TARGET – At least 33

MATHEMATICS MINIMUM LEARNING MATERIAL

CLASS - XII

Topics Marks

  1. Binary operation 6
  2. Inverse trigonometric function 4
  3. Properties of determinants 4
  4. Symmetric and skew Symmetric matrix 4
  5. Solution of equations using matrix method 6
  6. Continuity 4
  7. Logarithmic and Parametric differentiation 4
  8. Second order differentiation 4

9. Increasing & decreasing functions 4

10. Limit of sums 4

11. Vectors 4

12. Linear programming 6

TOTAL NUMBER OF QUESTION ( 85)

MINIMUM LEARNING MATERIAL

TARGET – AT LEAST 33

1. Binary Operation (1/6 marks) (5 questions)

  1. Let * be a binary operation, on the set of all non-zero real numbers, given by a * b =

for all a, b R –{0}. Find the value of x, given that 2 *. (x * 5) = 10.

  1. The binary operation * : R × R R is defined as a * b = 2a + b. Find (2 * 3) * 4
  2. If a binary operation ‘*’ on the set of integer Z , is defined by a * b = a + 3b2 Then find the value of 2 * 4
  3. Let * be a binary operation on N defined by a*b = Show that * is commutative, associative. Find the identity element for * if exists.
  4. A= R×R and * be a binary operation on A defined by (a , b) * (c , d) = (a+c , b+d) (a , b),(c , d) A Show that * is commutative & associative on A. Find the identity element for * on A. Also find inverse of every element (a,b) A
  1. Inverse trigonometric function (4 marks) ( 8 Questions)

Use

Sin-1x = cos-1 = tan-1
  1. Prove that: tan -1 + tan -1 + tan -1 =
  2. Prove that: tan -1 + tan -1 + tan -1+ tan -1 =
  3. Find the value of
  4. Prove that : cot-1
  5. Prove that:
  6. Prove that: .
  7. Solve tan -1 2x + tan -1 3x =
  8. If tan -1 + tan -1 = , then find the value of x
  1. Symmetric and Skew Symmetric Matrix ( 4 Marks) ( 2 Questions)

1 . Express the following matrix as the sum of a symmetric and a skew-symmetric matrix and verify your result

2. For what value of x, is the matrix A =

  1. Properties of determinants(4 Marks) ( 9 Questions)

USE Make three one’s then two zero

Using the properties of determinants, show that

1. = (x – y)(y –z)(z – x)

2. =(

3. = (a - b)(b - c)(c - a)(a + b + c)

4. = xyz(x – y)(y – z)(z – x)

5. = (()

6. = (5x + 4)(x – 4)2

7. = 9 (a + b) b2

8. = (a + b + c)

9. = 2(a + b + c)

  1. Solution of equations using matrix method (6 marks) ( 7 Questions)
  1. Using matrix method, solve the following system of equations :

2x – y + z = 3 , - x +2y – z = - 4 , x – 2y + 2z = 1

  1. Using matrix method, solve the following system of equations :

x + 2y – 3z = - 4 , 2x + 3y + 2z = 2 , 3x – 3y – 4z =11

  1. Solve the following system of equations, using matrices :
  1. Find A-1, where A = , Hence, solve the system of linear equations :

x – 2y = 10, 2x + y + 3z = 8, - 2y + z = 7.

  1. Given that A = and B = , find AB and use it to solve the system

of equations : x –y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.

  1. If A = and C = . Find AC and hence solve the

Equations : x - 2 y = 10, 2 x + y + 3 z = 8, - 2 y + z = 7.

  1. The perimeter of a triangle is 90 cm. The longest side exceeds the shortest side by 16 cm and the

sum of the lengths of the longest and shortest side is twice the length of the other side. Use thematrix method to find the sides of the triangle.

6 .Continuity (4 Marks) ( 6 Questions)

A function f is said to be continuous at x =a if LHL = RHL = f(a)

Find the unknown constant if the functions are continuous

(1) (2)

(3) (4)

(5) (

(6). The function is defined as follows:

If is continuous on [0,8] find the values of a and b.

7.Logarithmic & Parametric differentiation(4 marks) ( 12 Questions)

Differentiate w.r. to x

(1) +(sinx)x (2) + (3) ( cos x) y = ( cos y )x

(4) = e x – y Show that (5) (6) (sinx)x+sin-1√x

(7) (x)sinx +(logx)x (8) (x)cosx +(sinx)tanx

(9)

(10) If

(11) If x= a cos2t and y = b sin2t , then find

(12) If x = findat

8.Differentation of Second Order ( 4 Marks) ( 4 Questions)

1 If , show that

2 .If ,then show that

3.If then show that

4. If ,then show that

9. Increasing & decreasing functions(4 Marks) ( 7 Questions)

1.Find the intervals in which the function f(x) = 2x3– 9 x2+12x+15 is increasing anddecreasing.

2.Find the intervals in which the function f(x) = 2x3+ 9 x2+12x+20 is increasing and decreasing.

3.Find the intervals in which the function f(x) = 2x3 - 15x2+36x+17 is increasing and decreasing.

4.Find the intervals in which the function f(x) = x3- 12x2+36x+17 is increasing anddecreasing.

5.Find the intervals in which the function f(x) = 20 – 9x+6x2 – x3 is increasing and decreasing.

6.Find the intervals in which the function f(x) = (x – 1)3 (x – 2)2 is increasing anddecreasing.

7. Find the intervals in which the function f(x) = sin x – cos x,is increasingand decreasing

.

10 .Limit of sums (6 marks) ( 5 Questions)

1). where nh =b-a
2). 1+2+3+4+……..(n-1) =
3). 12+22+33+……..+(n-1)2=

1. . 2. 3. 4. 5.

11. Vectors ( 4 or 5 Marks) ( 10 Questions)

1.Find a vectorof unit 6 which is parallel to

2.Find projection of on the vector.

3.Write the value of p, for which are

a)parallel vectors b) perpendicular vectors

4.Find the value of p, if (2+p

5.Show that points (6,-7,0) ,(16,-29,-4), ( 0,3,-6) and (2,5,10) are coplanar

6.Find the volume of the parallelepiped whose sides are given by

7.If any three vectors , , , are coplanar, show that the vectors +, , and , are also coplanar

8.If = = 2,find a vector which is perpendicular to both and and . = 27.

9.Find a unit vectors which is perpendicular to each of the vectors and

.

10.If are three vectors such that = 1, =4 and = 2, and + + =0, find the value of

. + . + .

11. Linear programming problems (6 Marks) ( 10 Questions)

1. A manufacturer produces nuts and bolts. It takes 1 hr of work on machine A 3 hrs on machine B to produce package of nuts. It takes 3 hrs on machine A and 1 hr on machine B to produce a package of bolts. He earns a profit of Rs 17.50 per package on nuts and Rs 7 per package on bolts. How many packages of each should be produced each day so as to maximize his profits, if he operates his machines for atmost 12 hr a day. Formulate above as a Linear Programming Problem (LPP) and solve it graphically.

2. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hrs of machine time and 3 hrs of craftman’s time in its making while a cricket bat takes 3 hrs of machine time and 1hr of craftman’s time. In aday, the factory has the availability of not more than 42 hrs of machine time 24 hrs of craftman’s time. If the profit on a racket and on a bat are Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.

3. A cottage industry manufacture pedestal lance and wooden shades, each requiring the use of grinding / cutting machine and a sprayer. It take two hours on grinding / cutting machine and three hour on sparyer to manufacture a pedestal lamp. It takes one hour on the grinding / cutting machine and two hour on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most twenty hours and the grinding / cutting machines for at the most twelve hours. The profit from the sale of a lamp is Rs. 5/- and that from a shade is Rs. 3/- assuming that the manufacturer can sale all the lamps and shades that he produce, how should he schedule his daily production in order to maximize his profit? Make an LPP and solve it graphically

4.A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost

Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed

250 units. Determine the number of units of each type of computers which the merchant should stock to get

maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs

4500 and on portable model is Rs 5000.

5. One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Findthe maximum number of cakes which can be made from 5kg of flour and 1 kg of fat, assuming that there is no shortage of the other ingredients used in making the cakes.Make an LPP and solve it graphically

6. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5760/- to invest and has space for at most 20 items. A fan cost Rs. 360/- and a sewing machine cost Rs. 240/-. He can sale 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 should invest his money in order to maximize his profit? Formulate the problem as an LPP and solve it graphically.

7..A man has Rs. 1500/- for purchasing wheat and rice. A bag of rice and a bag wheat cost Rs. 180/- and Rs. 120/- respectively. He has a storage he capacity of only 10 bags. He earns a profit of Rs. 11/- and Rs. 9/- pre bag of rice and wheat respectively. Formulate the problem as an LPP to find the numbers of bags of each type he should buy for getting maximum and solve it graphically.

8. A dietitian wishes to mix two types of foods in such a way that the vitamin contents of mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food 1 contains 2 units/ kg. of vitamin A and 1 unit/kg. of vitamin C while food II contains 1 unit/kg. of vitamin A and 2 units /kg. of vitamin C. It cost Rs. 5 per kg. to purchase food I and Rs. 7/- per kg. to purchase food II find the minimum cost of such a mixture. Formulate above as LPP and solve graphically.

9. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs4 per unit and F2 costs Rs6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists if mixture of these two foods and also meets the minimal nutritional requirements.

10. 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 reserves 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?

THANKS …

MANISH KUMAR SHARMA

PGT MATHS (KV SEC 12 DWARKA)