Kendriya Vidyalaya (Bsf) Dabla

KENDRIYA VIDYALAYA (BSF) DABLA

MATHEMATICS

Class-XII

UNIT-I. RELATIONS & FUNCTIONS

(10 MARKS)

Define relation R on the set N of natural numbers by R={(x,y):y= x+7, xisanatural

Numberlessthan4;x, y∈N}.Write down the domain and the range.

Let R be the relation in the set N given by R = {(a,b); a = b – 2 , b > 6}

Whether the relation is reflexive or not ? Justify your answer.

3. Let R be the relation on R defined as (a , b) R iff 1+ ab > 0 a,bR.

Show that R is symmetric.
Show that R is reflexive.
Show that R is not transitive.

4. If f(x) = , then find f(1/x).

5. Let A = {-1, 0, 1} and B = {0 , 1}. State whether the function f: A → B defined by

f(x) = x2is bijective .

6.Show the function f:R→R defined by f(x)=, xR is one-one and onto function.

Also find the inverse of the function f.

7. If f(x) = , then show that

(a) = – f(x) , (b) =

8. If f: RR is defined by f(x) = . Find f -1

9. Show that the relation R on A ,A = { x| x Z , 0 ≤ x ≤ 12 } ,

R = {(a ,b): |a - b| is multiple of 3.} is an equivalence relation.

10. Let A = Set of all triangles in a plane and R is defined by

R={(T1,T2) : T1,T2 A & T1~T2 }. Show that the R is equivalence relation. Consider the right angled ∆s, T1 with size 3,4,5; T2 with size 5, 12,13; T3 with side 6, 8, 10; Which of the pairs are related?

11.Consider a function f :R+[-5, ∞) defined f(x) = 9x2 + 6x - 5. Show that f is invertible &

f -1(y) = , where R+ = (0 , ∞).

12. Let * be a binary operation on the set Q of rational numbers defined as a * b = .

Write the identity and inverse of *, if any.

13. On R – {1} a binary operation ‘*’ is defined as a * b = a + b – ab. Prove that ‘*’ is

commutativeand associative. Find the identity element for ‘*’.Also prove that every

element of R – {1} isinvertible.

Prove the following:\

Prove the following:.

Provethat:.

Prove the following : .

Solve for

Solve for x:, x>0.

Solve for x:.

Solve for x:

Prove that

UNIT-II. ALGEBRa

(13 MARKS)

1. If A=, prove that Hence find

2. A=and I= , find k so that .

3. Express the matrix A= as the sum of a symmetric and skew –symmetric matrix.

4.Using properties of determinants, prove that

5. If x, y and z are different and, show that xyz=-1.

6. Using properties of determinants, prove the following:

7. Using properties of determinants prove the following:

8. Using properties of determinants prove the following:

9. Using properties of determinants, solve the following for x:

10. Using properties of determinants, solve the following for x:

11. Using properties of determinants, show that

12. Using properties of determinants, prove the following

13. Using properties of determinants, show that

14. Solve the following system of equation by matrix method, where x 0, y  0, z  0;

; ;

Three shopkeepers A, B and C are using polythene, handmade bags (prepared by persons) and newspapers envelopes as carry bags. It is found that the shopkeepers A,B ,C are using(20,30,40) ,(30,40,20) ,(40,20,30) polythene ,handmade bags and newspapers envelopes respectively. The shopkeepers A, B,C spent Rs 250,Rs 220 and

Rs 200 on these carry bags respectively. Find the cost of each carry bags using matrices. Keeping in mind the social and environmental conditions, which shopkeeper is better and why?

In a survey of 25 richest person of three localities x, y ,z. It is found that in locality A 7 believes in honesty,12 in hard work and 6 in unfair means. While in Y 10 believes in honesty 11 in hard work , 4 in unfair means and in Z, 8 believes in honesty ,12 in hard work 5 in unfair means. If the income of 25 richest persons of locality x, y and z are respectively Rs.1,19,000,Rs.1,18,0001,18,000per day then find income of each type of person by matrix method. In your view which type of person is better for society and for nation.

Two schools A and B decided to award prizes ti their students for three values honesty, punctuality and obedience. School A decided to award a total of Rs. 11000 for the thee values to 5,4 and 3 students respectively while school B decided to award 10700 for the three values to 4 ,3 and 5 students respectively. If all the three prizes together amount to Rs. 2700, then

1)Represent the above situation by a matrix equation and form linear equations using

matrix multiplications.

2)Is it possible to solve the system of equations, so obtained using matrices. If possible solve it.

3) Which value you prefer to reward most and why?

UNIT-III. calculus

(44 MARKS)

If f(x)= is continuous at x=3. Find the values of a add b.

f(x)= is continuous at x=0. Find the value of k also discuss about continuity at x=1.

f(x)= is continuous function.

Show that f(x)= is continuous but not differentiable at x=3

f(x)= is continuous at x=0

x=, y=. Find .

if =0 then prove that
if then prove that is independent of a and b
If cos y = x cos(a+y) then prove that

If y = then show that (1-x2)- x- = 0

If y = a sin x + b cos x, then prove that + y = 0.

If y = 500 + 600, prove that - 49y = 0.

Find the second order derivative of + log x.

If y = , prove that ( 1 + ) + 2x = 0.

Find the intervals in which the function f given by f(x) = -2 - 9 - 12 +1 is strictly decreasing .

Find the intervals in which the function f given by f(x) = + , 0 x 2 is strictly increasing .

Find the value of x for which y = is an increasing function.

Prove that the function given by f(x) = cos x is

(a) strictly decreasing in (0,).

(b) strictly increasing in (, 2).

19. Find the equations of the tangent to the curve y =+ 4 + 1 at the point whose x-

coordinate is 3.

Find the equations of the normal to the curve y =+ 4 + 1 at the point whose x-coordinate is 3.

Find the equation of tangent to the curve y = , which is parallel to the line 4x -2y + 5 =0.

Find the equations of the tangent to the parabola at the point ( a

Find the equations of the tangent to the hyparabola - = 1 at the point (h,k.

Prove that the curves x = and xy = k cut at right angles if = 1.

The volume of a cube is increasing at the rate 10 increasing when the length of an edge is 15 cm?

A particle move along the curve 6y=+2. Find the points on the curve at which the

Y coordinate is changing 8 times as fast as the X –coordinate.

The radius of an air buble is increasing at the rate of cm/s. At what rate is the volume of the buble increasing when the radius is 1cm?

Sand is pouring from a pipe at the rate of 12.The falling sand forms, a cone on

the ground in such a way that the hight of the cone is always one –sixth of the

radius of the base .How fast is the height of the sand cone increasing when the

height is 4cm?

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximumpossible.

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top by cutting off squares from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.?

Show that the height of cylinder of maximum volume that can be inscribed in a sphere of radius ‘a’ is .

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is Sin-1(1/3).

Evaluate: as the limit of a sum.

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Using integration, find the area of the region bounded by the parabola and the circle 4+4=9.

Find the area between the curves y2 = x and x2 = y.

Find of the area of region enclosed between the two circles

=4 and

Find area of region bounded by y= and y=.

Find area of region bounded by triangle whose vertices are (-1,2), (1,5) and (3,4).

Using integration find area of the triangular region whose sides have the equations y=2x+1,y=3x+1 and x=4.

Solve the differential equation: .

Solve the differential equation:

Form a Differential equation representing the family of curves given by

(x- a)2 + 2y2 =a2 where a is an arbitrary constant.

Form a Differential equation representing the family of circles touching the Y-axis at origin.

Form a Differential equation representing the family of circles touching the X-axis at origin.

Solve the given differential equation = (1+x²) (1+y²).

Solve = 2.

UNIT-IV. VECTORS ANDTHREE DIMENSIONAL GEOMETRY

(17 MARKS)

Show that the vectors are and are orthogonal.

Show that the vectors and are orthogonal.

If , , and , show that is perpendicular to .

Find the angle between the vectors and using the scalar product of the vectors.

If 3i+j are such that is perpendicular to , then find the value of .

If are unit vectors such that .Find the value of

If be three vectors such that = 3, = 4 and = 5 and each one of them being perpendicular to the sum of the other two, find .

Three vectors satisfy the condition = . Evaluate the quantity , if = 1, =4 and = 2.

Find the shortest distance between the lines whose vector equations are

and

Find the shortest distance between the two lines and .

Find the equation of the plane passing through the points (0,-1,0),(1,1,1) and (3,3,0).

Find the equation of the plane determined by the points (3,-1,2),(5,2,4) and (-1,-1,6).

Find the equation of plane passing through the line of intersection of the planes x+2y+3z =4 and 2x+y-z+5 =0 and perpendicular to the plane 5x+3y-6z+8 =0.

Find the Cartesian as well as vector equations of the planes, through the intersection of planes which are at a unit distance from the origin.

Find the co-ordinate of the foot of the perpendicular and distance of the point from theplane. Find also the image of the point in the plane.

Find the foot of perpendicular and image of the point P(1, 2, 4) in the plane 2x + y – 2z + 3 = 0.

UNIT-V. LINEAR PROGRAMMING

(6 MARKS)

Solve the following system of equations by matrix method

A manufacturing company makes two models A and B of a product. Each piece of model A requires 9 labour hours for fabricating and 2 labour hours for finishing. Each piece of model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B. How many pieces of model A and Model B should be manufactured per week to realize a maximum profit? After income tax, sale tax, service tax, the net profit of the company is Rs 1,14,000. Should owner of the company close down the company? Discuss briefly.

A dealer in rural area wishes to purchase a number of sewing machine. He has only Rs. 5760 to invest and has space for at most 20 items. An electronic sewing machine costs him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic machine at a profit of Rs. 22 and manually operated machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy how should he invest his money in order to maximize his profit. Make it as a linear programming problem and solve it graphically. Keeping the rural background justify the values to be promoted for the selection of manually operated machine.

A manufacturer manufactures two types of steel trunks. He has two machines A and B. For completing the first type of trunk, it requires 3 hours on machine A and 1 hour on machine B; whereas the second type of trunk requires 3 hours on machine A and two hours on machine B. Machine A can work for 18 hours and B for 8 hours only per day. There is a profit of Rs.30 on the first type of the trunk and Rs.48 on second type per trunk. How many trunk of each type should be manufactured every day to earn maximum profit? Solve the problem graphically.

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs.5 and Rs.4 per unit respectively. One unit of the food A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of the food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost, but it must satisfy the requirements of the sick person. Form the question as LPP and solve it graphically. Explain the importance of balanced diet.

A cooperative society of farmers has 50 hectare of land to grow two crops Xand Y. The profit frot from crop Xand Y per hectare are estimated as Rs10,500 and Rs.9,000 respectively.aluquidherbicite has to be used for crop Xand Y at rates of 20 litres and 10 litres per hectare further no harms more than 800 litres of herbicicide should be used in order of protect fish and wildlife using pond which collects drainage from this land.How much land should be allocated to each crop so as to maximize the total profit of the society

UNIT-VI. PROBABILITY

(10 MARKS)

A fair die tossed thrice. Find the probability of getting an odd number at least once.

Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that (1) the problem is solved (2) exactly one of them solved the problem

Event A and B are such that P (A) = 1/2, P (B) =7/12 and P(not A or not B) =1/4. State whether A and B are independent.

Two cards are drawn at random and without replacement from a pack of 52 cards. Find the probability that both the cards are black.

The probabilities of A, B and C hitting a target are 1/3,2/7 and 3/8 respectively. If all the three tried to shoot the target simultaneously, find the probability that exactly one of them can shoot it.

A husband and a wife appear for interview for two vacancies for the same post. The probability of husband’s selection is 1/7 and that of wife’s is 1/5.What is the probability that (1) only one of them will be selected (2) at least one of them will be selected.

A coin is tossed once. If it shows a head,it is tossed again but if it shows a tail,then a die is thrown. If all possible outcomes are equally likely, find the probability that the die shows a number greater than 4, if it is known that the first throw of coin results a tail.

Three persons A B and C fire a target in turn,starting with A. Their probability of fitting the target are 0.5,0.3 and o.2 respectively. Find the probability of at most one hit.

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (1) both balls are red (2) one of them is black and other is red.

In an examination, an examinee either guesses or copies or knows the answer of multiple choice questions with four choices. The probability that he makes a guess is and probability that he copies the answer is . The probability that his answer is correct, given that he copied it, is . Find the probability that he knew the answer to the question, given that he correctly answered it.

A problem in statistics is given to three students A, B and C, whose chances of solving it are , and respectively. What is the probability that the problem will be solved.

CHAPTER INTEGRALS AND AREA BOUNDED UNDER THE CURVES

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ANSWERS (4 MARKS)

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