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## Tuesday, 18 August 2020

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Notes-1

1. (a) Show that
1 + a     1        1
1        1+b       1    = abc + bc + ca + ab.
1          1        1+c

(c) Use the principle of mathematical induction to show that :

(d) Find the 18th term of a G.P. whose 5th term is 1 and common ratio 2/3.
(e) If (a - ib) (x + iy) = (a2 + b2) i and a + ib ≠0,find x and y.
(f) Find two numbers whose sum is 54 and product is 629.
(h) Find the equation of the straight line through
(-2, 0, 3) and (3, 5, -2).

2. (a)               5  3  0
If A = 3  2  0 , find A-1.
0  0  1
(b) Solve the system of equations x + y + z = 5,y + z = 2, x + z = 3 by using Cramer's rule.
(c) Find the area of A ABC whose vertices are
A (1, 3), B (2, 2) and C (0, 1).

(d)
5  3  8
Reduce A = 0  1  1 to normal
1 -1  0
form by elementary operations.

3. (a) Find the sum to n terms of the series 0.7 + 0.77 + 0.777 + .......
(b) Find three terms in G.P. such that their sum is 31 and the sum of their squares is 651.
(c) If 𝛂 and 𝛃 are roots of x2 - 4x + 2 = 0, find the equation whose roots are 𝛂2 + 1 and 𝛃2+1.
(d) Solve the inequality
x2 - 4x - 21 ≤ 0

4. (a) Find the value of constant k so that

is continuous at x = 5.

(c) If a mothball evaporates at a rate proportional to its surface area 4𝜋𝖗2. 2, show that its radius decreases at a constant rate.

5. (a) Show that the three points with position

(b) Find the direction cosines of the line passing through (1, 2, 3) and (-1, 1, 0).
(c) Two -electricians, A and B, charge 400 and 500 per day respectively. A can service 6 ACs and 4 coolers per day while B can service 10 ACs and 4 coolers per day. For how many days must each be employed so as to service at least 60 ACs and at least 32 coolers at minimum labour cost ? Also
calculate the least cost.

Notes-2

1. Attempt any eight parts from the following :

show that A2 -4A+5I2 = 0 Also, find A4.
(c) Show that 133 divides 11n+2 + 12 2n+1 for every natural number n.
(d) If pth term of an A.P is q and qth term of the A.P. is p, find its rth term.
(e) If 1,𝛚,𝛚2 are cube roots of unity, show that
(2-𝛚) (2-𝛚2) (2-𝛚19) (2-𝛚23) = 49
(f) If 𝛂,𝛃 are roots of x2-3ax + a2 = 0, find the value(s) of a if 𝛂2+𝛃2=7/4.

2 -1 0
2. (a)        If A =  1  0 3 , show that
3  0 -1
2 -1 7
(b)  If A = 3 5 2 , show that A is row
1 1 3
equivalent to I3.

AB = 6 I3. Use it to solve the system of linear equations x-y = 3, 2x + 3y + 4z = 17,y+ 2z= 7.

3. (a) Find the sum of all the integers between 100 and 1000 that are divisible by 9.
(b) Use De Moivre's theorem to find (√3+i)3.
(c) Solve the equation x3-13x2 + 15x + 189 = 0,given that one of the roots exceeds the other by 2.
(d) Solve the inequality 2/|x-1|>5 and graph its solution

4. (a) Determine the values of x for which f(x) = x4-8x3 + 22x2-24x + 21 is increasing and for which it is decreasing
(b) Find the points of local maxima and local minima of
f(x) = x3-6x2 + 9x + 2014, x 𝓔 𝗥.

(d) Using integration, find length of the curve y = 3-x from (-1, 4) to (3, 0).

5.

(c) A tailor needs at least 40 large buttons and 60 small buttons. In the market, buttons are available in two boxes or cards. A box contains 6 large and 2 small buttons and a card contains 2 large and 4 small buttons.If the cost of a box is Rs 3 and cost of a card is Rs 2, find how many boxes and cards should be purchased so as to minimize the expenditure.

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