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Solution:

[3x²−a(2x−3a)]³=(3x²)³−3C1(3x²)².a(2x−3a)+3C2(3x²)a²(2x−3a)²−a³(2x−3a)³

⇒27x⁶−27x⁴a(2x−3a)+9x²a²(4x²−12ax+9a²)−a³(8x³−3×4×x²×3a+3×2x×9a²−27a³)

⇒27x⁶−54x⁵a+81a²x⁴−108a³x³+81a⁴x²−8a³x³+36a⁴x²−54a⁵x+27a⁶

⇒27x⁶−54x⁵a+117a²x⁴−116a³x³+117a⁴x²−54a⁵x+27a⁶

Concept:

The expansion of (x + y)ⁿ is given by the binomial expansion. The expansion will be 

(x + y)ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^n-1 y + ⁿC₂ x^n-2 y² + ... + ⁿC_r x^n-r y^r + ... + ⁿC_n yⁿ;

The general term is given by 

T_r =  nCr xn-r yr ;

Calculation:

Given expansion is \(\left(3x^2 + \frac 1 x\right)^{30}\)

Let the r^th term has x¹² and the coefficient be A;

⇒ T_r = A x¹²;

The general r^th term of the given expansion will be 

T_r = \(^{30}C_r (3x^2)^{30-r}(\frac {1}{x})^{r} \)

⇒ T_r = ³⁰C_r 3^30-r × x^60-2r × x^-r = 30Cr 330-r × x60-3r

Equating the power of x with 12,

⇒ 60 - 3r = 12 ⇒ r = 16

Now the coefficient will be 

A = 30Cr 330-r = ³⁰C₁₆ 3¹⁴;

Concept:

(1 + x)n =  ⁿC0 + ⁿC1 x + ⁿC2 x² +...+ ⁿCn xⁿ

Calculations:

We know that (1 + x)n = nC0 + nC1 x + nC2 x2 +...+ nCn xn

To find the sum of all the coefficients in the expansion of (1 + x)n , put x = 1

⇒(1 + 1)n = nC0 + nC1 1 + nC2 12 +...+ nCn 1n

⇒(2)n = nC0 + nC1 + nC2 +...+ nCn 

⇒ nC0 + nC1 + nC2 +...+ nCn= 2n

Hence, the sum of all the coefficients in the expansion of (1 + x)n = 2n

Concept:

The general term of the expansion of (x + a)n is usually denoted by Tr+1 =  nCr xn-r ar

Calculation:

For the given expression \(\rm \left ( bx^2+\frac{1}{bx} \right )^{13}\)  ,    n = 13

\(\rm T_{r+1}=^{13}\textrm{C}_{r}(bx^2)^{13-r}\left ( \frac{1}{bx} \right )^r\)

\(\rm T_{r+1}=^{13}\textrm{C}_{r}b^{13-r}(x^2)^{13-r}\left ( \frac{1}{b^rx^r} \right )\)

\(\rm T_{r+1}=^{13}\textrm{C}_{r}b^{13-2r}(x)^{26-3r}\)

Now, in order to find out coefficient of x⁸, 26 - 3r must be 8.

i.e.  26 - 3r = 8

r = 6

Hence putting r = 6 in equation (1) we get,

\(\rm T_{6+1}=^{13}\textrm{C}_{6}b^{13-12}(x)^{26-18}\)

\(\rm T_{6+1}=^{13}\textrm{C}_{6}bx^{8}\)

Therefore of coefficient of x⁸ is equal to ¹³C₆ b

Concept:

(1 + x)ⁿ  = [ⁿC₀ + ⁿC₁ x + ⁿC₂ x² + … +ⁿC_n x_n]

Calculation:

Given:  (1 - x)^k and the third term of the expansion is (-1/4)x²

Expansion of (1 - x)k =  [^kC₀ - ^kC1 x + ^kC2 x2 + …]

So the third term = ^kC2 x² = (-1/4)x²

\(\rm \Rightarrow \frac{k!}{2!(k-2)!}=\frac 1 4\)

\(\rm \Rightarrow \frac{k\times (k-1)\times (k-2)!}{2(k-2)!}=\frac 1 4\)

\(\rm \Rightarrow \frac{k\times (k-1)}{2}=\frac 1 4\)

\(\rm \Rightarrow k^2-k=\frac 1 2\)

\(\rm \Rightarrow 2k^2-2k- 1 =0\)                     .....(Multiply by 2 on both the sides)

\(\rm \Rightarrow k = \frac {2 \pm \sqrt {12}}{4} = \frac {1 \pm \sqrt {3}}{2}\)

\(\Rightarrow \rm k = \frac {1 \pm \sqrt {3}}{2}\)

Hence, option (1) is correct.

Concept:

General Term of binomial expansion = T_r+1 = ⁿC_r x^n-r . y^r 

Calculation:

Here, \(\rm \left(3x - \dfrac{x^3}{6}\right)^9 ={^9C}_r(3x)^{9-r}(-\frac{x^3}{6})^r\)

\(\rm ={^9C}_r(3^{9-r}x^{9-r})(\frac{(-x)^{3r}}{6^r})\)

\(\rm ={^9C}_r(3^{9-r}(-1)^{r}x^{9-r+3r})(\frac{1}{6^r})\)

Now we need x17 

So, 17 = 9 + 2r

⇒ 2r = 8

⇒ r = 4

Now, coefficient of x¹⁷ : \(\rm T_5={^9C}_4(3^{9-4}(-1)^{4})(\frac{1}{6^4})\)

\(\rm ={^9C}_4\times(\frac{3^{5}}{6^4})=\frac{9\times8\times7\times6\times 3}{4\times3\times2\times16}\)

=\(\frac{126\times 3}{16}\)

= 189/8

Correct option: c) Propeller shaft

Given in the question

Angular speed, ω = 20 rad/s

r = 10 cm = 0.10 m

Now for condition (a)

Here, The linear speed at rim = ωr

=ωr

= 20 x 0.10

= 2 m/s

Now for condition(b)

At radius middle point 

r' = 0.10/ 2 

r' = 0.05 m

Hence

The linear speed at middle point radius = ωr'

= 20 x 0.05

= 1 m/s .

When animals feel cold, they curl their bodies into the ball so as to decrease the surface area of their bodies. As total energy radiated by a body varies directly as the surface area of the body, the loss of heat due to radiation would be reduced.

(i) 1.8 × 10¹⁷ J/S

(ii) 7 × 10⁹ kg

(iii) 47.7 N/m².

All the variations in the species do not have equal chances of survival in the environment. The survival of the variations depends upon the nature of variation. Different individuals have different chances. Selection of variants by environmental factors forms the basis for evolutionary processes. These variations may lead to increased survival advantages of the individuals due to positive adoption of traits or may merely contribute to the genetic drift.