MINIMUM PROGRAMME FOR AISSCE

KENDRIYA VIDYALAYA DANAPUR CANTT MINIMUM PROGRAMME FOR AISSCE 8 SUBJECT : MATHEMATICS (CLASS XII) Prepared By : K. N. P. Singh Vice-Principal K.V. Danapur Cantt

() MATRIX. Find X and Y if 7 y & y 7 8 X,, 6. If find X (Ans. X = [ ]. Find such that. If A If A 5 & B 7 5 Epress the Matri 6 8 as the sum of a symmetric and skew symmetric 6 5 5 5 (Ans. = or ) n n V n N n n then Prove that A n verify that (AB)' = B'A' Matri. Using elementary row transformation find the inverse of Matri 5 7 5 Ans. 8 5 Show that A satisfies the equation + =. Hence find A 5 8 Ans. 8 A If show that A (adja) = (adj A) A = A I. Using Matri method, solve the following system of linear equations : y z y z y z (Ans. =, y = & z =

() DETERMINANT.. Evaluate a b b c c a b c c a a b c a a b b c Using properties of determinant prove that : (i) (ii) (iii) (iv) y y z yz y y z z y z a b c a a b b c a b a b c c c c a b a b c abc a b c a b ab b ab a b a a b b a a b (vii) b c c a a b a b c q r r p p q p q r y z y z z y (v) (vi) z a b c a b b c c a a b c a b c (viii) (i) If, y, z are different and A y z y y, z z then show that + yz =

() RELATION & FUNCTIONS. Prove that the relation R on the set N N defined by (a, b) R (c, d) a + d = b + c V (a, b), (c, d) N N is an equivalance relation.. Consider the set N N, the set of all ordired pairs of natural numbers. Let R be the relation in N N which is defined by (a, b) R (c, d) if and only if ad = bc. Prove that R is an equivalence relation.. Prove that the relation R on set Z of all integers defined by (a, b) R (a b) is divisible by 5 is an equivalence relation on Z.. Show that the function f : R R given by f () = a + b, a, b r, a o is a bijection. Show that the function f : R { } R {} given that f is a bi jective function. Let f : R+ [-5, ] given by f () = 9 + 6 Show that f is invertible and f y y 6 Let A be the set of all real numbers ecept Let an operation * be defined on A as a * b = a + b + ab V a, b A Prove that (i) A is closed under the given operation (ii) * is commutative as well as associative (iii) the number O is the identity element (iv) every element a A has a as inverse a 7 7 Let f : R R be defined as f and g : R R 5 5 5 7 5 5 be defined as g 7 5 Show that fog = IA and gof = IB where IA and IB are idnetity functions on A and B respectively.

(5) INVERSE TRIGONOMETRIC FUNCTIONS 6. Find the value of tan cos sin. Find the value of sec tan cos ec cot. tan Prove that tan tan 7 7. Prove that tan + tan + tan = Prove that tan tan 7 Prove that tan tan tan tan 5 7 8 a a b Prove that tan cos tan cos b b a sin sin Prove that cot, ], [ sin sin tan Prove that cos. If cos + cos y + cos z = Prove that + y + z + yz =. If tan + tan y + tan z = Prove that + y + z = yz. (a) Solve : tan tan (b) Solve : tan (cos ) = tan ( cosec ) tan. Solve : tan. Write Write tan tan in the simplest form. cos in the simplest form. sin (Ans. 5) 6

(6) CONTINUITY & DIFFERENTIABILITY., A function f is defined by f Find whether f is continuous at =. (Ans. continuous) A function f is defined by f 5 Show that the function is discontinuous at =.. cos, Discuss the continuity of the function f at =,. if if For what value of k is the function f k if continuous at =? (Ans. k = 5) k cos Find the value of k if f () is continuous at, where f,, (Ans. k = 6) a b if If the function f if 5a b if is continuous at =, find the values of a & b. (Ans. a = & b = ) cos f For what value of k, 8 k Prove that f () = is continuous at = but not differentiable at =,, is continuous at =? (Ans. k = )

(7) DIFFERENTIATION Find the derivative w.r. to of the following :. cos sin tan cos sin. a cos b sin tan b cos a sin. tan. tan 5 sin If If sin y = sin (a + y) Prove that If a sin If y = tan show that. If y log a Prove that a. If y. If y = A cos n + B sin n Prove that. If y = a cos (log ) + b sin (log ) prove that. If y = sin (m sin ) show that ( ) y y + my = If y = sin + (sin) find (Ans. ) sin sin y cot sin sin t cos and y a If (Ans. ) y... Prove that t n is independent of. sin a y sin a Prove that y d y d y Prove that ( + ) y + y n y = d y n y, then prove that y y log d y y

(8) APPLICATION OF DERIVATIVES. Verify Lagrange's Mean Value Theorem for the function f () = ( ) ( ) ( ) in the interval [, ] (Ans. ). Verify Rolle's Theorem for the function f sin cos,,. Ans. c A particle moves along the curve 6y = +. Find the points on the curve at which y co-ordinate is changing 8 times as fast as the -co-ordinate. Ans. (, ) &,. Find the intervals in which the function f () = + 6 + 7 is (a) increasing (b) decreasing Ans. (Increasing in (, ) U (6, ) decreasing in (, 6) Find the intervals in which the function f given by f () = sin cos (i) is increasing (ii) is decreasing (Ans. Increasing when 7 7 Decreasing when or ) Find the intervals in which the function f () = 7 + is increasing or decreasing (Ans. (i) & < or > ) At what points on the curve + y y + =, is the tangent parallel to y-ais? (Ans. (, ) & (, ) Prove that the curves = y & y = k cut at right angles if 8k =. Find the Co-ordinates of the points on the curve y = 6 + where the tangents are to -ais. (Ans. (, ) & (, 7)). Find the equation of the tangent & normal to the curve. = cos, y = sin at Equation of normal y Ans. Equation of tan gent y

(9). Using differentials, find the approimate value of (i).7 (Ans..95) (ii).8 (Ans..69). Find the approimate value of f (.), when f () = + 5 +. If y = and if changes from to.97, using differential, find the approimate change in y. (Ans..96). Find the points of local maima or local minima for the function f () = sin cos, < <. (Ans. ma value = at (Ans. ) 7, min value = at Show that the rectangle of ma. area that can be inscribed in a circle of radius r is a square of side r.. Find the point on the curve y = which is nearest to the point (, 8). (Ans. (, y) Two sides of a triangle have lengths 'a' & 'b' and the angle between them is. What value of will maimize the area of the triangle? Find the ma. area of the triangle also. (Ans. = /, ma area = ½ ab A wire of length 5 m is to be cut into two pieces. One of the two pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square & the circle is minimum? ( Ans. 5 m& m) An open 6, with a square base is to be made out of a given quantity of metal c sheet of area c. Show that the ma volume is. 6. A window is in the form of rectangle surmounted by a Semi circle. If the total perimeter of the window is m, find the dimensions of the window so that ma. m light in admitted.. Show that the height of the cylinder of ma. volume that can be inscribed in a sphere of radius R is R.

() INTEGRATION Evaluate :. sin sin sin e a sin b. e a cos b. tan. cot. tan cot. 5 sin. cos. cos cos. sin. cos tan sin sin tan log. sec sin sin..... cos cos cos cos e sin sin. tan cos e. n tan cos

() INTEGRATION AS A LIMIT OF SUM Evaluate following integrals as the limit of sums : 5.... 5 e 5.

() DEFINITE INTEGRALS. sin. sin. 5 tan cot Ans.. 5 sin. tan sin 5 sin cos5 log tan d 6 tan log sin log 8 log log sin log sin. 9 log sin cos sin cos Ans.. a cos b sin. sin cos ab

() AREA OF THE BOUNDED REGION y. 9. Find the area of the region bounded by the ellipse (Ans. 6 sq unit). Find the area of the region included between the Parabolas y = a & = ay. 6a sq unit sq unit 6. Find the area of the region {(, y) y }. Find the area of the region {(, y) + y + y} Find the area of the region enclosed between the two circles + y = and ( ) + y = sq unit sq unit sq unit Find the area of the region {(, y) : y } Find the area enclosed by the Parabola = y and the straight line = y 9 sq unit 8 Find the area of the region enclosed between the circles + y = and ( ) + y = Find the area of the smaller region bounded by the ellipse line. y a b. y & the a b ab sq unit Using integration, find the area of the triangle whose vertices are (, ), (, ) & (, ). 8 sq unit sq unit Using integration, find the area of ABC when A is (, ), B is (, 7) & C is (6, ) (Ans. 9 sq unit) Find the area of the region bounded by the lines + y =, y = & + y 7 = (Ans. 6 sq unit)

() DIFFERENTIAL EQUATION. m sin Verfiy that y e is a solution of the differential equation d y m y. F o r m t h e d i f f e r e n t i a l e q u a l c o r r e s p o n d i n g t o ( a ) a & b. + (y b) = r by eliminating (Ans. (+y) = r (y)) a y. Solve the differential equation y. Solve the differential equation ( + e) + e ( + y) = Given y =, when = Solve the differential equation Solve the differential equation y y y Solve the differential equation e e y Solve the differential equation y y Solve the differential equation y y. Solve the differential equation + (y + y) = given that y = when =. Solve the differential equation ( + y) = (tan y ). Solve the differential equation y. Solve the differential equation. Solve the differential equation Solve the differential equation Solve the differential equation cos y Given y = when = y y given y = when = y tan y y y log y sin given that y = when =

(5) VECTORS & DIMENSIONAL GEOMETRY. If a i ˆj k, b i k & c i ˆj find the value of such that a c is to b.. Find a unit vector in the direction of the sum of the vectors a i j k & b i j k i j k. Find the value of for which the vectors a i ˆj k & b i ˆj 8k are. parallel. (Ans. = 6) If a i ˆj k & b ˆj k, find the vector c such that a c b & a.c 5 Ans. c i ˆj k If a 5, b & a b 5, find a.b (Ans. 6) Find the area of the gm whose adjacent sides are determined by the vectors (Ans. 5 sq unit) a i ˆj k & b i 7ˆj k If a, b & a.b, find a b (Ans. 6). Show that the points (,, ), (, 5, 6) & (,, ) are collinear. If a b c Prove that a b b c c a Prove that a b b c c a a b c. Find the foot of the r from the point (,, ) on the line y z,. Find the shortest distance between the lines r i ˆj k i ˆj k & r i ˆj k i ˆj k. unit Find the shortest distance between the following lines Y 5 z 7 Y z & 7 6 Ans. 9 unit

(6). Find the equation of the plane Passing through the points (,, ), (,, ) & (,, ) (Ans. y + z + 5 = ) Find the equation of the plane passing through the points (,, ) & (,, ) and r to the plane y + = (Ans. + 5y + z = ) Find the equation of the plane passing through the points A (,, ), B (,, 5) & C (,, ) in vector form. Also epress it in Cartesian form. The Cartesian equations of a line are + = 6y = z. Find the fied point through which it passes, its direction rations and also its vector equation. Ans.,, ;,, 6 r i j k i ˆj 6 k Find the value of so that lines 7 y z 7 7 y 5 6 z & 5 7 are at right angle. y z. Also, write the equation of the line Joining the given point and its image and find the length of Find the image of the point (, 6, ) in the line the segment joining the given point and its image. y 6 z,, 7,, 6 y Ans.. Find the angle between the planes r. i j k 6 & r. i j k 5. Find the image of the point (,, ) in the plane y + z = (Ans.,, ). Prove that the lines. Find the equation of the plane passing through the intersection of the planes r. i ˆj k 6 & r. i ˆj k 5 and the point (,, ) y z 5 y z 6 & are coplanar. Also 5 7 7 find the plane containing these two lines. Ans. r. i ˆj 6 k 69

(7) LINEAR PROGRAMMING. Solve the following Linear Programming Problem graphically Maimize z = 5 + 5 y Subject to 5 + y + y 6, y (Ans. 5). A diet is to contain at least 8 units of vitamin A and units of minerals. Two foods F & F are available. Food F costs Rs. per unit and F costs Rs. 6 per unit, one unit of food F contains units of vitamin A and units of minerals. One unit of food F contains 6 units of vitamin A and units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of miture of these foods & also meets the mineral nutritional requirements. (Ans. Rs. ). One kind of cake requires gm of flour and 5 gm of fat, another kind of cake requires 5 gm of flour & gm of fat. Find the ma. number of cakes which can be made from 5 kg of flour and 6 gm of fat, assuming that there is no shortage of the other ingradients used in making the cakes. Make it as an LPP and solve it graphically. (Ans., ). There are two types of fertilisers 'A' and 'B' A' consists of % Nitrogen & 5% phosphoric acid where as 'B' consists of % nitrogen and 5% phospheric acid. After testing the soil conditions, farmers finds that he needs at least kg of nitrogen & kg of phosphoric acid for his crops. It 'A' costs Rs. per kg and 'B' cost Rs. 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. (Ans. A. kg, B :. kg) Solved the following linear programming problem graphically Maimize z = 6 + 5y Subject to constraints + y 5 + y 9, y (Ans. =, y =, z = 65)

(8) A farmer mies two brands P and Q of cattle feed. Brand P, costing Rs. 5 per bag, contains units of nutritional element A,.5 units of element B and units of element C whereas brand Q costing Rs. per bag contains.5 units of nutritional element A,.5 units of element B and units of element C. The minimum requirements of nutrients A, B and C are 8 units, 5 units and units respectively. Determine the number of bags of each brand which should be mied in order to produce a miture having a minimum cost per bag? What is the minimum cost of the miture per bag? (Ans. Rs. 95 at (, 6) An aeroplane can carry a maimum of passengers. A profit of Rs. is made on each eclusive class ticket and a profit of Rs. 6 is made on each economy class ticket. The airline reserves at least seats for eecutive class. However at least times as many passengers prefer to travel by economy class than by the eecutive class. Determine how many tickets of each type must be sold in order to maimise the profit for the airline. What is the maimum profit? (Ans. Ma profit Rs. 6 at (,6) Two godowns A and B have grain capacity of quintals and 5 qunitals respectively. They supply to ration shops D, E and F, whose requirements are 6, 5 and quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table : How should the supplies be transported in order that the transportation cost in minimum. What is the minimum cost? Transportation cost per quintal (in Rs.) From/To A B D 6 E F.5 (Ans. : Minimum cost Rs. 5 at (, 5))

(9) PROBABILITY. If A & B are two events such that P A, P B (i). P (A/B) (ii) P (B/A) (iii) P A / B & P A B (iv) P A /B Find 5,,, 8 A problem in Mathematics is given to students whose chances of solving it are,,. What is the probability that te problem is solved?. Two persons A & B throw a die alternately till one of them gets a 'three' and wins the game. Find their respectively probabilities of winning, if A begins.. Three persons A, B, C throw a die in succession till one gets a 'si'and wins the game. Find their respective probabilities of winning, if A begins. 6 5, 6 5, & 9 9 9 A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is and that of wife's selection is. 7 5 What is the probability that (i) both of them will be selected? (ii) only one of them will be selected? (iii) none of them will be selected? 5 7 5 In a bolt factory, machines A, B and C manufacture respectively 5%, 5% and % of the total bolts of their output 5, and percent are respectively defective bolts. A bold is drawn at random from the product. If the bolt drawn is found to be defective, what is the probability that it is manufactured by the machine B? 8 69 A Insurance company insured scooter drivers, car drivers and 6 truck drivers. The probabilities of an accident involving a scooter driver, car driver & a truck driver are.,. and.5 respectively. One of the insured person meet with an accident. What is the probability that he is a scooter driver? 5

() A card from a pack of 5 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be hearts. Find the probability of missing card 5 to be a heart. A man is known to speak truth out of times. He throws a die and reports that 8 it is a si. Find the probability that it is actually a si.. There are three coins. One is two headed coin, another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is choosen at random and tossed, it shows heads, what is the probability 9 that it was the two headed coin?. Three urns A, B and C contain 6 red an white ; red and 6 white; and red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from 6 6 urn A.. A random variable X has the following probability distribution : X : 5 6 7 P () : k k k k k k 7k+k Find each of the following : (i) k (ii) P ( < 6) (iii) P ( 6) (iv) P (< <5) 8 9,,, 5. Find the mean & variance of the number of tails in three tosses of a coin. Mean.5, var