Assignment (To Be Done After Studying Blocks 1 and 2) Course Code: MTE-04 Assignment Code: MTE-04/TMA/2021 Maximum Marks: 100
Assignment (To Be Done After Studying Blocks 1 and 2) Course Code: MTE-04 Assignment Code: MTE-04/TMA/2021 Maximum Marks: 100
Assignment (To Be Done After Studying Blocks 1 and 2) Course Code: MTE-04 Assignment Code: MTE-04/TMA/2021 Maximum Marks: 100
1) Which of the following statements are true? Justify your answers. (This means that if you
think a statement is false, give a short proof or an example that shows it is false. If it is
true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a
set’ is true, you need to say that this is true because it is a well-defined collection of 3
objects.)
ix) The Substitution Method for solving a linear system should be employed when the
Elimination Method fails.
x) If a monic polynomial of degree n has n roots in Z , then all its coefficients are in
Z. (20)
2) a) Prove that DO, DB and DE are the AM, GM and HM of a and b, as shown in
Fig. 1, Unit 6. (6)
b) Give an example, with justification, to show why 0 < a i < 1 in Theorem 6, Unit 6. (2)
c) Prove that
[1.2 + 2.3 + ... + n (n + 1)] ≥ n + 1 , for n ≥ 1 . (2)
n (n + 3) 4
c) Prove that if A and B are sets such that A × B ≠ « , then A ∪ C ≠ « for any set C. (2)
4) a) {
Show the geometric representation of the set z ∈ C z + 1 = 5 . } (2)
i) Ferrari’s method;
ii) Descartes’ method.
Are the cubics you get from (i) and (ii) above the same?
[Hint: To obtain a solution of the resolvent cubic, you can apply Theorem 5 of Unit
6.] (15)
6. Solve the following linear systems by the method given alongside each.
i) 2x – 3y + z = 1, x + y + z = 2, 3x – 4z – 17 = 0
(by Elimination Method). (5)
ii) 2x – 3y = 1, 5 – 2y = z
(by Substitution Method) (3)
7. a) A lady bought a plot of land for ` 30 lakhs. She wanted to landscape it. So she
bought 15 bushes and 18 trees from a nursery for ` 975/-. A month later she
bought 7 bushes and 5 trees from the same nursery for ` 470/-. She paid a
gardener ` 5000/- to plant them. How much did each bush and tree cost her? (4)
4
b) A collection of 58 coins, consisting of 25p, 50p, ` 1 and ` 2 coins are in a bag. The
` 1 coins number 5 times that of the 25p coins. The ` 2 coins are double the number
of ` 1 coins, and thrice the number of the 50p coins. If the total value of the coins is
` 80/75, how many coins of each kind are there? (4)
c) Create a meaningful problem related to your life that can be represented by the
equations x – 4 = y, 2x + y = 5. (2)
25 2.5 10 1
50
335 573 π e
8) a) If A = , find A . (1)
25 2.5 10 1
50
e π 357 573
1 0 0 0 1 0 1 0 0
b) Calculate 4 5 0 , 0 4 5 and 4 5 0 . (2)
1 −1 7 7 1 −1 20 − 20 140