QUESTION BANK

Tilak Maharashtra Vidyapeeth

(Deemed University)

QUESTION BANK

APPLIED MATHEMATICS - I

First Year Diploma (All)

UNIT 1 : ALGEBRA

I) ALGORITHM

^1.If 2x = 23y = 512 Find value of log zx3 where x = y3

2. Find value of a) log ₃₄₃7 b) log ₅_Ö250 c) log ₉27 d) log ₃61 e) log ₅125 f) log ₁₀ 1000 g) log ₂_Ö312

3. Simplify a) log₂14 – log₂7 b) (log ₃4) * (log ₄81) c) log ₅243

Log ₂₅9

d) log _ab * log_bc * log _da

e) log a²/bc + log b²/ca + log c²/ab

4. Solve for x

a) log ₂(x+5) + log₂(x-2) = 3

b) log x/ log 5 = log 25/ log 125

c) (4 log 3) (log x) = log 27

log 9

d) log _X = log 64

log 4 log 16

d) log_x² + log_x⁴+ log_x⁸= 6

5. Prove that

1. log_xa³ * log_cb³ * log_ac³ = 27

2. if log ( x+y ) = 1 ( log x + log y)

7 2

Prove that x + y = 47

Y x

3 4

3. log _y x * log _Z y * log _x z³= 1

4. log x² + log y²+logz²= 0

y^zz^x x^y

UNIT 2 DETERMINANTS

I) Solve determinant equations

2 3 1

a) 6 x 2 = 0

4 x -2

4 9 2 2 2x

b) 3 x 7 = 2 2x

8 1 6

4x + 2 2x + 1 1 2x 4x²

c) x + 1 2x - 1 = 0, in 1 4 16

1 1 1

1 x x² 1 1

d) 1 2 4 = 2 2

1 3 9

II) With out expanding prove that

4 5 7 5 3 11

a) 2 3 1 + 4 2 9 = 0

9 11 13 7 1 13

4 5 7 4 4 3

B) 2 3 1 = -2 1 5

9 11 13 -1 -3 7

5 1 3 10 1 3

c) 6 2 5 + 19 2 5 = 0

9 1 7 26 1 7

III) Without expanding find the value of

6 1 9

a) 9 4 7

18 3 27

x-y y-z z-x

b) y-z z-x x-y

z-x x-y y-z

6 9 12

c) 2 3 4

5 6 13

3 4 5

7 8 9

d) 1 1 1

61 54 57

e) 52 55 58

53 56 59

V) Solution of Simultaneous equation using CRAMER’S Rule

i) 4x - 3y = 2 ii) 2x - 3y + 1 = 0

3x + 5y = -13 5y – 4x = 3

ii) 1 + 3 = 5 iv) 3

x y x - 2

3 - 4 = 2

x y

V) 2x + y + z = 1 vi) x + y + z = 6

X + Y = 2z = 0 3x + 3y + z = 12

5x + 3y = 3z = 2 2x + 3y + z = 14

vii) x + y + z = 1 viii) x + y + z = 4

2x + 3y + z =4 2x + y + z = 1

4x + 1y + z = 16 x – y + z = -3

ix) x + y + z = 6 x) x – 2y + 3z = 4

2x + y + 2z = -2 2x + y – 3z = 5

x + y –3z = -6 x – y – 2z = -3

III) PARTIAL FRACTION

Find partial fraction

1) x+1 2) (x+3) (x+1)

(x-z)(x-3) x(x+4) (x+2)

3) x 4) x

(x-1) (x-2) (x-3) x² + x-z

5) x-5 6) 1

x³ + x² –6x x² + 3x + z

7) 5x + 1 8) 3x-z

x²+x-z (x-z)(x=3)

9) x²-2x+7 10) 3x+2

(x+1) (x2 –1)² (x + 1) (x²-1)

11) 3x²-4x 12) x²+1

(x-1)³ x³ + 1

Binomial Theorem

i) Expand using Binomial Theorem

a) (2x+3y)⁴ b) 2x - 3 c) (x-2y)²

3 2x

ii) Using Binominal thermo prove that

5 5

2 + 1 - 2 -1 = 82

5 5

b) 3 + 1 - 3 - 1 = 152

iii) a) Find 5^th term of (x = 2y)⁸

b) Find 4^th term of x³ - 2 9

2 x²²

c) Find the term independent of x in the expansion of x³ + m is 1320

x⁸

find m.

2x³ - b

iv) a) Find the middle term in expansion of a x

b) Find the middle terms in expansion of ( 3x – x³/ 9 )⁹

v) a) using binomial thermo find approximate value of 9. 18 .

b) Find approximate value of 1 using binominal theorem.

c) Find approximate value of 3 997 & 408

UNIT 2 TRIGNOMETRY

5) prove pollinating

i) Cosec²θ - Cos²θ . Cosec²θ = 1

ii) Sec² θ + Cosec² θ = Sec² θ Cosec² θ

1 - sin θ = Sec θ – Tan θ

iii)

1 + sin θ

iv) Sin θ 1 – Sin θ

+ = 2 Sec θ - Tan θ

1+ Cos θ Cos θ

v) Cos² θ 1 + Tan² θ = 1

vii) Sin θ + Cot θ Cos θ = Cosec θ

viii) Sec² θ – Sin² θ Sec² θ = 1

ix) Sec² θ + Cosec² θ = tan θ + Cot θ

x) Sec θ + 1 = Cot θ + Cosec θ

Sec θ - 1

6) Trigonometric Ratios of Allied, Compound, Multiple angle

2) Find the value of or show that

i) Tan² 45 + Cos 60 – Cosec² 30 + Sin² 0 = -5/2

ii) Cos 45⁰ Cos 60⁰ - Sin 45 Sin 60 = - 3 - 1

2 2

iii) tan 420 + tan 300 = 0

1- tan 420 tan 300

iv) sin²30 cos 60 + cos² 45 tan 45

v) tan²45 – cosec²30 + 2 sin 30 = -2

v) if tan θ = 1/ find value of cosec² θ – sec² θ