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CBSE Class 12 Pre Board 2021 : Mathematics - Set 3 (CBSE Gulf Sahodaya Qatar Chapter, Doha)

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PRE BOARD EXAMINATION - 1(2020-21) MATHEMATICS (041) Time: 3 hrs C LASS: XII Max.Marks:80 General Instructions: 1. This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks 2. Part-A has Objective Type Questions and Part -B has Descriptive Type Questions 3. Both Part A and Part B have choices Part A: 1. It consists of two sections- I and II. 2. Section I comprises of 16 very short answers type questions. 3. Section II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs. Part B: 1. It consists of three sections- III, IV and V. 2. Section III comprises of 10 questions of 2 marks each. 3. Section IV comprises of 7 questions of 3 marks each. 4. Section V comprises of 3 questions of 5 marks each. 5. Internal choice is provided in 3 questions of Section III, 2 questions of Section IV and 3questions of Section-V. You have to attempt only one of the alternatives in all such questions. Part A Section I All questions are compulsory. In case of internal choices attempt any one. 1.Let the relation R be defined in N by aRb if 2a + 3b = 30, then find R. 1 2.If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one- one from A nto A. OR For the set A = {1, 2, 3}, define a relation R in the set A as follows: 1 R = {(1, 1), (2, 2), (3, 3), (1, 3)} .Write the ordered pairs to be added to R to make it the smallest equivalence relation. 3. Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? Symmetric? Transitive? Justify. 4. If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I. 1 1 5. The area of a triangle with vertices ( 3, 0), (3, 0) and (0, k) is 9 sq. units then find the value of k. 1 1 6. If A is invertible matrix of order 3 3, then find |A |. OR 2 If | 8 5 6 5 |=| | find the value of x. 8 3 2 2 7. Find the value of ( 3 + + 5 + 1) . 1 8. Give the number of arbitrary constants in the particular solution of a differential equation of third order. 1 9. Findarea of the region bounded by the curve y = 4x, y-axis and the line y = 3. 1 10. If a and b be two unit vectors inclined to x-axis at angles 30 and 120 respectively, find the value of |a + b|. 1 11. Find a vector in the direction of a = 2 whose magnitude is 7 unit. 1 12. Show that the vector + + is equally inclined to axes. 1 13. A line makes angle , , with x-axis, y-axis and z-axis respectively then find the value of cos 2 + cos 2 + cos 2 1 14. Write the Cartesian equation of the following line given in vector form: 1 15. Three balls are drawn from a bag containing 2 red and 5 black balls, if the random variable X represents the number of red balls drawn, then write the values that X can take . 1 16. If A and B are two independent events such that P(A) = 17 and P(B) = 16 then find P(A B ). 1 Section II Both the Case study based questions are compulsory. Attempt any 4 sub parts from each question 17 and 18. Each question carries 1 mark 17. Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold handmade fans, mats and plates from recycled material at a cost of Rs. 25, Rs. 100 and Rs. 50 each. The number of articles sold is given. School / Article Hand-fans Mats Plates A 40 50 20 B 25 40 30 C 35 50 40 Based on the above information, answer the following questions: (Attempt any four) (i) Find the fund collected by school A if they sold 45 hand-fans, 40 mats and 25 plates. 1 (A) 6375 (B) 6735 (C)5635 (D)3635 (ii) Find the fund collected by school B and C. 1 (A) 14000 (B) 12000 (C) 10000 (D) 13000 (iii) Find the total fund collected by all the schools. (A)15000 (B) 21000 (C) 18000 (D) 17000 (iv) If the number of hand-fans and mats are interchanged for all the schools, what is the total fund collected by all schools. (A) 15000 (B) 16000 (v) (C) 18000 1 (D)12000 Find the total number of all articles sold. (A) 130 (B) 430 (C) 330 (D)230 18. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flap as shown in the figure. 1 Based on the above information, answer the following questions: (Attempt any four) ( i ) The polynomial function V(x) which represents the volume of the box is given by (A) 4 3 138 2 + 1080 (B) 4 3 + 138 2 + 1080 (C) 3 138 2 + 1080 (D) none (ii) The side of the square (x) to be cut off so that the volume of the box maximum is (A) 3 (B) 5 (C) 18 (D) 5 & 18 3 (iii) The maximum volume of the box in cm is 1 1 1 (A) 1250 (B) 4250 (C) 2000 (D) 2450 (iv) The second derivative of the volume function at the value of x for which the maximum volume occurred is 1 (A) 120 (B) 140 (C) 120 (D) none (v) What should be the side of the square to be cut off so that the volume of the box is maximum if a square sheet of length 18 cm is given (A) 3 (B) 4 (C) 5 (D) none Part B Section III 19. Prove that: 9 9 1 1 9 1 2 2 sin sin 8 4 3 4 3 1 20. Express the matrix [ 3 If[2 3] [ 2 2 ] as the sum of a symmetric and a skew symmetric matrix 4 OR 1 2 ] [ ] = find x. 3 0 8 2 21. Find the values of k so that the function is continuous at the indicated point 2 . 22. Find points at which the tangent to the curve y = x3 3x2 9x + 7 is parallel to the x-axis . 2 23.Integrate the following OR Evaluate 2 24. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x, and the circle x2+ y2 = 32. 2 25. Solve 2 26. Find the equation of the plane through the intersection of the planes 3x y + 2z 4 = 0 and x + y + z 2 = 0 and the point (2, 2, 1). 2 27. Find the position vector of a point which divides the join of points with position vectors a +b and 2a b in the ratio 1:2 internally and externally. 2 28. Prove that if E and F are independent events, then the events E and F' are also independent. OR Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number? 2 Section IV All questions are compulsory. In case of internal choices attempt any one 29. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1 , L2 ) : L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4. 3 30. Prove that the curves x = y and xy = k cut at right angles if 8k = 1. 3 31.A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum? OR Differentiate 3 32. Find the equation of all the tangents to the curve y = cos (x + y), 2 x 2 , that are parallel to the line x + 2y = 0. 3 33. Evaluate 3 34. Find the area of the region included between the parabola y = 3 2 4 and line 3x 2y + 12 = 0. OR Find the area of the region bounded by the parabola y2= 2x and the straight line x y = 4. 1 35. Solve ( 2 1) + 2 = 2 1 3 3 Section V All questions are compulsory. In case of internal choices attempt any one 36. Find the image of the point having position vector + 3 + 4 in the plane . (2 + ) + 3 = 0 OR Find the image of the point (1, 6, 3) in the line 1 = 1 2 = 2 3 5 37. 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. 5 38. Three machines E1 , E2 , E3 in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric tubes. It is known that 4% of the tubes produced one each of machines E1 and E2 are defective, and that 5% of those produced on E3 are defective. If one tube is picked up at random from a day s production, calculate the probability that it is defective. 5

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