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#### Important 5 mark questions chapter 4,5

11th Standard

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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 50

Part A

10 x 5 = 50
1. Prove that 2nCn =  $\frac { { 2 }^{ n }\times 1\times3\times ...(2n-1) }{ n! }$

2. Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination.

3. Prove 1.3+2.32+3.33+...+n-3n=$\frac{(2n-1)3^{n+1}+3}{4}$ for all n ∈ N

4. Prove that the sum of the first n non-zero even numbers is n2 + n,

5. n2 - n is divisible by 6, for each natural number n $\ge$ 2.

6. 1 + 5 + 9 + ... + (4n - 3) = n(2n -1), $\forall$n $\in$N.

7. Find the sum of the series $\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+3^{ 3 }+.... }{ 1+3+5 } to\quad n\quad terms$

8. Find the coefficient of x in the expansion of $log(\frac{1}{1-5x+6x^2})$.

9. Expand ${1\over(1+3x)^2}$ in powers of x. Find a condition on x for which the expansion is valid.

10. Find the value of $(a^{2}+\sqrt{a^{2}-1})^{4}+(a^{2}-\sqrt{a^{2}-1})^{4}$