KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION MODEL EXAMINATION MATHEMATICS CLASS:XII

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1 KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION MODEL EXAMINATION 0-3 MATHEMATICS CLASS:XII Time: 3Hours Max.Marks:00 General Instructions. All questions are compulsory.. The question paper consists of 9 questions divided into three sections A, B, C. Section A comprises of 0 questions of one mark each, section B comprises of questions of 4 marks each and section C comprises of o7 questions of six marks each. 3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 04 questions of four marks each and0 questions of six marks each. You have to attempt only one of the alternatives in all such questions. SECTION A. Let E={,,3,4} and F={,}. Then find the number of onto functions from E to F.. Find the numerical value of tan[tan - (/5)-π/4] 3. For what value of k, the matrix has no inverse? 4. If A= then A+A T =I, Find the value of. 5. Give an example of two nonzero x matrices A,B such that AB=O 6. Evaluate: 7. Evaluate: where f(x) = x-[x]; [x] is the integral part of x.

2 8. Let be vectors of magnitudes 3,4,5 respectively. Let Then, find 9. The points with position vectors are collinear. Find the value of a 0. Find the equation of the line parallel to x axis and passing through the origin SECTION B. Let a relation R on the Set N of natural numbers be defined as (x,y)εr if and only if x -4xy+3y =0 for all x,yεn. Verify that R is reflexive but not symmetric and transitive.. Find x if 3. Show that 4. Find the values of a and b so that the function f(x)= is continuous for 0 x π 5. If cos y =x cos(a+y), with cos a ±, prove that If x=a(t+sint) and y=a(+cost). Find

3 6. Find the intervals in which xe x(-x) is strictly increasing or decreasing. 7. Evaluate Evaluate 8. Form the differential equation of the function (a+bx) Form the differential equation of the family of circles in the second quadrant and touching the co-ordinate axes. 9. Solve the differential equation [x sin (y/x)-y] dx+xdy = 0 0. For any two vectors. State whether the lines intersect or not. If intersecting find the point of intersection.. There are 5 cards numbered to 5. One number on one card. Two cards are drawn at random without replacement. Find the probability distribution of the sum of the numbers on the two cards. SECTION C 3. If A= Find A - and hence solve the system of linear equations X+y+z=4,-x+y+z=0, x-3y+z= Using elementary transformations, find the inverse of the matrix 4. A window of perimeter(including the base of the arc ) is in the form of a rectangle surrounded by a semi circle. The semi-circular portion is fitted

4 with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much lighrt per square metre as the coloured glass does. Show that the ratio of the length and breadth of the rectangle is 6:6+π, sothat the window transmits maximum light. 5. Sketch the region bounded by the curves y= and y= and find its area 6. Evaluate Evaluate 7. Find the equation of the plane passing through the point(,,) and containing the line the line ) Also, show that the plane contains 8. Every gram of wheat provides 0. gm of proteins and 0.5gm of carbohydrates. The corresponding values for rice are 0.05gm and 0.5gm respectively. Wheat costs Rs.4 per kg and rice Rs.6 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 50gms and 00gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost. Frame an LPP and solve it graphically. 9. Bag A contains 3 red and 4 black balls and bag B contains 4 red and 5 black balls. One ball is transferred from bag A to bag B and then a ball is drawn from bag B. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

5 MODEL EXAMINATION 0-3 MATHEMATICS CLASS:XII BLUE PRINT S.No Topics VSA SA LA Total ( mark each) (4 marks each) (6 marks each).(a) (b) Relations and Functions Inverse Trigonometric Functions () () 4() 4() 0(4).(a) Matrices () 6() (b) Determinants () 4() 3(5) 3.(a) (b) (c) (d) Continuity and differentiability Applications of derivatives Integration Applications of integrals () 8() 4() 4() 6() 6() 6() (e) Differential equations 8() 44() 4.(a) (b) Vectors 3-Dimensional geometry () () 4() 4() 6() 7(6) 5. Linear programming 6() 6() 6. Probability 4() 6() 0() Total 0(0) 48() 4(7) 00(9) Marking Scheme Q.NO Value points/ans Marks 4-7/7 3 K=3/ 4 α=π/3 5 A= B= 6. log x + log sin x + c a= -40 0

6 For xεn, x -4xx+3x =0 (x,x)εr R is reflexive (3,)εR as =0 But So(,3)ε R R is not symmetric (9,3)εR as =0 (3,)εR as =0 But (9,) R as R is not transitive () Sin - (5/x)+sin - (/x)=π/ /x=sin(π/-sin - 5/x)=cos(sin - 5/x) Let sin - x=t 5/x=sint Cost= / / t=cos - /x= 44/x =(x -5)/x X=±3 But x=-3 does not satisfy the equation X=3 / / / /

7 3 Applying R ar,r br,r 3 cr 3 and multiply whole determinant by /abc Taking out common factors a,b,c from C,C,C 3 respectively Applying R R +R +R 3 Taking out from R Applying R R -R,R 3 R 3 -R Applying R R +R +R 3 On expanding along Ist row {-c (-b a )+b (c a )}=4a b c 4 f(x) is cont. at x=π/4 LHL=RHL π/4+a sin π/4=x π/4cot π/4+b a-b= π/ () f(x) is cont.at x= π/ x π/cot π/+b=a(-)b a+b= () soving ()&() getting a=3 π/&b=-3 π/4 5 Getting / Getting dx/dt=a(+cost) &dy/dt=-asint dy/dx+-asint/a(+cost)=-tan(t/) 6 f (x)=e x(-x) +x e x(-x) (-x) =-e x(-x) (x-)(x+) F (x)=0 gives x= or -/ In (-,-/),function is decreasing (-/,),function is increasing (, ),function is decreasing /

8 7 Put +xe x =t I= implies e x (x+)dx=dt / Let Getting A=,B=-,C=- I= / I= 8 Correct differentiation w.r.t. x Obtaining Again differentiating w.r.t x Obtaining eqn. of circle as (x+a) +(y-a) =a Diiferentiating w.r.t x Obtaining(x+y) (y +)=(x+yy )

9 9 Getting dy/dx=y/x - sin (y/x) Put y=vx then dy/dx=v+x(dv/dx) Getting -cosec vdv=dx/x Integrating cotv=logx+c Cot(y/x)-logx=c 0 R.H.S= Simplifying and put Let If they intersect, then for some and 3 -= +, () 5-3=3 +4, () 7-5= (3) Solving ()&() and getting =/, =-3/ Which satisfies (3) Hence lines intersect Point of contact(/,-/,-3/) Getting random variable as 3,4,5,6,7,8,9 X P(X) /0 /0 /5 /5 /5 /0 /0 3 A is invertible Finding A - =/0 Writing For each Correct Probability A T X=B WhereX=

10 X=(A T ) - B X= A=IA 4 a a b Perimeter P=a+4b+πb () Let the transmission rate of coloured glass be L and Q be total transmitted light Q=ab(3L)+/πb (L) /

11 Q=L/(6Pb-4b -5πb) using () dq/db=l/(6p-48b-0πb) applying dq/db=0 implies / / / so Q is maximum now (48+0π)b=6P=6(a+4b+πb) b:a=6:(6+π) 5 (- 5,0) - ( 5,0) Solving the equations and obtaining x=,- Required area= Evaluation and simplification Obtain area as 5π/4-/ 6 I=

12 ()+() I= Put π/cosx = t so sinx dx = -/π dx I = Integrating and simplifying I= = = 7 Let A(,,) P(-3,,5) pt on the line / /

13 8 Suppose xgms ofwheat and y gms of rice are mixed in the daily diet Constraints 0.x+0.05y x+0.5y 00 x/4+y/ 00, x 0,y 0 Objective function Minimize Z= / Feasible region is unbdd and has vertices A(0,000) B(800,0) P(400,00) Point Z A (0,000) 6 B (800,0) 3. P (400,00).8 Min. Z=.8 Wheat 400gms rice 00gms

14 9 E :Red ball is transferred from A to B E :Black ball is tranferred from A to B E:Red ball is drawn from B P(E )=3/7 P(E )=4/7 P(E/E )=5/0=/ P(E/E )=4/0=/5 3 P(E /E)= =6/3

Design of Question Paper Mathematics - Class XII

Design of Question Paper Mathematics - Class XII Time : 3 hours Max. Marks : 100 Weightage of marks over different dimensions of the question paper shall be as follows : A. Weightage to different topics/content