ࡱ> [ Jfbjbj 4ΐΐ|&#####7778ot7Ip6"6 7HHHHHHHLNH# 7/H6 7 7H##Hn@n@n@ 7##Hn@ 7Hn@n@:&F,FP78RF HH0I\FRCO<COFCO#FX 7 7n@ 7 7 7 7 7HH> 7 7 7I 7 7 7 7CO 7 7 7 7 7 7 7 7 7 : Guess Paper 2010 Class XII Subject Maths M.M. : 100 Time 3 hrs. General instuctions All questions are compulsory . The question paper consists of 29 questons divided into three sections A , B ,and C . Sectio A comprises of 10 questions of one mark each , Section B comprises of 12 questions of 4 marks each and Section C comprises of 7 questions 6 marks each All questions in section A are to be answered in one word , one answer as per the exact requirement of the question There is no overall choice . However , internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of alternatives in all such questions. Use of calculators is not permitted . You may ask for logarithmic tables. If required,  Sec -A 1 ) Solve for x : sin-1x cos-1x =  / 6 2 ) Determine whether the following relation are reflexive , symmetric and transitive : Relation R in the set N of all natural numbers defined as : R = (( x , y ) : y = x + 5 , x < 4 (  3 ) If A = 1 2 , then show that 2A = 4 A 4 2  4 ) If x 2 + 2 2x = 3 7 find the value of x & y y2 3y -3 5 ) If A is a square matrix of order 2 and | A | = - 5 find the value of | 3 A | 6 ) Find the point on the curve y = 3x2 12x + 5 at which the tangent is parallel to x axis ?  7 ) 1 dy / dx 2 3 / 2 = k (d 2 y / d x 2 ) , determine the order and degree of differential equation , state whether it is linear or non linear ? 8 ) Find the scalar projection of vector i^ - 2 j^ + k^ on the vector 4 i^ 4 j^ + 7 k^ 9 ) If a = 4i^ + 2j^ k^ and b = 5i^ + 2 j^ 3k^ , find the angle between a + b and a b 10 ) For what value of p will the line through ( 4 , 1 , 2 ) and ( 5 , p , 0 ) be perpendicular to the line through ( 2 , 1 , 1 ) and ( 3 , 3 , -1 ) Sec B 11) Let * be the binary operation on N defined by a * b = LCM of a and b i ) 5 * 7 ii ) 20 * 16 iii ) Is * commutative ? iv ) find identity element of * in N v ) Which element of N are invertible for the operation * ? 12 ) Consider the function f : R + [ 4 , " ) defined by f ( x ) = x2 + 4 , where R + is the Set of all non- negative real numbers . Show that f is invertible . Also find the inverse of f 13 ( b + c ) 2 a2 bc ( c + a ) 2 b2 ca = ( a b ) ( b c ) ( c a) ( a + b + c ) ( a2 + b2 + c2 ) ( a + b )2 c2 ab Prove by using the method of operation. 14 ) 1 log ( 1 + x )  d x , Evaluate 1 + x 2 0   15 ) Differentiate Sin x w . r . t . tan -1 1 x 2 1 + x 2 OR Find the coordinates of the point at which the tangents to the curves y = x2 6x + 1 is parallel to the chord joining to points ( 1 , - 4 ) and ( 3 , - 8 ) ? 16 ) If y = x2 + x + 1 and x changes from 2 to 2.01 , find the actual change and approximate change in the value of y 17 ) If y = Sin (m Sin-1 x ) , prove that ( 1 x2 ) d 2 y / d x 2 - x d y / d x + m 2 y = 0 18 ) Express the vector a = 5 i^ - 2 j^ + 5 k^ as the sum of two vectors such that one is  Parallel to the vector b = 3 i^ + k^ and the other is perpendicular to b 19 ) Find the image of point ( 2 , -1 , 5 ) in the line  r = ( 11 i^ - 2 j^ - 8 k^ ) +  ( 10 i^ - 4 j^ - 11 k^ ) 20 ) A die is thrown 20 times , getting a number greater than 4 is considered a success , find the mean and the variance of the number a successes . OR A coin is tossed until a head appears or until it has been tossed three times . Given that head does not appear on the first toss , what is the probability that coin is tossed three times 21 ) x2 , evaluate  ( xsinx + cos x)2 OR  4  x 1 + x 2 + x -3 dx , evaluate 0 22 ) Find the local maximum and local minimum values of f ( x ) = 2 Sin x + Sin2x in [ 0 , 2  ] OR Find the values of k for which f ( x ) = k x 3 - 9 k x 2 + 9 x + 3 is increasing on R Sec - C 23 ) An oil company requires 13000 , 20000 and 15000 barrels of high grade , medium grade and low grade oil respectively . Refinery A produces 100 , 300 and 200 barrels per day of high , medium and low grade oil respectively whereas the refinery B produces 200 , 400 and 100 barrels per day respectively .If refinery A costs 400 per day and B costs Rs. 300 per day to operate , how many days should each to run to minimize the cost of requirement ? 24 ) Find the equation of plane passing through the line of intersection of the planes  r . ( 4 i^ - j ^ + k^ ) = 0 , r . ( 3 i^ + j^ - k^ ) = 5 and parallel to the line  r . ( 4 i^ - 3 j^ + k ^ ) +  ( I ^ - 2 j^ + 3 k^ ) ( OR ) Find the shortest distance between the lines x - 1 y - 2 z 3 x 2 y - 4 z - 5  = = and = = 2 3 4 3 4 5  25 ) 1 - 1 0 2 2 - 4 If A = 2 3 4 B = - 4 2 - 4 , Find A B , 0 1 2 2 -1 5 Hence , solve the equation : x y = 3 , 2 x + 3 y + 4 z = 17 , y + 2 z = 7  26 ) Find the area of the region bounded by { ( x , y ) ; x2 d" y d" x }  27 ) dy 4 x y - tan-1 x  + = , solve the differential Equation dx x2 + 1 ( x 2 + 1 )3 ( OR ) Solve ( x 2 + 3xy + y 2 ) d x - x 2 d y = 0 , solve the differential equation . 28 ) A rectangle is inscribed in a semi - circle of radius r with of one of its sides on the diameter of the semi circle . Find the dimension of the rectangle so that its area is maximum . 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