KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION SAMPLE QUESTION PAPER CLASS XII. TIME: 3hrs MATHEMATICS M.M 100

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1 KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION SAMPLE QUESTION PAPER CLASS XII TIME: 3hrs MATHEMATICS M.M 00 GENERAL INSTRUCTIONS:. All questions are compulsory. The question paper consists of 9 questions divided into three sectionsa, B and C. Section A comprises of 0 questions of mark each.section B comprises of questions of marks each and Section C comprises of 7 questions of marks each. 3. All questions in Section A are to be wriitten in one word, one sentence or as per the exact requirement of questions.. There is no overall choice.however, internal choice has been provided in questions of marks and questions of marks each.you have to attempt only one of the alternatives in all such questions. 5. Use of calculator is not permitted.you may ask for logarithmic tables, if required. SECTION- A. If A= [ ], find the determinant -A.. If A= [ ] B=[ ],find ( ) 3. If f: R is a bijection given b y f(x) = +3, find (x).. Using principal value,evaluate: ( ) 5. Find the equation of the tangent to the curve y = at the point ( ). Find, if two vectors and are such that = and =3 and. =. 7. Evaluate 8. Evaluate: 9. Let [ ] [ ], then find the value of x and y. 0. Find projection of +3 - on = -3 +

2 SECTION B. Let A=QxQ.Let * be a binary operation on A defined by : (a,b)*(c,d) = (ac,ad+b).find i) Identity element of (A,*) & ii) the invertible element of (A,*).. Solve: ( ) ( ), x>0 Find the value of ( ) ( ) ( ) 3. Using properties, show that =0. Showthat f(x)= ( ) { 5. If y=[ ],then prove that is continuous at x=0 If y=acosmx+bsin mx, show that. Showthat ( ) is an increasing function of x for all values of x>- 7. Evaluate: Evaluate: ( )( ) 8. Solve the differential equation :, given that y=,when. Solve the differential equation:( )( ) 9. Solve the differential equation: ( ) ( ) 0. The scalar product of the vector î +ĵ+ with the unit vector along the sum of vectors î +ĵ-5 and λ î +ĵ+3 s equal to one.find the value of λ. Find the vector equation of the line passing through (,,3) and parallel to the planes ( î -ĵ+ ) ( 3î +ĵ+ ). Three students A, B and C are given a question to solve.their respective probability to solve the question are What is the probability that only one of them solves it correctly?

3 SECTION C 3. If [ ], find solve the following system of linear equations x-3y+5z=, 3x+y-z=5 and x+y-z= -3. Show that the height of the cylinder of maximum volume that can be inscribed in a shpere of radius R is.also find the maximum volume. Show that the right circular cone of least curved surface area and givenvolume has an altitude equal to times the radius of the base. 5. Evaluate: ( ) dx Evaluate: dx. Find the area of the region {( ) } 7. Find the equation of the plane passing through the point ( -,,) and perpendicular to the line joining the points (-3,,) and (,3,). Find also the perpendicular distance of the origin from the plane. 8. If an old man rides his motor cycle at 5 km/hr, he has to speend Rs. per km on petrol.if he rides at a faster speed of 0 km /hr, the petrol cost increases to Rs.5 per km. He has Rs 00 to spend on petrol and wishes to find maximum distance he can travel within one hour.express this as a linear programming problem and then solve it. 9. There are 3 bags each containing 5 white balls and 3 black balls.also there are bags, each containing white balls and black balls.a white ball is drawn at random.find the probability that this white ball is from a bag of the first group. ******************************************************************************

4 MARKING SCHEME SAMPLE PAPER MATHEMATICS-CLASS XII SECTION A Q.No. Value Points Marks Total 3 [ ] 3 ( ) 5 x-y+33=0 7 8 ( ) 9 X= and y=3 0 SECTION B Property of Id element Id element (,0) Prtoperty of Inverse Inverse doesn t Application of correct formula Correct equation Solving and x=8 Application of formula Second application Combining to get 3 Use of three properties and Conclusion Calculation of LHL Calculation of RHL Conclusion LHL=RHL=f(0)=3/ / /

5 5 Finding of dy/dx and rearranging Writing correct derivatives Rearrangement and proof Finding first derivative Simplification Expressing as perfect squares and conclusion 7 Expressing the denominator in cos or sin. Applying the correct integral Simplification.or any other alternate methods Using partial fraction and finding three constants Evaluating three integrals and simplification,an: log( ) Identification and I.F-linear and I.F.is Solving :y = Writing particular solution: ( ) Separating the variables Inegrating by substitution Solving and writing Particular solution: ( ) / / 9 Identification and substitution Separating variables Integrating and replacing v by x/y Solu. Is 0 Concept of unit vector Sum of î +ĵ-5 and λ î +ĵ+ Modulus of the above sum Dot product and value of λ= Equation of line is Concept of normal and line(perpependicular) Solving of equations a-b+c=0 and 3a+b+c=0 as a=-3,b=5 and c= Writing the final equatrion / ½ / / Probabilities for not soliving Required probability=p( C)+P( B )+P(A ) SECTION C / / 3 deta=- exists

6 adja=[ ]&B= [ ] X= ( ) gives x=,y=and z=3 Writing the correct relation in one variable Step wise application of differentiation Obtaining the correct relationand max. volume= ( ) Finding the correct relation in one variable Step wise application of differentiation Obtaining the correct relation 5 Giving substitution as and simplication Application of limits and simplification,an: ( ) Application of propertyand I Evaluating integral by substitution Appliying limits and simplification.an : Identification of the given curves and rough sketch Solving the equations and finding limit of integrals +x-9=0 gives x=/and Area= 7 Equation of plane passing through (-,,) is A(x+)+B(y-)+C(z-)=0 Equation of a line joining (-3,,) and (,3,) is Since the plane is perpendicular to the line,line and normal are parallel an A=5,B= and C= Equation of plane is 5x+y+z-=0 Distance from origin to the plane is units 8 Let the old man travel xkm at 5km/hr and y km at0km/hr Objective function is z=x+y Constraints : x x+5y and Graphing two lines with scale and shading of common region ( ) Locating the corner points as O(0,0),(5,0), ( ) and (0,0)and calculation of z(max)=30 at ( ) ½ ½ 9 Let Ebe the event of selecting a bag from the first group And E Ebe the event of selecting a bag from the second group A be the event of ball drawn is white /

7 Then, P(E)=3/5,P(E)=/5 and P(A/E)=5/8, P(A/E)=5/8 Therefore the required probability,p(e/a)= ( ) ( ) ( ) ( ) ( ) (( ) ) = /