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Mathematical Induction and Binomial Theorem
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Previous Years Questions
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MCQ (Single Correct Answer)
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The number of zeros at the end of $$\left| \!{\underline {\, {100} \,}} \right. $$ is
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The coefficient of a3b4c5 in the expansion of (bc + ca + ab)6 is
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For x$$\in$$R, x $$\ne$$ $$-$$1, if $${(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ..... + {x^{2016}} ...
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If c0, c1, c2, ......, c15 are the binomial coefficients in the expansion of (1 + x)15, then the value of $${{{c_1}} \ov...
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The number of irrational terms in the expansion of $${\left( {{3^{{1 \over 8}}} + {5^{{1 \over 4}}}} \right)^{84}}$$ is
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72n + 16n $$-$$1 (n$$ \in $$ N) is divisible by
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If n is even positive integer, then the condition that the greatest term in the expansion of (1 + x)n may also have the ...
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The number (101)100 $$-$$ 1 is divisible by
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Let $${(1 + x + {x^2})^9} = {a_0} + {a_1}x + {a_2}{x^2} + ... + {a_{18}}{x^{18}}$$. Then,
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If n > 1 is an integer and x $$\ne$$ 0, then (1 + x)n $$-$$ nx $$-$$ 1 is divisible by
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If A and B are coefficients of xn in the expansions of (1 + x)2n and (1 + x)2n $$-$$ 1 respectively, then A/B is equal t...
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The number (101)100 $$-$$ 1 is divisible by
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The coefficient of xn om the expansion of $${{{e^{7x}} + {e^x}} \over {{e^{3x}}}}$$ is
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If $${(1 - x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + \,\,....\,\,{a_{2n}}{x^{2n}}$$, then the value of $${a_0} + {a_...
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Sum of the last 30 coefficients in the expansion of (1 + x)59, when expanded in ascending powers of x is...
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If in the expansion (a $$-$$ 2b)n, the sum of the 5th and 6th term is zero, then the value of $${a \over b}$$ is...
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$$({2^{3n}} - 1)$$ will be divisible by $$(\forall n \in N)$$
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using binomial theorem, the value of (0.999)3 correct to 3 decimal places is
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If the coefficients of x2 and x3 in the expansion of (3 + ax)9 be same, then the value of a is...
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If C0, C1, C2, ......, Cn denote the coefficients in the expansion of (1 + x)n then the value of C1 + 2C2 + 3C3 + ........
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For each n $$\in$$ N, 23n $$-$$ 1 is divisible by here N is a set of natural numbers.
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Product of any r consecutive natural numbers is always divisible by
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The coefficient of x$$-$$10 in $${\left( {{x^2} - {1 \over {{x^3}}}} \right)^{10}}$$ is
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If $${}^{16}{C_r} = {}^{16}{C_{r + 1}}$$, then the value of $${}^r{P_{r - 3}}$$ is
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If the magnitude of the coefficient of x7 in the expansion of $${\left( {a{x^2} + {1 \over {bx}}} \right)^8}$$, where a,...
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MCQ (More than One Correct Answer)
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The remainder when $${7^{{7^{{7^{{{..}^7}}}}}}}$$ (22 time 7) is divided by 48 is
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Subjective
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Prove by induction that for all n $$\in$$ N, n2 + n is an even integer (n $$\ge$$ 1).
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Show that, for a positive integer n, the coefficient of xk (0 $$\le$$ k $$\le$$ n) in the expansion of 1 + (1 + x) + (1 ...
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$$A = \left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]$$ then by the principle of mathematical induction, pr...
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