HOLIDAY HOMEWORK FOR CLASS XII MATHS

ASSIGNMENTS

LEVEL I

  1. If a matrix has 5 elements, what are the possible orders it can have? [CBSE 2011]
  2. Construct a 3 × 2 matrix whose elements are given by aij = |i – 3j |
  3. If A = , B = , then find A –2 B.
  4. If A = and B = , write the order of AB and BA.
  5. There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommended daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using

matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the 2 families. What awareness can you create among people about the balanced diet from this question ?[CBSE 2015]

  1. For the following matrices A and B, verify (AB)T = BTAT,

where A = , B =

  1. Give example of matrices A & B such that AB = O, but BA ≠ O, where O is a zero matrix and A, B are both non zero matrices.
  2. If B is skew symmetric matrix, write whether the matrix (ABAT) is symmetric or skew symmetric.
  3. If A = and I = , find a and b so that A2 + aI = bA

LEVEL II

  1. .[CBSE 2014]
  1. Solve the following matrix eq. for x: [CBSE 2014]
  1. For what value of x, is the matrix A =a skew-symmetric matrix? [CBSE 2013]
  2. If matrix A=and A2= kA, then write the value of k [CBSE 2013]
  1. Write the element a12 of the matrix A= [aij]2X 2 , whose elements aij are given by

[CBSE 2015]

LEVEL III

1.If A = , then find the value of A2–3A + 2I

2.Express the matrix A as the sum of a symmetric and a skew symmetric matrix, where:

A =

3. If A = , prove that An = , nN

* 4. Three schools X, Y and Z organized a fete ( mela) for collecting funds for flood victims in which they sold hand- held fans , mats and toys made from recycled material , the sale price of each being Rs 25 , Rs 100 and Rs50 respectively. The following table shows the number of articles of each type sold :

Articles / School
X / y / Z
Hand held fans / 30 / 40 / 35
Mats / 12 / 15 / 20
Toys / 70 / 55 / 75

Find the total fund collected.

*5. A trust fund has Rs.30,000 that is to be invested in two different types of bonds. The first bond pays 5% p.a. interest which will be given to orphanage and second bond pays 7% interest p.a. which will be given to financial benefits of the trust. Using matrix multiplication, determine how to divide Rs.30,000 among two types of bonds if the trust fund obtains an annual total interest of Rs.1800. Why is it required to help orphan children ?

(ii) Cofactors &Adjoint of a matrix

LEVEL I

  1. Find the co-factor of a12 in A =
  2. Find the adjoint of the matrix A =

3.Verify A(adjA) = (adjA) A = I if . A =

(iii)Inverse of a Matrix & Applications

LEVEL I

  1. If A = , write A-1 in terms of A [ CBSE 2011]
  2. If A is square matrix satisfying A2 = I, then what is the inverse of A ?
  3. For what value of k , the matrix A = is not invertible ?
  1. If A = , show that A2 –5A – 14I = 0. Hence find A-1
  2. If A, B, C are three non zero square matrices of same order, find the condition

on A such that AB = AC B = C.

  1. Find the number of all possible matrices A of order 3 × 3 with each entry 0 or 1

.

LEVEL II

  1. If A= Then show that and hence find .[ CBSE 2015]

2. If A , then find using elementary row operations. [ CBSE 2015]

  1. Two schools P and Q want to award their selected students on the values of Discipline, Politeness andPunctuality. The school Pwants to award Rs. X each, Rs.Y each and RsZ each for the three respective values to its 3,2 and 1 students with a total award money of Rs. 1,000. School Q wants to spend Rs.1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards. [CBSE 2014Delhi]
  2. A typist charges Rs 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs 180. Using matrices , find the charges of typing 1 English and 1 Hindi page separately. However typist charged only Rs 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem.[ CBSE 2016 ]

LEVEL III

1 If A = , find A-1 and hence solve the following system of equations:

2x – 3y + 5z = 11, 3x + 2y – 4z = - 5, x + y – 2z = - 3

2. Using matrices, solve the following system of equations:

a. x + 2y - 3z = - 4

2x + 3y + 2z = 2

3x - 3y – 4z = 11 [CBSE 2011]

b. 4x + 3y + 2z = 60

x + 2y + 3z = 45

6x + 2y + 3z = 70 [CBSE 2011]

3. Find the product AB, where A = , B = and use it to

solve the equations x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7

4. Using matrices, solve the following system of equations:

- + = 4

+ - = 0

+ + = 2

5. Using elementary transformations, find the inverse of the matrix

*6. The RAW (Resident Welfare Association) of a colony has three different committees with total of 12 members.First committee is adult education committee, which looks afterthe literacy needs of the adults, the second committee is health and cleanliness committee, which looks after health and cleanliness needs of the colony and third committee is safety committee, which looks after the safety needs of the colony. The number of themembers of the first committee is half the sum of the members of the other two committees and the number of members of the second committee is the sum of the members of the othertwo committees. Reduce the information in the form of algebraic statements and solve using matrices.What are the benefits of keeping our surroundings clean?

(iv)Que based on

LEVEL I

  1. Evaluate [CBSE 2011]
  2. What is the value of , where I is identity matrix of order 3?
  1. If A is non singular matrix of order 3 and = 3, then find
  2. For what valve of a, is a singular matrix?

5. If A is a square matrix of order 3 such that = 64, find

6. If A is a non singular matrix of order 3 and = 7, then find

LEVEL II

  1. If ,write the value of x. . [CBSE 2014Delhi]
  2. [CBSE 2014 ALL INDIA]
  3. A is a square matrix of order 3 and | A|= 7. Write the value of | adj. A|.[CBSE 2014]

LEVEL III

  1. If A = and 3 = 125, then find a.
  2. A square matrix A, of order 3, has = 5, find

(v).Properties of Determinants

LEVEL I

1. Find positive valve of x if =

2. Evaluate

  1. Using properties of determinants, prove the following :

[CBSE 2012]

4.= (1 + pxyz)(x - y)(y - z) (z - x)

5. [CBSE 2012]

6.Using properties of determinants, prove that

[CBSE 2014]

LEVEL II

Q1 Using the properties of determinant s, solve the following for x: [ CBSE 2015]

2. .Using properties of determinants, prove that

.[CBSE 2014ALL INDIA]

3. Using properties of determinants, prove the following:

[CBSE 2013 All India]

4.Using properties of determinants, prove that[CBSE 2015]

LEVEL III

  1. Using properties of determinants, solve the following for x :
  1. = 0 [CBSE 2011]
  1. = 0 [CBSE 2011]
  1. = 0 [CBSE 2011]
  1. If a, b, c, are positive and unequal, show that the following determinant is negative:
  1. [CBSE 2013]
  2. [CBSE 2012]
  3. = 2abc( a + b + c)3
  1. If p, q, r are not in G.P and .
  2. If a, b, c are real numbers, and Show that either a + b +c = 0 or a = b = c.

*10. Using properties of determinants prove that is a perfect square.

QUESTIONS FOR SELF EVALUTION

1. Using properties of determinants, prove that :

2. Using properties of determinants, prove that : [CBSE 2015]

3. Using properties of determinants, prove that :

[CBSE 2015]

4. .Express A = as the sum of a symmetric and a skew-symmetric matrix.

5. Let A = , prove by mathematical induction that : .

6. If A = , find x and y such that A2 + xI = yA. Hence find .

7. Let A=. Prove that .

8. Solve the following system of equations : x + 2y + z = 7, x + 3z = 11, 2x – 3y = 1.

9. Find the product AB, where A = and use it to solve

the équations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.

10. Find the matrix P satisfying the matrix equation .