1、http:/ Educational P | | | Sample Paper 2008Class XIISubject - Mathematics_Application of DerivativesQ1 The volume of a cube is increasing at a constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube.Q2 Use differentials to find the approximate
2、 value of 037.Q3 It is given that for the function f(x) = x3 6x2 + ax + b on 1, 3, Rolles theorem holds with c = 2+ . Find the values of a and b if f(1)= f(3) = 031Q4 Find a point on the curve y = (x 3)2, where the tangent is parallel to the line joining (4, 1) and (3, 0).Q5 Find the intervals in wh
3、ich the function f(x) = x4 8x3 + 22x2 24x + 21 is decreasing or increasing.Q6 Find the local maximum or local minimum of the function.f(x) = sin4x + cos4x, 0111 if x = 1(5ax 2b if x1is continuous at x = 1, find the values of a and b.Q11 If y = x + n, then prove that 2a2anydxQ12 Prove : dx12sixQ13 Fi
4、nd when y = sec-1, nxQ14 If ex + ey = ex+y, prove that )1(xyedQ15 Given that cos -= prove that 8cos.42,si222 coe1sc xxQ16 If x=a( + sin), y= a(1+ cos), prove that 2yadxVECTOR ALGEBRAClass XIIQ.1. Find a vector in the direction of vector that has magnitude 7 units.2aijQ.2. Show that the points A, B a
5、nd C with position vectors, respectively, form the vertices of a right 34,bndc35ijkjkijk angled triangle.Q.3. Find , if two vectors are such that .aa2, b and4ahttp:/ Educational P | | | Q.4. Find the area of the parallelogram whose adjacent sides are determined by the vectors 3andb27aijkijkQ.5. If
6、 a unit vector makes angles with , and acute angle with , then find and hence3iwthj4kthe components of .Q.6. Let , and be three vectors such that and each one of them being perpendicular bc, b, c5ato the sum of other two, Find .bcQ.7. Find the value of .().().()ijkikijQ.8. The scalar product of the
7、vector with a unit vector along the sum of vectors jFind the value of .245andi23isequlton.ijQ.9. If the sum of two unit vectors is a unit vector, Prove that the magnitude of their difference is .3Q.10. If are position vectors of points A and B respectively, then find the position vector of points of
8、 trisection of bAB.Q.11. Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.Q.12. ABCD is a parallelogram. If the coordinates of A, B, C are (-2, -1), (3, 0) and (1, -2) respectively, Find the co-ordinate of D.Q.13. Sho
9、w that the points A, B, C with position vectors 235,23and7respctivly areolinrbccQ.14. If a vector makes , , with OX, OY and OZ respectively, prove that sin2+sin2+sin2=2. Q.15. If inclined at an angle , then prove that sin = .ndreuitvors 1bQ.16 If .0, a3, b5, 7, findthe agltwen ad bccQ.17. If .2, 5 n
10、d8, ibaThree Dimensional GeometryClass XIIQ.1. Find the direction cosines of X, Y and Z-axis.Q.2. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector .32ijkQ.3. Find the value of p so that the lines,http:/ Educational P | | | 17437-x56 and aretigh n
11、les.32p1xyzyzpQ.4. Find the shortest distance between the lines whose vector equations are:-r()(2)1(tijtksssQ.5. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x 3y + 4z -6 = 0.Q.6. Find the vector equation of the plane passing through the intersection of p
12、lanes r.()6andr.(2i34)5andthepoit(1,)ijkjkQ.7. Find the angle between the line 1 l0x 2y-z 36xyzQ.8. Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then 221bcpQ.9. Show that the angles between the diagonals of a cube is cos-1 .13Q.10. Find the equati
13、on of the line passing through the point (-1, 3, -2) and perpendicular to the lines x+21and123-5xyzyzQ.11. Find the foot of the perpendicular drawn from the point (0, 2, 3) on the line 314,52xyzAlso, find the length of the perpendicular.Q.12. Find the shortest distance between the following pairs of
14、 lines whose cartesian equations are :1x+12and,z=233xyyzQ.13. A plane meets the coordinate axis in A, B, C such that the centroid of triangle ABC is the point (p,q,r). Show that the equation of the plane is .3pqrQ.14. Find the equation of the plane passing through the point (1, 1, -1) and perpendicu
15、lar to the planes x + 2y + 3z 7=0 and 2x 3y + 4z = 0.Q.15. Find the distance between parallel planes, .(236)5andr.(6918)20rijkijkQ.16. Show that the lines :are coplanar. 43ij ikAlso, find the plane containing these two lines.Class-XIIPROBABILITYhttp:/ Educational P | | | Q.1. A die is thrown twice
16、 and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once.Q.2. Assume that each born child 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
17、given that at least one is a girl.Q.3. If A and B are two independent events, show that the probability of occurrence of at least one of A and B is given by :1 P(A).P(B)Q.4. Probability of solving specific problem independently by A and B are and 1/3 respectively. If both try to solve the problem in
18、dependently, find the probability that the problem is solved.Q.5. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the pr
19、obability that the second ball is red.Q.6. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings.Q.7. In an examination, 20 questions of true-false type are asked. Suppose
20、a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers true, if it falls tails, he answers false. Find the probability that he answers at least 12 questions correctly.Q.8. A die is thrown again and again until three sixes are obtained. Find the pro
21、bability of obtaining the third six in the sixth throw of the die.Q.9. If P(A) = 3/8, P(B) = and P (AB) = , find P and P ._A/B_/Q.10. The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3 and the probability that he will buy a shirt given th
22、at he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.Q.11. A can solve 90% of the problem given in a book and B can solve 70%. What is the probability that at least one of them w
23、ill solve the problem, selected at random from the book.Q.12. Three persons A, B, C throw a die in succession till one gets a six and wins the game. Find their respective probabilities of winning, if A begins.Q.13. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it i
24、s a six. Find the probability that it is actually a six.Q.14. An urn contains 4 white and 3 red balls. Find the probability distribution of the number of red balls in a random draw of three balls.Q.15. In a meeting 70% of the members favour a certain proposal,30% being opposed. A member is selected
25、at random and let X=0 if he opposed and X =1 if he is in favour. Find E(x) and Var(x).Q.16. Find the probability distribution of the number of doublets in 4 throws of a pair of dice.LINEAR PROGRAMMINGhttp:/ Educational P | | | Q.1. One kind of cake requires 200gm of flour and 25gm of fat, and anot
26、her kind of cake requires 100gm of flour and 50gm of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.Q.2. A dietician wishes to mix together two kinds of food X and Y in suc
27、h a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The Vitamin contents of one Kg food is given below :-One Kg of food X costs Rs. 16 and one Kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required d
28、iet.Q.3. Maximise and Minimise:Z = x + 2ySubject to constraints x + 2y 100, 2x y 0, 2x + y 200, x, y 0Q.4. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserve
29、s at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit.Q.5. Two godowns A and
30、 B have grain capacity of 100 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 godowns to the shops are given in following table :Transportation Cost Per Quinta
31、l (in Rs.)From / To A BD 6 4E 3 2F 2.50 3How should the supplies be transported in order that the transportation cost is minimum. What is the minimum cost.Q.6. Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stitch 6 shirts and 4 pants per day, while B can stitch 10 shirts a
32、nd 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.Q.7. Solve the following L.P.P graphically:Maximise: Z = 60x + 15ySubject to constraints x + y 503x + y 90, x, y 0Q.8. A dealer wishes to purchase a number of fans and sewing
33、machines. He has only Rs 5,760 to invest and has space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how s
34、hould he invest his money in order to maximise his profit.Q.9. If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2/km on petrol. If he drives it at a faster speed of 40km/hr, the petrol cost increases to Rs. 5/km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an L.P.P. and solve.PREPARED BY :- 9811363962Food Vitamin A Vitamin B Vitamin CX 1 2 3Y 2 2 1http:/ Educational P | | | Anil Kumar TondakM.Sc.(Maths), M.Ed.,M.A.(Eco),M.B.A.(Mktg)
