INDIAN SCHOO MUSCAT QUESTION BANK DEPARTMENT OF MATHEMATICS SENIOR SECTION. Relations and Functions

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1 INDIAN SCHOO MUSCAT QUESTION BANK 07-8 DEPARTMENT OF MATHEMATICS SENIOR SECTION Relations and Functions mark (For conceptual practice) Which of the following graphs of relations defines a transitive relation on A = {,, 3, 4 }? R = { (, ), ( 3, 4 ), (, 3 ), (, 4) } R = { (, ), ( 3, 4 ), (, 4) } If f : R R is given by f() = 3 +, find f ( ) 3 Let f() =, then what is the value of ( fofof)() 4 Let f() =, what is the value of such that f(f()) =? 5 Let be a binary operation on N defined by a b = a + b + 0 for all a, b N write the identity element for in N 6 ab Let be a binary operation defined on Q + by a b = for all a b 3 Q + What is the inverse of 4 6? 7 If f : R R is given by f() = 3 5 Find f ( ) 4 MARK QUESTIONS 8 Prove that the function R on the set Z of all integers defined by: {(, y ) R y is divisible by 4}, is an equivalence relation 9 Discuss the commutativity and the associativity of the binary ab operation on R defined by a b =, a, b R 4 0 Let L be the set of lines in the XY plane and R be a relation in L defined as R = { (L, L) : L is parallel to L} Show that R is an equivalence relation Find the set of all lines related to the line y = +4 Show that the function f : Q Q defined by f() = 3 + 5, for all Q

2 is one and onto Hence or otherwise, find f Let be a binary operation on N set of, the set of natural numbers defined by a b = a b, a,b N Show that is neither commutative nor associative 3 The binary operations and o on R defined by a b = a b and a o b = a Show that i) is commutative but not associative ii) o is associative but not commutative iii) is distributive over o 4 n,if n is even Let f : W W, be defined by f(n) = n,if n is odd Show that f is invertible and f = f 5 Let A be the set of all numbers ecept and o be an operation defined on A by aob = a + b + ab, a, b A Prove that i) A is closed under the given operation ii) o is commutative as well as associative iii) the number 0 is an identity element iv) a each element a of a has as its inverse a 6 Marks Questions 6 Consider f : R+ [ 5, ) given by f() = Show that f is 7 invertible with 6 ( ) f 3 Show that f : R { } R { } given by f() = Also find f is invertible 8 Let A = Q Q and let be a binary operation defined on A by ( a, b ) ( c, d ) = ( ac, ad + b ) i) Is commutative? ii) Is associative? iii) Find the identity element of ( A, ), if eists iv) Find the invertible elements of ( A, ) 9 Let X be a non empty set P(X) be its power set Let be a binary operation defined on P(X) by A B = A B for all A, B P(X) then prove that i) is a binary operation on P(X)

3 ii) Is associative? iii) Is commutative? iv) Find the identity element in P(X) wrt v) Find all the invertible elements of P(X) If o is another binary operation defined on P(X) by AoB = A B, then prove that is distributive over o 0 Let A = N N and let be a binary operation defined on A by ( a, b ) ( c, d ) = ( ad + bc, bd ) i) Show that is associative ii) Is commutative? iii) Show that ( A, ) has no identity element INVERSE TRIGONOMETRIC FUNCTIONS marks Write the principal value of cos (cos(680 o )) If sin (sin 5 + cos ) =, then find the value of 3 Find the value of cot ( π cot 3) 4 Write the value of cos ( ) + sin ( ) 5 Using principal values, Write the value of [cos ( ) + sin ( )] 6 Evaluate: cot [ cos ( sin )] 7 Write cot ( ), > in the simplest form 8 If tan + tan y = π, y < then write the value of + y + y 4 9 Find the value of sec (tan y ) 0 If sin + sin y = π 3,then find the value of cos + cos y Marks Write in simplest form : tan [ cos +sin ], [ π, π ] Show that tan tan 3 5 tan 8 9 = π 4 3 Sow that tan y 4 Evaluate 4 tan 5 tan y +y = π 4 5 Show that tan = tan tan ( 3a 3 a 3 3a 3) = 3 tan ( ) a 7 Prove that : sin + cos = π, if [,]

4 8 Solve the equation tan ( + ) = tan 9 Write in the simplest form sin [ + ], < < a 0 If sin + +a sin b = +b tan then show that = a+b ab 4Marks Solve the equation for : sin + sin ( ) = cos Prove that tan +cos + cos ( ) = π +cos cos 4 3 If sin(cot ( + )) = cos(tan ), then find 4 Prove that cos(tan {sin(cot )}) = Show that tan ( sin 3 4 ) = Find the greatest and least values of (sin ) + (cos ) 7 Solve for :cos[tan ()] = sin [cot 3 4 ] 8 Simplify:tan a cos b sin [ ], if a tan > b cos +a sin b 9 Find the value of the epression sin ( tan 3 ) + cos(tan ) 0 Solve for : tan (sin) = tan ( sec ), π

5 MATRICES AND DETERMINANTS One mark questions If matri A = [ ] and A = k A, then write the value of k Construct a matri, A = [aij], whose elements are given by aij (i+j) = Find from the matri [ ] [ ]= [5 6 ] If A = [ 4 3 ], find B such that A B +3I = O Find the number of all possible matrices of order, with each entry as or Using determinants, find the area of triangle whose vertices are (,7), (,) and (0,8) For what value of, the matri [ 6 4 ] is a singular matri 3 + If = 4, then write the value of If A is a non-singular matri and A = I, then find A - If the determinant of the matri A, of the order 3 3, is 4 then find the value of 3A Two mark questions 3 4 If A and B are symmetric matrices then show that BA AB is neither symmetric nor skew symmetric Epress the matri A = [ 3 ] as sum of symmetric and skew symmetric 4 5 matrices If A = [ 3 4 ], then show that (A A ) is a skew symmetric matri k 0 For what value of k, the matri [ 3 ] is singular? 4 5 Find the inverse of matri [ cos sin sin cos ] Using elementary row transformation find the inverse of matri [ ] Show that the points (a + 5, a 4 ), (a, a + 3 ) and (a, a) do not lie on a straight line for any value of a

6 a 3 If matri [ b ] is skew symmetric, then find the values of a, b and c c Solve the matri equation [ ] [ 0 ] [ ] = O 0 If A = diag [, ], B = [ y 0 ], C = [5 6 ] and A + B = C, find and y 8 FOUR MARK QUESTIONS Use matri multiplication to divide Rs in two parts such that the total annual interest at 9% on the first part and % on the second part amounts Rs 3060 Three schools A, B and C organised a mela for collecting funds for helping rehabilitation of earthquake victims They sold hand-made fans, mats and plates from recycled material at a cost of Rs 5, Rs 00 and Rs 50 each respectively The number of articles sold by each school is given below Articles/ A B C Schools Hand-made fans Mats Plates Find the funds collected by each school separately by the above sale Also find the total fund generated Write one value depicted in the above situation Using elementary row transformation, find the inverse of the matri 3 3 [ 5 3 ] Solve the matri equation [ 5 ] [ 0 ] [ 4] = O 0 3 If A = [ ] and B = [k ] such that AB = BA k 5k 3 Then show that k + 7k = 0

7 For questions 6 to 8 without actual epanding show that 4 5 b b c a b c c c a b = (a + b + c) 3 a b c a a b + c c + a a + b a b c q + r r + p p + q = p q r y + z z + + y y z (b + c) a a b (a + c) b = abc(a + b + c) 3 c c (a + b) If = 0, then find the value of a 0 If f() = a a, using properties of determinants find the value a a of f() f() a Si mark questions Find A -, where A = [ 3 ] Hence solve the following system of linear equations y 3z = - 4, + 3y +z = and 3 3y 4z = 3 Find A -, where A = [ 3 ] Hence solve the following system of linear equations y +3z = 4, + 3y 3z = - and 3 y + 4z = If A = [ ] and B = [ 3 solve the following system of linear equations, y + z = 4, y z = 9 and + y +3z = ], find AB Use this answer to

8 cosc cosb If A + B + C = 0, then show that cosc cosa = 0 or cosb cosa If A, B and C are the interior angles of a triangle, cosc cosb then show that cosc cosa = 0 cosb cosa Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness The school A wants to award Rs each, Rs y each, Rs z each for the three respective values to 3, and students respectively with a total award money of Rs 600 School B wants to spend Rs 300 to award its 4, and 3 students on the respective values(by giving the same award money to the three values as before)if the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value For the matri A = [ 3], show that A 3 6A + 5A + I = O 3 Hence find A - 0 Find the inverse of a matri [ 3] using elementary row operations 3 0 y z Find the value of, y and z, if A = [ y z] satisfies A = A - y z For matri A = [ ], find values of a and b such that A + a I = b A Using properties of determinants, prove the following : a bc ac + c ab + a b ac = 4 a b c ab bc + b c 36 CONTINUITY AND DIFFERENTIABILITY: MARK QUESTIONS : Discuss the continuity of the function f for R 5 f at Eamine the continuity of the function 5

9 3 dy Find, if y Differentiate sin, with respect to 5 dy Find, if sin y log 6 Differentiate e, with respect to 7 sin Differentiate 5, with respect to 8 If y sin 3, find y 9 d y 3 Find, if y e 0 Differentiate log log, with respect to MARK QUESTIONS : For what value of k is the function defined by sin cos, 0 f continuous at = 0? k, 0 3 3, 0 Show that the function given by f is not, 0 continuous at = 0 3 Find y 5 4 If y tan 5 dy If y 9, find 6 y If a b 7 If 3log y e dy, find dy, find dy, then find y e log sin 8 Differentiate 9 If y with respect to e e, prove that y y 0 Verify the Rolle s Theorem for the function f sin in, * 4- MARK QUESTIONS : Find the value of k so that the function defined by k, f is continuous at cos, ( foreign 0) a y tan a dy Find, if 0

10 3 Differentiate sin with respect to 4 Differentiate the function y y 5 If 7 6 y 0) 9 y, prove that 6 Find the derivative of tan with respect to dy y (foreign 7 Verify the Rolle s Theorem for the function f in [,] 8 dy Find, if log y log (Delhi 03) 9 If acos log tan and y asin, find the value of dy at 4 30 Verify Mean Value Theorem for the function f 3 in [4,6] MARK QUESTIONS 3 sin, if 3cos Let f a, if If f be continuous at b sin, if find a and b Delhi 06) 3 Show that the function 3 33 differentiable at 3 Delhi 03), f, R is continuous but not 4, 4 f at 0, 4 Eamine the continuity of the function dy Find, if 35 Find (AI 03) y tan 3 y sin 36 dy, if ( (

11 36 Differentiate cos with respect to ( Delhi 0) 37 If sin t and y sin pt, prove that d y dy p y 0 38 If p q pq dy y d y y y, prove that (i) and (ii) 0 ( foreign 0, 4) 39 If y sin dy cos sin, prove that y 40 If y y dy 0, for - < <, show that ( foreign 0) APPLICATION OF DERIVATIVES: MARK QUESTIONS Find the slope of the tangent to the curve y 3 6 at the point on it whose -coordinate is ( Delhi 009) The total cost C() associated with provision of free mid-day meals to students of a school in primary classes is given by 3 C If marginal cost is given by rate of dc change of the total cost, find the marginal cost of food for 300 students (Delhi 03) 3 The radius of a circle is increasing at the rate of 07 cm/sec what is the rate of increase of its circumference? 4 Find the point on the curve y = 8 for which the abscissa and ordinate change at the same rate 5 If the rate of change of volume of a sphere is equal to the rate of change of its radius, then find the radius 6 Show that the function y 4 9 is increasing for all R 7 At what point on the curve y 4, tangent is parallel to the X- ais? 8 It is given that at =, the function f 4 6 a 9 attains its maimum value on the interval [0,] Find the value of a 9 Find the maimum and minimum values if any of the function given by f Show that the function y 7 is strictly decreasing for R

12 0 MARK QUESTIONS : A circular disc of radius 3cm is being heated Due to epansion, its radius increases at the rate of 005cm/s Find the rate at which its area is increasing when radius is 3cm The length of a rectangle is increasing at the rate of 35 cm/sec and its breadth is decreasing at the rate of 3cm/sec Find the rate of change of the area of the rectangle when length is cm and breadth is 8 cm 3 Find the points on the curve y 3 0 at which tangent is parallel to X- ais 4 For what values of is the rate of increase of twice the rate of increase of? 5 Prove that the function given by f is increasing in R 6 Show that the function given by f sin is strictly decreasing in, 7 At what point on the curve y does the tangent make an angle of 45 with the X- ais? 8 If y 5 and changes from to 3, then find the approimate change in y 9 Find the maimum and minimum values, if any, of the function given by g 3 0 A particle moves along the curve y 3 3 Find the points on the curve at which y-coordinate is changing twice as fast as - coordinate 4- MARK QUESTIONS : A stone is dropped into a quiet lake and waves move in circles at a speed of 5cm/s At the instant when the radius of the circular wave is 8cm, how fast is the enclosed area increasing? A spherical balloon is being inflated by pumping in 6cm 3 /s of gas At the instant when balloon contains 36 cm 3 of gas, how fast is its radius increasing? 3 Find the intervals in which the function given by f 3 36 is increasing or decreasing 7

13 4 Show that y log throughout its domain (foreign 0) 5 Find the points on the curve y, > - is an increasing function of, where tangent is parallel 9 4 to the y- ais 6 Find the equation of tangent to the curve sin 3t, y cost, at t 4 7 Using differential, find the approimate value of 49 5 ( Delhi 0) 8 Find the absolute maimum value and the absolute minimum value for the function f 4, in the given interval 9, f 0, where 9 Using differential, find the approimate value of 3 f Find the local maima and local minima, if any, of the function given by, f sin cos, 0 < < 6 - MARK QUESTIONS : 3 Sand is pouring from a pipe at the rate of cm 3 /s The falling sand forms a cone on the ground in such a way that the height of the cone is always one- sith of the radius of the base How fast is the height of the sand cone increasing when the height is 4cm? ( AI 0) 3 Find the values of for which the function f is an increasing function Also, find the points on the curve, where the tangent is parallel to the X- ais (Delhi 00) 9 p 9 y and 33 Find the value of p for which the curves py cut each other at right angles (AI 05) 34 An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water Show that the cost of the material will be the least when the depth of the tank is half of its width 3 35 At what points will the tangent to the curve y 5 36 be parallel to the X- ais? Also find the equations of the tangents to the curve at these points 36 Of all the rectangles each of which has perimeter 40m, find one

14 which has maimum area Find the area also 37 Manufacturer can sell items at a price of Rs 5 each The 00 cost price of items is Rs 500 Find the number of items he 5 should sell to earn maimum profit 38 A wire of length 36cm is cut into two pieces One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle Find the length of each piece so that the sum of the areas of the two be minimum 39 Show that the right circular cylinder of given volume and open at the top, has minimum total surface area, provided its height is equal to radius of its base ( foreign 04) 40 Find two positive numbers whose sum is 6and sum of whose cubes is minimum Integrals Mark Questions cos * 9 sin * Marks Questions 9 5 0* ( )( )( 3) 3* 4* ( ) ( 3) ( )( 4)

15 5* 3sin 5 cos cos 4sin 6 ( )( 3) cos 4 sin sin 9 e e 6e 5 0* 4 ( ) tan 3 sec 3* 3 log 4 4 5* 7 sin 4 6* 4 cos cos 8 5 / cos tan sin 3cos 4 sin 3sin 5 9 e 30 3sin tan 5 cos cos 4sin 3 cos 3 cos3 4e 9e 6e 4e 33 sin 4 sin cos 36 5 / cos sin 3cos 4 sin 3sin 5 APPLICATION OF INTEGRALS 6 Marks Using integration, find the area of the region {(, y): + y a, y > a,, y 0} Using integration, find the area of the triangle formed by positive -ais and tangent and normal to the circle + y = 4 at (, 3) 3 Sketch the region bounded by the curves y = 5 and y =

16 find its area using ntegration 4 Using integration, find the area of the region bounded by the curves:y = + +, = 3 5 Prove that the curves y = 4 and = 4y divide the area of the square bounded by = 0, = 4, y = 4 and y = 0 into three equal parts 6 Using integration, find the area bounded by the tangent to the curve 4y = at the point (,) and the lines whose equations are = y and = 3y 3 7 Find the area of the region enclosed between the two circles + y = and ( ) + y = 8 Find the area of the region {(, y): y 4, 4 + 4y 9} using method of integration 9 Using integration, find the area of the triangle ABC, where A is (,3),B is (4,7) and C is (6,) 0 Draw the graph of y = + and using integration find the area below y = + above ais and between = 4 to = DIFFERENTIAL EQUATIONS mark Write the order and degree of the differential equation y ( d y ) + ( dy ) y dy = 0 If cos dy + ysin = tan is a differential equation, then find its order and degree 3 Write the differential equation formed from the equation y = m + c, where m and c are arbitrary constants 4 Write the differential equation obtained by eliminating the arbitrary constant C in the equation representing the family of curves y = C cos 5 If m and n are the order and degree, respectively of the differential equation y ( dy )3 + 3 ( d y ) y = sin, then write the value of m + n 6 Find the differential equation of the family of lines passing through the origin

17 7 Find the solution of the differential equation dy = 3 e y 8 Write the integrating factor of the differential equations dy + y = e 9 Solve the differential equation dy = e y + 3 e y 0 Find the differential equation representing the family of curves V = A + B, where A and B are arbitrary constants r marks Show that the solution of the differential equation y = ( ) + ce is dy + y 43 = 0 Form the differential equation of the equation y = a cos + b sin, where a and b are constant Obtain the differential equation of the family of circles passing through the points (a, 0) and (-a, 0) dy 3 Solve the differential equation = cos +cos 4 Find the general solution of the differential equation dy = e3 4y 5 Find the general solution of the differential equation log ( dy ) = + 6 Find the general solution of differential equation dy + y = e 7 Find the sum of the order and degree of the following differential equation d y + 3 dy + ( + ) = 0 8 Solve the differential equation dy = (y 3 ) 9 Solve the differential equation e tan y + ( e ) sec ydy = 0 0 Find the differential equation of all lines in XY plane 6marks Form the differential equation of the family of Circles in the second quadrant and touching the coordinate aes If cos(a + y) = cos y, then prove that dy Hence sow that = cos (a+y) sin a sina d y + sin (a + y) dy = 0 3 Solve the following differential equation ( + y ) = (tan y )dy 4 Find the particular solution of the differential equation ( + ) dy = (em tan y) given that y = when = 0

18 5 Solve the differential equation dy + (y + y ) = 0 given y =, when = 6 Solve the following differential equation cos ( y ) (y + dy) = y sin (y ) (dy y) 7 Show that the differential equation (e y + y) = dy is homogeneousfind the particular solution of this differential equation, given that = when y= 8 Find the general solution of the differential equation ( ) dy = + \ 9 Find the particular solution of the differential equation : e y y sin ( y dy ) + sin (y ) = 0 for =, y = 0 0 Show that the differential equation [sin ( y ) y] + dy = 0 is homogeneous Find the particular solution of the differential equation, given that y = π when = 4 3D GEOMETRY Mark ) Find the Cartesian equation of the line which passes through the point (-,4,5) and parallel to the line given by +3 = y 4 = z = ) Find the vector equation of a plane at which is at a distance of 7 units from the origin and the normal to the vector (3i + 5j 6k ) 3) Find the intercepts cut off by the plane +y-z+5=0 4) Find the distance of a point (,5-3) from the plane r (6i 3j + k ) = 4 5) Find the equation of the plane with intercept 3 on the y-ais and parallel to ZOX plane marks 6) Find the vector and the Cartesian equation of the line that passes through the points (,3,-,-5), (3,-,6) 7) Find the angle between the pair of the lines given by +3 = 3 y = z+3 + and = y 4 = z ) Find the angle between the two planes + y +3z-=0 and -

19 y +5 =0 9) Find the Cartesian equation of the line which passes through the point (,-,-) and parallel to the line given by 6 = 3y + = z 0) Find the vector and Cartesian equations of the plane that passes through the point (0,,-)and having normal vector i + j k 4marks ) Find the shortest distance and the vector equation of the line of shortest distance between the lines given by r = 3i 5j + 9k + λ(i 7j + 5k)and r = (μ )i + ( + μ)j + (9 3μ)k ) Find the distance of the point (,-,3) from the plane -y+z=5, measured along a line parallel to = y = z 3 6 3) Find the foot of the perpendicular from the point (,3,4) to the plane y+z+3=0 also find the image point 4) Find the valves of p so that the line = 7y 4 = z and = y 5 = 6 z are right angles 3 p 3p 5 5) Find the equation of the plane which contains the line of intersection of the planesr (i + j + 3k ) 4 = 0: r (i + j + k ) + 5 = 0 and which is perpendicular to the plane r (5i + 3j 6k ) + 8 = 0: VECTOR ALGEBRA marks ) Find the magnitude of the following vector:- a = i 7j 3k ) Find the unit vector in the direction of the vector a = i + j + k 3) Find the Vector Joining the points P(5,3,0) and Q(-,-,-4) Q to P 4) Find the position vector of the midpoint of the vector joining the points P(,3-,0) and Q(,-,) 5) Find the projection of the vector a =i + 3j + k on the vector b = i + j + k Marks 6) If a is a unit vector and ( a ) ( + a ) = 8, then find 7) Find the area of a triangle having the points A(,,3), B(-,)

20 and C(-,,3,) as its vertices 8) Find the area of a parallelogram whose adjacent sides are determined by the vectors a=i j + 3k and b = i 7j + k 9) Given a = 3, b = 5, and a b = 60 find a b 0) Find λ and μ if (i + 6j + 7k ) ( i + λj + μk ) = 0 4 Marks ) If a and b are Unit vectors Inclined at an angleθ, then prove that sin θ = a b ) If with reference to the right handed system of mutually perpendicular unit vectors i, j and k, α = 3i j,β = i + j 3k, then epress β in the form β = β + β where β is parallel to α and β is perpendicular to α 3) a = i + 4j +k, b =3 i - j +7k, c = i - j +4k,Find a vector d which is perpendicular to both a and b & c d = 5 4) If a, b are two vectors then prove that ( + a ) ( + b ) = { a b } + [a + b + (a b )] 5) Show that the angle between two diagonals of a cube is cos / 3 Marks LINEAR PROGRAMMING PROBLEMS ) A company sells two different products A and B The two products are produced in a common production process which has a total capacity of 500 man hours It takes 5 hours to produce a units of A and 3 Hours to produce a unit B, the demand in the market shows that the maimum number of units of A that can be sold is 70 and that for B is 5 Profit on each unit of A is Rs0 and that on B is Rs 5Form the constrains to solve this problem ) A manufacturing Company makes two models A and B of a product Each piece of model A requires 9 labor hours for fabricating and lab our hour for finishing Each piece of Model

21 B requires lab our hours of fabricating and 3 hours for finishing For fabricating and finishing, the maimum lab our hours available are 80 and 30 respectively The company makes a profit of Rs8000 on each piece of model A and Rs000 on each piece of model B Form the constrains to solve this problem 3) A manufacture has three machines I,II, and III installed in his factory Machines I and II are capable of being operated for at most hours where as machine III must be operated for at least 5 hours a day She produces only two items M and N each requiring the use of all the three machines The number of hours required for producing I unit of each of M and N on the three machines are given in the following table Items Numbers of hours required on the machine I II III M N 5 She makes a profit of Rs600 and Rs400 on items M and N respectively Form constrains to find how many of each items should she produce so as to maimize her profit assuming that she can sell all the items that she produced? 4Marks 4) Solve the following linear programming problem graphically: Maimize Z=60X +5Y subject to constraints + y 50, y 90 and 0 y 0 5) Solve the following linear programming problem graphically: Minimize Z=X - 5Y+0 subject to constraints y 0, + y and 3 y 4 0 y 0 6Marks 6) A oil company requires3000,0,000,and 5,000 barrels of high grade, medium and low grade oil respectively Refinery A produces 00, 300 and 00 barrels per day of high grade, medium and low grade oil respectively, where as refinery B produces 00, 400 and 00 barrels per day respectively If A Cost Rs400 per day and B cost rs300 per day to operate, how Many days should each refinery be run to minimize the cost, meeting the requirements (Ans: 70/3, 0/3) 7) Two go downs A and B have gain capacity of 00 quintals and 50 quintals respectively They supply to 3 ration shops, D, E and F whose requirements are 60,50 and 40 quintals respectively The cost of transportation per quintal from the go downs to the shops

22 are given in the following table: Transportation cost per quintal ( In Rs) From/To A B D E F How should be the supply be transported in order the transportation cost is minimum? What is the minimum cost? 8) An aeroplane can carry a maimum of 00 passengers A profit of Rs 500 is made on each eecutive class ticket out of which 0% will go to the welfare fund of the employees Similarly a profit of Rs 400 is made on each economy ticket out of which 5% will go for the improvement of facilities provided to economy class passengers In both cases, the remaining profit goes to the Air line fund The Air line reserves at least 0 seats for eecutive class However at least 4times 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 maimize the net profit of the Airline Make the above as an LPP and solve graphically Do you think more passengers would prefer to travel by such an Air line than by others 9) If a young man rides his motor cycle at 5km/h, he had to spend Rsper km on petrol If he rides at a faster speed of 40km/h, the petrol cost increases Rs 5 per kmhe has Rs00 to spend on petrol and wishes to find what is the maimum distances he can travel within one hour Epress this as on LPP and solve it graphically 0) Beena wishes to mi two types of food P and Q in such a way that the vitamin content of the miture contains at least units of vitamin A and units of vitamin B food P costs Rs60 kg and food Q cost 80 Kg Food P contains 3 units /kg of vitamin A and 5 unit /kg of vitamin B while food Q contains 4units /kg of vitamin A and units/kg of vitamin B Determine the minimum cost of the miture ) A dietician wishes to mi altogether two kinds of food X and Y in such a way that the miture contains at least 0 units of vitamin A, units of vitamin B and 8 Units of vitamin C The vitamin content of one kg food is given below Food Vitamin A Vitamin B Vitamin C

23 X 3 Y One kg of food cost Rs 6 and one kg of food Y costs Rs 0 Find the least cost of the miture which will produce the required diet PROBABILITY: MARK QUESTIONS : Given P(A) = 0, P(B) = 03 and PA B = 0 Find P(A/B) Given P(A) = 04, P(B) = 07 and P(B/A) = 06 Find PA B 3 Given P A, P B and P A B Are the events A and 3 6 B independent? MARK QUESTIONS : 4 cards numbered to are placed in a bo, mied up thoroughly and then a card is drawn at random from the bo If it is known that the number on the drawn card is more than 3, find the probability that it is an even number 5 Given that two numbers appearing on throwing two dice are different Find the probability of the event the sum of numbers on the dice is 4 4- MARK QUESTIONS : 6 A family has children Find the probability that both are boys, if it is known that (i) atleast one of the children is a boy (ii) the elder child is a boy ( AI 00) 7 Assume that each child born is equally likely to be a boy or a girl If a family has two children, what is the conditional probability that both are girls, given that (i) the youngest is a girl (ii) at least one is a girl ( Delhi 04) 8 A husband and a wife appear in an interview for two vacancies for the same post The probability of husband s selection is /7 and that of wife s selection is /5 What is the probability that (i) both will be selected? (ii) only one of them will be selected? (iii) none will be selected? 9 Three machines E, E and E3 in a certain factory producing electric bulbs, produce 50%, 5% and 5% respectively, of the total daily output of electric bulbs It is known that 4% of the bulbs produced by each of the machines E and E are defective and that 5% of those produced by machine E3 are defective If one bulb is picked up at random from a day s production, calculate the probability that it is defective ( foreign 05) 0 A card from a pack of 5 cards is lost From the remaining cards of the pack, two cards are drawn at random and are found to be both clubs Find the probability of the lost card being of clubs

24 (Delhi 00) In answering a question on a multiple choice test, a student either knows the answer or guesses Let 3/5 be the probability that he knows the answer and /5 be the probability that he guesses Assuming that a student who guesses at the answer will be correct with probability /3, what is the probability that the student knows the answer, given that he answered it correctly? ( foreign 00) Two cards are drawn simultaneously ( or successively without replacement) from a well- shuffled pack of 5 cards Find the mean and variance of the number of red cards ( AI 0) 3 A random variable X has the following probability distribution: X P(X) 0 k k k 3k k k 7k +k Find (i) k (ii) P(X< 3) (iii) P( X> 6) (iv) P(0 < X < 3) ( AI 0) 4 How many times must a man toss a fair coin so that the probability of having at least one head is more than 80%? ( Delhi 0) 5 Five cards are drawn one by one with replacement from a wellshuffled deck of 5 cards Find the probability that (i) all the five cards are diamonds (ii) only 3 cards are diamonds (iii) none is diamond ( foreign 04)

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