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KENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION CLASS XII- FIRST COMMON PRE-BOARD EXAMINATION-2013-2014 Answer key (Mathematics) Q No. Value points Marks 1 to 10 4 6 2 1) .2) 3) 3 6 9 4) a=4,b=2; 3 4 4 8 12 dy y 7 13 1 3 6 2 5) 6)k= , 7) (3iˆ 6 ˆj 2kˆ), , , 8)Show that dx 2 x 2 2 7 7 7 7 Each 1 mark 9) 1 10)1.5 cm/sec 11 12 For proving commutative 1.5 For proving associative 1.5 For proving 0 is the identity and it is not in N 1 For cross multiplying by 2 0.5 For applying the formula and getting 2=6x2 2 For rejecting x<0 0.5 For getting x = 1 1 3 OR For tan-1 = and the remaining term in terms of cot-1 4 1.5 Converting into cot-1 to tan-1 1 For applying tan-1x+tan-1y 1 For getting the value π .5 Page 1 of 6 13 For applying R1 R1-R2 and R2 R2-R3 2 a 1 1 0 For getting (a-1)2 2 1 0 3 3 1 1 1 For getting (a-1) 14 For each differentiation 1.5 mark each For 15 3 3 dy tan x x tan x sec 2 x log x sin x cos x (cot x cos x sin x log sin x) dx x For substituting x tan in each of the term and simplifying For differentiating and getting the answer 16 For converting the limit x For getting a 2 1 3 1 2 3 to h 0 .5 mark each 1 3 1 and b 4 for each 1.5 mark 2 (OR) 17 For finding first derivative 1 For cross multiplication,squaring and substituting each 0.5mark 1.5 For finding second derivative and proving the result 1.0+0.5 1.5 For dividing by cos4x in the Nr.and Dr. 1 Substituting tan2x=t and getting the answer as 1 tan 1 (tan 2 x) C 2 (OR) 1 For splitting (6x-5) as -3(3-2x)+4 1.5+1.5 For getting correct integrated value for each 2 8 3x x 2 3 2 (2 x 3) 8 3x x 2 Page 2 of 6 1.5+1.5 41 1 2 x 3 sin +C 2 41 18 Put x+b=t and expanding cos(t-b+a) 1 Splitting in to two terms and integrating 19 Ans.cos(a-b)log sin(x + b) - sin(a-b).(x+b)+c 1+1+1 For expressing in terms of cos and sin 1 For applying F(a +b-x)=f(x) 1 For I+I and getting the answer 20 1+1 12 For getting a b c 3 a 3 b 3 c 1 1 For getting angle between a, b , c with a b c as cos 1 3 21 Let A be the point (2,3,4) and let B(x1,y1,z1)be the point in the plane. To find the equation of AB To find (x1,y1,z1)=(3λ+2,6 λ+3,2 λ+4) 1+1+1 1 1 Substituting the point in the plane and getting λ=-1 (x1,y1,z1)=(-1,-3,2) and d=7 1 1 OR For finding N 10iˆ 18 ˆj 22kˆ 2 For finding the equation 5x+9y+11z=8 22 23 For P(E)=0.6,P(F)=0.4,P(E F)=0.2 2 3x0.5 1.5 P(F/E)=0.33 .5 For each question on VBQ 1+1 For correct diagram and SA=πr2+ 2 2 v 4v r r 2 2 3 For differentiating and equating to 0 and to get π r =v(π+2) For deciding minimum 1 For proving h:2r=π : π+2 1 (OR) Page 3 of 6 Let C . Then AC= a sec +b cosec 2 1 b 3 For finding first derivative and getting tan a 2 For second derivative and showing minimum 2 3 2 3 For proving the minimum length of AC= a b 1 3 2 1 24 1 For correct diagram and shading 1 a b For getting { a 2 x 2 (a x)}dx a0 For correct integration and getting the answer as 25 2 ab 4 1 ab 2 For identifying homogeneous and putting y=vx For getting up to 1 1 v dx dv 2 x 4v 2 2v 1+1+1 4 For integrating and simplifying to get the answer y2+2xy= Page 4 of 6 2 c x3 26 1 For finding (x1,y1,z1)=(10λ+11,-4 λ-2,-11 λ-8) For finding d.rs. of AB 10λ+9,-4 λ-1,-11 λ-13 1 Substituting in the equation of plane and to find λ=-1 1 For (x1,y1,z1)=(1,2,3) and distance= 14 1 For image 0,5,1 (OR) 1 For equation of plane x+2y+3z-4+ λ(2x+y+z+5)=0 1 Using Dr’s of normal to the planes, finding λ=-1 1 Equation plane: x-y-2z+9=0 4 1 27 Let E1: the number greater than 4, E2: the number not greater than 4,A: he reports it is greater than 4 1 3 P(E1)= ,P(E2)= 2 3 2 , P(A/ E1)= ,P(A/ E2)= 3 5 5 1 3 3 7 1 Value based Answer 1 P(E1/A)= Page 5 of 6 2 28 2 1 1 Let x no. of packets of food P and Y no. of food Q be prepared. Constraints: x, y 0,4 x y 80, x 5 y 115 and 3x 2 y 150 Objective function: Z=6x+3y Corner points: (2,72),(15,20),(40,15) It is when x=15,y=20 minimum .Minimum amount of Vit.A=150units 29 2 2 4 1 For det.A=10, A 1 5 0 5 10 1 2 3 Using ( A ) ( A) 1 To get x 1 3 1 2 18 4 14 ,y ,z 10 10 10 Page 6 of 6