Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

80k Users

60 mins Expert Faculty Ask Questions

Question

If coefficient of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of x are both zeros, then is $(a, b)$ equal to

Moderate

Solution

$\begin{array}{l} S = (1 + a x + b x^{2}) (1 - 2 x)^{18} \\ = (1 + a x + b x^{2}) [1 +^{18} C_{1} (- 2 x) +^{18} C_{2} (- 2 x)^{2} +^{18} C_{3} (- 2 x)^{3} +^{18} C_{4} (- 2 x)^{4} + \dots .]] \end{array}$
Coefficient of in $x^{3}$ the expansion of $^{18} C_{3} (- 2)^{3} + a (^{18} C_{2} (- 2)^{2}) +^{18} C_{1} (- 2) b = 0$
Divide by $^{18} C_{1} (- 2)$ to obtain
$\frac{304}{3} - \frac{19}{3} a = 0 \Rightarrow a = 16$
From $(1), b = 17 \times 16 - \frac{544}{3} = \frac{272}{3}$

Ready to Test Your Skills?

Check Your Performance Today with our Free Mock Tests used by Toppers!

Talk to our academic expert!

Create Your Own Test
Your Topic, Your Difficulty, Your Pace

free study material

JEE

JEE Mock Tests

JEE study guide

JEE Revision Notes

JEE Important Questions

JEE Sample Papers

JEE Previous Year's Papers

NEET

NEET previous year’s papers

NEET important questions

NEET sample papers

NEET revision notes

NEET study guide

NEET mock tests

CUET

CUET previous year’s papers

CUET important questions

CUET sample papers

CUET revision notes

CUET study guide

CUET mock tests

CBSE

CBSE previous year’s papers

CBSE important questions

CBSE sample papers

CBSE revision notes

CBSE study guide

CBSE mock tests

NCERT solutions

Class 12 NCERT Solutions Class 11 NCERT Solutions Class 10 NCERT Solutions Class 9 NCERT Solutions Class 8 NCERT Solutions Class 7 NCERT Solutions Class 6 NCERT Solutions

NCERT exempler

Class 12 NCERT Exemplar Class 11 NCERT Exemplar Class 10 NCERT Exemplar Class 9 NCERT Exemplar Class 8 NCERT Exemplar Class 7 NCERT Exemplar Class 6 NCERT Exemplar

State Board

Rajasthan State Board West Bengal State Board Gujarat State Board Telengana State Board Andhra Pradesh State Board Uttar Pradesh State Board

Subject

Maths Science Physics Chemistry Economics Accounting Business Studies

2023 | www.infinitylearn.com

Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

If coefficient of x3 and x4 in the expansion of 1+ax+bx2(1−2x)18 in powers of x are both zeros, then is (a, b) equal to

S=1+ax+bx2(1−2x)18=1+ax+bx21+18C1(−2x)+ 18C2(−2x)2+18C3(−2x)3+18C4(−2x)4+…. Coefficient of in x3 the expansion of 18C3(−2)3+a 18C2(−2)2+18C1(−2)b=0Divide by 18C1(−2) to obtain 3043−193a=0⇒a=16From (1),b=17×16−5443=2723

Ready to Test Your Skills?

free study material

If coefficient of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of x are both zeros, then is $(a, b)$ equal to