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Class 12 ISC Model 2017 : Mathematics

3 pages, 38 questions, 25 questions with responses, 32 total responses,    0    0
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-4Question IX (a) A bag X contains 3 white balls and 2 black balls and another bag Y contains 2 white balls and 4 black balls. A bag and a ball out of it are picked at random. What is the probability that the ball is white? [5] dx (b) Evaluate: 5 4sin x [5] SECTION B [20 Marks] Question X (a) In any triangle ABC, prove by vector method that a b c = = sin A sin B sin C [5] (b) Find the distance between the point ( 1, 5, 10) and the point of intersection of the line x 2 y +1 z 2 = = with the plane 3 4 12 x y+z =5 [5] Question XI (a) The vector equations of two lines are given by [5] (b) Find the cartesian equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes x + 2 y + 3 z 7 = 0 and between these two lines. [5] Question XII (a) If the mean and variance of a binomial distribution are 9 and 6 respectively, find the distribution. [5] (b) There are 4 boys and 2 girls in room A and 5 boys and 3 girls in room B. A girl from one of the two rooms laughed loudly. What is the probability that the girl who laughed was from room B? [5] MARKS:100 TIME:3 Hours Candidates are allowed additional 15 minutes for only reading the question paper. They must not start writing during this time. SECTION A Answer Question I compulsory and any other 5 questions. SECTION B & SECTION C Answer 2 questions either from Section B or from Section C. All work, including rough work, should be done on the same sheet as and adjacent to the rest of the answer. Lack of essential steps will result in loss of marks. SECTION A [30 Marks] Question I Each question carries 3 marks ( ) ( [3X10=30] ) 2 1 2 1 1. Prove that sec tan 2 + cos ec cot 3 =15. [3] e ( tan x + log sec x ) dx [3] 2. Evaluate: r , + (t 2) r = (1 t )i j + (3 2t )k r . Find the shortest distance + (2s 1) r = (s+ 1)i j (2s + 1) k 2x 3 y + 4z = 0 . MATHEMATICS CLASS XII 3. Evaluate: x Lim ( cos x log tan x ) x 2 [3] 4. Two regression lines are represented by 2 x + 3 y = 10 , 4 x + y = 5 . Find the regression line of y on x. [3] 3 5. Evaluate: x 2 dx [3] 3 6. Find the derivative of sin x 2 with respect to x 3 [3] 7. Using determinants, determine whether the system x 3 y + 2 z = 4 , 2 x + y 3 z = 2 and 4 x 5 y + z = 5 is consistent or not. [3] 8. A fair die is thrown once. What is the probability that an even number or a number greater than three will turn up? [3] -2- 9. The coefficient of rank correlation of marks obtained by 10 students in English and Economics was found to be 0.5. It was later discovered that the difference in ranks in the two subjects obtained by one of the students was wrongly taken as 3 instead of 7. Find the correct coefficient of rank correlation. [3] 10.Using Boolean Algebra prove that ( A + B ) ( A + C ) + B ( B + C ) = AC + B [3] Question II (a) Using properties of determinants prove that bc b 2 + bc c 2 + bc 3 a 2 + ac ac c 2 + ac = ( ab + bc + ca ) a 2 + ab b 2 + ab ab [5] 1 1 + tan 1 = 3 7 [5] Question V a x a+x 1 (a) Using a suitable substitution, find the derivative of tan [5] (b) Evaluate: Lim x 2 tan x tan 3x [5] Question VI 1 (a) Evaluate: xe x ( 1+ x) 2 dx [5] 0 ( A + A) ( B + D ) ( C + A) ( C + D ) . Simplify the expression and construct switching circuit for the simplified expression. [5] Question III x2 + 1 x 4 + 1 dx 1 (b) Prove that 4 2 tan with respect to x. (b) Construct switching circuit for the Boolean expression (a) Evaluate: -3- (b) An article manufactured by a company consists of two parts A and B. In the process of manufacturing of part A, 9 out of 104 parts are defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part B. Calculate the probability that the article manufactured will not be defective. [5] [5] (b) Ten students got the following percentage of marks in Mathematics and Physics. Calculate Spearman s Rank Correlation Coefficient. [5] Maths 56 6 75 85 85 87 91 95 97 98 4 Physics 66 72 56 66 7 78 7 88 90 89 4 4 Question IV (a) For the data given below, find the regression equation of X on Y. Using it, find the value of X when Y=15. [5] X 20 25 30 35 4 45 0 Y 12 1 16 20 22 25 4 Question VII (a) Evaluate: x tan 1 x dx [5] (b) Using Cramer s Rule, solve the following system of equations: 2 3 10 4 6 5 6 9 20 + + = 4 , + = 1 and + =2 x y z x y z x y z [5] Question VIII (a) Evaluate: Lim ( 1 + sin x ) cot x x 0 (b) Evaluate: 2 sin 2 x dx 1 + sinxcosx 0 [5] [5] -5SECTION C [20 Marks] -5SECTION C [20 Marks] Question XIII (a) Two tailors P and Q can earn Rs 150 and Rs 200 per day respectively. P can stitch 6 shirts and 4 trousers a day while Q can stitch 10 shirts and 4 trousers per day. How many days should each work to produce at least 60 shirts and 32 trousers at minimum labour cost? [5] (b) A Bill of Rs 1000 drawn on 7th May, 2011 for six months was discounted on 29th August, 2011 for cash payment of Rs 988. Find the rate of interest charged by the bank. [5] Question XIV (a) Given the total cost function for x units of a commodity as Question XIV (a) Given the total cost function for x units of a commodity as x3 C ( x) = + 3 x 2 7 x + 16 . 3 C ( x) = [5] [5] Question XV (a) Calculate the index number for the year 1979 with 1970 as the base year by using the weighted average of price relative method: . [5] Price in Rs Commodity Weight 1970 1979 A 22 2.50 6.20 B 48 3.30 4.40 C 17 6.25 12.75 D 13 0.65 0.90 [5] (b) Calculate the three yearly moving averages of the data given below and plot them on a graph paper. [5] 1983 137 84 140 85 134 86 137 87 151 88 121 89 124 90 159 91 157 92 169 93 172 [5] marginal average cost is given by {x MC C ( x)}x 2 . (b) If the difference between banker s discount and True discount of a bill for 73 days at 5% per annum is Rs 10, find the Banker s discount and the amount of the bill. [5] Year Production x3 + 3 x 2 7 x + 16 . 3 Find (i) the marginal cost (ii) the average cost (iii) show that the Find (i) the marginal cost (ii) the average cost (iii) show that the marginal average cost is given by {x MC C ( x)}x 2 . Question XIII (a) Two tailors P and Q can earn Rs 150 and Rs 200 per day respectively. P can stitch 6 shirts and 4 trousers a day while Q can stitch 10 shirts and 4 trousers per day. How many days should each work to produce at least 60 shirts and 32 trousers at minimum labour cost? [5] (b) A Bill of Rs 1000 drawn on 7th May, 2011 for six months was discounted on 29th August, 2011 for cash payment of Rs 988. Find the rate of interest charged by the bank. [5] 94 150 [5] (b) If the difference between banker s discount and True discount of a bill for 73 days at 5% per annum is Rs 10, find the Banker s discount and the amount of the bill. [5] Question XV (a) Calculate the index number for the year 1979 with 1970 as the base year by using the weighted average of price relative method: . [5] Price in Rs Commodity Weight 1970 1979 A 22 2.50 6.20 B 48 3.30 4.40 C 17 6.25 12.75 D 13 0.65 0.90 [5] (b) Calculate the three yearly moving averages of the data given below and plot them on a graph paper. [5] Year Production 1983 137 84 140 85 134 86 137 87 151 88 121 89 124 90 159 91 157 92 169 93 172 94 150

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