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Prove that if p is a prime number greater than 2, then the difference `[(2+sqrt(5))^p]-2^(p+1)` is divisible by p, where [.] denotes greatest integer.

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Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

show that [(sqrt(3) +1)^(2n)] + 1 is divisible by 2^(n+1) AA n in N , where [.] denote the greatest integer function .

Statement-1: the highest power of 3 in .^(50)C_(10) is 4. Statement-2: If p is any prime number, then power of p in n! is equal to [n/p]+[n/p^(2)]+[n/p^(3)] + . . ., where [*] denotes the greatest integer function.

A die is thrown three times. The chance that the highest number shown on the die is 4 is p, then the value of [1//p] is where [.] represents greatest integer function is _________.

A die is thrown three times. The chance that the highest number shown on the die is 4 is p, then the value of [1//p] is where [.] represents greatest integer function is _________.

Prove that a positive integer n is prime number, if no prime p less than or equal to sqrtn divides n.

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Let the sum sum_(n=1)^(9)1/(n(n+1)(n+2)) written in the rational form be p/q (where p and q are co-prime), then the value of [(q-p)/10] is (where [.] is the greatest integer function)

Let PN be the ordinate of a point P on the hyperbola x^2/(97)^2-y^2/(97)^2=1 and the tangent at P meets the transverse axis in T, O is the origin, then [(ON.OT)/2010]=------------, where [.] denotes the greatest integer function.

ARIHANT MATHS ENGLISH-BIONOMIAL THEOREM-Exercise (Subjective Type Questions)
  1. Find the value of (18^3+7^3+3xx18xx7xx25) /(3^6+6xx243xx2+15xx18xx4+20...

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  2. Determine the term independent of a in the expansion of ((a+1)/(a^(2/...

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  3. If in the expansion of (1+x)^n ,a ,b ,c are three consecutive coeffici...

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  4. In (3 3+1/(3 3))^n if the ratio of 7th term from the beginning to the ...

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  5. if Sn = C0C1 + C1C2 +...+ C(n-1)Cn and S(n+1)/Sn = 15/4 then n is

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  6. C1/C0+2C2/C1+3C3/C2+............+nCn/C(n-1)=(n(n+1))/2

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  7. Find the term in(3sqrt(((a)/(sqrt(b))) + (sqrt((b)/ ^3sqrt(a))))^(21) ...

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  8. The coefficient of x^r[0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+...

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  9. Prove that if p is a prime number greater than 2, then the difference ...

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  10. Integer just greater tehn (sqrt(3)+1)^(2n) is necessarily divisible by...

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  11. Solve the equation ""^(11)C(1) x^(10) - ""^(11)C(3) x^(8) + ""^(11)...

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  12. If (1 + x)^(n) = C(0) = C(1) x + C(2) x^(2) + …+ C(n) x^(n) , find...

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  13. Evaluate sum(0 le i le j le 10) ""^(21)C(i) * ""^(21)C(j).

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  14. Find the coefficient of x^4 in the expansion of (1+x+x^2+x^3)^(11)dot

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  15. Find the coefficient of x^4 in the expansion of (2-x+3x^2)^6dot

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  16. If for z as real or complex, (1+z^2+z^4)^8=C0+C1z2+C2z4++C(16)z^(32)t ...

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  17. If for z as real or complex such that (1+z^(2) + z^(4))^(8) = C(0)...

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  18. If a0,a1,a2,.... be the coefficients in the expansion of (1+x+x^2)^n ...

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  19. If a(0), a(1), a(2),… are the coefficients in the expansion of (1 ...

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  20. If a(0), a(1),a(2),… are the coefficients in the expansion of (1 +...

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