90) (8,10) (10,150) (1,180) y x k k k 15 Thus, the variation is a direct variation. i) When x 7,; y 15 y y 105 (7,105) ii) When y 165,; x 165, x 11 x 15 (11,165) (Oct-015) ways100@gmail.com

89 10 th Maths Graphs Way to Success 4. Ex.10. (3) x y Draw the graph for the above table and hence find (i) the value of y if x 4 (ii) the value of x if y 1 (March-01, June-01, March 013) Solution: Points (1,) (3,6) (5,10) (7,14) (8,16) y k x k Ex.10. (6) k Thus, the variation is a direct variation. y i) When x 4, y 8 4 (4,8) 1 ii) When y 1, x x 1, x 1 6 (6,1) No. of workers x No. of days y Draw graph for the data given in the table. Hence find the number of days taken by 1 workers to complete the work (March-016) Solution: Points (3,96) (4,7) (6,48) (8,36) (9,3) (16,18) xy k k (3 96) (4 7) (6 48) (8 36) (9 3) (16 18) k 88 Thus, the variation is a indirect variation. When x 1, y? 1xy 88 y 88 4 (1, 4) 1 ways100@gmail.com

90 10 th Maths Graphs Way to Success We prepared tabular column for the following 6. Example 10.9 A bank gives 10% S.I on deposits for senior citizens. Draw the graph for the relation between the sum deposited and the interest earned for one year. Hence find (i) the interest on the deposit of Rs. 650 (ii) the amount to be deposited to earn an interest of Rs.45 (Oct-013, March-015) Solution: Let us form the following table Deposit Rs. x S.I. earned Rs. y From the table y 1 x 10 The graph is a straight line. So direct variation. i) The interest for the deposit of Rs.650 is Rs.65 y (650, 65) 10 ii) The amount to be deposited to earn an interest of Rs.45 is Rs x x 450 (450, 45) Ex.10. (1) A bus travels at a speed of 40km/hr. Write the distance-time formula and draw the graph of it. Hence, find the distance travelled in 3 hours. (June-013,June-014) Solution: Let us form the following table Time x Distance in kms y Points : (1,40) (,80) (3,10) (4,160) (5,00) y x k k40 Thus, the variation is a direct variation. When x 3,; y 3 40 y10 km (3,10) ways100@gmail.com

91 10 th Maths Graphs Way to Success 8. Ex.10. (4) The cost of the milk per litre Rs.15. Draw the graph for the relation between the quantity and const. Hence find (i) the proportionality constant, (ii) the cost of 3 litres of milk (Apr -014) Solution: Let us form the following table No. of litres x Cost in Rs. y Points ;: (1,15) (,30) (3,45) (4,60) (5,75) y k x Thus, the variation is a direct variation i) The proportionality constant k 15 ii) cost of 3 litres of milk y Ex.10.- (5) y 45 > Rs. 45 (3,45) Draw the Graph of xy 0, x, y > 0. Use the graph to find y when x 5, and to find x when y 10 (Oct-01, June-015) Solution: Let us form the following table: x y xy 0 Points (1,0) (,10) (4,5) (5,4) (10,) Thus, the variation is indirect variation. k 0 When x 4, 4 y 0 y 0 5 (4,5) 4 When y 10, x 10 0 x 0 (,10) 10 ways100@gmail.com

92 10 th Maths Points to Remember Way to Success Commutative Property POINTS TO REMEMBER 1. SETS AND FUNCTIONS i) A B B A ii) A B B A Associative Property i) A B C (A B) C ii) A B C (A B) C Distributive Property i) A B C A B (A C) ii) A B C A B (A C) De Morgan s laws for set difference i) A\ B C A\B A\C ii) A\(B C) (A\B) (A\C) De Morgan s laws for complementation i) A B A B ii) A B A B Formulae for the cardinality of union of sets i) n A B n A + n B n(a B) ii) n A B C n A + n B + n C n A B n B C n A C n(a B C). SEQUENCES AND SERIES OF REAL NUMBERS i) The formula for the nth term of an A.P. is t n a + (n 1)d ii) The sum S n of the first n terms of an arithmetic sequence with first term a and common difference d is given by S n n a + n 1 d or n [a + l] where is l is the last term iii) The sum of the first n terms of a geometric series is given by S n a[r n 1] r 1 S n na (if r 1) a[1 r n ] 1 r (if r 1) iv The sum of the first n natural numbers n n(n+1) v The sum of squares of first n natural numbers n vi The sum of the first n odd natural numbers n-1) n vii The sum of first n odd natural numbers When the last term l is given) n l l+1 viii The sum of cubes of the first n natural numbers n 3 ways100@gmail.com n(n+1) n n+1 (n+1) 6

93 10 th Maths Points to Remember Way to Success 3. ALGEBRA i) The basic relationships between zeros and coefficients of a quadratic polynomial p x ax + bx + c are Sum of zeros b, Product of zeros c a a ii) A quadratic equation ax + bx + c 0 has b 4ac iii) If b 4ac 0, the two roots are equal iv) If b 4ac > 0, the two roots are distinct v) If b 4ac < 0, no real roots (or) roots are imaginary Let us write some results involving α and β (i) α β (α + β) 4αβ (ii) α + β [(α + β) αβ ] (iii) α β α + β α β (α + β)[ (α + β) 4αβ ] only if α β (iv) α 3 + β 3 (α + β) 3 3αβ(α + β) (v) α 3 β 3 (α β) 3 + 3αβ(α β) (vi) α 4 + β 4 (α + β ) α β [(α + β) αβ ] (αβ) (vi) α 4 β 4 α + β α β (α + β ) 4. MATRICES i A matrix having m rows and n columns, is of the order m n ii) Addition or subtraction of two matrices are possible only when they are of same order. iii) If A is a matrix of order m n and B is a matrix of order n p, then the product matrix AB is defined and is of order m p iv) Matrix multiplication is not commutative in general AB BA v) Matrix multiplication is associative (AB)C A(BC) vi) (A T ) T A, (A + B) T A T + B T, (AB) T B T A T ways100@gmail.com

94 10 th Maths Points to Remember Way to Success 5. COORDINATE GEOMETRY i) Mid point of the line segment joining the points (x 1, y 1 ), and (x, y ) ii) Centroid of a triangle ABC, (x 1, y 1 )(x, y ), and (x 3, y 3 ) is G x 1+x +x 3, iii) Section formula (internally) lx +mx 1, ly +my 1 l+m l+m x 1 +x iv)the area of the triangle formed by the points (x 1, y 1 ), x, y, and (x 3, y 3 ) is 1 (x 1y + x y 3 + x 3 y 1 (x y 1 + x 3 y + x 1 y 3 )} sq.units ways100@gmail.com , y 1 +y y 1 +y +y 3 3 v) The area of the quadrilateral formed by the points (x 1, y 1 ), x, y, (x 3, y 3 )and (x 4, y 4 ) is 1 x 1 y + x y 3 + x 3 y 4 + x 4 y 1 x y 1 + x 3 y + x 4 y 3 + x 1 y 4 Sl.No Straight line Equation 1 x axis y 0 y axis x 0 3 Parallel to x-axis y k 4 Parallel to y-axis x k 5 Parallel to ax + by + c 0 ax + by + k 0 6 Perpendicular to ax + by + c 0 bx ay + k 0 Given Equation sq. units 1 Passing through the origin y mx Slope m, y-intercept c y mx + c 3 Slope m, a point x 1, y 1 y y 1 m(x x 1 ) y y 1 4 Passing through two points (x 1, y 1 ), (x, y ) x x 1 y y 1 x x 1 x 5 x -intercept a, y-intercept b a + y b 1 vi) Slope of the horizontal line is 0 and slope of the vertical line is undefined. vii) Two lines are parallel if and only if their slopes are equal viii) Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. That is m 1 m 1 1) sin θ + cos θ 1 ) sin θ 1 cos θ 3) sinθ 1 cos θ 4) cos θ 1 sin θ 5) cosθ 1 sin θ 7. TRIGONOMETRY 6) sec θ tan θ 1 7) sec θ 1 + tan θ 8) secθ 1 + tan θ 9) tan θ sec θ 1 10) tanθ sec θ 1 11) cosec θ 1 + cot θ 1)cosecθ 1 + cot θ 13) cosec θ cot θ 1 14) cot θ cosec θ 1 15) cotθ cosec θ 1

95 10 th Maths Points to Remember Way to Success 8. MENSURATION i) Curved Surface Area of a cylinder πrh sq. units ii) TSA of a cylinder πr(h + r) sq. units iii) Volume of a cylinder V πr h cu. Units iv) Slant height of the cone l r + h v) Height of the cone h l r vi) Radius of the cone r l h vii) CSA of the cone πrl sq. units viii) TSA of the cone πr(l + r)sq. units ix) Volume of the cone V 1 3 πr h cu. units x) Volume of the frustum V 1 3 πh(r + r + Rr) cu.units xi) CSA of the sphere 4πr sq. units xii)volume of the sphere V 4 3 πr3 cu. Units xiii) CSA of the Solid hemisphere πr sq. units xiv) TSA of the Solid hemisphere 3πr sq. units xv) Volume of the solid hemisphere V 3 πr3 cu. units xvi) CSA of right circular hollow cylinder πh(r + r) sq. units xvii) TSA of right circular hollow cylinder π (R + r) (R r + h) sq. units xviii)volume of right circular hollow cylinder V π h (R + r) (R r) cu. units xix)volume of the hollow sphere V 4 3 π(r3 r 3 ) cu. Units ways100@gmail.com

96 10 th Maths Points to Remember Way to Success i ) Range L S ii) Coefficient of Range L S L+S 11. STATISTICS iii) Standard deviation for an ungrouped data 1. σ 3. σ x x n n d d n n iv) Standard deviation for a grouped data 1. σ 3. σ fd f. σ (d x A) 4. σ (d x x ). σ fd fd f f x A c ( d ) ways100@gmail.com c d n d d n n fd fd f f (d x x ) x A c ( d ) (d x A) v) Standard deviation of a collection of data remains unchanged when each value is added or subtracted by a constant vi) Standard deviation of a collection of data gets multiplied or divided by the quantity k, if each item is multiplied or divided by k. vii) Standard deviation of the first n natural numbers σ viii) C. V σ x 100 n 1 1 c

97 10 th Maths Points to Remember Way to Success - Notes - ways100@gmail.com

98

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