Simple Applications: JEE Questions & Solutions

The coefficient of x⁴⁸ in (1+x) + 2(1+x)² + 3(1+x)³ + … + 100(1+x)¹⁰⁰ is equal to

Let C_r denote the coefficient of x^r in the binomial expansion of (1+x)ⁿ, n∈ℕ, 0≤ r≤ n. If Pₙ = C₀ - C₁ + 2²/3C₂ - 2³/4C₃ + … +…

The value of (100 C₅₀)/51 + (100 C₅₁)/52 + … + (100 C₁₀₀)/101 is:

The sum of the coefficients of x⁴⁹⁹ and x⁵⁰⁰ in (1+x)¹⁰⁰⁰+x(1+x)⁹⁹⁹+x²(1+x)⁹⁹⁸+…+x¹⁰⁰⁰ is:

If Σ_(r=0)⁵ ¹¹C_(2r+1)/(2r+2) = m/n, gcd(m, n) = 1, then m - n is equal to:

The sum of all rational terms in the expansion of ( 1 + 2^(1/3) + 3^(1/2))⁶ is equal to

The least value of n for which the number of integral terms in the Binomial expansion of ( 3-root √7 + 12-root √11)ⁿ is 183, is:

The remainder, when 7¹⁰³ is divided by 23, is equal to:

The sum of all rational terms in the expansion of ( 2 + √3)⁸ is

If Σ_(r=1)⁹ ( (r+3)/2^r) · ⁹C_r = α ( 3/2)⁹ - β, α, β ∈ ℕ, then (α + β)² is equal to

Let (1 + x + x²)¹⁰ = a₀ + a₁ x + a₂ x² + … + a₂₀ x²⁰. If (a₁ + a₃ + a₅ + … + a₁₉) - 11a₂ = 121k, then k is equal to

For an integer n ≥ 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)²ⁿ⁻³ is 16, then the distance of the…

If 1² · C(15,1) + 2² · C(15,2) + 3² · C(15,3) + … + 15² · C(15,15) = 2^m · 3ⁿ · 5^k, where m, n, k ∈ ℕ, then m + n + k is equal to:

The sum of the series 2 × 1 × ²⁰C₄ - 3 × 2 × ²⁰C₅ + 4 × 3 × ²⁰C₆ - 5 × 4 × ²⁰C₇ + … + 18 × 17 × ²⁰C₂₀, is equal to

The product of the last two digits of (1919)¹⁹¹⁹ is:

If A denotes the sum of all the coefficients in the expansion of (1 - 3x + 10x²)ⁿ and B denotes the sum of all the coefficients in the…

Remainder when 64^32³² is divided by 9 is equal to:

Let α = Σ ⁿC_k/(k + 1) and β = Σ (ⁿC_k ⁿC_(k+1))/(k + 2). If 5α = 6β, then n equals:

Let a be the sum of all coefficients in the expansion of (1-2x+2x²)²⁰²³(3-4x²+2x³)²⁰²⁴. If the equations cx²+dx+e=0 and 2bx²+ax+4=0 have a…

The sum of all rational terms in the expansion of (2^(1/5) + 5^(1/3))¹⁵ is equal to:

The coefficient of x⁷⁰ in: x²(1 + x)⁹⁸ + x³(1 + x)⁹⁷ +... + x⁵⁴(1 + x)⁴⁶ is (99C_p) - (46C_q). A possible value of p + q is:

If (³⁰C₁)² + 2(³⁰C₂)² + 3(³⁰C₃)² + … + 30(³⁰C₃₀)² = (α · 60!)/(30!)², then α is equal to:

The constant term in the expansion of (2x + 1/x⁷ + 3x²)⁵ is.

The remainder when (2023)²⁰²³ is divided by 35 is:

The coefficient of x³⁰¹ in (1 + x)⁵⁰⁰ + x(1 + x)⁴⁹⁹ + x²(1 + x)⁴⁹⁸ + … + x⁵⁰⁰ is:

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of (4-root √2+1/(4-root √3))ⁿ is √6:1,…

Among the statements: (S1): 2023²⁰²² - 1999²⁰²² is divisible by 8 (S2): 13(13)ⁿ - 11n - 13 is divisible by 144 for infinitely many n ∈ ℕ…

If the coefficients of three consecutive terms in the expansion of (1 + x)ⁿ are in the ratio 1: 5: 20, then the coefficient of the fourth…

The coefficient of x⁷ in (1 - x + 2x³)¹⁰ is:

Let the number (22)²⁰²² + (2022)²² leave the remainder α when divided by 3 and β when divided by 7. Then (α² + β²) is equal to:

The number of integral terms in the expansion of ( 3^(1/2) + 5^(1/4))⁶⁸⁰ is equal to:

The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)ⁿ⁺², which are in the ratio 1: 3: 5, is equal to:

The sum of the coefficients of the first 50 terms in the binomial expansion of (1-x)¹⁰⁰, is equal to

If 1/(n+1) ⁿCₙ + 1/n ⁿCₙ₋₁ + … + 1/2 ⁿC₁ + ⁿC₀ = 1023/10 then n is equal to:

Fractional part of the number 4²⁰²²/15 is equal to:

The remainder, when 7¹⁰³ is divided by 17, is.

Let (a+bx+cx²)¹⁰=Σ_(i=0)²⁰ pᵢ xⁱ, a,b,c∈ℕ. If p₁=20 and p₂=210, then 2(a+b+c) is equal to