PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

Using binomial theorem, prove that  \[3^{2n + 2} - 8n - 9\]  is divisible by 64, \[n \in N\] .

Find the coefficient of: 

(i) x¹⁰ in the expansion of  \[\left( 2 x^2 - \frac{1}{x} \right)^{20}\]

Name the octants in which the following points lie: (5, 2, 3)

Find the coefficient of: 

(ii) x⁷ in the expansion of  \[\left( x - \frac{1}{x^2} \right)^{40}\]

Find the coefficient of: 

(iii)  \[x^{- 15}\]  in the expansion of  \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]

Find the coefficient of: 

(iv)  \[x^9\]  in the expansion of  \[\left( x^2 - \frac{1}{3x} \right)^9\]

Find the coefficient of: 

(v)  \[x^m\]  in the expansion of  \[\left( x + \frac{1}{x} \right)^n\]

Name the octants in which the following points lie:

(–5, 4, 3) 

Name the octants in which the following points lie: 

(4, –3, 5)

Name the octants in which the following points lie: 

 (7, 4, –3)

Name the octants in which the following points lie: 

(–5, –4, 7) 

Name the octants in which the following points lie: 

(–5, –3, –2) 

Name the octants in which the following points lie:

 (2, –5, –7) 

Name the octants in which the following points lie: 

(–7, 2 – 5)

Find the image  of: 

 (–2, 3, 4) in the yz-plane.

Find the image  of: 

 (–5, 4, –3) in the xz-plane. 

Find the image  of:

 (5, 2, –7) in the xy-plane.

Find the coefficient of: 

(vi) x in the expansion of  \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]

Find the coefficient of: 

(vii) \[a^5 b^7\]  in the expansion of  \[\left( a - 2b \right)^{12}\]

Find the coefficient of: 

(viii) x in the expansion of \[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16}\]