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This is a Pre Board Question paper of Kendriya Vidyalaya,Kolkata Region

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- 1. BLUE PRINT -2nd pre board- 2013 XII mathematics VSA SL. NO TOPIC (1 Mark) (a) SA LA (4 Mark) (6 Mark) Relations & Functions 1(1) 4(1) - Inverse Trigonometric Functions 1(1) 4(1) - Matrices 2(2) - 6(1) Determinants 1(1) 4(1) - Continuity & differentiability 8(2) - Applications of Derivatives 4(1) 6(1) 4(1) TOTAL 6(1) 10(4) (b) (a) 13(5) (b) (a) (b) Integrals 2(2) Applications of Integrals (c) 6(1) Differential Equations 44(11) 4(2) (d) (a) Vectors 2(2) 4(1) - Three Dimensional Geometry 1(1) 4(1) 6(1) Linear Programming - - 6(1) Probability - 4(1) 6(1) TOTAL 10(10) 48(12) 42(7) 17(6) (b) CBSE –Annexure –F 2013-14 Pratima Nayak,KV,Fort William 6(1) 10(2) 100(29)
- 2. MATHEMATICS (041) CLASS XII Time allowed : 3hours Max Marks: 100 General Instructions 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each. 3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators in not permitted. You may ask for logarithmic tables, if required. Section A Q1. Let A N N and let * be the binary operation on A is defined by a*b= Find 3 * 2 . 5 Q2. Find the principal value of cos 1 cos . 3 Q3. For what value of k, the matrix is skew symmetric? Q4. A is a non- singular square matrix of order 3 and Q5. Find x and y, if Pratima Nayak,KV,Fort William + = A 4 .Find adjA .
- 3. Q6. If are two unit vectors inclined to x–axis at angles 300 and 1200 and respectively, write the value of | sec 2 (log x ) dx x Q7. Evaluate: . Q8. Find the projection of - + on -2 + . Q9. Evaluate Q.10 Find the value of p for which the following two lines are perpendicular to each other. x3 y 5 z 7 x 1 y 1 z 1 and 1 2 1 7 p 1 Section B given by f(x) = x2 + 4.Show that f is invertible with the Q11. Consider f : R + inverse f-1 of given by , where R + is the set of all non negative real numbers. Q12. Prove that x 1 1 x 1 tan 1 + tan = x 2 x 2 OR + = Q13. Using the properties of determinants, prove that – 1 x 1 y 1 z x3 y 3 ( x y )( y z )( z x)( x y z ) z3 Q14. Show that the function f (x) = |x + 2| is continuous at every x differentiable at x = –2. Q15. If , find OR Pratima Nayak,KV,Fort William R but fails to be
- 4. If x sin ( a + y ) + sin a cos ( a + y) = 0 prove that = Q16. Using differentials, find the approximate value of Q17. . Evaluate: dx n 1) x( x OR x 1 Evaluate: Q18. ( x 3) 3 e x dx Using vectors, fine the area of a triangle ABC whose vertices are A (1, 1, 2), B (2, 3, 5) and C(1, 5, 5) OR If a , b and c are unit vectors such that a is perpendicular to the plane of b , c and the angle between b , c is then find a b c 3 Q19. Solve the differential equation x dy y y x tan dx x Q20. Solve the following differential equation OR Form the differential equation of the family of circles of radii 3. Q21. Find the shortest distance between the lines ˆ ˆ j ˆ r i ˆ (2i ˆ k ) and j ˆ r 2ˆ ˆ - k (3ˆ - 5ˆ 2k) i j ˆ i j Pratima Nayak,KV,Fort William
- 5. Q22. A target is displayed as ‘’ Be truthful ‘’ the probability of A’s hitting a target is 4/5 and that of B’s hitting is 2/3.They both fire the target .Find the probability at least one of them will hit the target Only one of them will hit the target . Which value is emphasized in the question? OR Assume that each child born 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 the youngest is a girl ? At least one is a girl ? Pre-natal sex determination is a crime. What will you do if you come to know that some of our known is indulging in pre-natal sex determination? SECTION -C Q23. Two schools A and B want to award prizes their students for the values of honesty (X) , punctuality ( Y ) and obedience( Z ) .The sum of all the awardees is 12.Three times of the sum of awardees for obedience and punctuality added to two times of the number of awardees for honesty is 33.The sum of the number of awardees for honesty and obedience is twice the number of awardees for punctuality, using matrix method, find the number of awardees for each category. Apart from these values suggest one more other value which could be considered for award? Q24 A window in the form of a rectangle is surmounted by a semi circular opening. The total perimeter of the window is 30 m. find the dimensions of the rectangle part of the window to admit maximum light through the whole opening. OR Show that the volume of greatest cylinder that can be inscribed in a cone of height h and semi vertical angle α is, Pratima Nayak,KV,Fort William 4 3 h tan 2 . 27
- 6. Q25. Using properties of definite integrals, evaluate: Q26. Find the equation of plane passing through the line of intersection of the planes x + 2y + 3z = 4 and 2x + y – z + 5 =0 and perpendicular to the plane 5x + 3y - 6z + 8 = 0. Q27. Find the area bounded by the curves OR Find the area of the region bounded by the two parabolas y = x2 and y2 = x. Q28. In a group of 400 people,160 are smokers and non vegetarians,100 are smokers and vegetarians and remaining are non smokers and vegetarian. The probability of getting a special chest pain disease are 35% , 20% and 10% respectively. A person is chosen from the group at random and found to be suffering from the disease. What is the probability is that the selected person is smoker and no vegetarian? What value is reflected in the question? Q29. A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain 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. Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’. Formulate this problem as a linear programming problem to minimize the cost of such a mixture. In what way a balanced and healthy diet is helpful in performing your day-to-day activities? Pratima Nayak,KV,Fort William
- 7. Answer ans Marking Scheme Q1. 4 / 3 Q2. . Q3. K = .Q4 16 (sin 1 x) 2 C Q6. 2 Q7. Tan ( log x) Q8. 7 Q9. 2 Q5. . x = 3, y = 3 Q10. P = - 4 Q11. For one- one For onto 1 Use the formula for tan-1x + tan-1y 1 Correct solution 12 3 OR 1 1 tan-1x + tan-1y 13. Applying R1 R1 – R2 , R2 and result R2 – R3 2 1 For taking common 1 For expansion along C1 1 For getting required result 1 Q14. L.H.L , R.H.L , L.H.L =, R.H.L , Continuous L.H.D, RHD ½ + ½ +½ ½ LHD ≠ RHD Not differentiable Q15 u = xy v = yx ½ + ½ +½ ½ =0 Putting the values and simplifying 1 1+1 Answer 1 OR x = -[sin a cos ( a + y) ]/ sin ( a + y ) 1 dx/dy = sin a/ sin2(a+y) 2 = Q16. Given Pratima Nayak,KV,Fort William 1
- 8. 1 Ans. 1 =54+68(0.01)=54.68. 2 Ans.17 I= dx n 1) x( x Substitute xn = t dx = I= 1 dt n x n 1 1 1 dt t (t 1) n 1 1 xn c = log n n x 1 2 OR I= x 1 ( x 3) 3 e x dx = x 32 x e dx ( x 3) 3 1 1 2 x ( x 3) 2 ( x 3) 3 e dx = 1 = e f ( x) f ( x) dx x = ex f(x) + c = ex Q18. 1 1 +c ( x 3) 2 1 =1 +2 +3 =0 +4 +3 Area of triangle= 1 =-6 -3 +4 ans. 1+1 OR a b c 1 a.b 0 ; a.c 0 , b .c b c cos 3 2 a b c (a b c ).(a b c ) Pratima Nayak,KV,Fort William Or cos 3 = 1 2 . ½ + ½ +½ +1/2
- 9. =1+1+1+2 1 2 . =4 1+1 Q19. dy y y = tan( ) x dx x dy dv Put y = vx =v + x dx dx dv v+x = v+tan v dx dx cot vdv x & 1 =cx 1+1 Q20. dy 1+1 Or, 1+1 OR ( x - a )2 + ( y - b )2 = 9 1 Formation of equation 3 Q21. ˆ ˆ k i j ˆ ˆ ˆ a 2 a1 i k , b1 b2 2 - 1 1 3 -5 2 ˆ ˆ j 3i ˆ 7k b1 b 2 59 shortest distance = Ans 22. P(A) =4/5 3 (b1 b2 ).(a 2 a 2 ) b1 b2 = 10 59 P(B) = 2/3 1 1 (i) P(at least one) =1 – P(0)= 1- P( . .) 1 (ii) P (only one) = P(A + . B) 1 (iii) Truthfulness 1 OR Ans22. (i). A: Both are girls ={GG}, Pratima Nayak,KV,Fort William B: Youngest is the girl BG, GG 1
- 10. 1 A p( A B) 1 P( ) 4 2 B p( B) 2 4 (ii). A: Both are girls B: At least one of them is girl BG, GG, GB 1 A p( A B) 1 P( ) 4 3 B p ( B) 3 4 1 (iii) Every valuable answer given by the student 1 Q23. x + y + z = 12, 3(y + z ) + 2x = 33 , x + z -2y =0 Matrix multiplication form 1 1 |A| =3 Cofactors 2 x = 3,y = 4 z = 5 1½ One appropriate value ½ Q24.Figure 1 30= 2x+2y+2y+ 1 A=2x 1 = 1 =-( Length = 1 m ,breath = 1 OR Can be marked in similar way. Q25. Use of property ,I= 2I = Use of property Pratima Nayak,KV,Fort William 1 1½ 1
- 11. 2I = 1½ tanx = t sec2xdx =dt & Correct result I = 2 Ans26 The required plane is (x+2y + 3z ) + k (2x + y – z +5 )= 0 1 Or (1+2k)x +(2+k)y +(3-k)z-4+5k=0 1 5(1+2k) +3 (2+k) -6 (3-k)=0, i.e k= 7/19, 3 The equation of the plane is : 33x+45y +50z = 41 Q27. 1 Correct figure 1½ Point of intersection Required area = 2( Area of shaded portion) 1+2 Finding integral and getting answer sq.unit 2½ OR Figure 1 Intersection points ( 0,0) and ( 1,1). 1 Area of the shaded region = dx = dx= [ 2/3 x3/2 – x3 /3 ] 3 = 1/3 sq. unit. Ans28. P(E1) = 2/5 P ( E2) = ¼ 1 p(E3) = 7/20 P(A/E1) = 35% P(A/E2) = 20% P(A/E3) = 10% Formula for P( E1/A) and expression Correct answer 1 1 1+1 1 Ans 29 Let the mixture contain x kg of Food ‘I’ and ‘y’ kg of Food ‘II’ Min Z 50 x 70 y 2x y 8 x 2 y 10 x0 y0 Drawing the graph feasible region has no point in common. x=2 & y=4 Pratima Nayak,KV,Fort William ½ +1/2 +1/2+1/2 2
- 12. MinZ=380 Marking may be done for all alternative correct answer. Pratima Nayak,KV,Fort William 2

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