INDIAN SCHOO MUSCAT QUESTION BANK DEPARTMENT OF MATHEMATICS SENIOR SECTION. Relations and Functions
|
|
- Dustin Haynes
- 5 years ago
- Views:
Transcription
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)
25
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,,
More informationEINSTEIN CLASSES. C B S E XIIth Board PRACTICE ASSIGNMENT
EINSTEIN CLASSES P R E S E N T S C B S E XIIth Board PRACTICE ASSIGNMENT MATHEMATICS NOTE THE FOLLOWING POINTS : Einstein Classes is primarily concerned with the preparation of JEE-ADVANCE /JEE-MAIN/BITS/PMT/AIIMS
More informationCBSE Examination Papers
CBSE Eamination Papers (Foreign 0) Time allowed: hours Maimum marks: 00 General Instructions: As given in CBSE Sample Question Paper. Set I SECTION A Question numbers to 0 carry mark each.. Write the principal
More information12 th Class Mathematics Paper
th Class Mathematics Paper Maimum Time: hours Maimum Marks: 00 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 9 questions divided into four sections A, B, C
More informationCBSE 2018 ANNUAL EXAMINATION DELHI
CBSE 08 ANNUAL EXAMINATION DELHI (Series SGN Code No 65/ : Delhi Region) Ma Marks : 00 Time Allowed : Hours SECTION A Q0 Find the value of tan cot ( ) Sol 5 5 tan cot ( ) tan tan cot cot 6 6 6 0 a Q0 If
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks : 100
MATHEMATICS Time allowed : hours Maimum Marks : General Instructions:. All questions are compulsory.. The question paper consists of 9 questions divided into three sections, A, B and C. Section A comprises
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. Type of Questions Weightage of Number of Total Marks each question questions
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (Matrices and Determinants) (iii) Calculus 44 (iv) Vector and Three dimensional Geometry 7 (v) Linear Programming
More information3. Total number of functions from the set A to set B is n. 4. Total number of one-one functions from the set A to set B is n Pm
ASSIGNMENT CLASS XII RELATIONS AND FUNCTIONS Important Formulas If A and B are finite sets containing m and n elements, then Total number of relations from the set A to set B is mn Total number of relations
More informationC.B.S.E Class XII Delhi & Outside Delhi Sets
SOLVED PAPER With CBSE Marking Scheme C.B.S.E. 8 Class XII Delhi & Outside Delhi Sets Mathematics Time : Hours Ma. Marks : General Instructions : (i) All questions are compulsory. (ii) The question paper
More informationCBSE Board Paper Foreign 2013
CBSE Board Paper Foreign 03 Set - I Time: 3 Hours Max Marks: 00 General Instructions (i) All questions are compulsory (ii) The question paper consists of 9 questions divided into three sections A, B and
More informationoo ks. co m w w w.s ur ab For Order : orders@surabooks.com Ph: 960075757 / 84000 http://www.trbtnpsc.com/07/08/th-eam-model-question-papers-download.html Model Question Papers Based on Scheme of Eamination
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. (ii) Algebra 13. (iii) Calculus 44
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total
More informationPRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES PRADEEP SHARMA. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 1
PRADEEP SHARMA PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page Chapter Relation and Functions Mark Questions A relation R in a Set A is called..., if each element of A is related to every element
More informationDIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI
456789045678904567890456789045678904567890456789045678904567890456789045678904567890 456789045678904567890456789045678904567890456789045678904567890456789045678904567890 QUESTION BANK 456789045678904567890456789045678904567890456789045678904567890456789045678904567890
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 017-18 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500
More informationBASIC MATHEMATICS - XII SET - I
BASIC MATHEMATICS - XII Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Candidates are required to give their answers in their own words as far as practicable. The figures in the margin
More informationSAMPLE QUESTION PAPER
SAMPLE QUESTION PAPER CLASS-XII (201-17) MATHEMATICS (01) Time allowed: 3 hours Maximum Marks: 100 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 29 questions.
More informationCBSE Examination Paper, Foreign-2014
CBSE Eamination Paper, Foreign-4 Time allowed: hours Maimum marks: General Instructions: As per given in CBSE Eamination Paper Delhi-4. SET I SECTION A Question numbers to carr mark each.. Let R = {(a,
More informationMathematics Class X Board Paper 2011
Mathematics Class X Board Paper Solution Section - A (4 Marks) Soln.. (a). Here, p(x) = x + x kx + For (x-) to be the factor of p(x) = x + x kx + P () = Thus, () + () k() + = 8 + 8 - k + = k = Thus p(x)
More informationAPPLICATIONS OF DERIVATIVES OBJECTIVES. The approimate increase in the area of a square plane when each side epands from c m to.0 cm is () 0.00 sq. cm () 0.006 sq. cm () 0.06 sq. cm () None. If y log then
More informationCDS-I 2019 Elementary Mathematics (Set-C)
1 CDS-I 019 Elementary Mathematics (Set-C) Direction: Consider the following for the next three (03) items : A cube is inscribed in a sphere. A right circular cylinder is within the cube touching all the
More informationDesign of Question Paper Mathematics - Class XII
Design of Question Paper Mathematics - Class XII Time : 3 hours Max. Marks : 100 Weightage of marks over different dimensions of the question paper shall be as follows : A. Weightage to different topics/content
More informationCBSE MATHS 2010 YEAR PAPER
CBSE MATHS YEAR PAPER Important Instructions: (i) The question papers consists of three sections A B and C. (ii) All questions are compulsory. (iii) Internal choices have been provided in some questions.
More information1. SETS AND FUNCTIONS
. SETS AND FUNCTIONS. For two sets A and B, A, B A if and only if B A A B A! B A + B z. If A B, then A + B is B A\ B A B\ A. For any two sets Pand Q, P + Q is " x : x! P or x! Q, " x : x! P and x b Q,
More informationMockTime.com. NDA Mathematics Practice Set 1.
346 NDA Mathematics Practice Set 1. Let A = { 1, 2, 5, 8}, B = {0, 1, 3, 6, 7} and R be the relation is one less than from A to B, then how many elements will R contain? 2 3 5 9 7. 1 only 2 only 1 and
More informationI K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists
LEVEL I. A circular metal plate epands under heating so that its radius increase by %. Find the approimate increase in the area of the plate, if the radius of the plate before heating is 0cm.. The length
More informationFILL THE ANSWER HERE
HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5
More informationMathematics. Guess Paper: 2014 Class: XII. Time Allowed: 3Hours Maximum Marks: 70. Section A
Mathematics Guess Paper: 04 Class: XII Time llowed: Hours Maimum Marks: 70 General Instructions:. The question paper consists of 9 questions divided into three sections, B and C.. Section comprises of
More informationBasic Mathematics - XII (Mgmt.) SET 1
Basic Mathematics - XII (Mgmt.) SET Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Model Candidates are required to give their answers in their own words as far as practicable. The figures
More informationMathematics. Class - XII. Chapter Assignments
Mathematics Class - XII Chapter Assignments Chapter 1 Relations and Functions 1 mark Questions 1. If R= {(a, a 3 ): a is a prime number less than 5} be a relation. Find the range of R. 2. If f:{1,3,4}
More informationNATIONAL QUALIFICATIONS
Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -
More informationPROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( ) Water seeps out of a conical filter at the constant rate of 5 cc / sec. When the height of water level in the cone is 5 cm, find the rate at which the height
More informationMaths-III. Important Types in Maths III. Prepared By : Sameer V. shaikh { }
Mhs-III Important Types in Mhs III Prepared By : Sameer V. shaikh {Engr.sameer@gmail.com} {9765158158} MINIMUM Imp TYPES FOR MATHS III Types of Problems No Type of Problem Min/max marks Locion in Q.P
More informationTime allowed : 3 hours Maximum Marks : 100
CBSE XII EXAMINATION-8 Series SGN SET- Roll No. Candiates must write code on the title page of the answer book Please check that this question paper contains printed pages. Code number given on the right
More informationCLASS 12 SUBJECT : MATHEMATICS
CLASS 2 SUBJECT : MATHEMATICS CBSE QUESTION PAPER 27(FOREIGN) General Instructions: (i) All questions are compulsory. (ii) Questions 4 in Section A carrying mark each (iii) Questions 5 2 in Section B carrying
More information1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3
[STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)
More information4) If ax 2 + bx + c = 0 has equal roots, then c is equal. b) b 2. a) b 2
Karapettai Nadar Boys Hr. Sec. School One Word Test No 1 Standard X Time: 20 Minutes Marks: (15 1 = 15) Answer all the 15 questions. Choose the orrect answer from the given four alternatives and write
More informationGuess Paper 2013 Class IX Subject Mathematics
Guess Paper 01 Class IX Subject Mathematics A 1. A man goes out at 16:4 and arrives at a post bo, 6 km away, at 17:0. He walked part of the way at 5 km/hr and then, realizing the time, he ran the rest
More informationWelcome to Advanced Placement Calculus!! Summer Math
Welcome to Advanced Placement Calculus!! Summer Math - 017 As Advanced placement students, your first assignment for the 017-018 school year is to come to class the very first day in top mathematical form.
More informationGURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY
GURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY Class :- XII ASSIGNMENT Subject :- MATHEMATICS Q1. If A = 0 1 0 0 Prove that (ai + ba)n = a n I + na n-1 ba. (+) Q2. Prove that (+) = 2abc
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationSYLLABUS. MATHEMATICS (041) CLASS XII One Paper Three Hours Marks: 100
SYLLABUS MATHEMATICS (041) CLASS XII 2012-13 One Paper Three Hours Marks: 100 Units Marks I. RELATIONS AND FUNCTIONS 10 II. ALGEBRA 13 III. CALCULUS 44 IV. VECTS AND THREE - DIMENSIONAL GEOMETRY 17 V.
More informationMODEL PAPER - I MATHEMATICS. Time allowed : 3 hours Maximum marks : 100
MODEL PAPER - I MATHEMATICS Time allowed : 3 hours Maimum marks : General Instructions. All questions are compulsy.. The question paper consists of 9 questions divided into three sections A, B and C. Section
More informationMATHEMATICS. metres (D) metres (C)
MATHEMATICS. If is the root of the equation + k = 0, then what is the value of k? 9. Two striaght lines y = 0 and 6y 6 = 0 never intersect intersect at a single point intersect at infinite number of points
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII
SAMPLE QUESTION PAPER MATHEMATICS (01) CLASS XII 017-18 Time allowed: hours Maimum Marks: 100 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 9 questions. (iii)
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks: 100
MATHEMATICS Time allowed : 3 hours Maimum Marks: 00 General Instructions:. All questions are compulsory.. This question paper contains 9 questions. 3. Questions 4 in Section A are very short-answer type
More informationSET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100
General Instructions. All questions are compulsor.. This question paper contains 9 questions.. Questions - in Section A are ver short answer tpe questions carring mark each.. Questions 5- in Section B
More informationAPPLICATION OF DERIVATIVES
94 APPLICATION OF DERIVATIVES Chapter 6 With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature. WHITEHEAD 6. Introduction In Chapter 5, we have learnt
More informationMockTime.com. (b) 9/2 (c) 18 (d) 27
212 NDA Mathematics Practice Set 1. Let X be any non-empty set containing n elements. Then what is the number of relations on X? 2 n 2 2n 2 2n n 2 2. Only 1 2 and 3 1 and 2 1 and 3 3. Consider the following
More informationINTERNATIONAL INDIAN SCHOOL, RIYADH. 11cm. Find the surface area of the cuboid (240cm 2 )
INTERNATIONAL INDIAN SCHOOL, RIYADH CLASS: IX SUBJECT: MATHEMATICS 1. SURFACE AREAS AND VOLUMES 1. The diagonal of a cube is 12cm. Find its volume. 2. If the lateral surface area of a cube is 1600cm 2,
More informationAll Rights Reserved Wiley India Pvt. Ltd. 1
Question numbers to carry mark each. CBSE MATHEMATICS SECTION A. If R = {(, y) : + y = 8} is a relation of N, write the range of R. R = {(, y)! + y = 8} a relation of N. y = 8 y must be Integer So Can
More informationSAMPLE QUESTIONS CLASS X
SAMPLE QUESTIONS SUMMATIVE ASSESSMENT II 2014 2015 CLASS X Mathematics VSA: 1 MARKS 1. If the common difference of an AP is 3, then what is a15 - a 9? 2. If the ratio between the length of the shadow of
More informationSAMPLE QUESTION PAPER MATHEMATICS CLASS XII ( ) BLUE PRINT. Unit VSA (1) SA (4) LA (6) Total. I. Relations and Functions 1 (1) 4 (1) 5 (2)
SAMPLE QUESTION PAPER MATHEMATICS CLASS XII (2013-14) BLUE PRINT Unit VSA (1) SA (4) LA (6) Total I. Relations and Functions 1 (1) 4 (1) 5 (2) Inverse Trigonometric Functions 1 (1) *4 (1) 5 (2) II. Matrices
More informationSaturday, March 27, :59 PM Annexure 'F' Unfiled Notes Page 1
Annexure 'F' CLASS-XII SAMPLE QUESTION PAPER MATHEMATICS CLASS XII (2013-14) BLUE PRINT Unit VSA (1) SA (4) LA (6) Total I. Relations and Functions 1 (1) 4 (1) 5 (2) Inverse Trigonometric Functions
More information12 STD BUSINESS MATHEMATICS
STD BUSINESS MATHEMATICS 6 MARK FAQ S: CHAPTER :. APPLICATION OF MATRICES AND DETERMINANTS. Given, A verify that A AdjA (J 7; O 7 ; O ). Find the inverse of, b a A and verify that I AA. (J 6). Verify A
More informationLIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS. Sl. No. Name Designation
LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS Sl. No. Name Designation. Sh. S.B. Tripathi R.S.B.V., Jheel Khuranja (Group Leader) Delhi. (M. 98086). Sh. Sanjeev
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationMULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maximum Marks : 100. [ Q. 1 to 60 carry one mark each ] A. 0 B. 1 C. 2 D.
M 68 MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maimum Marks : [ Q. to 6 carry one mark each ]. If sin sin sin y z, then the value of 9 y 9 z 9 9 y 9 z 9 A. B. C. D. is equal
More informationMATHEMATICS. (Three hours) (Candidates are allowed additional 15 minutes for only reading the paper.
[ For more model sample papers visit : MATHEMATICS www.4ono.com (Three hours) (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.) ---------------------------------------------------------------------------------------------------------------------
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS / UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value.
More informationCHAPTER-1. SETS. Q.4 Write down the proper subsets of { a, b, Q.5 Write down the power set of { 5,6,7 }? Verify the following result :
CHAPTER-. SETS Q. Write the following sets in roster form (i) A = { : is an integer and 5 5 } (ii) B = { : is a natural number and < < 4} (iii) C= { : is a two- digit natural number such that sum of digit
More informationHEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)
HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum
More informationMATHEMATICS Paper & Solutions
CBSE-XII-8 EXAMINATION Series SGN MATHEMATICS Paper & Solutions SET- Code : 6/ Time : Hrs. Ma. Marks : General Instruction : (i) All questions are compulsor. (ii) The question paper consists of 9 questions
More informationMATHEMATICS. SECTION A (80 Marks)
MATHEMATICS (Maximum Marks: 100) (Time allowed: Three hours) (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.) ---------------------------------------------------------------------------------------------------------------------
More informationAPPLICATIONS OF DERIVATIVES
ALICATIONS OF DERIVATIVES 6 INTRODUCTION Derivatives have a wide range of applications in engineering, sciences, social sciences, economics and in many other disciplines In this chapter, we shall learn
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII Time allowed : 3 Hours MAX.MARKS 100 Blue Print. Applicatoin.
Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation
More informationX- MATHS IMPORTANT FORMULAS SELF EVALUVATION 1. SETS AND FUNCTIONS. 1. Commutative property i ii. 2. Associative property i ii
X- MATHS IMPORTANT FORMULAS SELF EVALUVATION 1. SETS AND FUNCTIONS 1. Commutative property i ii 2. Associative property i ii 3. Distributive property i ii 4. De Morgan s laws i ii i ii 5. Cardinality of
More informationMATHEMATICS CHAPTER I : RELATIONS AND FUNCTIONS
VIDYA DEVI JINDAL SCHOOL DELHI ROAD,HISAR HOLIDAY HOMEWORK (2016-2017) CLASS XII [COMMERCE] ENGLISH Q.1. Read The Invisible Man (the novel prescribed by the CBSE)and make a project highlighting the following:
More informationWBJEEM Answer Keys by Aakash Institute, Kolkata Centre MATHEMATICS
WBJEEM - 05 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. μ β γ δ 0 B A A D 0 B A C A 0 B C A * 04 C B B C 05 D D B A 06 A A B C 07 A * C A 08 D C D A 09 C C A * 0 C B D D B C A A D A A B A C A B 4
More information10 th MATHS SPECIAL TEST I. Geometry, Graph and One Mark (Unit: 2,3,5,6,7) , then the 13th term of the A.P is A) = 3 2 C) 0 D) 1
Time: Hour ] 0 th MATHS SPECIAL TEST I Geometry, Graph and One Mark (Unit:,3,5,6,7) [ Marks: 50 I. Answer all the questions: ( 30 x = 30). If a, b, c, l, m are in A.P. then the value of a b + 6c l + m
More informationPRE BOARD EXAMINATION CODE : E SESSION CLASS : X MAXIMUM MARKS: 80 SECTION A
PRE BOARD EXAMINATION CODE : E SESSION 017-018 CLASS : X MAXIMUM MARKS: 80 SUBJECT : MATHEMATICS TIME : HOURS General Instructions: (i) All questions are compulsory. (ii) The question paper consists of
More informationThe matrix: "Is an organization of some elements written in rows
st term st Sec. Algebra. The matrix: "Is an organization of some elements written in rows Ex: and columns between brackets in the form ( ) ". st column nd rd - st row - nd row 7 rd row * The order of any
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationSURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS
SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -
More information(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.
Calculus Review Packet 1. Consider the function f() = 3 3 2 24 + 30. Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point,
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 02 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More informationFree Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
Free Response Questions 1969-010 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions
More informationAPPLICATIONS OF DERIVATIVES
9 APPLICATIONS OF DERIVATIVES In the previous lesson, we have learnt that the slope of a line is the tangent of the angle which the line makes with the positive direction of x-axis. It is denoted by the
More informationCBSE CLASS X MATH
CBSE CLASS X MATH - 2011 Q.1) Which of the following cannot be the probability of an event? A. 1.5 B. 3 5 C. 25% D. 0.3 Q.2 The mid-point of segment AB is the point P (0, 4). If the Coordinates of B are
More informationMath Bank - 6. What is the area of the triangle on the Argand diagram formed by the complex number z, -iz, z iz? (a) z 2 (b) 2 z 2
Math Bank - 6 Q.) Suppose A represents the symbol, B represents the symbol 0, C represents the symbol, D represents the symbol 0 and so on. If we divide INDIA by AGRA, then which one of the following is
More informationDELHI PUBLIC SCHOOL BLUE PRINT WEEKLY TEST CLASS XI (MATHEMATICS)
DELHI PUBLIC SCHOOL BLUE PRINT WEEKLY TEST CLASS XI (MATHEMATICS) S. N0. TYPES OF QUESTIONS NO. OF QUESTION MARKS TOTAL 1. VERY SHT ANSWER 6 1 6 2. SHT ANSWER 5 4 20 3. LONG ANSWER WITH ONE 4 6 24 VALUE
More informationSAMPLE PAPER 3 (SA II) Mathematics CLASS : X. Time: 3hrs Max. Marks: 90
1 SAMPLE PAPER 3 (SA II) MRS.KIRAN WANGNOO Mathematics CLASS : X Time: 3hrs Max. Marks: 90 General Instruction:- 1. All questions are Compulsory. 1. The question paper consists of 34 questions divided
More informationSSLC MODEL QUESTION PAPER-1
SSLC MODEL QUESTION PAPER-1 MATHEMATICS Time Allowed :2.30 hrs Maximum Marks : 100 SECTION I Note : (i) Answer all the 15 questions. 15x1 = 15 (ii) Choose the correct answer from the given four alternatives
More information( ) 7 ( 5x 5 + 3) 9 b) y = x x
New York City College of Technology, CUNY Mathematics Department Fall 0 MAT 75 Final Eam Review Problems Revised by Professor Kostadinov, Fall 0, Fall 0, Fall 00. Evaluate the following its, if they eist:
More informationRao IIT Academy/ ISC - Board 2018_Std XII_Mathematics_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS. XII - ISC Board
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)
More informationMAXIMA AND MINIMA - 2
MAXIMA AND MINIMA - GREATEST AND LEAST VALUES Definition: Let f be a function defined on a set A and l f( A ). Then l is said to be (i) the maimum value or the greatest value of f in A if f( ) l A. (ii)
More informationAP Calculus Free-Response Questions 1969-present AB
AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions
More information( ) 2 + ( 2 x ) 12 = 0, and explain why there is only one
IB Math SL Practice Problems - Algebra Alei - Desert Academy 0- SL Practice Problems Algebra Name: Date: Block: Paper No Calculator. Consider the arithmetic sequence, 5, 8,,. (a) Find u0. (b) Find the
More informationMATHEMATICS. r Statement I Statement II p q ~p ~q ~p q q p ~(p ~q) F F T T F F T F T T F T T F T F F T T T F T T F F F T T
MATHEMATICS Directions : Questions number to 5 are Assertion-Reason type questions. Each of these questions contains two statements : Statement- (Assertion) and Statement- (Reason). Each of these questions
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 05 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More informationMathematics Class XII
Mathematics Class XII Time: hour Total Marks: 00. All questions are compulsory.. The question paper consist of 9 questions divided into three sections A, B, C and D. Section A comprises of 4 questions
More informationChapter 27 AB Calculus Practice Test
Chapter 7 AB Calculus Practice Test The Eam AP Calculus AB Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour and 45 minutes Number
More informationD - E - F - G (1967 Jr.) Given that then find the number of real solutions ( ) of this equation.
D - E - F - G - 18 1. (1975 Jr.) Given and. Two circles, with centres and, touch each other and also the sides of the rectangle at and. If the radius of the smaller circle is 2, then find the radius of
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More information02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by =
0. π/ sin d 0 sin + cos (A) 0 (B) π (C) 3 π / (D) π / (E) π /4 0. If (, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) + y = 0 (B) b ay = 0 (C) a by = 0 (D) b + ay = 0 (E) a + by = 0 03.
More informationChapter 8: Radical Functions
Chapter 8: Radical Functions Chapter 8 Overview: Types and Traits of Radical Functions Vocabulary:. Radical (Irrational) Function an epression whose general equation contains a root of a variable and possibly
More informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationSAMPLE QUESTION PAPER Class-X ( ) Mathematics. Time allowed: 3 Hours Max. Marks: 80
SAMPLE QUESTION PAPER Class-X (017 18) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided
More informationGrade XI Mathematics
Grade XI Mathematics Exam Preparation Booklet Chapter Wise - Important Questions and Solutions #GrowWithGreen Questions Sets Q1. For two disjoint sets A and B, if n [P ( A B )] = 32 and n [P ( A B )] =
More information