A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH CBSE Mathematics 06 Solved paper for Class XII(0+) Section - A Q. For what value of k, the system of linear equations. y z y z y kz 4 has aunique solution? Ans. For any System of equations to have unique solution. A 0 0 k (k+)-(k+) + (4-) 0 k+-k-+ 0 -k 0 k 0 Q.. If a 4i ˆj kˆ and b i ˆj kˆ, parallel to the vector a b then find a unit vector Ans. a b 4iˆ Jˆ k ˆ iˆ ˆj k ˆ a b = 6iˆ ˆj kˆ a b = 6 9 4 49 7 Sol : (6iˆ ˆj kˆ ) 7 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Q. Find and if i ˆ ˆj kˆ X iˆ ˆj kˆ 9 0. X iˆ ˆj kˆ Ans. iˆ ˆj 9kˆ iˆ ˆj kˆ 9 0 iˆ 9 ˆj 7 kˆ 9 i.e. 9 0 7 0 7 9 0 9 0iˆ 0 ˆj 0kˆ Q. 4. Write the sum of intercepts cut off by the plane rˆ. iˆ ˆj kˆ 5 0 on the three aes. Ans. 4 z 5 4 y z 5 5 5 5 4 5 5 Sum of intercepts cut offby plane on three aes = 5 5 5 = 5 cos sin Q. 5 If. A=, find satisfying o< < when A+A T = sin cos I where A T is transpose of A. A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH cos sin Ans. A sin cos Condition given to us, T A A I cos asin a cos d sin a 0 sin a cos a sin a cos a 0 cos cos 4 0 Q. 6 if A is a X matri and A =k A, then write the value of K. Ans. A = k A n Using KA = k A, Whereistheorder of square Matine A A k=7 7 A k A A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH SECTION B Q. 7. Find ( ) 4 d. Ans. f 4 d d 4 4 Comparing coefficients of & constants 4,i.e. ( 4 ) 4 d t dt ( 7) ( ) d, Where t 4 4 4 7 sin c 7 Q. 8 Evaluate: d. 5 Sol. f ( ) 5 d A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 4
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH f ( ) 5 5 5 Using property a a a f d f f d 0 5 o 5 5 d d ] 0 0 = 8. Ans Q. 9. Find the equation of tangents to the curve y= + -4, which are perpendicular to line + 4y +=0. Ans.Let coordinates of point of contact be, y as it lies on y 4 A y 4 B Differentiating A w.r.t. dy d dy d, y Since the tangent at, y is perpendicular to line 4y 0 slope of tangentat, y X Slope of line =- A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 5
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH X 4 X = Now = y 4 8 y ( ) ( ) 4 8 4 4 6 Coordinates of point of contact are (,8) & (, 6) dy d,8 () 4 dy d, 6 ( ) 4 Equationof tangent at (,8) y-8= 4 (-) y =4 +6 Equiv of tangent at (-,-6) y+6 =4 (+) y+6=4 +8 y =4+ sin ( ) sin Q. 0 If f()={ a,<0,=0 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 6
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH b,>0 is Continuous at =0, then find the values of a and b Since f () is continuous at =0 Ans. lim f lim f f 0 0 0 sin ( a ) sin lim ( a ) lim 0 ( a ) 0 a++= a=- b b lim 0 b b lim b 0 b b=4 Q..Solve for X: tan - (-)+ tan - + tan - (+) = tan -. Ans. tan tan tan tan tan tan tan A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 7
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 4 4 4 OR Prove that tan - 6 8 4 tan tan ;. 6 8 4 tan tan tan 4 tan 6 8 4 4 6 8 4 4 6 8 4 4 48 5 tan 4 48 4 4 4 5 6 tan 4 6 8 tan Q. if cos (a+y) = cosy then prove that Hence show that sina d y dy sin ( a y) 0. d d dy dy sin( a y) cos( a y) sin y Sol. d d dy a y. d sin a cos ( ) cos ( a y) dy d sin y sin( a y) A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 8
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH dy cos ( a y) d cos y sin y sin( a y) cos( a y) dy d a y y a y y a y cos sin cos( ) cos sin( ) a y cos ( a y) dy cos sin( a y y ) d sin a 6 4 4 Find dy if y sin d 5 Or Sol. Put = sin sin 4cos Sin- 5 4 sin sin cos 5 5 sin cos d sin sin d cos consider sin d 5 sin sin( d) y d y sin sin dy d 4 Q. A bag X contains 4 white balls and black balls, while another bag y contains white balls and black balls. Two balls are drawn (Without replacement) at random from of the bags A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 9
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH and were found to be one white and one black. Find the probability that the balls were drawn from bag y. Ans. E =Select Bag X E= Select Bag Y A= Selecting white & black P( E ) P( E) A c X c 8 P( ) E 5 4 6 c A c X c P( ) E 5 6 c E P(Balls drawn are from bag Y)= P( ) A A p( E) P ( ) E A A p( E ) P ( ) p( E ) P ( ) E E * / 5 8 ( * * ) 5 5 0 8 0 0 0 8 9 0 = 9 ans. 7 7 Or A and B throw a pair of dice alternately, till one of them gets a total of 0 and wins the game. Find their respective probabilities of winning. If A starts first Sol. E person Agets a 0 F person B gets a0 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 0
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH P( E) 6 p( E) P( F) p( F) A wins if he throws0 in st, rd &5 th throws. a. '0' in st throw b. 0 in rd throw = P E F E P( E) P( F) P( E) * * ( ) * c. 0 in 5 th throw = P E F E F E * * * * 4 = * And so on. (Probability of A wins)= ( )... P E E F E E F E F E 4 * *... 44 44 X A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH P (B wins) = = Q. 4 Find the coordinates of the foot of perpendicular drawn from the point A (-, 8, 4) to the line joining the points B (0,-, ) and C (, -,-). Hence find the image of A in line BC. Ans. A.(,8.4) B(0,,) y s Equation of BC C(,,) y s 0 y z 0 y z 4 Coordinate of D (,, 4 ) Direction Ratios of AD, 9, 4 S in ce AD is preperdicular tobc ( ) ( 9) 4( 4 ) 0 4 4 8 6 4 0 4 4 0 Coordinates of D(,, 7) Image of A in line BC= A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 8 y 4 z y 6 7 z 0 Image ofa in line BC is (-,-6,0) Q5.Show that the four points (0,-,-),(-4,4,4), (4,5,) and (, 9, 4)are coplanar. Find the equation of the plane containing them. SOLUTION The equation of plane passing through (0,--) is a( 0) b( y ) c( z ) 0 i If it passesthrough (-4, 4, 4) and (4, 5, ) than a( 4) b(5) c(5) 0 ii and, a(4) b(6) c() 0 or, a() b() c() 0 Solving ii and iii by cross multiplication, we obtain a b c 5 5 0 4 0 a b c ( say) 5 7 a 5, b 7 and c Substituting the values of a, b and c in i, we get 5 7 ( y ) ( z ) 0 5 7y z 4 0 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Q. 6. A typist charges Rs 45 for typing 0 English and Hindi pages, while charges for typing English and 0 Hindi pages are Rs. 80. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. per from a poor student shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem? Sol.Let the charge of typing one englishpage = Rs. the change of typing one hindipage =Rs. y 0+y=45 +0y=80 X B Where 0 45 0 y 80 A 00 9 9 0 given system has unique solution given by = A B 0 adj A 0 0 A adja A 9 0 X 0 45 9 0 80 0*45 *80 9 *45 0*80 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 4
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 90 0 9 85 5 Charge for typing English page = Rs.0 Charge for typing Hindi page = Rs.5 Shyam was charged Rs. less than usual. Q. 7. Find the particular solution of the differential equation. y e /y d+(y- e /y ) dy =0 Given that =0 when y=. Sol.We have, y y ye d ( y e ) dy o d dy e y y e y y Clearly, the given differential equation is a homogeneous differential equation. As the right hand side of as a function of y So, we put =vyand d v y dv to get dy dy v dv ve v y dy e v v dv ve y v v dy e dv y dy e v ye v dv dy v ye dv dy y A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 5
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH v y e dv f dy v e log y log C v c e log y y c e log y, y c Hence e log given the general solution of the given different y equation. y c e log y =0, y = e 0 c = log =log C y e log y is the particular solution Q. 8. Find the particular solution of differential equation dy y cos given that y = when =0. d sin SOLUTION We have, dy y cos dy cos y d sin d sin sin X This is a linear differential equation with P= cos and Q sin sin X A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 6
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH I.F.=e cos d sin log(sin ) ( sin ) e e Multiplying both sides of i by I.F. = + sin, we get dy (+ sin ) y cos d Integrating with respect to, we get y( sin ) d C y( sin ) C C y, Which gives the generalsolution. ( sin X ) Put y= and =0, c= y ( sin X ) Q. 9 Find : 5 e d Ans. Put t, d dt t ( t 5) e dt ( t ) t t / e dt ( ) ( ) t t t t e dt e dt ( t ) ( t ) A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 7
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH t t e e dt et dt dt ( t ) ( t ) ( t ) t e e c c ( t ) ( ) Find : d Or Sol: let A B C ( ) A A B B C C C om paring cofficients of, & cons tan ts = A c A B B c A 5 C 5 B 5 d 5 5 d 5 = log( ) tan log ( ) c 5 5 5 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 8
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH SECTION-C Q. 0 Prove that F in 0, Solution.we have, 4sin ( ) is an increasing function of cos 4sin f ( ) cos cos cos 4cos sin f '( ) 8cos 4 f '( ) cos 4cos cos f '( ) cos f cos (4 cos ) '( ).0 0,. cos 0,4 cos 0 cos 0 cos for all and Hence, f '( ) is incear sin g on 0,. OR show that the semi- vertical angle of a cone of maimum volume and given slant height is tan or co s. Solution let a be the semi show that the semi- vertical angle of a cone VAB of given slant height t. in AOV, A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 9
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH VO OA cos a and sin a VA VA VO OA cos a and sin a l l VO l cos a, OA l sin a Let be the volume of the cone, then, ( ) ( ) V OA VO ( sin ) cos V l a l a V l sin a sin a cos a dv The critical points of are given by 0 da Check RD Sharma part XII for further solution. Q. Using properties of determinants, prove that ( ) y z zy z ( z y) y yz y z zy y z y Or b c ba ca ab c a cb abc( a b c) ac bc a b Solution b c ba ca / abc ab c a cb MultiplyingR, R andr bya, b, andc ac bc a b A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 0
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Respectively, we get abc a b c ba ca ab c a b cb ac bc a b a b c ba ca abc ab c a b cb Taking a, b and ccommon from c, c andcrespectively abc ac bc a b b c a a b c a b abc( a b c) c c a b b c a a c c a b b c a b AppliyingC C C and C C C we get b c a 0 a 0 c a b b c ( a b) c ( a b) a b b c a 0 a a b c 0 c a b b Taking ( a b c) common From C & C c a b c a b a b b c a 0 a a b c 0 c a b b Appliying R R R R b a ab OR abc( a b c) A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 0 If A = 0 and A 6A 7A KI 0 Find k. 0 Sol. A 5 0 8 4 5 8 0 A 0 4 8 4 0 55 K= Q.. Using the method of integration, find the area of the triangular region whose vertices are A(, -), B(4,) and C(,) Ans.Equation of AB Y ( ) 4 5 Y ( ) 5 Y 7 ( y 7) 5 eqn of BC y 4 4 y 4 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH 5 5 y y equl of AC Y 4 Y y 6 y 4 6 4 4 BC AC d BC AB d 4 5 5 5 8 d 5 d 4 0 5 4 d d 6 6 5 4 * * 8 54 5 6 6 9 6 6 squnits Q.. Let A =R R and * be a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH Show that * is commutative and associative. Find the identify element for* on A. Also find the inverse of every element (a, b) E A. Eample Let A N 0 N 0 and let '*' be binary operation on A defined by a, b * c, d a c, b d For all( a, b),( c, b) A. Show that (i) * is commutative on A. (ii) * is associative on A. Also find the identity element, if any, in A. SOLUTION (i) Commutativity: Let (a,b), (c,d)a. Then, ( a, b) * ( c, d) ( a c, d b) and ( c, d) * ( a, b) ( c a, d b) a c c a and b d d b for all a, b, c, d N ( a, c) * ( c, d) ( c, d) * ( a, b) for all ( a, b),( c, d) NN A '* ' is Commutative ona. Please Refer RD Sharma page.0 Eample 9 Q. 4. Three numbers are select at random (Without replacement) from first si positive integers. Let X denotes the largest of the three numbers obtained. Find the probability distribution of X. Also. Find the mean and variance of the distribution. Ans. we observe that X on take value, 4, 5 6. A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 4
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH P (X=)=Probability that largest of three number is. X X * 6 5 4 P (=4)= Probability that largest of no. is 4 4 * 6 5 4 4 X X c P( 5) Pr obalitity that l arg est of three no is5 5 4 5 * * * c 6 5 4 6 P ( 6) c X 4 5 6 P(X) 0 4 5 To Be solved later.. Q. 5. A retired person wants to invest an amount of Rs. 50,000 his broker recommends investing in two types of bonds. A and B yielding 0% and 9% return respectively on the invested amount. He decides to invest at least Rs. 0, 000 in bond A and at least R.s 0, 000 in bond B. he also wants to invest at least as much in bond A as in bond B solve this linear programming problem graphically to maimize his returns. Q.5. Let his investment on Bond ' A' Rs. Bond ' B ' Rs. Y A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 5
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH y 50, 000 0, 000 y 0, 000 0 9 z y 00 00 Q. 6. Find the equation of the plane which contains the line of intersection of the planes. r. (i-j+k) -4= and r. (-i +j+k)+5=0 And whose intercept on - ais is equal to that of on Y-ais. y z 4 Sol. 6 y z 5 equation of intersection of planes y z 4 ( y z 5) 0 y z 4 5 y z 4 5 4 5 4 5 4 5 4 5 A ONE INSTITUTE OF COMPETITIONS, PH 9876608, 9876464 Page 6