Exams › JEE Advanced › Maths
Correct answer: 27
The coefficient of x² in (x³ + 2x² + x + 4)¹⁵ is 15 * 2 * 4¹⁴ = 30 * 4¹⁴ = 30 * 2²⁸. Check divisibility: 8=2³ (yes), 25=5² (no, 30=2*3*5 so 5 divides once, 5² does not divide), 27=3³ (30 has only one factor 3, so 27 does not divide either), 64=2⁶. Let's count powers of 2: 30*2²⁸ = 2*3*5*2²⁸ = 2²⁹*15. So 64=2⁶ divides it. 25 does not divide (only one factor of 5). 27 does not divide (only one factor of 3). 8=2³ divides. So coefficient is not divisible by 25 and not divisible by 27. Among the options, 64 divides it (since 2²⁹ has 2⁶). The one NOT dividing: between 25 and 27, the question asks which is NOT a divisor. Since both 25 and 27 don't divide, but the standard answer for this question is 64... Re-examine: coefficient = C(15,1)*2*4¹⁴ + C(15,1)*1*(terms for x from others giving x)... Actually x alone from one factor and x from another gives x²: C(15,2)*1*1*4¹³. So coeff = 15*2*4¹⁴ + C(15,2)*1*4¹³ = 15*2*4¹⁴ + 105*4¹³ = 4¹³*(15*2*4 + 105) = 4¹³*(120+105) = 4¹³*225 = 2²⁶ * 225 = 2²⁶ * 9 * 25. Divisible by 8 (2³), 25, 27? 225 = 9*25, only 3² not 3³, so not divisible by 27. Divisible by 64=2⁶? 2²⁶ yes. So NOT divisible by 27. But answer listed as 64? 2²⁶*225: 225 is odd, so total power of 2 is 26. 64=2⁶ divides. 27=3³: only 3² in 225, so 27 does not divide. Answer should be 27.