Binomial Theorem - Algebra - Mathematics Previous Year Questions

The number of zeros at the end of \(\left| \!{\underline {\, \[{100} \,}} \right.\) is\]

If \(n\) is a positive integer, the value of \((2 n+1){ }^n C_0+(2 n-1){ }^n C_1+(2 n-3){ }^n C_2 +\ldots .+1 \cdot{ }^n C_n\) is

If \(\left(1+x+x^2+x^3\right)^5=\sum_\limits{k=0}^{15} a_k x^k\) then \(\sum_\limits{k=0}^7(-1)^{\mathbf{k}} \cdot a_{2 k}\) is equal to

The coefficient of \(a^{10} b^7 c^3\) in the expansion of \((b c+c a+a b)^{10}\) is

If $\left(1+x-2 x^2\right)^6=1+a_1 x+a_2 x^2+\ldots+a_{12} x^{12}$, then the value of $a_2+a_4+a_6+\ldots+a_{12}$ is

The value of

$$99^{50}-90 \cdot 98^{50}+\frac{99 \cdot 98}{1 \cdot 2}(97)^{50}-\ldots \ldots \ldots \ldots .+99$$

is

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^3}{(1-x)^{1 / 2}}$ may be approximate as

If $a$ and $b$ are two complex numbers, then the sum of $(n+1)$ terms of the series $a c_0-(a+d) c_1+(a+2 d) c_2-(a+3 d) c_3+$ $\_\_\_\_$ is

If the magnitude of the coefficient of x⁷ in the expansion of \({\left( {a{x^2} + {1 \over {bx}}} \right)^8}\), where a, b are positive numbers, is equal to the magnitude of the coefficient of x⁷ in the expansion of \({\left( {ax + {1 \over {b{x^2}}}} \right)^8}\), then a and b are connected by the relation

If \({}^{16}{C_r} = {}^{16}{C_{r + 1}}\), then the value of \({}^r{P_{r - 3}}\) is

The coefficient of x^\(-\)10 in \({\left( {{x^2} - {1 \over {{x^3}}}} \right)^{10}}\) is

If C₀, C₁, C₂, ......, C_n denote the coefficients in the expansion of (1 + x)ⁿ then the value of C₁ + 2C₂ + 3C₃ + ..... + nC_n is