Let (p, q) and (P, Q) be two pairs of canonical variables. The transformation Q = qα cos βp, P = qα sin βp is canonical for

The value of the expression:sin²1° + sin²11° + sin²21° + sin²31° + sin²41° + sin²45° + sin²49° + sin²59° + sin²69° + sin²79° + sin²89° is?

A

0

B

$${\text{5}}\frac{1}{2}$$

C

$${\text{4}}\frac{1}{2}$$

D

5

Evaluate the following:
$$\frac{{\cos 2\theta \cdot \cos 3\theta - \cos 2\theta \cdot \cos 7\theta + \cos \theta \cdot \cos 10\theta }}{{\sin 4\theta \cdot \sin 3\theta - \sin 2\theta \cdot \sin 5\theta + \sin 4\theta \cdot \sin 7\theta }}$$

A

cot6θ.cot5θ

B

cos6θ.cos5θ

C

cos6θ.cot5θ

D

-cot6θ.cot5θ

Find the value of the following 3(sin⁴θ + cos⁴θ) + 2(sin⁶θ + cos⁶θ) + 12sin²θ.cos²θ = ?

A

3

B

2

C

0

D

5

Find the ∫∫x3 y3 sin⁡(x)sin⁡(y) dxdy. ]) b) (-x3 Cos(x) – 3x2 Sin(x) – 6xCos(x)-6Sin(x))(-y3 Cos(y) + 3]) c) (-x3 Cos(x) + 3x2 Sin(x) + 6xCos(x)-6Sin(x))(-y3 Cos(y) + 3y2 Sin(y) + 6yCos(y) – 6Sin(y)) d) (–x3 Cos(x) + 6xCos(x) – 6Sin(x))(-y3 Cos(y))

A

(x3 Cos(x) + 3x2 Sin(x) + 6xCos(x)-6Sin(x))(y3 Cos(y) + 3]) ) c) (-x3 Cos(x) + 3x2 Sin(x) + 6xCos(x)-6Sin(x))(-y3 Cos(y) + 3y2 Sin(y) + 6yCos(y) – 6Sin(y)) d) (–x3 Cos(x) + 6xCos(x) – 6Sin(x))(-y3 Cos(y)) View Answer]

B

y2 Sin(y) – 2[-yCos(y) + Sin(y)

C

y2 Sin(y) – 2[-yCos(y) + Sin(y)

D

y2 Sin(y) – 2[-yCos(y) + Sin(y)

sin²5° + sin²10° + sin²15° + ...... sin²85° + sin²90° is equal to?

A

$${\text{7}}\frac{1}{2}$$

B

$${\text{8}}\frac{1}{2}$$

C

9

D

$$9\frac{1}{2}$$

What is the value of {[sin (x + y) – 2 sin x + sin (x – y)]/[cos (x – y) + cos (x + y) – 2 cos x]} × [(sin 10x – sin 8x)/(cos 10x + cos 8x)]?

A

0

B

tan2x

C

1

D

2 tan x

Silicon is doped with arsenic (concentration 10²⁰ atoms.m^-3). At room temperature, the electron and hole mobilities in Si are 0.14 m².v^-1.s^-1 and 0.05 m²v^-1.s^-1 respectively. The conductivity in (?m)^-1, at room temperature for Si doped with As is

A

0.11

B

0.96

C

2.24

D

2.72

If the contents of exchangeable ions (cmol kg^-1) determined in soil are Na⁺ = 5, K⁺ = 5, Ca²⁺ = 10, Mg²⁺ = 2, H⁺ = 1, Al⁺³ = 1, Cl^- = 1.6 and HCO^-₃ =0.4, the CEC of soil is

A

26 cmol (P⁺) kg^-1

B

20 cmol (P⁺) kg^-1

C

24 cmol (P⁺) kg^-1

D

None of these

If a² sec²x - b² tan²x = c², then the value of sec²x + tan²x is equal to (assume b² ≠ a²)

A

$$\frac{{{b^2} - {a^2} + 2{c^2}}}{{{b^2} + {a^2}}}$$

B

$$\frac{{{b^2} + {a^2} - 2{c^2}}}{{{b^2} - {a^2}}}$$

C

$$\frac{{{b^2} - {a^2} - 2{c^2}}}{{{b^2} + {a^2}}}$$

D

$$\frac{{{b^2} - {a^2}}}{{{b^2} + {a^2} + 2{c^2}}}$$

Raw material purchased:
1^st January, 600 units @ Rs. 12 per unit
12^th January, 500 units @ Rs. 14 per unit
21^st January, 300 units @ Rs. 13 per unit
Raw material issued for manufacture:
3^rd January 300 units
5^th January 124 units
15^th January 250 units
16^th January 300 units
Raw material returned to stores from manufacturing department on 14^th January, 50 units. The material is issued on First-in-First out method.
The value of material remaining in store on 21^st January will be:

A

5,775

B

6,100

C

6,350

D

6,600

Let (p, q) and (P, Q) be two pairs of canonical variables. The transformation Q = q^α cos βp, P = q^α sin βp is canonical for

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