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L Pre Co~ncil ExaDJinatton - 2019 ClassXII . Time :-3hrs M.l\1. 100 Mathematics AL-50 Candidate are al/owed additional 15 n,i,mtesfor 011/y reading the paper They must 11ot start writing during the time. SECTION A (80 MARKS) 1. (i) Find the identity element in set of all ralional numbers except - 1 with respect to * defined by a b = a + b + nb. 1 a b+c (iQ $how that : 1 b c + a = 0. 1 c a+b (iii) If sin ( cos- 1 ~ + sin- 1 1 x) = 1, find the value of x. (iv) Find X and Y, if X+ Y= G~] and X - Y = [~ 11 ~ Evaluate: f 1 + ~cotx . 0 (vi) If y = 2 esin x , find ~~ . (viz) Salve the differential equation : dy - e-' Y = e-v - Y. . dx (viii)Prove that the function j (x) = x3 - 6x2 + l2x - 18 is increasing on R. (ix) The probability that a boy will not pass M .B.A examination is ~ and that a girl . 5 will not pass is i . Calculate the probability that atleast one of them passes the examination. 5 (x) If A and B are two events such that P(A) = 1 , P(B) = 4 1 2 and P(A P(not A and not B). 2 2. If f(x) = 3 x- , prove that / /j(x)l 2x-3 = x for all (!: sing properties of determinants, prove that n - . B) = ]_, find 8 [lQ X 2) (4) x e R - {~} . 2 b2 + c2 a2 a2 b2 c2+a2 b2 c2 c2 a2 +b2- = 4a2 b2c2 [4) -i . 4. Solve .the eq ation; sin~{ (6x) .'+ sin-1 (6 3 x) = S. (a) Show that the function f (x) = 2x _ 1x I is continuous at x [4] = O. (4] OR (b) Verify Lagrange's Mean Value theorem for the functiort / (x) = (x - 3)(x - 6) (x - 9) :. on the interval [3, 5) . .. d2u d 6. If Y = (sin::' 1 x) 2, prove that (1 - x2) ~-x~ - 2 = 0. 2 dx 7. (a) Evaluate: f x-4+3x+2 x+ 1 dx. [4] dx [4 ) 2 OR 2 (b) ~Evaluate: J(x2 +3) dx as a limit of sum. 0 8. (a) Find the equation of the tangent to the curve y = -Sx2 + 6x + 7 at the point ( ,~) . [41 OR ~ (b) The volume of a cube is increasing at a rate of 7 cm3 /sec. How fast is the surface area increasing when the lengt,h of an edge is 12 cm? {&so1ve the differential equation: y dx - (x + 2y2) dy = 0. v [4] 10. Three shots are fired at a moving target. The probabilities of hitting the target at first, second and third shorts are -~ 11. (a) Given two matrices A = A r~ -: ~] .!. , ~ 2 10 ..!.. . Calculate the probability that the target gets 10 and w and B ~-7~ -5-11 -14]62 and B = [ 1 -3 2 Find AB and ~ this result to solve the following system of equations: X - 2y + 3z = 6, X + 4y + Z = 12, X - 3y + 2z = 1. [6] OR (b) Find the inverse of the matrix A = [- ~ transformation. 0 2 _3 -~l] by 2 using elementary _row ri..{ 12. (a) The sum of the f , . L '1 t!H-r..., L . sur ace- are --.,,.--,.--:---;- . h 'd 2x and :.. d . as cf a rectanguJa para elopiped w1t s1 es x, all a sphere is is tnin' 3 . . given tobe constant. Prove that the sum of their volumes imum, if x is equal t O minimum value of the s the ~hree times the radius of the sphere . Find the . um of the vo}um!;!s. (b) S~ow that the volume 0 f h OR t e greatest cylinder which can be inscribed in a cone of :.. he1ght h ~nd semivertical angle a. is ...!. .1th3 tan2 a . [6]13, Evaluate: x -4x-9 dx 27 (x -2)(x2 +9) [6] 14 There are two identic l b 3 whit d a oxes containing respectively 4 white and 3 red balls, and 7 b . e~ red ~alls. A box is chosen at random and a ball is drawn frorn it. lf the 11 15 a w ite, what 15 the probability that it is from the first box? J SECTION B (20 MARKS) lS. (a) The dot products of a vector with the vectors are 0, 5 and 8 respectively. Find the vector. i + J - 3k, i + 3J- 2k 3 2 2i +] + 4k [6] (b) Find the angle between the line ~ = Y - 7 = 2 - 7 and the plane x and -2 + y + 2z = 0. '(c) Find the equations of the line which passes through the point (1 , 2, 3) and is parallel to the line x = ay + b, z = cy + d. [6] 16. (a) Find the area of triangle the coordinates of ~hose-.vertices are (1, 2, 4), (3, 1, - 2) and (4, 3, 1). OR (b) Show that if four points a, b, c , d are coplanar, then ' -+ -+ ~ -+ [b c d]+[c a d] + [a b d] = [a b c] . ./ (4] 17. (a) Find th~ .shortest distance between the lines x - 8 = y + 9 = z -10 and x - 15 = y - 29 =5 - z . : 3 7 -16' 3 8 5 l ~J -s-- [4] OR ..,"~ ,. (b) Find the equation of the plane passing through the line of intersection of the planes x + 2y + 3z - 5 = 0 and 3x - 2y - z + 1 = 0 and cutting off equal intercepts on the x and z-axes. 18. Find the area enclosed by the curves y2 = x and y2 = 4 - 3x. [6] SECTION C (20 MARKS) 19. (a) The average cost function associated with prod ucing and marketing x units of an item is given by 20 AC = 5x - 13 + - . .X r finct ~ (i) the total cost function and the_marginal cost function. ,I (ii) the range of values of the output x, for which AC is increasing. (b) Given the two regression _lines 4;X + 3y + 7 = o and 3x + 4y + B = 0, determine the coefficient of correlation. ~ manufacturer's total cost function is given by C = 2000 + 25x + 3x2. Find the average cost function . the mafginal cost function. the marginal cost when 25 units are produced. f6] 10. (a)':For the following data, find the regression equation of y on x. (c) If the (i) .:(ii) (iii) ~;_-= -- : -J_ _ -:~~_J --:- 1 - -:--T- 1-_ -J OR (b) From the o inven data: x = 25, y- =30, byx (i) the regression equation y on x. = 1 6 and bxy = 0 4, find ~(i1) what is the most likely value of y when x = 60? (iii} the coefficient of correlation. 21L (.:1) The rnarginal cost of a product is given by MC = soo <2500. Find 2x+25 and the fixed cost is the total cost and the average cost of producing 72 units. OR (b) A company wants to launch a new product. It invested ~ 37,500 as fixed cost and ~ 200 per unit as the variable cost of production. The revenue function for the sah: of x units is given by 4825x - 125x2 Find the break even point(s). [4) 22. Mrs. Jolly wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains atleast 10 units of vitarnin A, 12 units of vitamin B and 8 units of vHamin C. Th.e vitamin contents of one kg of food is given below: Vitamin A 1 Vit~,in,,B 2 Vitartti,n._ G _ _J 1 3 ----- - -- --- - - - - - - - - - - - - - - 2 1 2 i (Jne kg of food X costs ~ 6 and one kg of food Y costs ~ 10. Find the least cost of the rnixture which will produce the diet. [6] - - - - . / /
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