Can you please solve this question along with telling general approach for such pattern of questions Q 5 - Maths - Probability

Question

Can you please solve this question along with telling general
approach for such pattern of questions?
Q 5 If 1 + x + x 2 + x 3 x 5 = a k k = 0 15      
    x k   t h e n   a 2 k k = 0 7    
          is equal to
(a) 128
(b) 256
(c) 512
(d) 1024

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

Each question requires different apporach
 But expansions withmore than 2 terms are rarely asked even in exams like JEE Advanced
Now1+x+x2+x35=∑k=015 akxk⇒1+x+x2+x35=a0+a1x+a2x2+a3x3+
+a14x14+a15x15 iPut x=1⇒1+1+1+15=a0+a1+a2+a3+
+a14+a15⇒45=a0+a1+a2+a3+ +a14+a15
iiPut x=-1 in i⇒1-1+1-15=a0-a1+a2-a3+
+a14-a15⇒0=a0-a1+a2-a3+ +a14-a15 iiiii+iiia0+a1+a2+a3+
+a14+a15+a0+a1+a2+a3+ +a14+a15=45⇒2a0+a2+
+a14=1024⇒a0+a2+
+a14=512⇒∑k=07 a2k=512 Answer

Hope this information will clear your doubts about this topic

If you have any doubts just ask here on the ask and answer forum
and our experts will try to help you out as soon as possible
Regards

Can you please solve this question along with telling general approach for such pattern of questions? Q.5. If 1 + x + x 2 + x 3 x 5 = a k k = 0 15 x k t h e n a 2 k k = 0 7 is equal to (a) 128 (b) 256 (c) 512 (d) 1024

Can you please solve this question along with telling general approach for such pattern of questions?
Q.5. If $(1 + x + x^{2} + x^{3}) x^{5} = {}_{k = 0}^{15}a_{k} x^{k} t h e n {}_{k = 0}^{7}a_{2 k}$ is equal to
(a) 128
(b) 256
(c) 512
(d) 1024