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केन्द्रीय विद्यालय संगठन KENDRIYA VIDYALAYA SANGATHAN जयपुर-संभाग Jaipur REGION Question Bank 2016-17 कक्षा: 12 CLASS: 12 गणित (०४१) MATHEMATICS(041) Question bank FOR CLASS XII MATHEMATICS (2016-17) PATRON DR. JAIDEEP DAS Deputy Commissioner KVS Jaipur Region COORDINATOR Shri R C BHURIA Principal KV No. 5, Jaipur Prepared Reviewed and Updated by Navratan Mittal & Vinod Kumar PGT (Maths) K V No. 5 (I Shift), Jaipur PREFACE It gives me immense pleasure to present the question bank of Class XII Mathematics for session 2016-17 by KVS Jaipur Region. This question bank is prepared strictly as per the question pattern framed by the CBSE of Mathematics for Class XII. I am confident that the question bank for Class XII Mathematics will help the students immensely to understand the board based questions pattern and will improve the quality performance of the students. Wish you all the best. (Dr. Jaideep Das) Deputy Commissioner KVS RO, Jaipur This question bank is prepared reviewed under the patronage of Dr. Jaideep Das, Deputy Commissioner, KVS (RO) JAIPUR Smt. V. Gowri - Asst. Commissioner, KVS (RO) JAIPUR Sh. Jyothy Kumar - Asst. Commissioner, KVS (RO) JAIPUR Smt. Sukriti Raiwani - Asst. Commissioner, KVS (RO) JAIPUR Under the guidance of Sh. R C BHURIA Principal, K V No. 5, JAIPUR Prepared Reviewed and Updated by Navratan Mittal & Vinod Kumar PGT (Maths) K V No. 5 (I Shift), Jaipur TOPIC -1: RELATIONS & FUNCTIONS 1/2 Mark questions: Q.1 If f :R→ R is given by f(x) = (3 − 𝑥 3 )1/3 ,determine f{f(x)}. Q.2 If the binary operation * defined on Q, is defined as a*b = 2a + b – ab for all a,b ∈Q,find the value of 3*4. Q.3 If f is an invertible function, defined as f(x) = 3𝑥−2 5 , write f¯¹(x). Q.4 Let * be a binary operation on N given by a*b = LCM(a,b) for all a,b,∈ N. Find 5*7. Q.5 State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2) ,(2,1) } not to be transitive. Q.6 If f(x) = x + 7 and g(x) = x – 7, x∈ R, find fog(7). Q.7 Let A= {1,2,3}, B ={4,5,6,7} and let f = {(1,4), (2,5), (3,6) } be a function from A to B. State whether f is one-one or not. 4/6 Marks Questions: Q.1 Let A = NXN and * be a binary operation on A defined by (a,b) * (c,d) = (a+c ,b+d). Show that * is commutative and associative .Also, find identity element for * on A,if any. Q.2 Consider f: R+ → [-5,∞) given by f(x) = 9𝑥 2 + 6x – 5. Show that f is invertible and f-1(y) = ( √𝑦+6 − 1 3 ). Q.3 Is the binary operation defined on set N, given by a*b = 𝑎+𝑏 2 for all a, b ∈ N, commutative? Is the above binary operation associative? Q.4 Let A = R – {3} and B = R – {1}. Consider the function f :A→ B defind by f(x) = 𝑥−2 . 𝑥−3 Show that f is one-one and onto and hence find f-1 Q.5 Prove that the relation R in the set A ={5,6,7,8,9} given by R ={(a,b) :Ia-bI is divisible by 2} is an equivalence relation .Find all elements related to the element 6. Q.6 Show that the relation R on the set A = {x ∈ Z : 0 ≤ 𝑥 ≤ 12 }, given by R = {(a,b) : |a-b| is a multiple of 4} is an equivalence relation. Q.7 Let R0 denote the set of all non-zero real numbers and let A = R0 x R0. If * is a binary operation on A defined by : (a,b) * (c,d) = (ac, bd) for all (a,b), (c,d) A. a) Show that * is both commutative and associative on A. b) Find the identity element in A. c) Find the invertible element in A. Q.8 Let N denote the set of all natural numbers and R be the relation on N X N defined by (a, b) R (c, d) if ad (b + c) = bc(a + d). Show that R is an equivalence relation. Q.9 Determine whether the relation R defined on the set of all real numbers as R = {(a,b): a,b R and a- b+√3 S, Where S is the set of all irrational numbers }, is reflexive, symmetric and transitive. Q.10 Show that the relation R on the set A of points in a plane, given by R= {(P, Q): Distance of the point P from the origin = Distance of Q from origin} is an equivalence relation. Further show that the set of the points related to a point P≠ (0,0) is the circle passing through P with origin as centre. TOPIC- 2: INVERSE TRIGONOMETRIC FUNCTIONS 1/2 Mark questions: 𝜋 1 1. Find the value of: 𝑠𝑖𝑛 [ − 𝑠𝑖𝑛−1 (− )]. 3 2 1 2. If sin(sin-1 + cos-1x) = 1, then find the value of x. 2 3. Find the Principal value of: 𝑡𝑎𝑛−1 (√3) − 𝑠𝑒𝑐 −1 (−2). 1 1 2 2 4. Write the principal value of: 2sin-1 + cos-1 . 5. Prove that : 3 sin-1x = sin-1(3x – 4x3) 6. Find the Principal value of 𝑐𝑜𝑠 −1 (𝑐𝑜𝑠 7𝜋 ) + 𝑡𝑎𝑛−1 (𝑡𝑎𝑛 3 3 7. Evaluate: sin (2sin-1 ). 5 √3 8. Write the value of 𝑡𝑎𝑛−1 (2𝑠𝑖𝑛𝑥(2𝑐𝑜𝑠 −1 ( )). 2 9. Express in simplest form: 𝑡𝑎𝑛−1 [ 1 x2 𝑐os 𝑥 1−𝑠in 𝑥 tan 1 10. Prove: tan 1 2 2 x 2 1 x . 2x ]. 3𝜋 6 ). 4 Marks questions:5 12 13 13 11. Simplify 𝑠𝑖𝑛−1 ( 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 ). 12. Write the simplest form of the function; 𝑡𝑎𝑛−1 ( 1 1 31 2 7 17 13. Prove that: 2𝑡𝑎𝑛−1 + 𝑡𝑎𝑛−1 = 𝑡𝑎𝑛−1 1 3 2 4 14. Show that: tan( sin-1 )= √1+ 𝑥 2 −1 ) , x ≠ 0. 𝑥 4− √7 3 𝜋 15. Solve for x: tan-12x + tan-13x = 1 4 𝜋 16. Evaluate: 𝑡𝑎𝑛 (2𝑡𝑎𝑛−1 ( ) + ). 5 4 1 1 1 1 𝜋 5 7 8 8 4 17. Prove that : 𝑡𝑎𝑛−1 + 𝑡𝑎𝑛−1 + 𝑡𝑎𝑛−1 𝑡𝑎𝑛−1 = 18. Solve for x: 2 𝑡𝑎𝑛−1 (𝑐𝑜𝑠𝑥) = 𝑡𝑎𝑛−1 (2 𝑐𝑜𝑠𝑒𝑐 𝑥) 19. Prove that: 9𝜋 4 + 9 4 sin-1 1 3 = 9 sin-1 4 x 1 2√2 3 x 1 20. Solve for x: tan 1 tan 1 x 2 x 2 4 1 cos x 1 cos x x ,0 x 2 1 cos x 1 cos x 4 2 21. Prove that: tan 1 22. Solve for x: tan-1(x - 1) + tan-1x + tan-1(x + 1) = 𝑡𝑎𝑛−1 3𝑥 4 5 5 13 23. Prove that : 𝑠𝑖𝑛−1 + 𝑠𝑖𝑛−1 24. Solve for x : 𝑡𝑎𝑛−1 [ 1+𝑥 25. Prove that : 𝑡𝑎𝑛−1 [ 3 sin 2𝑥 1−𝑥 ]= 5+3 cos 2𝑥 𝜋 4 + 𝑠𝑖𝑛−1 16 65 = 𝜋 2 + 𝑡𝑎𝑛−1 𝑥 1 ] + 𝑡𝑎𝑛−1 [ tan 𝑥] = 𝑥 4 sin 𝑥 26. Write in simplest form: 𝑡𝑎𝑛−1 ( cos 𝑥 ) + 𝑡𝑎𝑛 −1 (1−𝑠𝑖𝑛𝑥 ) 1−cos 𝑥 27. Prove that : 𝑐𝑜𝑠 −1 12 13 + 𝑠𝑖𝑛−1 3 5 = 𝑠𝑖𝑛−1 1 5√3 5 7 28. Prove that: 2 𝑡𝑎𝑛−1 + 𝑠𝑒𝑐 −1 65 + 2𝑡𝑎𝑛−1 3 3 6 5 2 5√13 29. Prove that: cos (𝑠𝑖𝑛−1 + 𝑐𝑜𝑡 −1 ) = 30. Prove that: : 𝑡𝑎𝑛−1 [ 56 6𝑥−8𝑥 3 1−12𝑥 2 ] - 𝑡𝑎𝑛−1 [ 4𝑥 1−4𝑥 2 1 8 = 𝜋 4 ] = 𝑡𝑎𝑛−1 2𝑥 ; |2x| < 1 √3 . 31. Prove that: : 𝑡𝑎𝑛−1 ( 𝑥 2−1 𝑥 2+1 ) + 𝑡𝑎𝑛−1 [ 2𝑥 2𝜋 ]= 𝑥 2 −1 3 𝑥 𝑥 𝑥2 2𝑥𝑦 𝑎 𝑏 𝑎 𝑎𝑏 32. If 𝑐𝑜𝑠 −1 + 𝑐𝑜𝑠 −1 = α then show that 2 cosα + 𝑦2 𝑎2 = sin2α. 3 33. Solve for x; cos(𝑡𝑎𝑛−1 𝑥) = sin (𝑐𝑜𝑡 −1 ) . 4 𝑥 34. If y = 𝑐𝑜𝑡 −1 (√cos x)− 𝑡𝑎𝑛−1 (√cos x), prove that siny = tan2 . 2 35. Prove that cos[𝑡𝑎𝑛−1 {sin(𝑐𝑜𝑡 −1 𝑥)}] = √ 1+ 𝑥 2 2+ 𝑥 2 . 𝜋 1 𝑎 𝜋 1 𝑎 2𝑏 4 2 𝑏 4 2 𝑏 𝑎 36. Show that: tan( + 𝑐𝑜𝑠 −1 ) + tan( - 𝑐𝑜𝑠 −1 ) = . TOPIC-3: ALGEBRA OF MATRICES 1/2 Mark questions: 1 1. if A=[ 1 2 3 −1 2 4 ] and B=[ ] .Find the value of 2A-B. 4 5 3 −2 6 6 2 ]=[ 5 𝑎𝑏 2. If [ 𝑎+𝑏 5 3. If [ 1 2 3 1 7 ][ ] =[ 3 4 2 5 𝑘 7𝑦 4. If [ 2𝑥 − 3𝑦 2 ] .Find the value of a and b. 8 11 ], then find the value of k. 23 5 −21 5 ], then find the value of x & y. ]= [ −3 11 −3 𝑐𝑜𝑠𝛼 5. Simplify: 𝑐𝑜𝑠𝛼 [ −𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛼 ] + sin𝛼 [ 𝑐𝑜𝑠𝛼 𝑐𝑜𝑠𝛼 −𝑐𝑜𝑠𝛼 ]. 𝑠𝑖𝑛𝛼 3 4 −1 2 1 6. If AT = [−1 2] and B = [ ], then find AT – BT 1 2 3 0 1 7. If 2[ 𝑦 1 3 ]+[ 0 𝑥 1 0 5 6 ]=[ ] ,find x and y. 1 8 2 1 8. If A is a 3 x 3 matrix, whose elements are given by 𝑎𝑖𝑗 = |-3i + j| then write the value of 3 a23 + a31 9. if A=[ 𝑥−1 𝑦+2 1 ] and B=[ 2 2 𝑧+1 2 ]. Find the value of x, y and z if A=B. 4 10.Find X and Y if X+Y=[ 7 0 3 0 ] and X-Y=[ ] 2 5 0 3 4 Marks questions: −1 1. if A=[ 2 ] and B=[−2 −1 −4] ,verify that(𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 3 3 2 0 2. if A= [1 4 0] , show that 𝐴2 − 7𝐴 + 10𝐼 = 0 0 0 5 1 5 3. Find the number x and y such that 𝐴2 + 𝑥𝐴 + 𝑦𝐼 = 0. Where A=[ ] 4 3 1 𝑎 1 𝑚𝑎 4. For A=[ ], show that 𝐴𝑚 = [ ] , for all positive integers m. 0 1 0 1 1 1 1 3𝑛−1 3𝑛−1 3𝑛−1 𝑛 5. 𝐴 = [1 1 1] , 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝐴 = [3𝑛−1 3𝑛−1 3𝑛−1 ],for all positive integers n. 1 1 1 3𝑛−1 3𝑛−1 3𝑛−1 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛 𝜃 6. If A=[ ] , thenprove by principle of mathematical induction that 𝐴𝑛 = −𝑠𝑖𝑛𝜃 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠𝑛𝜃 𝑠𝑖𝑛𝑛𝜃 [ ] for all 𝑛 ∈ 𝑁. −𝑠𝑖𝑛𝑛𝜃 𝑐𝑜𝑠𝑛𝜃 7. 2 3 4 4 0 5 8. if A=[ 5 7 9], B= [1 2 0], verify that (𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 . −2 1 1 0 3 1 9. 4 2 −1 10. Express the matrix A=[3 5 7 ] as the sum of a symmetric and skew –symmetric 1 −2 1 matrix. 1 3 5 11. Express [−6 8 3] as a sum of symmetric and skew –symmetric matrices. −4 6 5 0 6 7 0 1 1 2 12.if A=[−6 0 8] , B=[1 0 2] , C=[−2], verify (𝐴 + 𝐵)𝐶 = 𝐴𝐶 + 𝐵𝐶 . 7 −8 0 1 2 0 3 13.find the matrix X so that 𝑋 [ 14.Find the value of [1 𝑥 1 2 4 5 3 −7 −8 −9 ]=[ ] 6 2 4 6 1 3 2 1 1] [ 2 5 1] [2] = 0. 15 3 2 𝑥 2 0 1 15.Find 𝐴 − 5𝐴 + 6𝐼, 𝑖𝑓 A= [2 1 3]. 1 −1 0 2 −3 16.Show that A= [ ] satisfies the equation x2 – 6x +17=0. Hence find A-1 3 4 17. Find the inverse of the following matrix by using elementary transformations. 2 3 (i) [ 5 1 ] 2 2 (ii) [ 1 −6 ] −2 (iii) 2 [ −1 −3 ] 2 (iv) [ 5 2 ] 2 1 1 (v) [ −3 6 ] 5 0 1 2 2 3 1 1 1 2 1 3 −2 2 −1 3 (vi) [1 2 3] (vii) [2 4 1] (viii) [3 1 1] (ix) [−3 0 1 ] (x) [1 2 4] 3 1 1 3 7 2 2 3 1 2 1 0 3 1 1 Value based questions: 18.The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs.80, Rs.60 and Rs.40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra. Write the importance of the books in our life. 19. A farmer posses 30 acre cultivated land that must be in two different mode of cultivations organic and inorganic. The yield for organic and inorganic system of cultivations is 11 quintals/acre and 14 quintals/acre respectively. Using matrix method, determine how to divide 30 acre land among two mode of cultivation to obtained yield 390 quintals. Which mode of cultivation you prefer most and why? 20. For well being of an orphanage, three trust A, B, and C has donated 10%, 15% and 20% of their total fund 2,00,000, 3,00,000 and 5,00,000 respectively. Using matrix multiplication, find the total amount of money received by orphanage by three trust. By such donations, which values are generated? 21.A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs.180. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem? 22. There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in a family B. The recommended daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the 2 families. What awareness can you create among people about the balanced diet from this question? TOPIC-4: DETERMINANTS AND ITS PROPERTIES 1/2 Mark questions: 41 1 5 1. Without expanding evaluate the determinant:|79 7 9| 29 5 3 2. If A is a square matrix of order 3 such that |𝑎𝑑𝑗 𝐴| = 64, then find the | 𝐴 | 3.If A = [ 1 2 ] then find the |2 𝐴 | 4 2 4. For what value of 𝑥, the matri𝑥 :- 5−𝑥 [ 2 𝑥+1 ] is singular? 4 5. If A is a square matri𝑥 of order 3 such that |adj A| = 64, find |3A| 6 √5 6.Evaluate : | √ | √20 √24 7.If A is a square matrix of order 3 and |3 𝐴| = 𝐾|𝐴|, then write the value of K 8. Write the value of the determinant : 0 |𝑐𝑜𝑠150 𝑠𝑖𝑛75 𝑠𝑖𝑛150 | 𝑐𝑜𝑠750 1 2 9. If A=[ ], then find the value of |2A| 4 2 2 10. Find the value of 𝑥, 𝑖𝑓 | 5 4 2𝑥 |= | 1 6 4 | 𝑥 3 − 2𝑥 11. For what value of x the given matrix[ 2 12. Evaluate | 𝑥+1 ] is singular ? 4 𝑠𝑖𝑛30° 𝑐𝑜𝑠30° | −𝑠𝑖𝑛60° 𝑐𝑜𝑠60° 13. If A is a square matrix of order 3 such that |𝑎𝑑𝑗 𝐴| = 25, find |𝐴| 14. If A is a square matrix of order 3 and |𝐴| = 5 find A(adjA). 41 1 15. Without expanding evaluate the determinant:|79 7 29 5 5 9|. 3 16. For what values of k, the system of linear equations: x + y + z =2, 2x +y –z = 3 and 3x + 2y + kz = 4 has a unique solution ? 17. If A = [ 𝑐𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼 18. Evaluate| 𝑠𝑖𝑛𝛼 ], then for any natural number n, find the value of Det(An). 𝑐𝑜𝑠𝛼 𝑎 + 𝑖𝑏 −𝑐 + 𝑖𝑑 𝑐 + 𝑖𝑑 |. 𝑎 − 𝑖𝑏 2 −3 19.Find the co-factor of the element a12 of the determinant [6 0 1 5 2−𝑘 3 20.For what value of K, the matrix [ ] is not invertible? −5 1 4 Marks questions:- 5 4] −7 −3 −2 ], find A.(Adj A) 1 −4 −2 0 0 2. If A= [ 3 4 0], find |𝑎𝑑𝑗 𝐴| 10 −7 3 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥 1 0 3. Given that A=[ ] and A(adjA)= k[ ], find the value of k. −𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 0 1 1. If A=[ 4.Using properties of determinants, Prove that: a2 1 ab ab b2 1 ca cb ac bc 1 a2 b2 c2 c2 1 (𝑏 + 𝑐)² 𝑎𝑏 𝑐𝑎 4. Prove that| 𝑎𝑏 (𝑎 + 𝑐)² 𝑏𝑐 | 𝑎𝑐 𝑏𝑐 (𝑎 + 𝑏)² =2abc(a+b+c)³, Using properties of determinants. −𝑏𝑐 5. | 𝑎² + 𝑎𝑐 𝑎² + 𝑎𝑏 𝑏² + 𝑏𝑐 −𝑎𝑐 𝑏² + 𝑎𝑏 𝑐² + 𝑏𝑐 𝑐² + 𝑎𝑐 | −𝑎𝑏 4 ] ,then𝑎𝑑𝑗(𝐴𝐵) = (𝑎𝑑𝑗𝐴)(𝑎𝑑𝑗𝐵) 0 3 −1 7. Find the inverse of the matrix A= [ ].Also verify that 𝐴−1 𝐴 = 𝐼2 = 𝐴−1 𝐴 2 6 2 −3 8. Given A=[ ], compute 𝐴−1 and show that 2𝐴−1 = 9𝐼 − 𝐴 −4 7 4 5 9. If A=[ ], show that A-3I= 2(I+3A-1) 2 1 2 𝑥 10.If A=[ ], x≠ 1,calculate (i) A2 (ii) (A2)-1 4 2 6. If A=[ 1 −1 2 ] and B=[ 0 3 −3 =(bc+ca+ab)³ 𝑎 11.If a + b +c ≠ 0, and |𝑏 𝑐 𝑐 𝑎| = 0 , then show that- a = b = c 𝑏 𝑥+𝑦 𝑦+𝑧 𝑧+𝑥 12.Using properties of the determinants: evaluate | z 𝑥 𝑦 | 1 1 1 13.If a, b, and c are positive and unequal, show that the value of the determinant 𝑎 |𝑏 𝑐 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑐 𝑎| is negative. 𝑏 14. Using properties of determinants, solve for 𝑥. 𝑎+𝑥 |𝑎 − 𝑥 𝑎−𝑥 𝑎−𝑥 𝑎+𝑥 𝑎−𝑥 𝑎−𝑥 𝑎 − 𝑥| = 0 𝑎+𝑥 1 𝑎 |1 15.Without expanding the determinant, evaluate: |𝑏 1 𝑐 𝑎 𝑏𝑐 𝑏 𝑐𝑎 || 𝑐 𝑎𝑏 𝑏+𝑐 𝑎 𝑎 16.15. Prove that | 𝑏 𝑐+𝑎 𝑏 | = 4𝑎𝑏𝑐 𝑐 𝑐 𝑎+𝑏 𝑎 𝑎 + 𝑏 𝑎 + 2𝑏 17.16. Prove that |𝑎 + 2𝑏 𝑎 𝑎 + 𝑏 | = 9𝑏 2 (𝑎 + 𝑏) 𝑎 + 𝑏 𝑎 + 2𝑏 𝑎 𝑥 2 3 18.If X= -4 is a root of |1 𝑥 1| = 0 then find other two roots. 3 2 𝑥 2 1 𝑎 𝑏𝑐 1 𝑎 𝑎 2 19.Show that |1 𝑏 𝑏 |=|1 𝑏 𝑐𝑎 | 1 𝑐 𝑐 2 1 𝑐 𝑎𝑏 𝑎 𝑏 𝑐 20.If a + b +c=0 and |𝑏 𝑐 𝑎| = 0 then prove that a= b =c 𝑐 𝑎 𝑏 20. By using properties of determinant, show that: 1 𝑥 𝑥2 |𝑥 2 1 𝑥 | = (1 − 𝑥 3 )2 𝑥 𝑥2 1 𝑥 𝑥² 1 + 𝑥³ 21.If x,y,z are different and |𝑦 𝑦² 1 + 𝑦³| = 0, then 1+xyz =0 𝑧 𝑧² 1 + 𝑧³ 1+𝑎 22.Prove that | 1 1 1 1+𝑏 1 1 1 | = abc+ab+ bc+ca, Using Properties of determinants. 1+𝑐 6 Marks questions: 1. Using properties of determinants, prove that 1 + 𝑎2 − 𝑏 2 2𝑎𝑏 −2𝑏 2 2 | | = (1 + 𝑎2 + 𝑏 2 )3 2𝑎𝑏 1−𝑎 +𝑏 2𝑎 2𝑏 −2𝑎 1 − 𝑎2 − 𝑏 2 2. Using the properties of determinants , prove that: 𝑥 𝑥 2 1 + 𝑝𝑥 3 3. |𝑦 𝑦 2 1 + 𝑝𝑦 3 | = (1 + 𝑝𝑥𝑦𝑧)(𝑥 − 𝑦)(𝑦 − 𝑧)(𝑧 − 𝑥 ) 𝑧 𝑧 2 1 + 𝑝𝑧 3 15 − 2𝑥 11 − 3𝑥 7 − 𝑥 4. Solve for x:| 11 17 14 | = 0 10 16 14 2 2 (𝑎 + 𝑏 )/𝑐 𝑐 𝑐 2 2 5. Solve for x:| | = 4𝑎𝑏𝑐 𝑎 (𝑏 + 𝑐 )/𝑎 𝑎 𝑏 𝑏 (𝑐 2 + 𝑎2 )/𝑏 𝑎2 𝑏2 𝑐2 𝑎2 𝑏 2 𝑐 2 6. Prove that |(𝑎 + 1)2 (𝑏 + 1)2 (𝑐 + 1)2 | = 4 | 𝑎 𝑏 𝑐 | (𝑎 − 1)2 (𝑏 − 1)2 (𝑐 − 1)2 1 1 1 1 cos(𝛽 − 𝛼) cos(𝛾 − 𝛼) 1 cos(𝛾 − 𝛽)| = 0 7. Prove that:|cos(𝛼 − 𝛽) cos(𝛼 − 𝛾) cos(𝛽 − 𝛾) 1 1 1 1 8. In a triangle ABC,if| 1 + sin 𝐴 1 + sin 𝐵 1 + sin 𝐶 | = 0 then prove 2 2 sin 𝐴 + sin 𝐴 sin 𝐵 + sin 𝐵 sin 𝐶 + sin2 𝐶 that ∆ABC is an isosceles triangle. 𝑝 𝑏 𝑐 9. If |𝑎 𝑞 𝑐 | = 0 then find the value of p/(p-a) + q/(q-b) + r/(r-c) 𝑎 𝑏 𝑟 1 𝑎2 + 𝑏𝑐 𝑎3 10. Prove that|1 𝑏 2 + 𝑐𝑎 𝑏 3 | = −(𝑎 − 𝑏)(𝑏 − 𝑐)(𝑐 − 𝑎)(𝑎2 + 𝑏 2 + 𝑐 2 ) 1 𝑐 2 + 𝑎𝑏 𝑐 3 11. Find the inverse of the matrix A=[ 1 12.If [ 𝑡𝑎𝑛𝑥 −𝑡𝑎𝑛𝑥 1 ][ 1 −𝑡𝑎𝑛𝑥 𝑎 𝑐 𝑡𝑎𝑛𝑥 ] 1 −1 𝑏 1+𝑏𝑐 ] and show that: aA-1=(a2+bc+1)I-aI. 𝑎 =[ 𝑎 𝑏 −𝑏 ],then find the values of a and b. 𝑎 1 0 1 13. 12.If A=[1 2 3 𝑎 2 2 −1 3] and 𝐴 = [−4 5 1 2 −1 1 2 2 3 𝑏],then find the values of a and b. −3 1 2 2 1 −1 2 14. 13.Find the adjoint of the matrix A=[ 2 3 5],also verify that A(adjA)= −2 0 1 |𝐴|𝐼3 =(adjA)A 2 1 3 15. 14. Find the inverse of the matrix A=[ 4 −1 0], also verify that 𝐴−1 𝐴 = 𝐼3 −7 2 1 1 2 2 16. 15. If A=[2 1 2], prove that 𝐴2 − 4𝐴 − 5𝐼 = 0. Hence find 𝐴−1 2 2 1 6 Marks Questions: Solve the system of linear equations, using matrix method: 1. x+3y+4z =8, 2x+y+2z =5, 5x+y+z =7 2. 8x+4y+3z =18, 2x+y+z =5, x+2y+z =5 1 1 1 𝑥 𝑦 𝑧 2 1 3 1 1 1 = 4, + + = 0, + + = 2, x≠ 0, y≠ 0, z≠ 0 𝑥 𝑦 2 𝑥 𝑦 𝑧 1 −1 1 4. If A = [2 1 −3] find A-1 and hence solve. x+2y+z =4, -x+y+z =0, x-3y+z =2. 1 1 1 3. - + −4 4 4 1 −1 1 5. If A = [−7 1 3 ] , B = [1 −2 −2] , find AB and use Hence to solve the 5 −3 −1 2 1 3 following equations:x-y+z =4, x-2y-2z = 9, 2x+y+3z =1 3𝑥 − 8 3 6. Solve for x :| 3 3𝑥 − 8 3 3 3 3 | 3𝑥 − 8 =0 1 −1 0 2 2 −4 7. Given that A=[2 3 4] and B=[−4 2 −4] find AB.Use this to solve the 0 1 2 2 −1 5 following system of equations. x-y=3, 2x+3y+4z=17,y+2z+7. 1 −1 2 −2 0 1 8. Use the product [0 2 −3] [ 9 2 −3]to solve the system of equations x3 −2 4 6 1 −2 y+2z=1,2y-3z=1 and 3x-2y+4z=2. 9. An amount of Rs.5000 is put into three investments at the rate of interest of 6%,7% and 8% per annum respectively. The total annual income is Rs.358. If the combined income from the first two investments is Rs.70 more than the income from the third, Find the the amount of each investment by matrix method. 10.A school wants to award its students for the values of honesty,regularity and hardwork with a total cash award of Rs.6000. Three times the award money for hard work with a total cash award of Rs.6000.Three times the award money for hardwork added to that given for honesty amounts to Rs.11000.The award money given for honesty and hard work together is double the one given for regularity. Represent the above situation algebraically and find the award for each value using matrix method . Apart from these values namely honesty ,regularity and hard work suggest one more value which the school must include for awards. 11.For keeping fit X people believes in morning walk, Y people believe in Yoga and Z people join Gym. Total number of people is 70, further 20%, 30% and 40% people are suffering from any disease who believe in morning walk, Yoga and Gym respectively. Total number of such people is 21. If morning walk costs Rs.0, Yoga costs Rs.500 per month and gym cost Rs.400 per month and total expenditure is Rs.23,000. (1) Formulate a matrix problem . (2) Calculate the number of each type of people. (3) Why exercise is important for health. 12.An amount of Rs.600 crores is to be spend by the government in three schemes. Scheme A is for saving girl child from the cruel parents who don’t want girl child and get the abortion before her birth. Scheme B is for saving of newlywed girls from death due to dowry. Scheme C is planning for good health for senior citizen. Now twice the amount spent on scheme C together with amount spent on scheme A is Rs.700 Crores .And three times the amount spent on scheme A together with amount spent on scheme B and scheme C is Rs.1200 Crores. Find the amount spent on each scheme using matrices. What is importance of saving girl child from the cruel parents who don’t want girl child and get the abortion before her birth. 13.Two schools A & B want to award their selected teachers on the values of honesty ,hardwork and regularity. The school A wants to award Rs. x each, Rs.y each and Rs. Z each for three respective values to 3,2and 1 teachers with a total award money of Rs.1.28 lakhs. School B wants to spend Rs.1.54 lakhs lakhs to award its 4, 1 and 3 teachers on the respective values (by giving the same award money for the three values as before). If the total amount of award for one prize on each value is Rs.57,000 using matrices find the award money for each value. 14.A School wants to reward the students participating in co-curricular activities (category I) and with 100% attendance (category II) brave students (category III) in a function.The sum of the numbers of all the three category student is 6. If we multiply the number of category III by 2 and added to the number of category I to the result , we get 7. By adding II and III category would to three times the first category we get 12. Form the matrix equation and solve it. *********************** TOPIC-5: CONTINUITY AND DIFFERENTIABILITY 4 Marks Questions: Differentiate the following. 1. y = sin3(8x) 2. y = log(cot(ex+ 1)) 3. y = sin(sin(sin(x))) 4. y = cosec (log (2x4)) 5𝑥+12√1−𝑥 2 5. y = 𝑠𝑖𝑛−1 [ 13 ] 6. y = sin−1 (𝑥 2 √1 − 𝑥 2 + 𝑥 √1 − 𝑥 4 ) 7. If 𝑦 = 𝑒 𝑎𝑐𝑜𝑠 −1 𝑥 , −1 ≤ 𝑥 ≤ 1 then show that: (1 − 𝑥 2 ) 𝑑2𝑦 𝑑𝑦 𝑑𝑥 𝑑𝑥 −𝑥 2 8. If y = tan 1 2 then show that 2 2 dx a 9 x 2 a 2 4 x 2 a 6x dy 5ax 3a 2a d2y dy 2x 1 x 2 2 9. If y = (tan x) , show that (1 x ) 2 dx dx -1 2 10. If y = tan−1 ( 2 2 2𝑥 1+ 𝑥 2 𝑑𝑦 ) + sec −1 ( 1−𝑥 2 ), then prove that 𝑑𝑥 = 1− 𝑥 2 4 1+ 𝑥 2 − 𝑎2 𝑦 = 0 Topic: Logarithmic and parametric differentiation Basic Level Questions: 1. Find 2. Find 3. Find 4. Find 5. Find 6. Find 7. Find 8. Find 9. Find 10. Find dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx when x = 2at2, y = at4 when x = acos3t, y = bsin3t when x = a (Ɵ + sin Ɵ ), y = a(1 + cos Ɵ ) when x = et(sin t + cos t), y = et (sint – cos t) when x = cos Ɵ – cos 2 Ɵ, y = sin Ɵ – sin 2 Ɵ if y = e3logx if y = 2log2 𝑥 if y = 5𝑥 2 5 if y = xx if y = esinx Average Level Questions: 2 2 dy 2t 1. Find when x = t 2 , y = 2 dx 1 t 1t 2. Find 3. Find 4. Find 5. Find 6. Find 7. Find 8. Find 9. Find 10.Find dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx when x = a sin pt and y = b cos pt at t = π/4. when y = a cos3θ, x = b sin3θ when x = eθ(1 + 1 ) and y = e-θ (1 – 1 ) when x = eθ( 2sin θ + sin2 θ) , y = eθ( 2cos θ + cos2 θ) when xyyx = ab when y = (logx)x + xlogx when y = xx + x1/x when y = sin(xx) dy when y = xx + yx = 1 dx Above Average Level Questions: 1t dy 2bt when x = a ,y= 2 2 dx 1 t 1 t 2 1. Find 2. If x = asin 1t acos and y = 1t , show that dy y =dx x 3. If x = sec Ɵ – cos Ɵ and y = sec n Ɵ - cos n Ɵ, then show that (x 2 + 4) ( dy 2 ) = n2 (y2 + 4) dx 1 t 4. If x = log t and y = , prove that y2 + y1 = 0. 5. If y = sin pt and x = sin t , then prove that (1 – x2) y2 – x y1 + p2 y =0. 6. Differentiate w.r.t. x, esin x + (tan x ) x 7. Differentiate w.r.t. x, xcos x + (sin x ) tan x 8. Differentiate w.r.t. x, xlog x + (log ) x 9. If xy = e x – y , prove that 10.Find dy = dx log x (1 log x) 2 dy when xy + yx = (x + y)x + y dx Topic: Mean Value theorem Basic Level Questions: 1. 2. 3. 4. 5. 6. 7. 8. 9. Find the slope of chord A ( 0, 1) B ( 2,5) Is f(x) = tanx on [ -1 , 1] is continuous ? Is f(x) = |x| on [ -1 , 1] is continuous and differential ? Is logx is continuous on for all x>0 Find the derivative of function f(x) = (x-2) (x-3) (x-4) Find the derivative of function cos 2( x- π/4) If (cosx –sinx) =0 then what is x ? If sin2x = 0 then what is x ? Is f(x) = 2+(x-1)2/3 on (0,2) is differentiable ? 10. Find the derivative of 𝑠𝑖𝑛𝑥 𝑒𝑥 Average Level Questions: Verify mean value theorem for the following function: 1. 2. 3. F(x) = x (x-2) on [ 1,3] F(x) = x3-2x2-x+3 on [0,1) F(x) = (x-3) (x-6) (x-9) on [3,5] 4. F(x) = 𝑒 1−𝑥 0n [-1,1] 5. F(x) = 6. F(x) = cosx on [0,π/2 ] 2 1 on [1,4] 4𝑥−1 7. F(x) = √25 − 𝑥 2 0n [1,5] 8. F(x) = 𝑥 1/3 on [-1,1] 9. F(x) =𝑒 𝑥 on [0,1 ] 10. F(x) = x (x+4)2 on [0,4] Above Average Level Questions: Verify mean value theorem for the following function 1) 2) 3) 4) F(x) = log 𝑒 𝑥 on [ -1 ,1] F(x) = x-2sinx on [ -π,-π] F(x) = 2sinx +sin2x on [0,π] Find c of mean value theorem of the function F(x) = x (x-1) (x-2) on [0,1/2] 5) Using mean value theorem find a point p on the curve y=√𝑥 2 − 4 defined in the interval [2,4] where tangent is parallel to the chord joining the points on the curve . Discuss the applicability of mean value theorem for the function Is f(x) = |x| on [ -1 , 1] 6) 7) Find c so that f’(c) = 𝑓(6)−𝑓(4) 6−4 where f(x) = √𝑥 + 2 and c € (4,6). Topic: Second order derivatives Basic Level Questions: 1. Find second order derivative of x3. 2. If y = cot x, find 𝑑2𝑦 𝑑𝑥 2 at = 𝜋 2 . 3. If y = 5 cosx -3 sinx , prove that 𝑑2𝑦 4. If √𝑥 +√𝑦 = √𝑎, find 𝑑𝑥 2 𝑑2𝑦 𝑑𝑥 2 + y = 0. at x=a. 5. If y=500𝑒 7𝑥 + 600𝑒 −7𝑥 , prove that 𝑑2𝑦 𝑑𝑥 2 = 49y. 𝑥 𝑦 𝑑2𝑦 𝑎 𝑏 𝑑𝑥 2 6. If + =1, find . 𝑑2𝑦 7. If 𝑥 = 𝑎 𝑠𝑖𝑛 𝑝𝑡 and 𝑦 = 𝑏 𝑐𝑜𝑠 𝑝𝑡, find the value of 8. If 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡 find 9. If 𝑦 = 𝑡𝑎𝑛𝑥, prove that 10.Find 𝑑2𝑦 𝑑𝑥 2 𝑑2𝑦 𝑑𝑥 2 2 𝑑 𝑦 𝑑𝑥 2 at x=0. 𝑑𝑥 2 . 𝑑𝑦 = 2y . 𝑑𝑥 when 𝑦 = 𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥 Average Level Questions: 1. If y= (cos −1 𝑥)2 , prove that (1-x2)y2 –xy1=2. 1. If y=log(x+√𝑥 2 + 𝑏 2 ) prove that ( x2+ b2) y2 + xy1 =0. 2. If y=𝑒 tan 3. If x = −1 𝑥 2𝑎𝑡 2 1+𝑡 prove that (1+x2) y2 + (2x-1)y1 = 0. and y = 3𝑎𝑡 1+𝑡 find 4. If 𝑥𝑦 = 𝑠𝑖𝑛𝑥, prove that 𝑑2𝑦 . 𝑑𝑥 2 2 𝑑 𝑦 2 𝑑𝑦 𝑑𝑥 2 5. If x = asin3t , y = bcos3t , find + + y =0. 𝑥 𝑑𝑥 𝑑2𝑦 𝑑𝑥 𝜋 at t= . 2 4 𝑑2𝑦 6. If y = 3 cos (logx) + 4 cos (logx) prove that x2 7. If y =𝑒 𝑥 ( sinx + cosx ), prove that 8. If y=A𝑒 𝑚𝑥 + B 𝑒 𝑛𝑥 , show that 9. If 𝑒 𝑦 (x+1) =1 ,show 𝑑2𝑦 𝑑2𝑦 𝑑𝑥 2 𝑑𝑦 -2 - (m+n) 𝑑𝑦 + x + y =0 𝑑𝑥 + 2y =0 𝑑𝑥 𝑑𝑦 𝑑𝑥 2 2 𝑑 𝑦 𝑑𝑦 that 2 = ( )2 𝑑𝑥 𝑑𝑥 𝑑𝑥 2 𝑑𝑥 mny=0. Above Average Level Questions: 1. If y= (1+√𝑥 2 − 1)m , prove that (𝑥 2 − 1) 𝑦2 + x y1 = m2 y 2. y = 𝑠𝑖𝑛 − 1 𝑥 √1− 𝑥 2 , show that (1-x2)y2 -3xy1 –y =0. 3. If x= asin2t (1+cos2t) and y= bcos2t( 1-cos2t) , find 𝑎 ),prove that 𝑎+𝑏𝑥 4. If y=x log( 𝑑2𝑦 𝑑𝑥 2 = 1 𝑎 𝑑2𝑦 𝜋 𝑑𝑥 4 at t= . 2 2 ( ) 𝑋 𝑎+𝑏𝑥 2 5. If y = [𝑙𝑜𝑔(𝑥 + √𝑥 2 + 1)] , Show that (1+x2) y2+xy1-2=0 6. If y= A 𝑒 −𝑘𝑥 cos( pt+c) prove that 𝑑2𝑦 𝑑𝑥 2 +2k 𝑑𝑦 𝑑𝑥 + ( p2 +k2)y =0. 7. If y = sin(𝑚 sin−1 𝑥), prove that (1-x2) y2-xy1 + m2y =0 8. If = 𝑎( 𝑐𝑜𝑠𝑡 + 𝑡𝑠𝑖𝑛𝑡), 𝑦 = 𝑎 (𝑠𝑖𝑛𝑡 − 𝑡𝑐𝑜𝑠𝑡) , 𝑓𝑖𝑛𝑑 9. If 𝑠𝑖𝑛(𝑥 + 𝑦) = 𝑘𝑦 ,prove that y2 +y(1+y1)3 =0. 10.If y= 𝑒 𝑎 sin −1 𝑥, prove that (1-x2) y2 –xy1-a2y =0 𝑑2𝑦 𝑑𝑥 2 . TOPIC-6: APPLICATION OF DERIVATIVES 1/2 MARK QUESTIONS RATE OF CHANGE OF QUANTITES 1. Find the rate of change of volume of the cone of constant height, w.r.t. radius of the cone. 2. Find the rate of change of area of a circle w.r.t. the radius x. 3. The side of an equilateral triangle is increasing at the rate of 0.5cm/sec. Find the rate of increase of its perimeter. 4. The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of perimeter of the square. 5. The radius of the circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference? 6. If the radius of a soap bubble is increasing at the rate of ½ cm sec. At what rate its volume is increasing when the radius is 1 cm. 4 MARKS QUESTIONS 7. An edge of a variable cube is increasing at the rate of 5 cm/sec. How fast the volume of the cube is is increasing when edge is 10cm long? 8. A balloon which always remains spherical is being inflated by pumping in gas at the rate of 900cm3/sec. Find the rate at which the radius of the balloon is increasing when the radius of the balloon is 15 cm. 9. The volume of cube is increasing at the rate of 6cm3/sec. How fast is the surface area increasing when the length of an edge is 15cm? 10. A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 5 times as fast as x-coordinate. 11. The total revenue received from the sale of x units of a product is given by R(x) = 3x 3 + 12x +10. Find the marginal revenue when x = 3. 12.A spherical balloon is being inflated by pumping in 16cm3/sec of gas. At the instant when balloon contain 36π cm3of gas, how fast is its radius increasing? INCREASING AND DECREASING 13.Find the value of a for which the function f(x) = x2 – 2ax + 6, x > 0 is strictly increasing. 14.Write the interval for which the function f(x) = cos x, 0 ≤ 𝑥 2𝜋 is decreasing. 15. If f (x) = ax + cos x is strictly increasing on R, find a. log 𝑥 16. Find the interval in which f(x)= , 𝑥 ∈ (0, ∞) is increasing ? 𝑥 4 17. For which values of x, the functions y =𝑥 4 − 𝑥 3 is increasing? 3 18. Find the interval in which function is increasing and decreasing: f(x) = – 2x3 – 9x2 – 12x + 1 19. Prove that the function f(x) = x2 – x + 1 is neither increasing nor decreasing in [0, 1]. 20. Find the interval in which the function f given by f(x) = sin x - cos x, 0 x < 2𝜋 is increasing or decreasing. TANGENTS AND NORMAL 21. Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4. 𝑥−1 22. Find the slope of tangent to the curve y = , x ≠ 2 at x = 10. 𝑥−2 23. Find the slope of the tangent to the curve y = x 3 – x + 1 at the point whose x –coordinate is 2. 𝑥2 24. Find the equations of tangent and normal to the curve 𝑦 2 = at (3, - 3) 4−𝑥 25. Find the equations of the normal lines to the curve y = 4x 3 + 3x + 5 which are parallel to the line 9y + x + 3 = 0. 26. Find the slope of the normal at the point (am3, am2) to the curve ax2 =y3 27. At what points on the curve x2 + y2 – 2x – 4y +1 = 0, the tangents are parallel to x – axis? 28. Find the equations of normal lines to the curve y = x3 – 3x which are parallel to the line x + 9y = 14. 29. Find the equation of the tangent and normal to the curve 3x2 – y2 = 3, which are perpendicular to the line x+ 3y = 2. 30. Find the points on the curve y = x3 at which the slope of the tangents is equal to ordinate of that point. 31. For the curve y = 4x3 – 2x5, find all the points at which the tangents passes through the origin. 32. If the tangent to the curve y = x3 + ax + b at point (1, - 6 )is parallel to the line y – x = 5. Find the value of a and b. 33. Find the equation of tangents and normal to the curve x1/2 + y1/2 = at the point (a2/4, a2/4). APPROXIMATION BY DIFFERENTIATION 34. If the error in the side of a square is 0.5%, find the percentage error in its area. 35. Using differentials find the approximate value of 1 (i) (82)4 (ii) (26)1/2 (iii) (127)1/3 (iv) (37)1/2 (v) (401)1/2 36.Find the appropriate change in volume of a cube when side increases by 1%. 37.Find the percentage error in calculating the volume of a cubical box if an error of 0.1% is made in measuring the length of edges of the cube. 38.Find the approximate value of f (5.001) where f(x) = x3 – 7x2 + 15. 39. If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating in the volume. MAXIMA AND MINIMA (6 MARKS QUESTIONS) 40. Find the points of local maximum or local minimum for the function f(x) = 2x36x2+6x+5 41. Find the maximum and minimum value of the function f(x) = 3-2sinx. 42. Find two positive numbers whose sum is 24 and whose product is max. 43. Show that of all the rectangle of given area, the square has the smallest perimeter. 44. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10m. Find the diameter of the window to get maximum light through the whole opening. 45. Find the point on the curve y2=4x which is the nearest to the point (2,-8). 46. Find the area of the greatest isosceles triangle that can be inscribed in a given ellipse x2/a2 + y2/b2=1 with its vertex coinciding with one extremity of the major axis. 47. Show that the height of a closed right circular cylinder of given surface and maximum volume is equal to the diameter of the base. 48. A piece of wire 28 units long is cut into two pieces. One piece is bent into the shape of a circle and other into the shape of a square. How the wire should be cut so that the combined area of the two figures is as small as possible. 49. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8 of the volume of sphere. 27 50. Show that the semi vertical angle of the cone of the maximum volume & of given slant height is 𝑡𝑎𝑛−1 √2 . 51. If length of 3 sides of trapezium other than base are equal to 10cm each, then find the area of trapezium when it is maximum? 52. A manufacturer can sell x items at a price of Rs 5 –(x/100) each. The cost price of x pens is Rs (x/5) + 500.What is the number of items, the manufacturer should sell to earn maximum profit? 53. If the performance of the student ‘y’ depends on the number of hours ’x’ of hard work done per day is given by the relation y = 4x – (x2/2). Find the number of hours, the student work to have the best performance. “Hours of hard work are necessary for success” Justify. 54. Profit function of a company is given as P(x) = (24x/5) –(x2/100) – 500 where x is the number of units produced. What is the maximum profit of the company? Company feels its social responsibility and decided to donate 10% of his profit for the orphanage. What is amount contributed by the company for the charity? Justify that every company should do it. 55. An expensive square piece of golden color board of side 24cm is to be made into a box without top by cutting each corner and folding the flaps to form a box. What should the side of the square piece to be cut from each corner of the board to hold maximum volume and minimum wastage? What is the importance of minimizing the wastage in utilizing the resources? SOME EXTRA QUESTIONS TOPIC : APPLICATIONS OF DERIVATIVES 1. Using differentials find the approximate value of 4 82 2. Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long. 3. Find the local maximum and local minimum values, if any for f(x) = sinx + cos x for 0<x< 2 4. Find two positive numbers whose sum is 16 and sum of whose cubes is maximum. 5. Find the equation of the tangent to the curve x + 3y – 3 = 0 which is parallel to the line 4x – y – 5 = 0 6. Find all the points of local maxima and minima and the corresponding maximum and 3 4 minimum values of the function: f(x) = x 4 8 x 3 45 2 x 105 2 7. Find all the points of local maxima and minima and the corresponding maximum and minimum values of the function: f(x) = sin x + 1 cos 2x where 0 x 2 2 8. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius ‘a’ is a square of side 2a 9. A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum. 10. Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm. 11. Show that the semi-vertical angle of a right circular cone of given surface area and 1 maximum volume is sin 1 3 12. Show that the volume of the greatest cylinder which can be inscribed in a cone of height ‘h’ and semi-vertical angle ‘’ is 4 h 3 tan 2 27 TOPIC-7: INTEGRATION 1/2 MARK QUESTIONS Find the following Integrals: 1. 2. 3. sec x. cos ecx dx log tan x x e cos(e )dx e (cot x log sin x) dx 2 x3 x3 x 4x 3 4. 5. e x (1 x) dx (2 x) 2 6. sec 2 x tan 2 x 2 tan x 10 dx 7. 8. (x 9. x a x sec x.dx 10. 3 3x 2 x 2 dx dx ( x 1)( x 5) 2 2 x dx 1)( x 2 3) 2 6 6 dx 4 sin 2 x 11. (a b cos 2 x)dx 12. xe dx 13. 2x x dx 2 6 3x 2 4 x 5 ( x3 2x2 5x 1)2 dx 1 )dx 15. e x (tan 1 x 1 x2 dx 16. ( x 2) 2 1 14. 17. x sec( x 2 ) dx cos x dx x sin xdx 19. 3 4 cos 2 x ( x 3) x 20. e dx ( x 4) 2 18. e x (1 x)dx 21. cos 2 ( xe x ) dx 22. x x log x 23. sin 24. d 3 f ( x) 4 x3 4 such that f 2 0, find f x dx x 1 (cos x) dx tan 4 x sec2 x dx x 25. 4 MARK QUESTIONS 1 26. sin x sin 2 x dx 27. 1 x x 28. tan 1 xdx (1 x)2 29. 30. 31. 32. 33. 34. 35. dx 2 (2 x 5) sec xdx x3 x 2 4 x 3dx 3 x2 1 ( x 2 2)(3x 2 4)dx cos 2 x cos 4 x cos 6 xdx ( tan x cot x ) dx e sin 3x dx sec x 1 dx 2x dx x1/ 3 dx 37. sin x(1 2 cos x) 36. x 38. 39. 1/ 2 dx 9 8x x2 6x 5 dx 3x 5 x 1 dx 40. x( x 4 1) dx 41. xa xb dx 42. x(x n 1) 2 43. sin 1 x cos 1 x sin 1 x cos1 x dx 44. e 2x cos 3 xdx 2sin 2 x cos x dx 2 x 4sin x dx 46. 5sin x 12 cos x 2 sin 2 x 47. e x dx 1 cos 2 x 45. 6 cos 48. x2 1 x4 x2 1dx 49. (tan log x sec log x )dx 2 x3 x 1 x2 1 dx 3x 5 dx 51. 3 x x2 x 1 dx 52. 4 x 1 50. 53. 54. x cot xdx cot xdx 2 1 sin x 55. sin 4 xdx 56. (3x 2) 57. ( x 4)( x 5)( x 6)dx 58. 2 5𝑥 2 𝑑𝑥 ∫1 𝑥 2 +4𝑥+3 1+cot 𝑥 59. ( x 1)( x 2)( x 3) ∫ 𝑥+log sin 𝑥 𝑑𝑥 𝜋 60. x 2 x 1dx 3+5 cos 𝑥 ∫02 𝑙𝑜𝑔 [ 3+5 sin 𝑥 ] 𝑑𝑥 6 MARKS QUESTIONS 61. 62. 1 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: ∫0 𝑥 (𝑡𝑎𝑛−1 𝑥)2 𝑑𝑥 𝜋 2 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: ∫0 log 𝑠𝑖𝑛𝑥 𝑑𝑥. 𝜋 2 63. 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: ∫0 64. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ 𝑠𝑖𝑛2 𝑥 = 𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥 𝜋 𝑥 𝑡𝑎𝑛 𝑥 ∫0 𝑠𝑒𝑐 𝑥+𝑡𝑎𝑛 𝑥 𝑑𝑥 1 √2 𝑙𝑜𝑔 (√2 + 1) 𝜋 𝑠𝑖𝑛 2𝑥 65. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫02 66. Prove that ∫−2|𝑥 𝑐𝑜𝑠𝜋𝑥| 𝑑𝑥 = 𝜋 67. 𝑠𝑖𝑛4 𝑥+𝑐𝑜𝑠 4 𝑥 2 𝑑𝑥 8 π 3 sin x+cos x π √sin 2x 6 Evaluate: ∫ 𝑑𝑥 2 x 1 log 2 x dx 68. Evaluate 1 69. Evaluate x tan x (sec x tan x) dx 0 70. Evaluate : (x 0 2 dx a )( x 2 b2 ) 2 TOPIC-8: APPLICATION OF INTEGRALS 6 MARKS QUESTIONS 1. Find the area of the region bounded between the line x 4 and the parabola y 2 16 x. 2. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 {(𝑥, 𝑦) ∶ 0 𝑦 𝑥 2 + 1 ,0 𝑦 𝑥 + 1, 0 𝑥 2}. 3. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 {(𝑥, 𝑦): 𝑥 2 + 𝑦 2 2𝑎𝑥, 𝑦 2 𝑎𝑥, 𝑥 0, 𝑦 0}. 4. 𝑈𝑠𝑖𝑛𝑔 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠: 𝑥 + 2𝑦 = 2, 𝑦 − 𝑥 = 1 𝑎𝑛𝑑 2𝑥 + 𝑦 = 7. 4 5. Sketch theGraph of y x 3 . Evaluate x 3 dx. What does this value of the int egral represent on 0 on the graph. 6. Find the area of the region bounded by y x2 2, y x, x 0 and x 3 . 7. Find the area of the region included between the parabolas y 2 4ax and x 2 4ay, where a 0 8. Find the area of the region{( x, y) : x2 y x }. 9. Find the area of the region {( x, y) : y 2 4 x, 4 x2 4 y 2 9}. 10. Sketchthe region common tothe circle x 2 y 2 16 and the parabola x 2 6 y. Also find the area of the region u sin g int egration. 11. Find the area bounded by the lines y 4 x 5, y 5 x and 4 y x 5. 12. Find the area of the region common tothe circles x2 y 2 4 and ( x 2)2 y 2 4. 13. u sin g int egration find the aea of the region bounded by the triangle whose vertices are (2,1), (3, 4) and (5, 2). 14. Find the area of the region x, y : 0 y x 2 1, 0 y x 1, 0 x 2. Find the area of the smaller region bounded by the ellipse 15. x 2 a2 y2 x y 1 andthe straightline 1. 2 b a b 16. Find the area enclosed by the given curves 𝑦 = 𝑥 2 , 𝑥 = 𝑦 . 17. Find the area enclosed by the given curves 𝑦 = |𝑥 + 3| 𝑥 = −6, 𝑥 = 0. 𝜋 18. Find the area enclosed by the given curves𝑦 = sin 𝑥 , 𝑥 = 0, 𝑥 = . 2 19. Find the area enclosed by the curves y = cos x , x = 0 and x = π 2 4 and x − axis. Find the area enclosed by the given curves 𝑦 = 𝑥 2 , 𝑦 = √𝑥 Find the area enclosed by the given curves 𝑦 2 = 4𝑎𝑥 and its letus rectum. Using integration find the area enclosed by a circle of radius 𝑟. Find the area of the region bounded by the curves𝑦 2 = 2𝑦 − 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑦 − 𝑎𝑥𝑖𝑠. Using integration find the area enclosed by the curves y = x, and the curves y = x 3 . Find the area of the region bounded by the curves 𝑦 2 = 4𝑎𝑥and x 2 = 4ay. Using integration find the area of the region in the first quadrant enclosed by 𝑥 − 𝑎𝑥𝑖𝑠, 𝑥 = √3𝑦 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑥 2 + 𝑦 2 = 4. 27. Draw the rough sketch of the curves 𝑦 = sin 𝑥, y = cos 𝑥 as 𝑥 varies from 0 to π⁄2 , 20. 21. 22. 23. 24. 25. 26. and find the area enclosed by them and x − axis. 28. Find the area enclosed by the given curves|𝑥| + |𝑦| = 1. 29. Find the area bounded by the curves 𝑥 = 𝑦 2 and 𝑥 = 3 − 2𝑦 2 . 30. Find the area of the region bounded by the curves y = x 2 + 2, and the lines y = x, x = 0 and x = 3. TOPIC-9: DIFFERENTIAL EQUATIONS 1/2 MARK QUESTIONS Determine order and degree of the following differential equations: 𝑑𝑠 4 𝑑2𝑠 1. ( ) + 3𝑠 2 = 0 𝑑𝑡 𝑑𝑡 3 2. 𝑑 2 𝑦/𝑑𝑥 2 = [ 𝑎 + (𝑑𝑦/𝑑𝑥)2 ]2 3. 4. 5. 6. 𝑦 = 𝑥 𝑑𝑦/𝑑𝑥 + √𝑎2 (𝑑𝑦/𝑑𝑥)2 + 𝑏 2 𝑦 = 𝑥(𝑑𝑦/𝑑𝑥) 3 + 𝑑 2 𝑦/𝑑𝑥 2 (√𝑎 + 𝑥) (𝑑𝑦/𝑑𝑥) + 𝑥 = 0 𝑥(𝑑𝑦/𝑑𝑥) + 2/(𝑑𝑦/𝑑𝑥) = 𝑦 2 d2y d2y dy 2 3 x log 7. 2 dx 2 dx dx 2 d4y dy 8. c 4 dx dx 9. y x 2 3 dy dy a 1 dx dx 2 2 d2y dy d2y 10. 2 3 x 2 log( 2 ) dx dx dx 11.Find the solution of differential equation: 𝑥√1 + 𝑦 2 𝑑𝑥 + 𝑦 √1 + 𝑥 2 𝑑𝑦 = 0. 12.Write the differential equation representing the family of curves 𝑦 = 𝑚. 13. Form the differential equation representing the family of curves : y A cos( x B) Where A and B are parameters. 14. Form the differential equation for the curve y cx 2c 2 c3 . 15. Solve the Differential Equation (1 e2 x )dy e x (1 y 2 )dx 0, Given thaty(0) 1. d2y dy 16. Showthat : y e ( A cos x B sin x) is the solution of the D.E. 2 2 2 y 0. dx dx x 17. Solvethe Diff .Equation x(1 y 2 )dx y(1 x2 )dy 0, giventhat y 0 when x 1. 18. Solve sin 1 dy x y. dx 19. Solve : 3e x tan ydx (1 e x )sec2 ydy 0. 20. Find the Integrating factor of (1 x 2 ) 4 MARKS QUESTIONS 21. Solve the D.E.( x2 y 2 )dx 2 xydy 0; given that y 1 when x 1. 22. By e lim inating the cons tan t Aand B obtainthe D.E. forwhich xy Ae x be x x 2 is a solution. d 2 y dy 2 dy 23. Showthat Ax By 1is the solution of x y 2 y . dx dx dx 2 2 24. Solve : ydx xdy 3x2 y 2e x dx 0. 3 25. Solve : sin 1 xdy y 1 x2 dx 0. dy e x (sin 2 x sin 2 x) . dx y(2log y 1) dy 27. Solve : 1 x 2 y 2 x 2 y 2 xy 0. dx dy dy 28. Solve : y x 2 1 x 2 , when x 1. dx dx 26. Solve : 2 1 29. Solve : 1 y dx (sin y x)dy, given that y(0) 0. 30. Solve : x 2 ydy x3 x 2 y 2 xy 2 y 3 dx 0. 31. Solve : ( y x) dy y x. dx dy 2 xy x 2 2 x 2 1 dx 32. Solve : dy 4x 1 2 y 2 . dx x 1 ( x 1)3 dy y tan x cos3 x. dx dy 34. Solve :(1 x)2 2 xy x 1 x 2 given that y(0) 0. dx 33. Solve : 35. Solve :(1 y 2 )dx (tan 1 y x)dy, giventhat y(0) 0. 36. Solve : ye y dx ( y3 2 xe y )dy, y(0) 1. 37. Solve : ( x3 y3 )dy x2 ydx 0. 38. Solve the differential equation √1 + 𝑥 2 + 𝑦 2 + 𝑥 2 𝑦 2 + 𝑥𝑦 𝑑𝑦 𝑑𝑥 =0 39. Solve the differential equation 𝑥 𝑑𝑦/𝑑𝑥 = 𝑦 − 𝑥 𝑐𝑜𝑠 2 (𝑦/𝑥). 40. Solve the differential equation (1 + 𝑦 2 )𝑑𝑥 = (tan−1 𝑦 − 𝑥)𝑑𝑦 . 6 MARKS QUESTIONS 41. 42. 43. 44. Solve the differential equation 𝑥 2 𝑑𝑦/𝑑𝑥 = 𝑦 2 + 2𝑥𝑦 given that 𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 1. Solve the differential equation 𝑑𝑦/𝑑𝑥 + 2𝑦 𝑡𝑎𝑛𝑥 = 𝑠𝑖𝑛𝑥, given that 𝑦 = 0 𝑖𝑓 𝑥 = 𝜋/3. Find the particular solution of the differential equation 𝑥 2 𝑑𝑦 + (𝑥𝑦 + 𝑦 2 )𝑑𝑥 = 0; 𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 1. Show that differential equation (𝑥 𝑠𝑖𝑛2 (𝑦/𝑥) − 𝑦)𝑑𝑥 + 𝑥𝑑𝑦 =0 is homogeneous.find the Particular solution of this differential equation, given that 𝑦 = 𝜋/4 𝑤ℎ𝑒𝑛 𝑥 = 1. 45. 46. Solve the following differential equation ( 𝑒 −2√𝑥 /√𝑥 – 𝑦/√𝑥 )𝑑𝑥/𝑑𝑦 = 1, 𝑥 ≠ 0. Show that the differential equation 𝑥𝑑𝑦/𝑑𝑥 𝑠𝑖𝑛(𝑦/𝑥) + 𝑥 − 𝑦 𝑠𝑖𝑛(𝑦/𝑥) = 0 is Homogeneous. Find the particular solution of this differential equation, given that 𝑥 = 1 𝑤ℎ𝑒𝑛 𝑦 = 𝜋/2. 𝐴𝑛𝑠. 𝑐𝑜𝑠 (𝑦/𝑥) = 𝑙𝑜𝑔|𝑥|. 47. Solve the following differential equation 48. Solve; 49. 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥 = 𝑥(2𝑦−𝑥) 𝑥(2𝑦+𝑥) 𝑖𝑓𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 1. 𝜋 + 𝑦 cot 𝑥 = 4 𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥, given that 𝑦 = 0 𝑤ℎ𝑒𝑛 𝑥 = . Find the particular solution of the differential equation 𝑑𝑦 𝑑𝑥 = 2 𝑥𝑦 𝑥 2 +𝑦 2 , given that 𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 0. 50. Find the particular solution of the differential equation(1 + 𝑥 2 )𝑑𝑦/𝑑𝑥 = 𝑒 𝑚 tan 𝑦. Given that 𝑦 = 1, 𝑤ℎ𝑒𝑛 𝑥 = 0. −1 𝑥 − TOPIC-10: VECTOR ALGEBRA 1/2 MARK QUESTIONS 1. Find the projection of 𝑖̂ − 𝑗̂ on𝑖̂ + 𝑗̂. 2. If |𝑎⃗| =2,|𝑏⃗⃗| =√3 and 𝑎⃗ .𝑏⃗⃗ =√3.Find the angle between 𝑎⃗ and 𝑏⃗⃗ . 3. If 𝑎⃗ is a unit vector and (𝑥⃗ - 𝑎⃗) . (𝑥⃗ + 𝑎⃗) =15. Find |𝑥⃗| . 4. Find the value of 𝑝 so that 𝑎⃗ =2𝑖̂ +𝑝𝑗̂ +𝑘̂ and 𝑏⃗⃗ =𝑖̂ -2𝑗̂+3𝑘̂ are perpendicular to each other. 5. Find the value of λ if 𝑎⃗ = λ𝑖̂ + 𝑗̂ + 4𝑘̂ is parallel to 𝑏⃗⃗ = 5𝑖̂ + 2𝑗̂ + 8𝑘̂. 6. Write the direction cosines of the line equally inclined to the three co-ordinate axes. 7. Find a unit vector in the direction of the vector b 6iˆ 2 ˆj 3kˆ . 8. Find a vector in the direction of vector a iˆ 2 ˆj whose magnitude is 7. 9. Find the area of a parallelogram whose adjacent sides are î − ĵ + k̂and 3î + 4ĵ − 5k̂. 10.Find the direction cosines of a line whose direction ratios are 2, -3, 6. 11.What is the cosine of angle which the vector 2iˆ ˆj kˆ makes with Y-axis? 12.If a and b are two vectors such that a b a , then prove that vector 2a b is perpendicular to vector b . 13.If a.a 0 and a.b 0 , then what can be concluded about the vector b ? 14.If a and b are two vectors such that a.b a b , then find the angle between a and b . 15.Write the value of (r.iˆ)iˆ (r. ˆj ) ˆj (r.kˆ)kˆ. 4 MARKS QUESTIONS θ 16.If â and b̂ are two unit vectors and θ is the angle between them, then prove that sin = 2 1 2 b|. |a⃗⃗ − ⃗⃗ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 17.If (i j k ), (2i 5 j 3k ), (3i 2 j 2k ) & (i 6 j k ) are the position vectors of points A, B, C & D respectively, then find the angle between AB & CD. Deduce that AB& CD are parallel. 1 1 1 18.Show that the vectors (2î +3ĵ+6k̂), (3î - 6ĵ+2k̂) and (6î +2ĵ-3k̂) are mutually 7 7 7 perpendicular unit vectors. ˆ ˆ ˆ ˆ ˆ 19.The two adjacent sides of a parallelogram are (2i 4 j 5k ) & (i 2 j 3k ) .Find the unit vectors parallel to its diagonals. Also find its area. 20.If a⃗⃗ and ⃗⃗ b are two vectors such that |a⃗⃗ + ⃗⃗ b| =|a⃗⃗| then prove that the vector 2a⃗⃗ + ⃗⃗ b is perpendicular to ⃗⃗ b. 21.If vectors a, b and c are such that a b c 0 and |a | 3,| b | 5 and | c | 7, find the angle between a and b . 22.Find a vector of magnitude 5 units, perpendicular to each of the vectors (a b ) and (a b ) where a iˆ ˆj kˆ and b iˆ 2 ˆj 3kˆ 23.The scalar product of the vector iˆ ˆj kˆ with a unit vector along the sum of vectors 2iˆ 4 ˆj 5kˆ and iˆ 2 ˆj 3kˆ is equal to one. Find the value of . 24.If a 2iˆ 2 ˆj 3kˆ, b iˆ 2 ˆj kˆ and c 3iˆ ˆj are such that a b is perpendicular to c . Find the value of . 25.Express the vector a 5iˆ 2 ˆj 5kˆ as the sum of two vectors such that one is parallel to the vector b 3iˆ kˆ and the other is perpendicular to b . 26.If a , b, c are the lengths of the opposite sides respectively to the angles A, B, C of a triangle a 2 b2 c2 ABC, show that : cos C 2ab . 27.If = 3iˆ 4 ˆj 5kˆ and 2iˆ ˆj 4kˆ , then express in the form = 1 2 , where 1 is parallel to and 2 is perpendicular to . 2 a.a a.b 28.Prove that a b a.b b .b . 29.For three vector a , b and c if a b c and a c b , then prove that a , b and c are mutually perpendicular vectors , with b a and a 1 30.If a,b and c are three vectors such that a b c 0, then show that a b b c c a . 31.If a and b are unit vectors inclined at an angle 𝜃, then prove that cos 2 1 aˆ bˆ . 2 32.If a, b and b are three unit vectors such that a.c a.c 0 and angle between b and c is 6 prove that a 2(b c) . 2 2 2 2 33.For any vector a , prove that a iˆ a ˆj a kˆ 2 a . 34.Write the value of 𝑝 for which a 3iˆ 2 ˆj 9kˆ and b iˆ pjˆ 3kˆ are parallel vectors. 35. Find the value of p if (2iˆ 6 ˆj 27kˆ) (iˆ 3 ˆj pkˆ) 0 . 36. If p is a unit vector and ( x p).( x p) 80 find | x | . 37. If a b c d and a c b d , show that a d is parallel to b c where a d and b c . 38. The scalar product of the vector iˆ ˆj kˆ with a unit vector along the sum of vectors 2iˆ 4 ˆj 5kˆ and iˆ 2 ˆj 3kˆ is equal to one. Find the value of . 39. If a.b and c are vectors such that a.b a.c and a b a c , a 0, then prove that b c . 40. If vectors a, b and c are such that a b c 0 and |a | 3,| b | 5 and | c | 7, find the angle between a and b . TOPIC-11: THREE DIMENTIONAL GEOMETRY 1/2 MARK QUESTIONS 1. Write the equation of a line parallel to the line x2 y3 z 5 and passing through the 3 2 6 point (1,2,3). 2. Write the direction cosines of the line equally inclined to the three co-ordinate axes. 3. Reduce the equation of the plane 3x + 4y – z + 7 = 0 in the normal form and hence find its distance from origin. 4. Find the distance between the parallel planes : r .(2iˆ ˆj 3kˆ) 4 and r .(6iˆ 3 ˆj 9kˆ) 13 5. Using direction ratio’s, show that the points (2, 3, 4), (–1, –2, 1) and (5, 8, 7) are collinear. 6. Cartesian equations of line AB are 2x 1 4 y z 1 . Write the direction ratio’s of a line 2 7 2 parallel to AB. 7. Write the vector equation of the line whose Cartesian equations are x 3 y+7 z 1 = . 2 3 5 8. Find the direction cosines of a line passing through the points (– 1, 0, 2) and (3, 4, 6). 9. Find the equation of the plane passing through the point (1,7,0) and perpendicular to the ˆ vector 4iˆ ˆj 5k 10. Find the length of the perpendicular from origin to the plane x-2y+2z-9=0. 11. The foot of the perpendicular drawn from the origin to a plane is (2, 5, 7). Find the equation of the plane. 12. Find the equation of the plane through (2, 3,4 ) and parallel to the plane 5x – 6y + 7z = 3. 13. Find the intercepts cut off by the plane 2x + y – z = 0. 14. Find the angle between the two planes 2x + y – 2z = 5 and 3x – 6y – 2z = 7. 15. Find the distance of a point (2, 5, -3) from the plane r. 6 iˆ 3 ˆj 2 kˆ 4 ˆ 16. If the points (1, 1, p) and (–3, 0, 1) be equidistant from the plane r .(3iˆ 4 ˆj 12k ) 13 0 then find the value of p. 17. If a line makes angle , , with positive directions of x- axis, y-axis and z-axis respectively, then find the value of sin 2 sin 2 sin 2 . 18. Find the value of , if line x 2 y 1 z 5 , is perpendicular to the plane 3x – y – 2z = 7. 6 4 19. Find the distance of the point (a, b, c) from X-axis. 20. Can a line have direction cosines 1 1 1 , , ? 5 5 5 21. If the equation of a line is x ay b, z cy d , then find direction ratios of the line and a point on the line. 22. Find the vector equation of a plane whose Cartesian equation is 2x+3y-4z+7=0. 23. Find the distance of a point (2, 5, -3) from the plane r. 6 iˆ 3 ˆj 2 kˆ 4 24. Reduce the equation of the plane 3x + 4y – z + 7 = 0 in the normal form. 25. Find the angle between the pair of lines: 4 MARKS QUESTIONS 26. Find the shortest distance between the lines l1 and l2 given by the following : l1 : x 3 y 5 z 7 , 1 2 1 27. Find the point on the line 28. l2 x 1 y 1 z 1 7 6 1 x 2 y 1 z 3 at a distance 3 2 from the point (1, 2, 3). 3 2 2 1 x 7 y 14 z 3 2p 2 Find the value of p so that the lines 3 7 7x y 5 6 z 1 5 and 3 p are at right angles. 29. A line makes an angle of π/4 with each of x-axis and y-axis. Find the angle between this line and the z-axis. 30. Find the equation of the plane through the points (-2, 6, -6), (-3, 10, -9) and (-5, 0, -6). 31. Find the length and the foot of the perpendicular from the point P(7, 14, 5) to the plane 2x + 4y – z = 0. 32. Find the angle between the line x 2 y 1 z 3 and the plane 3x + 4y + z +5 = 0. 3 1 2 33. Find the vector equation of the plane through the intersection of the planes r. 2iˆ 2 ˆj 3kˆ 7 r. 2iˆ 5 ˆj 3kˆ 9 and through the point (2, 1, 3). 34. A plane meets the co-ordinate axes in points A,B,C and the centroid of the triangle ABC is ( , , ), find the equation of the plane. 35. Find the angle between two diagonals of a cube. 36. Find the equation of plane which contains the line of intersection of planes r . ( iˆ ˆj kˆ ) 1 and r . (2 iˆ 3 ˆj kˆ ) 4 0 and parallel to X – axis. 1 37. Show that the angle between any two diagonals of a cube is cos 1 . 3 38. Find the equation of the plane passes through the points ( 1, 1 , 0 ) , ( 1, 2 ,1 ) and ( -2 , 2 ,-1 ). 39. Show that the lines r (3iˆ 2 ˆj 4kˆ) (iˆ 2 ˆj 2kˆ) and r (5iˆ 2 ˆj ) (3iˆ 2 ˆj 6kˆ) are intersecting. Hence, find their point of intersection. 40. Find the equation of line passing through points A(0, 6, -9) and B(-3, -6, 3). If D is the foot of perpendicular drawn from the point C (7, 4, -1) on the line AB, then find the coordinates of point D and equation of line CD. 41. Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured parallel to the line x 1 y 3 z 2 . 2 3 6 42. A plane meets the X , Y and Z axes at A, B and C respectively, such that the centroid of the triangle ABC is (1, 2, 3). Find the vector and Cartesian equation of the plane. 43. A vector n of magnitude 8 units is inclined to axis at 45 , y-axis at 60 and an acute angle with z-axis. If a plane passes through a point 2 ,1,1 and is normal to n, 2 ,1,1, find the equation in vector form. 44. Find the equation of plane(s) passing through the intersection of planes x + 3y + 6 = 0 and 3x – y - 4z = 0. and whose perpendicular distance from origin is unity. 45. Find the vector equation of the plane passing through the intersection of two planes 𝑟⃗⃗⃗. ( 𝑖̂ + 3 𝑗̂ − ̂𝑘 ) = 5 𝑎𝑛𝑑 𝑟⃗ . ( 2𝑖̂ – 𝑗̂ + 𝑘̂ ) = 3 and the point ( 2 ,1, 3). 6 MARKS QUESTIONS 46. Find the equation of the plane containing the lines r (iˆ ˆj ) (iˆ 2 ˆj kˆ) and r (iˆ ˆj ) (iˆ ˆj 2kˆ) . Find the distance of this plane from the origin and also from the point (1, 1, 1). 47. Find the equation of the plane passing through the intersection of the planes (2x 3 y z 1) ( x y 2z 3) 0 and perpendicular to the plane 3x y 2 z 4 0 . Also find the inclination of this plane with the xy plane. 48. Find the equation of a line passing through the point (2,1,3) and perpendicular to the line x 1 y 2 z 3 x y z . and 1 2 3 3 2 5 49. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to each of the following planes: 2x 3 y 3z 2 and 5x 4 y z 6 . 50. Find the equation of the plane passing through the point (0, 1, 0) and (3, 4, 1) and parallel to the line x 3 y 3 z 2 . 2 2 5 51. From the point P (1, 2, 4), a perpendicular segment is drawn on the plane 2x +y –2z+3=0, find its equation and its length. Also, find the co-ordinates of the foot of the perpendicular. 52. Find the equation of the plane determined by the points (3, –1, 2) , B (5,2,4) and C (–1, –1, 6). Also find the distance of the point P (6, 5, 9) from the plane. 53. Find the distance of the point (1, –2, 3) from the plane x y z 5 measured along a line parallel to x y z 2 3 6 54. Find the equation of the line passing through the point (–4, 3, 1) parallel to the plane x + x 1 y 3 z 2 2 1 2y –z = 0 and intersecting the line : 3 55. Find image of the line x 1 y 3 z 4 3 1 5 in the plane 2x – y + z + 3 = 0. 56. Find the distance of the point (– 2, 3, – 4) from the line x 2 2 y 3 3z 4 measured 3 4 5 parallel to the plane 4 x 12 y 3z 1 10 . 57. Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the line r (2iˆ ˆj 2kˆ) (3iˆ 4 ˆj 2kˆ) and the plane r .(iˆ ˆj kˆ) 5 . 58. Find the image of the point (3,5,3) in the line x z 2 y 1 . 1 3 2 59. Find the angle between the lines whose direction cosines are given by the equations 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0. 60. A line with direction ratios (2, 7, –5) is drawn to intersect the lines. x 5 y 7 z 2 x 3 y 3 z 6 3 1 1 , 3 2 4 Find the co-ordinates of the points of intersection and the length intercepted on it. 61. Find the coordinates of the foot of perpendicular and perpendicular distance of point P(3,2,1) from the plane 2x – y + z +1 = 0. Find also image of the point in the plane. 62. Find the image of the point (1, 6, 3) on the line x y 1 z 2 . Also, write the equation of 1 2 3 the line joining the given points and its image and find the length of segment joining given point and its image. 63. Find the distance of the point (2, 3, 4) from the line x3 y2 z measured parallel to 3 6 2 the plane 3x + 2 y +2 z – 5 = 0. 64. Prove that the lines x 4 y 3 z 1 x 1 y 1 z 10 intersect each other and and 1 4 7 2 3 8 find the point of intersection. 65. A line makes angle , , and with the four diagonals of a cube. Prove that 4 cos 2 cos 2 cos 2 cos 2 . 3 66. If l1, m1, n1 and l2 , m2 , n2 are direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of them are m1n2 m2n1 , n1l2 l1n2 , l1m2 l2m1 . 67. Find the equation of a line passing through the point (2,1,3) and perpendicular to the line x 1 y 2 z 3 x y z . and 1 2 3 3 2 5 68. Find the length and the equation of the line of shortest distance between the lines: x 3 y 8 z 3 3 1 1 and x3 y7 z6 3 2 4 69. If the points (1, 1, µ) and (-3, 0, 1) be equidistant from the plane r . ( 3i +4j -12 k ) + 13 =0. Find the value of µ. 70. Find the vector equation of the plane passing through the intersection of two planes 𝑟⃗⃗⃗. ( 𝑖̂ + 𝑗̂ + ̂𝑘 ) = 1 and 𝑟⃗⃗⃗. (2 𝑖̂ + 3 𝑗̂ − ̂𝑘 ) + 4 = 0 and parallel to x-axis. TOPIC-12: LINEAR PROGRAMMING 6 MARKS QUESTIONS 1. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awards is 12. Three times the sum of the awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards. 2. A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs. 5 per kg to purchase Food I and Rs. 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically. 3. A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods X and Y are available at a cost of Rs. 4 and Rs. 3 per unit respectively. One unit of food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas 1unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of food X and Y should be used to have least cost, satisfying the requirements. Make it an LPP and solve it graphically. 4. A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs. 5760.00 to invest and has space for at most 20 items. An electronic sewing machine costs him Rs.360.00 and a manually operated sewing machine Rs. 240.00. He can sell an Electronic Sewing Machine at a profit of Rs. 22.00 and a manually operated sewing machine at a profit of Rs.18.00. Assuming that he can sell all the items that he can buy how should he invest his money in order to maximize his profit. Make it as a linear programming problem and solve it graphically. Keeping the rural background in mind justify the ‘values’ to be promoted for the selection of the manually operated machine. 5. A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs. 5 per kg to purchase Food I and Rs. 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically. 6. A merchant plans to sell two types of personal computers --- a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make an L.P.P. and solve it graphically. 7. A company sells two different products A and B. The two products are produced in a common production process which has a total capacity of 500 man hours. It takes 5 hours to produce a unit of A and 3hours to produce a unit of B. the demand in the market shows that the maximum number of units of that can be sold is 70 and that of B is 125. Profit on each of A is Rs. 20 and on B is Rs. 15. How many unites of A and B should be produced to maximize the profit. Form an LPP and solve it graphically. 8. An aero plane can carry a maximum of 200 passengers. A profit of Rs. 1,000 is made on each executive class ticket and a profit of Rs. 600 is made on a economy class ticket. The air line reserves at least 20 tickets for executive class. However, at least 4 times as many passengers prefer to travel by economy class, than the executive class. Determine how many tickets of each type must be sold, in order to maximize profit for the airline. What is the maximum profit? Make an LPP and solve it graphically. 9. One kind of cake requires 300 g of flour and 15 g of fat, another kind of cake requires 150 g of flour and 30 g of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600 g of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an L.P.P. and solve it graphically. 10. A library has to accommodate two different types of books on a self. The books are 6 cm 1 and 4 cm thick and weigh 1 kg and 1 kg each respectively. The self is 96 cm long and can 2 support a weight of 21 kg. How should the self to be filled with the books of two types in order to include the greatest number of books? Make it as LPP and solve it graphically. 11. A school wants to award its students for the values of honesty, regularity and hard work with a total cash award of Rs.6000. Three times the award money for hard work with a total cash award of Rs.6000.Three times the award money for hard work added to that given for honesty amounts to Rs.11000.The award money given for honesty and hard work together is double the one given for regularity. Represent the above situation algebraically and find the award for each value using matrix method. Apart from these values namely honesty, regularity and hard work suggest one more value which the school must include for awards. 12. For keeping fit X people believes in morning walk, Y people believe in Yoga and Z people join Gym. Total number of people is 70, further 20%, 30% and 40% people are suffering from any disease who believe in morning walk, Yoga and Gym respectively. Total number of such people is 21. If morning walk costs Rs.0, Yoga costs Rs.500 per month and gym cost Rs.400 per month and total expenditure is Rs.23,000. (1) Formulate a matrix problem. (2) Calculate the number of each type of people. (3) Why exercise is important for health. 13. An amount of Rs.600 crores is spent by the government in three schemes. Scheme A is for saving girl child from the cruel parents who don’t want girl child and get the abortion before her birth. Scheme B is for saving of newlywed girls from death due to dowry. Scheme C is planning for good health for senior citizen. Now twice the amount spent on scheme C together with amount spent on scheme A is Rs.700 Crores .And three times the amount spent on scheme A together with amount spent on scheme B and scheme C is Rs.1200 Crores. Find the amount spent on each scheme using matrices. What is importance of saving girl child from the cruel parents who don’t want girl child and get the abortion before her birth. 14. Two schools A & B want to award their selected teachers on the values of honesty, hardwork and regularity. The school A wants to award Rs. x each, Rs. y each and Rs. Z each for three respective values to 3,2and 1 teachers with a total award money of Rs.1.28 lakhs. School B wants to spend Rs.1.54 lakhs to award its 4, 1 and 3 teachers on the respective values (by giving the same award money for the three values as before). If the total amount of award for one prize on each value is Rs.57,000 using matrices find the award money for each value. 15. A School wants to reward the students participating in co-curricular activities (category I) and with 100% attendance (category II) brave students (category III) in a function. The sum of the numbers of all the three category student is 6. If we multiply the number of category III by 2 and added to the number of category I to the result, we get 7. By adding II and III category would to three times the first category we get 12. Form the matrix equation and solve it. 16. A dealer in rural area wishes to purchase a number of sewing machines. He has Rs. 5760 to invest and has space for at most 20 items. An electronic machine cost him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic machine at a profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy how should he invest his money in order to maximize his profit? Make it as a linear programming problem and solve it graphically. Keeping the rural background in mind justify the values to be promoted for the selection of manually operated machines. 17. A retired person has Rs. 70,000 to invest and two types of bonds are available in the market for investment . First type of bonds yields an annual income of 8% on the amount invested and the second type of bonds yields 10 % per annum. As per norms, he has to invest a minimum of Rs.10,000 in the first type and not more than Rs. 30,000 in the second type . How should he plan his investment, so as to get maximum return, after one year of investment? 18. A manufacturer produces pizza and cakes. It takes 1 hour of work on machine A and 3 hours on machine B to produces a packet of cakes. He earns a profit of Rs. 17.50 per packet on pizza and Rs. 7 per packet of cakes. How many packets of each should be produced each day so as to maximize his profit if he operates his machines for at the most 12 hours a day? Form the above a linear programming problem and solve it graphically. Why pizza and cakes are not good for health? 19. An aero plane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each execute class ticket and a profit of Rs. 600 is made on each economy class ticket . The airline reserves at least 20 seats for executive class, however, at least 4 times as many passengers prefer to travel by economy class than by the executive class . Determine how tickets of each type must be sold in order to maximize the profit for the airline . What is the maximum profit? Do you feel that air travel is safer now then in older days? Discuss briefly. 20. A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs. 5 per kg to purchase food I and Rs. 7 per kg to purchase food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically. What is meant by balanced food? 21. Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligram of iron per kilogram, find the minimum cost of producing this new cereal if bran cost Rs. 5 per kg and rice cost Rs. 4 per kg. 22. An oil company requires 12000, 20000, and 15000 barrels of high-grade , medium and low grade oil , respectively ,Refinery A produces 100 , 300 and 200 barrels per day of highgrade , medium grade and low grade , respectively , while refinery B produces 200 , 400 and 100 barrels per day of high-grade , medium grade and low grade , respectively . If refinery A cost Rs. 400 per day and refinery B costs Rs. 300 per day to operate , how many days should each be run to minimize costs while satisfying requirements. 23. A Gardner has supply of fertilizers of type I which consist of 10% nitrogen and 6% phosphoric acid and type II fertilizers which consist of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop. If the type I fertilizer costs 60 paisa per kg and type II fertilizers costs 40 paisa per kg , determine how many kilograms of each fertilizer should be used so that nutrient requirement are met at a minimum cost. What is the minimum cost? 24. A small firm manufacturer items A and B. the total number of items A and B that it can manufacturer in a day is at the most 24. Item A takes one hour to make while item B takes only an hour. The maximum time available per day is 16 hours. If the profit on one unit of item A be Rs. 300 and one unit of item available per day is 16 hours. if the profit on one unit of item A be Rs. 300 and one unit of item B 160 , how many of each type of item be produced to maximize the profit ? Solve the problem graphically. 25. A cooperative society of farmers has 50 hectare of land to grow two crops X and Y .The profit from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 liters and 10 liters per hectare. Further, no more than 800 liters of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so at to maximize the profit of the society? TOPIC-13: PROBABILITY 1/2 MARK QUESTIONS 1. The probability that at least one of A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3. Find P( A )+P( B ). 2. 4. A fair coin is tossed 100 times. Find the probability of getting tails an odd number of times. Two dice are rolled one after the other. Find the probability that the number on the first is smaller than the number on the second. The mean and variance of a Binomial distributions are 6 and 4. Find the parameter n. 5. For a random variable X. E(X)=3, E( X )=1. Find Var(X). 6. Probability that A hits a target is 7. probability that the target will be hit if both A and B shoot at it? 3 1 2 p ( A B) p ( A B) p( A ) 4, 4, 3 , find p( A B) . If A and B are events such that 3. 2 1 3 2 and the probability that B hits is . What is the 5 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. The mean and variance of a binomial distribution are 4 and 2 respectively. Find the probability of 2 successes. Three houses are available in a locality. Three persons applied for the house. Each applies for one house without consulting others. Find the probability that all the three apply for the same house. A coin is tossed three times. Find the probability of getting head and tail alternatively. Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P(E|F). Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32 If P(A) = 0.8, P (B) = 0.5 and P(B|A) = 0.4, find P(A ∩ B). Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and p(A/B) =2/5 . Given that E and F are events such that P(E) = 0.4, P(F) = 0.5 and P(E ∩ F) = 0.2, find P(F|E). If P(A) = 0.8, P (B) = 0.5 and P(B|A) = 0.4, find (i) P(A|B) . If P(A) = 0.7, P (B) = 0.3 and P(B|A) = 0.4, find (i) P(A ∪ B). If P(A) = 6/11 , P(B) = 5/11 , P(A ∪ B) = 7/11, find P(A∩B) . If P(A) = 4/11 , P(B) = 3/11 , P(A ∪ B) = 5/11, find P(A|B) . If P(A) =4/7 , P(B) = 2/7 , P(A ∪ B) =5/7, find P(B|A) . If P(A) =3/5 ,and P(B) =1/5 ,find P(A ∩ B) if A and B are independent events. Let E and F be events with P(E) =3/5 ,P(F) =3/10 ,P(E ∩ F) =1/5 .Are E and F independent ? Given that the events A and B are such that P(A) =1/2 ,P(A ∪ B) =3/5 and P(B)=K .Find K if they are independent. Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(A ∩ B). Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(A ∪ B). Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(A / B). Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(B / A). If A and B are two events such that P(A) =1/4 ,P(B) =1/2 and P(A ∩ B) =1/8 . Find P(not A and not B) . Events A and B are such that P(A) =1/2 ,P(B) =7/12 and P(A not A or not B ) =1/4 .State whether A and B are independent . Given two independent events A and B such that P(A) =0.3 ,P(B)= 0.6 .find P(A and not B). 4 MARKS QUESTIONS 31. Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither 9 nor 11. 32. There are 5% of defective bulbs in a large bulk of bulbs. What is the probability that a sample of 10 bulbs will include not more than one defective bulb? 33. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of a scooter, a car and a truck meeting an accident are 0.01, 0.03 and 0.15 respectively. If one of the insured person meets with an accident, find the probability that he is a scooter driver. 34. A discrete random variable X has mean score equal to 6 and variance equal to 2. What is the probability when 5 X 6 . 35. Find the probability distribution of the number of kings drawn when 2 cards are drawn one by one without replacement from a pack of 52 playing cards. 36. Six dice are thrown 729 times. How many times do you expect at least 3 dies to throw a 5 or 6? 37. A bag contains 5 white, 7 red and 8 block balls. If 5 balls are thrown one by one with replacement. Find the probability distribution exactly 5 red balls drawn. 38. If x follows binomial distribution with mean 4 and variance 2, find p( x 4 ) 2 . 39. A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps, he is one step away from the starting point. p( A) p( A ) 1 p( B ) 1 B 4 and A 2 , then verify which of the 40. If for the events A and B, following statements are correct. i) A and B are mutually exclusive events. ii) A and B are independent. p( A ) 3 B 4 iii) p( B ) 1 2 A iv) 41. A determinant of second order is made with the elements 0 and 1 .What is the probability that the determinant made is non negative? 42. A, B, C and D cut a pack of 52 cards successively in the order given. If the person who cuts a spade first receives Rs.350. What are their respective expectations? 43. There are two bags I and Ii containing 3 red and 4 white balls, 2 red and 3 white balls respectively. A bag is selected at random and a ball is drawn from it. If it is found to be a red ball, find the probability that it is drawn from the first bag. 44. A and B play a game of Tennis. The situation of the game is as follows. If one scores two consecutive points after a deuce he wins, if less of a point is followed or preceded by win of a point, it is deuce. The chance of a server to win a point is 2/3. The game is at a deuce and A serving what is the chance that A will win the game? 45. There are three bags which contain 2 white, 3 black; 4 white, 1 black; 3 white, 7 black balls respectively. A ball is drawn of random from one of the bags and is found to be black. Find the probability that it was drawn from the bag containing maximum number of black balls. 46. In an examination, an examinee either guesses or copies or knows the answer of multiplechoice questions with four choices. The probability that he makes a guess is 1/3 and probability that he makes a guess is 1/3 and probability that he copies the answer is 1/6. The probability that his answer is correct, given that he copies it, is 1/8. Find the probability that he knew the answer to the question, given that he correctly answered it. 47. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact present; however the test also yields as false positive result for 0.5% of the healthy person tested. If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive? 48. Given that box I, II and III each containing two coins. In Box I both are Gold coins, in box II both are silver coins and in III there is one gold and one silver coin. One coin is taken out from one of the boxes. If it is gold coin what is the probability that the other coin in the box is not of gold? 49. A bag contains 5 white and 6 black balls and another bag contains 4 white and 3 black balls. A ball is drawn from the first bag and without seeing its colour is put in the second bag and then a drawn from the second bag and it is found to be black in colour. Find the probability that the transferred ball is white? 50. A letter is known to have come from either GANGANAGAR or SRINAGAR. On the envelope just two consecutive letters GA are visible. What is the probability that the letter has come from (i) GANGANAGAR (ii) SRINAGAR 6 MARKS QUESTIONS 51. If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up the machine produces 2 acceptable items, find the probability that the machine is incorrectly setup. 52. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and found to be hearts. Find the probability of the missing card to be a diamond? 53. A is known to speak truth 4 times out of 7 times. He takes a card from 52 playing cards and reports that it is a spade. Find the probability that it is actually spade? 54. There are three coins. One is two headed coin, another is a biased coin that comes up heads 80% of the time and third is an unbiased coin. One of the coins is chosen at random and tossed, it shows tail, what is the probability that it was a biased coin? 55. In a factory which manufactures bolts, machines, A, B, C manufacture respectively 25%, 35%, 40% of the bolts. Of their outputs, 5, 4, 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine either B or C? 56. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl? 57. An instructor has a question bank consisting of 300 easy True/False questions, 200 difficult True/False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question? 58. Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’. 59. Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II. 60. A die is tossed thrice .find the probability of getting an odd number at least once. 61. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls .find the probability that one of them is black and other is red. 62. Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently. Find the probability that (I) the problem is solved. (ii) Exactly one of them solves the problem. 63. One card is drawn at random from a well shuffled deck of 52 cards .Check whether the following events E and F are independent. Where E: the card drawn is a spade, F: the card drawn is an ace. 64. In a hostel, 60% of the students read Hindi news paper and 40% read English newspaper and 20% read both Hindi and English newspaper. A student is selected at random. (a)Find the probability that the student reads neither Hindi nor English newspaper.(b)if she reads English newspaper ,find the probability that she reads Hindi newspaper. 65. The probabilities that a husband and wife will be alive 20 years from now are 0.8 and 0.9 respectively .find the probability that in 20 years (a) both (b) neither (c) at least one will be alive . 66. The odds against A solving a certain problem are 4 to 3 and the odds in favour of solving the same problem are 7 to 5 .find the probability that the problem will be solved. 67. Two cards are drawn at random from a pack of 52 cards .find the probability that the cards are either both red or both aces. 68. A class consists of 80 students; 25 of them are girls and 55 boys; 10 of them are rich and the remaining poor; 20 of them are fair complexioned .what is the probability of selecting a fair complexioned rich girl. 69. A scientist has to make a decision on each of the two independent events I and II. Suppose the probability of error in making decision on event I is 0.02 and that on event II is 0.05. Find the probability that the scientist will make the correct decision on (i) both the events (ii) only one event. 70. In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B? 71. A and B throw a pair of dice turn by turn .The first to throw 9 is awarded a prize .if A starts the game, show that the probability of A getting the prize is 9/17. 72. A has two fire extinguishing engines functioning independently .The probability of availability of each engine, when needed, is 0.95. What is the probability that 1. Neither of them is available when needed 2. An engine is available when needed? 3. Exactly one engine is available when needed? 73. A speaks truth in 55 percent cases and B speaks truth in 75 percent cases .Determine the percentage of cases in which they are likely to contradict each other in stating the same fact . 74. The probability that a teacher will give an unannounced test during any class meeting is 1/5 .if a Student absent twice, find the probability that the student will miss at least one test. 75. A machine operates if all its three components function .The probability that the first component 76. Fails during the year is 0.14, the second component fails is 0.10 and the third component fails is 0.05. 77. What is the probability that the machine will fail during the year? 78. Three critics receive a book. Odds in favour of the book are5:2, 4:3 and 3:4 respectively for three critics. Find the probability that the majority are in favour of the book. 79. The probability of the student A passing an examination is 3/5 and of the student B passing is 4/5. Assuming the two events: ‘A passes’, ‘B passes’ as independent find the probability of: 1. Both students passing the examination. 2. Only A passing the examination. 3. Only one the two passing the examination. 4. Neither of the two passing the examination. 80. A and B decide to meet at Hanuman Temple between 5 to 6 p.m. with the condition that no one .Would wait for the other for more than 15 minutes. What is the probability that they meet? 81. Three persons A, B, C throw a die is successful till one gets a ‘six’ and win the game. Find their respective probabilities of the winning, if begin. 82. 30. A and B take turn in throwing two twice dice. The first to throw the 9 being awarded. Show that If A has the first throw; their chances of winning are in the ratio 9:8. 83. In a bulb factory, machines A,B and C manufacture 60% , 30% and 10% bulbs Respectively 1% , 2% and 3% of the bulbs produced respectively by A,B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that the bulb was produced by the machine A. 84. Coloured balls are distributed in three bags as shown in the following table Bag Colour of the ball Red Black White I 1 2 3 II 2 4 1 III 4 5 3 A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that they came from bag I? 85. A fair die is rolled. If 1 turns up, a ball is picked up at random from bag A, if 2 or 3 turns up, a ball is picked up at random from bag B, otherwise a ball is picked up from bag C. Bag A contains 3 red and 2 white balls, bag B contains 3 red and 4 white balls and bag C contains 4 red and 5 white balls. The die is rolled, a bag is picked up and a ball is drawn from it. If the ball drawn is red, what is the probability that bag B was picked up? Probability Distribution & Binomial Distribution 86. Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the (distribution) probability distribution of the number of aces. 87. Find the probability distribution mean and variance of the number of kings drawn when two cards are drawn one by one without replacement from a pack of 52 cards. 88. Find mean μ and variance for the probability distribution. X 0 1 P(X) 1/8 3/8 2 3 3/8 1/8 89. A pair of dice is thrown four times .getting a doublet is considered as a success. Find the probability distribution of number of success. 90. An urn contains 4 white and 3 red balls. Let X be the number of in red balls in a random draw of three balls. Find the mean and variance of X. 91. On a multiple choice examination with three possible answers (out of which one is correct) for each of the five question, what is the probability of the candidate would get four or more correct answers just by guessing. 92. A random variable X has the following probability distribution X P(X) Find: 0 0 1 k 2 2k 3 2k 4 3k 5 k2 6 2k 2 7 7𝑘 + 𝑘 2 1. K 2. P(X<3) 3. P(X>6) 4. P(0<X<3) 93. A die is thrown 10 times. If getting an even number is success, find the probability of getting at least 9 success. 94. Find the probability of getting at most two sixes in 6 throws of a single die. 95. A die is rolled once, find the probability distribution for the number obtain on it and find its mean and variance 96. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining a third six in the sixth throw of die. 97. From a lot of 30 bulbs, which includes 6 defective bulbs, a sample of three bulbs is drawn at random with replacement. Find the probability distribution of number of the defective bulbs. 98. A bag contains 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from bag, find the probability that none is marked with digit 6. 99. In a meeting 65% of the members favour and 35% oppose a certain proposal. A member is selected at random and we take X=0 if he oppose and X=1 if he favours. Find the mean and variance. 100. Find the mean variance and standard deviation of the number of heads in a simultaneous loss of three coins.