Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS XII - ISC Board MATHEMATICS - QP + SOLUTIONS Date: 6..8 Ma. Marks : Question SECTION - A (8 Marks) (i) The binary operation *: R R R is defined as a* b a b Find (*)*4. Given a* b a b (*)*4 (4 )*4 7*4 4 4 8 Topic: Relation & Function_Subtopic:Relations_ Level: _ISC Board / Mathematics (ii) 5 a If A b and A is symmetric matri, show that a b 5 a A b given A is symmetric matri A A T 5 a 5 a 5 b b b a T a b Topic: Algebra_Subtopic: Matri_ Level: _ISC Board / Mathematics (iii) Solve : tan cot tan cot tan tan cot tan tan
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions tan ( ) 4 tan 4 Topic: Relation & Function_Subtopic:ITF_ Level: _ISC Board / Mathematics (iv) Without epanding at any stage, find the value of: a b c a b y c z y z a b c a b y c z y z R / R R, R / R R a b y c z a b y c z y z R R Topic: Algebra_Subtopic: Determinant_ Level:_ISC Board / Mathematics (v) Find the value of constant k so that the function f ( ) defined as:, f ( ) k, is continuous at. Given f ( ) is continuous at f ( ) lim f ( ) k lim ( )( ) lim ( ) 4 K 4 Topic: Calculas_Subtopic:Differentiation_ Level: _ISC Board / Mathematics
(vi) Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions Find the approimate change in the volume V of a cube of side metres caused by decreasing the side by %. Volume of a cube V dv d Change in volume dv V d V V ( ) changein volume decrease by % Topic:Calculas_Subtopic: Rate measure_ Level: _ISC Board / Mathematics (vii) Evaluate : I 5 4 d. 5 4 d 5 4 d 4 d 5 5 4log c Topic: Calculas_Subtopic: Integration_ Level: _ISC Board / Mathematics (viii) Find the differential equation of the family of concentric circles y a Finally of concentric circles is y a Differential w.r.t. dy y d dy y d Topic: Calculas_Subtopic:Differentail Equation_ Level: _ISC Board / Mathematics
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions (i) If A and B are events such that P( A), P( B) and (a) P( A/ B) (b) P( B / A) P( A) P( B) P( A B) 4 P( A B ) 4 P( A / B) P( B) 4 P( A B), then find: 4 P( A B ) 4 P( B / A) P( A) Topic: Probabiltiy_Subtopic: Conditional_ Level: _ISC Board / Mathematics () In a race, the probabilities of A and B winning the race are and 6 respectively. Find the probability of neither of them winning the race. Let A win the race be E B win the race be E P( E ), P( E) 6 P( E E) P( E) P( E) P( E ) P( E ) 6 5 5 6 9 Topic: Probabiltiy_Subtopic: Independent Event_ Level: _ISC Board / Mathematics Question If the function f ( ) is invertible then find its inverse. Hence prove that ( fof )( ) Let y y y f ( ) 4 4
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions Now, L H S fof f f.. ( ) [ ( )] f ( ) fof ( ) Topic: Relation & Function_Subtopic:Function_ Level: _ISC Board / Mathematics Question If tan a tan b tan c, prove that a b c abc. tan a tan b tan c tan b tan c tan ( a) tan b c tan bc b c bc b c bc b c a bc tan( tan a) tan(tan a) b c a abc a a b c abc Topic:Relation & Function_Subtopic: ITF_ Level: _ISC Board / Mathematics Question 4 Use properties of determinants to solve for : a b c c b a and. a b c a b c c b a and. a b c C / C ( C C ) a b c b c a b c b a a b c b c 5 5
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions ( a b c) b a R R R b c b c ( a b c) b a b c ( a b c) ( b b) ( a b c) ( ) or a b c but ( a b c) Topic: Algebra_Subtopic:Determinant_ Level: _ISC Board / Mathematics Question 5 (a) Show that the function Continuity at f lim f ( ) lim lim f ( ) lim, f ( ), f ( ) lim f ( ) lim f f ( ) is continuous at Now differentiable at f ( ) f () ( R. H. D. at ) lim lim ( ) lim ( ) is continuous at but not differentiable. 6 6
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions f ( ) f () ( L. H. D. at ) lim L. H. D. R. H. D. lim f ( ) is not differentiable at Topic: Calculas_Subtopic:Differentiation_ Level: _ISC Board / Mathematics (b) OR Verify Rolle s theorem for the following function: f ( ) e sin on [, ] f ( ) e sin (i) f ( ) is continuous on [, ] because (ii) e and sin is differentiable on (, ) (iii) f () e sin f ( ) e sin (iv) Let c be number such that f ( c) f ( ) e cos sin e ( ) c f ( c) e (cos c sin c) f ( c) c e (cos c sin c) c e cos C sin C tan C 5 C,,,... 4 4 4 e &sin are continuous function on its domain. [, ] 4 Rolle s theorem verify Topic: Calculas_Subtopic: AOD Level: _ISC Board / Mathematics Question 6 If tan log, prove that d y dy y a a d d tan log y a 7 7
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions log y tan a differentiating both sides w.r.t. y e dy d a tan a a tan e dy ay d Again differentiating both sides w.r.t. d y dy a dy d d d d y dy a d d Topic: Calculas_Subtopic:Differentiation_ Level: _ISC Board / Mathematics Question 7 Evaluate : tan I tan d d Put t d dt d dt d t dt I t tan t dt t t I tan t dt t t t I tan t dt t t I t tan t t tan t c I t tan t t tan t c I tan c Topic: Calculas_Subtopic:Integration_ Level: _ISC Board / Mathematics Question 8 (a) Find the points on the curve y 4 5 at which the equation of the tangent is parallel to the -ais. y 4 5...(i) 8 8
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions dy d Given that lines is parallel to -a is dy d 4 Put in equation (i) then, y 4 5 4 Point, 4 when then y 4 5 6 Points, y, 4 amd,6 Topic: Calculas_Subtopic: AOD_ Level: _ISC Board / Mathematics OR (b) Water is dripping out from a conial funnel of semi-verticle angle 4 at the uniform rate of cm /sec in the surface, through a tiny hole at the verte of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water. Let r be the radius, h be the height and V be the volume of the funnel at any time t. V r h Let I be the slant height of the funnel Given : Semi-vertical angle = 45 in the triangle ADE : DE r sin 45 AE l AD h cos 45 AE l...(i) 9 9 B D A 45 E C
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions r and h...(ii) therefore the equation (i) can be rewritten as : I I V I V I 6 Differentiate w.r.t. t : dv dt dv dt l 6 l dl dt dl dt...(iii) dl dv...(iv) dt l dt Since it is given that rate of change (decrease) of volume of water w.r.t. t is dv dt therefore cm / sec dl 4 dt l l dl dt at l 4 4 cm / sec 4 4 Topic: Calculas_Subtopic: AOD_ Level:_ISC Board / Mathematics Question 9 (a) Solve : sin dy y sin tan d dy y cosec tan d dy y cosec tan d...(i) Compare dy Py Q d P cosec, Q tan I.F. e Pd I.F. e cosec d
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions loge cosec cot I.F. e loge cosec cot I.F. e I.F. cosec cot cosec cot solution of the linear differential equation y. IF. Q. IF. d cos y cosec cot tan d sin sin cos y cosec cot d d cos sin cos y cosec cot c Topic:Calculas_Subtopic: Differential Equations_ Level: _ISC Board / Mathematics (b) OR The population of a town grows at the rate of % per year. Using differential equation, find how long will it take for the population to grow 4 times. Here d dt [Since increase in population speeds up with increase in population] and let be the population at anytime t. d r dt (where r is proportionality constant) d r dt integrating both sides ln rt c, (where c is the integration constant) rt c e K e rt, where K e c Here r is the rate of increase and K is the initial population let then t = ke k o Given to find the time t taken to attain 4 times population, so 4 So, K e 4 rt e.t.5t e Taking log on both sides ln ln e.t.694.t t = 6.94 Topic:Calculas_Subtopic: Differential Equations_ Level:_ISC Board / Mathematics
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions Question (a) Using matrices, solve the following system of equations : (a) y 5z y 4z 5 y z Using this let us solve the system of given equation y 5 y 4z 5 y z This can be written in the form AX = B 5 4 y 5 z where 5 A 4 X y B 5 z we know A adj A A A 4 4 5 4 4 6 4 5 6 5 Hence it is a non - singular matri Therefore A eists Let us find the (adj A) by finding the minors and cofactors M M 4 4 4 4 6 4 M M 5 6 5 5 M 4 5 9
M M Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions 5 5 4 5 M 8 5 4 M 4 9 A A A A A 9 A 5 A A A 9 9 5 5 A We know AX = B, then X = A B y 9 5 Therefore z 5 Matri multiplication can be done by multiplying the rows of matri A with the column of matri B. 5 6 y 45 69 Therefore, z 5 9 Hence =, y= and z = Topic: Algebra_Subtopic:Matrices_ Level: _ISC Board / Mathematics OR (b) Using elementary tranformation, find the inverse of the matri : Let A A [ ] [ 9] [ 6]
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions 8 ( 5) 8 A eists. A IA A R R A R / R R A 5 8 R / R 5R A 5 R / R R 4 A 5 R / R R and R 8 5 4 A 5 R / R R 4 4
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions 4 A 5 8 6 5 A Topic: Algebra_Subtopic: Matrices_ Level: _ISC Board / Mathematics Question A speaks truth in 6% of the cases, while B is 4% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? 6 4 A speaks truth P A, P A' 4 6 B speaks truth PB, PB ' they contradict each other P A B' P A' PB 6 6 4 4 6 6 5 5 5 % of cases they likely to contradict each other 5% Topic: Probability_Subtopic: Probability_ Level: _ISC Board / Mathematics Question A cone is inscribed in a sphere of radius cm. If the volume of the cone is maimum, find its height. Let VAB be a cone of greatest volume inscribed in a sphere of radius. It is obvious that for maimum volume the ais of the cone must be along a diameter of the sphere. Let VC be the ais of the cone and O be the centre of the sphere such that OC =. Then, VC VO OC R height of cone Applying Pythagoras theorem, OA AC OC AC 5 5
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions AC 44 Let V be the volume of the cone, then V AC VC 44 V R 78 44...(i) dv 44 4 d A R= O C B d V d 4 6 6 4 4 dv d Now, gives 44 4 i.e., 44 4 i.e., 8 48 i.e., 4 i.e., or 4 d d V 4 4 4 6 Thus, V is maimum when = 4 Putting = 4 in (), we obtain Height of the cone = R 4 6cm Topic: Calculas_Subtopic:AOD_ Level: _ISC Board / Mathematics Question (a) Evaluate : d Let I d d d A ( ) B A( ) B A A A B B B 6 6
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions ( ) d d I d d ( ) d d log C log C Topic: Calculas_Subtopic: Integration_ Level: _ISC Board / Mathematics OR (b) Evaluate : I / / cos d sin cos cos d sin cos...() Using I / a a f ( ) d f ( a ) d cos d sin cos / sin d cos sin...() Adding eq. () & () I / / cos sin d sin cos sin cos d / sec sec tan d 7 7
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions I / sec d tan tan Put tan t, sec d dt when, t when, t I dt t t. 4 4 dt t t tan t tan I I 6 6 6 Topic: Calculas_Subtopic: Definite Integral_ Level: _ISC Board / Mathematics Question 4 From a lot of 6 items containing defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find : (a) The probability distribution of X (b) Mean of X (c) Variance of X In 6 items defective and 4 non-defective Let P is the probability of defective items Let number of defective items,, 8 8
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions C 4 4 P 6 C4 4 6 C4 5 C C 8 P 5 C C 6 P 5 4 6 C4 X P( ) P( ) P( ) 5 8 5 6 5 8 5 5 8 5 4 5 (b) X Mean Pi X i 4 5 (c) Variance P i X P i i X i 4 6.5 5 45 Topic: Probabiltiy_Subtopic:Probability distribution_ Level: _ISC Board / Mathematics SECTION - B ( Marks) Question 5 (a) Find if the scalar projection of a iˆ ˆj 4 kˆ on b iˆ 6 ˆj kˆ a b Projection of a on b is 4 b i ˆ ˆj 4 kˆ i ˆ 6 ˆj kˆ 6 6 4 49 4 4 is 4 units. 9 9
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions 8 4 7 8 8 5 Topic: Vector_Subtopic: Dot Product_ Level: _ISC Board / Mathematics (b) The Cartesian equation of line is : y 5 6z. Find the vector equation of a line passing through (7, 5, ) and parallel to the given line. Cartesian equation of a line is y 5 6z 5 i.e., y 6 z 6 Dividing by 6 throughout i.e., 5 y 6 D.r.s of the above line is,, Now, equation of a line passing through point (7, 5, ) and parallel to the above line whose d.r.s. is,, is r 7iˆ 5 ˆj iˆ ˆj kˆ r 7iˆ 5 ˆj i ˆ ˆj kˆ Topic: D geometry_subtopic: Line_ Level: _ISC Board / Mathematics (c) Find the equation of the plane through the intersection of the planes r iˆ ˆj kˆ 9 and r iˆ ˆj k and passing through the origin. Equation of I plane is r i ˆ ˆj kˆ i.e., iˆ yj ˆ zkˆ i ˆ ˆj kˆ 9 y z 9 9 y z 9...(i) Equation of II plane is r iˆ ˆj kˆ i.e., iˆ yj ˆ zkˆ iˆ ˆj kˆ i.e., y z i.e., y z...(ii) Now, equation of a plane passing through intersection of given planes is y z y z 9 y z 9
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions Since plane is passing through the origin (,, ) 9 9 Topic: D geometry_subtopic: Plane_ Level: _ISC Board / Mathematics Question 6 (a) If A, B, C are three non- collinear points with position vectors a, b, c, respectively, then show that a b b c b a the length of the perpendicular from Con AB is b a Let ABC be a triangle and let a, b, c be the position vectors of its vertices A, B, C respectively. Let CM be the perpendicular from C on AB. Then, Area of ABC AB CM AB CM Also, Area of ABC AB AC Area of ABC a b b c c a CC AB CM abbcca a b b c c a C M AB C M a b b c c a b a Aa Topic: Vector_Subtopic: Cross Product_ Level: _ISC Board / Mathematics (b) OR Show that the four points A,B, C and D with position vectors 4i 5j k, j k, i 9j 4k and a 4i 5j k b i j c i 9j 4k d 4i 4j 4k AB b a 4i 6j k 4 i j k M Bb respectively, are coplanar.
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions AC c a i 4j k AD d a 8i j k AB, AC & AD are coplanar if AB AC AD 4 6 4 8 4 6 4 45 6 66 6 6 66 = AB, AC & AD are coplanar Points A, B, C and D are coplanar i.e., AB AC AD. Topic: Vector_Subtopic:Scalar Triple Product_ Level: _ISC Board / Mathematics Question 7 (a) Draw a rough sketch of the curve and find the area of the region bounded by curve y = 8 andtheline =. Given equation is y 8 Comparing with y 4a We get 4a 8 i. e. a Given y y 8 4 y 8, S,,4 y 8 Also, meets y 8 y 6 y 4,4 and, 4 are their point of intersection., 4 Required are A 8 d / 8 d
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions / 4 / 8 / 8 8 8 sq.units Topic: Calculas_Subtopic: AODI_ Level: _ISC Board / Mathematics OR (b) Sketch the graph of y = + 4. Using integration, find the area of the region bounded by the curve y = + 4 and = 6 and =. y 4, if 4 & y 4, if 4 y = - - 4 = -6 = y (,4) y 4 ( 4,) For y 4 For y 4, when, y 4 when, y 4 & when y, 4 when y, 4 Points are Point are, 4 & 4,, 4 & 4, Required area 4 4 4 d d 6 4 4 4 4 6 4 4 6 4 44 46 44 4
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions 6 6 6 6 4 6 8 6 8 8 sq. unit Topic: Calculas_Subtopic: AODI_ Level: _ISC Board / Mathematics Question 8 Find the image of a point having position vector : i j k in the plane r. i j 4k. 4 Let B be root of pont r i j k i j k y z 4 A i j k, y, z 4 subtitute, y & z in plane y 4z 4 4 9 9 6 4 6, y, z, y, z by midpt formula, A B M (,y,z ) y z,, z in the plane r i j 4k, y, z z z Topic: D Geometry_Subtopic: Plane Level: _ISC Board / Mathematics *** can of AB : is 4 4