SRECTION A- 65 MARKS Question 1 In subparts (') to (xi) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed. | IfAand B are two skew-symmetric matrices , then (AB#BA) is [1] a) A skew-symmetric matrix. b) Asymmetric matrix, ©) Anull matrix, ) An Identity matrix. ii. f= dx is equal to a) log(e* +1) +e b) loge* + ce) e*+1te d) e+e The given function f:N > N defined by f(x) = x? +x a 1is 1) a) One-one function, b) Onto function. ¢) One-one and onto function, d) Neither one-one nor onto. (1) iii. Assertion (A): The degree of the differential equation 11] £2 + sin (2) + y = 0 is not defined. : Reason(R): sin (22) is not a polynomial in $. Which of the following is correct? : a) Both A and R are true and R is the correct explanation for A. b) Both A and R are true but R is not the correct explanation for A. c) Ais true and R is false. d) Ais false and R is true. Aproblem in mathematics is given to ieee a AB,C and their chances of solving the problems are > 5 and > + respectively. The {1 probability that the problem solved is &£sop SII ats —vi vil vill IfAis a square matrix of order 3 such that a 14] = —2, then tue value of |A adja] is a4 b) 0 c) -2 d) -8 The signum function is 7 {1} 8) Continuous for all xeR. _ >) Continuous for all xeR {0}, * 4 ©) Not continuous at x d) Not continuous at x Let f(x) = 2tan* (#4) + sin? (S).xer- ro} Statement 4: /“(x) = 0 Statement 2: f(x) is a constant function Which one of the following is correct? a) Statement 1 implies Statement 2. b) Statement 2 implies Statement 1 ©) Statement 1 is true only if statement 2 is true. 4) Statement 1 and 2 are independent of each other. Let, A be a non- singular matrix of order 2. tt) Statement 1: adj(agjA)=A Statement 2: jadja|=|A] Which one of the following is correct? | a) Statement 1 implies Statement 2. b) Statement 2 implies Statement 1 ©) Statement 1 is true only if statement 2 is true, d) Statement 1 and 2 are independent of each other, ate Assertion: f, log (2) dx = 0 rT im Reason: f°" f(x)dx = Oif f(2a—x) =-f(x) ‘Which of the following is correct? a) Both Assertion and Reason are true and Reason is the correct ‘explanation for Assertion. b) Both Assertion and Reason are true and Reason is not the correct explanation for Assertion. c) Assertion is true and Reason is false. d) Assertion is false and reason is true, If P(A) = 2, P(B) = and P(ANB) = “then P(A'N B)is equal to [1] asxi, Three houses are available ina local. Three persons apply for he houses. Each applies for one house without consulting others. The probability that all the three apply for the same house ,is a) 7/9 _b) 8/9 ©) 19 4) 20 xii, Ris a relation on the set A=(1,2,3) given by R={(1,2)}, then Ris jp a) Reflexive , b) Symmetric ©) Transitive 7 4) Equivalence “ ‘xiv. The adjoining diagram shows that a) fis. function from Ato B. 7 ») fis a one-one function from Ato B. By ) fis an onto function from A to B. ro 4) fis not a function from A to B. xv. HA=[ay],, where ay fifi + J tt) 0, if «= j, then A? is equal to al by A 2 oO d) None of these Quasfiorr2: a J Diflerentiate tan“ (75) with respect to sin-* (25), OR Il. Find the maximum and minimum value of f(z) = 4 —|cos3x| Question 3: ‘normal to the curve x= 11—asine,y = boos? att = %Question 5; ra} (1) i. Evaluate: fas e*ax OR ii, Evaluate: Question 6: a Solve: 2 tan™*(cosx) = tan“*(2cosecx) Question 7: (4) Solve: sin“*(6x) + sin-*(6V3x) Question ai Evaluate: Jeanna dx Questi i. Find the interval in which the function f(x) = neg strictly increasing, where 0 < x <1. OR a ify = sin(msin-*x), prove that (1 ~ x2)£2—x24 my = 0 a 40: 4] ah The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today , the probability that it will rain tomorrow is 0.7, if P41: denotes the probability that it doesn't rain todey. it will not rain tomorrow, if it rains today. Pz: denotes the probability that Ps: denotes the probability that it will rain tomorrow, if it does not rain today. Ps; denotes the probability that it will not rain tomorrow, if it does not rain today. DD Find PixPs—P2xPs # by/Caleulate the probability of raining to tomorrow. ORTh ail ® reliability of a COVID PCR test is specified as follows eople hay Otneonie reach esa 90% of the test detects the disease but 10% goes undetectey a8 showing COVID sao 22% 9 the testis judged COVID negative but 1% are diagnos Positive. From a large population of which only 0.1% have COvip one person is selected at ran ar ithologist hana eit pecad ar given the COVID PCR test, and the pathologist repo, Based on the given information, solve the following questions: 3) Whats the probability ofthe ‘person to be tested as COVID positive’ given that hg 's actually having COVID'? ©) What is the probability of the ‘person to be tested as COVID positive’ given that he is actually not having COVID'? ©) What is the probability that the ‘person is actually not having COVID'? 9) What is the probability that the ‘person selected will be diagnosed as COVip Positive’? al 8 The €quation of the path traversed by the ball headed by the footballer is y = ax? 4. bx +c; (where 0 < x < 14nd a,b,ceR and a#0) with respect to a XY- coordinate ‘system in the vertical plane. The ball passes through the points (2,15),(4,25) ang (14,15). Determine the values of a, b and c by solving the system of linear equations in a, b and c, using matrix method. Also find the equation of the path traversed by the ball. Question 12: @ i. Solve the differential equation: : 2 5 (1-2 SEE o(t ; dy +(1+e¥)dr=0 Find the particular solution satisfying the condition that y = 1 when x = 0. OR yo Acme Sart 13} ms ‘s Evaluate: P23] fe tu on ce b ul pg. 6 Qaor Question 13: [6] 4 Show that the rectangle of maximum perimeter which can be inscribed in a Circle of radius 10 cm is a square of side 10¥2 om. OR ii, Prove that the volume of the largest cone that can be inscribed ina sphere of radius R is 8/27 of the volume of the sphere. 4 44: A aaa sided die with faces labelled 1,2,3 and 4 is rolled and recorded. Let X be the result obtained when the die is rolled. The probability distribution for X is given in the following table, where p and q are constants. x 4 2 3 4 P(X=x) 0.3 ~ q 04 For the probability distribution, itis known that E(X)= 2. Find p and q Also, find P(X>2). Ajay plays a game with this rolls. His score is calculate score is at least 10. After 3 die are independent, find the Po.7 four- sided die. In this game he is allowed a maximum of five by adding the results of each roll. He wins the game if his rolls, Ajay has score of four points. Assuming that rolis of the probability that Ajay wins the game.SECTION B - 15 MARKS Question 15: 1} In subparts (i) and (ii) choose the correct options and in subparts () to(v), answer th questions as instructed. i. Consider the following statements and choose the correct option: Statement 4: If d= f+ pj + 2k and b = 21 + 3f + gk are parallel vectors if p= Zand q=4. Statement 2: as i by a,1 + a,f + ask and b = byt + bpf + b3k are parallel 5 Which of the following is correct? a) Only statement 1 b) Only statement 2 ) Both statement 1 and 2 d) Neither statement 1 nor Statement 2 ii. The planes 2x — y + 4z = 5 and 5x —2.Sy + 10z = 6 are a) Parallel b) Intersection on Y-axis ©) Perpendicular d) Pass through (0,0,5/4 ) ili, The equation of plane through the origin containing the line a) 2x+Sy—6z=0 b) x+5y—5z=0 ©) x-Sy+3z=0 d) xt+y+z=0 iv Find the angle between the lines =" = 2-2 — 24 ang z#2 : V. Find the vector equation of the plane which is at a distance ot y 8nd the direction ratio of the normal to this plane are <2,2,-1 >. "= {fom the origin ‘Question 16: @ i, Wa Band 2 are three vectors such that dx = 2,5 x 2 = a. then foove tat 2B and are mitiely ata ered ad [5] =, 2 = OR Pg.8 Qu Ing que—frk. [4] 1 =< 24% do not intersect q Find the angle @ between the vectors d= t+ f— and b Question 17; "Show that the lines ®ach other. OR iv ; Find the equation of the plane passing through the point (~1,~1,2) and perpendicular to the planes 3x + 2y ~ 32 = 1and Sx— 4ytr=5. Question 18: i i. e Sketch the region enclosed bounded by the curve y = —x? and the line U+yY+2=0, ii Evaluate: 2 (x? —x—2)ax iii, Hence find the area bounded by the curve, y = —x? and the line x + y+2=0. SECTION C ~ 45 MARKS Question 49: a In subparts (i) and (ii) choose the correct options and in subparts (ii) to (v) , answer the questions as instructed. By using the data x = 5, 7 = 13 and by, = 2.5, then the regression equation y on xis a) 25x-05 b) 0.5x+2.5 ¢) 0.5x-25 BT 2.5x +05 ; Ji Ip = 300 ~ 8x when p is monopolistic demand function, then marginal revenue at x=0 is a) 200 b) 500 c) 300 50 x For the given lines of regression, 2x ~ 3y =6 and 5x —4y = 20, find regression coefficients by and byx. revenue function, respectively as C(x) = 2x + 40 and ic id iv, Btven the cost function ant u R ue 1x — 0.2x?, find the break-even point.2/ ina factory, its found that the number of units (x) produced in a day depends upon the number of workers (n) is obtained by the relation x = 2%. The demand function of a hg ois 7 product is p = = + x. Determine the marginal revenue, when n=44, Question 20: a YL Given the total cost function for x units of commodity as 3 x Ce) = ar 3x? — 7x +16. Find af the marginal cost phe average cost = o fi. Given, the total cost function for x units of commodity, C(x) = ax? + bx? — cx +d, where a>0,b<0,c>0, show that average variable cost and marginal cost curves intergectat minimum average variable cost. Question 21: [4] j Given the observations (10, 5), (10, -3), (11, -2), (11,0), (12,1), (15,6), (16,4), (11, —2) Find La, brand bry. Predict the value of y corresponding to the value 14 of x. 9) Predict the value of x, when the value of y is 3. OR ji, Find the line of best fit to the following data using x as independent variable and y as dependent variable. x 4 3.14 6 (8 9 1 | 14 fy it_l2 [4 [4 [5 17 9 auf 22: 4] Maximise z = 8x + 9y subject to the constraints given below 2x +3y $ 6,3x-2y $6,y<1andx,y20 pg. 10