AMU MCA Binomial Theorem Previous Year Questions (PYQs)
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Showing 5 questions
If the sum of the coefficient in the expansion of $(x + y)^n$ is $1024$, then the value of the greatest coefficient in the expansion is
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Solution
The coefficient of $x^{53}$ in the expansion
$\sum_{m=0}^{100} {^{100}C_m}(x-3)^{100-m}\cdot2^m$ is
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Solution
The coefficients of three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1:7:42$, then the value of $n$ is
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Solution
Let coefficients be
$ ^nC_{r-2},\ ^nC_{r-1},\ ^nC_r $
Given ratio:
$ \frac{^nC_{r-2}}{^nC_{r-1}}=\frac{1}{7} $
$ \Rightarrow \frac{r-1}{n-r+2}=\frac{1}{7} $
$ \Rightarrow 7r-7=n-r+2 $
$ \Rightarrow n-8r+9=0 \quad ...(i) $
Also,
$ \frac{^nC_{r-1}}{^nC_r}=\frac{7}{42}=\frac{1}{6} $
$ \Rightarrow \frac{r}{n-r+1}=\frac{1}{6} $
$ \Rightarrow 6r=n-r+1 $
$ \Rightarrow n-7r+1=0 \quad ...(ii) $
Solving (i) and (ii),
$ r=8,\ n=55 $
If $49^n + 16n + \lambda$ is divisible by $64$ for all $n \in N$, then the least negative integral value of $\lambda$ is
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Solution
$49 \equiv -15 \pmod{64}$
Check small $n$ ⇒ condition satisfied at $\lambda = -1$
A polygon has $44$ diagonals. The number of its sides are
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Solution
$\frac{n(n-3)}{2} = 44$
$n(n-3)=88 \Rightarrow n=11$
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