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Singular and Non Singualar Matrices

Let Ck=nCk ...

Question

Let $C_{k} =^{n} C_{k}$ for $0 \leq k \leq n$ and $A_{k} = [\begin{matrix} C_{k - 1}^{2} & 0 0 & C_{k}^{2} \end{matrix}]$ for $k \geq 1$ and $A_{1} + A_{2} + . . . . A_{n} = [\begin{matrix} k_{1} & 0 0 & k_{2} \end{matrix}]$ then

A

k_{1} = k_{2}

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B

k_{1} + k_{2} =^{2 n} C_{2 n} + 1

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C

k_{1} =^{2 n} C_{n - 1}

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D

k_{2} =^{2 n} C_{n - 1}

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Solution

The correct option is A $k_{1} = k_{2}$
$\begin{matrix} W e h a v e A_{k} = ∣ ∣ ∣ \begin{matrix} {c^{2}}_{k - 1} & 0 0 & c_{2}^{2} \end{matrix} ∣ ∣ ∣ A_{n} = [\begin{matrix} c_{0}^{2} & 0 0 & c_{1}^{2} \end{matrix}] + [\begin{matrix} c_{1}^{2} & 0 0 & c_{2}^{2} \end{matrix}] + . . . . . + [\begin{matrix} c_{n - 1}^{2} & 0 0 & c_{n}^{2} \end{matrix}] = [\begin{matrix} c_{0}^{2} + c_{1}^{2} + . . . + c_{n - 1}^{2} 1 p t & 0 0 & c_{1}^{2} + c_{2}^{2} + . . . + c_{n}^{2} 1 p t \end{matrix}] = [\begin{matrix} ^{2 n} c_{n} - 1 & 0 0 & ^{2 n} c_{n} - 1 \end{matrix}] {(1 + x)}^{n} =^{n} c_{0} +^{n} c_{1} x +^{n} c_{2} x^{2} + . . . . . . +^{n} c_{n} x^{n} {(x + 1)}^{n} =^{n} c_{0} x^{n} +^{n} c_{1} x^{n - 1} +^{n} c_{2} x^{x - 2} + . . . . . . . +^{n} c_{n} {(1 + x)}^{2 n} = [^{n} c_{0} +^{n} c_{1} x + . . . . +^{n} c_{n} x^{n}] [^{n} c_{0} x^{n} +^{n} c_{1} x^{n - 1} + . . . . +^{n} c_{n}] c o e f f i c i e n t o f x^{n} :^{2 n} c_{n} =^{n} c_{0}^{2} +^{n} c_{0}^{2} + . . . . +^{n} c_{n}^{2} K_{1} = K_{2} =^{2 n} c_{n} - 1 H e n c e, o p t i o n A i s t h e c o r r e c t a n s w e r . \end{matrix}$

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Similar questions

C_{k} =

^{n} C_{k}

0 \leq k \leq n

A_{k} = [\begin{matrix} C_{k - 1}^{2} & 0 0 & C_{k}^{2} \end{matrix}]

k \geq 1

A_{1} + A_{2} + . . . + A_{n} = [\begin{matrix} k_{1} & 0 0 & k_{2} \end{matrix}]

\frac{x^{2}}{a^{2} + k_{1}} + \frac{y^{2}}{b^{2} + k_{1}} = 1, \frac{x^{2}}{a^{2} + k_{2}} + \frac{y^{2}}{b^{2} + k_{2}} = 1

k_{1} \neq k_{2}

intersect at an angle

Equilibrium constants $K_{1}$ and $K_{2}$ for the following equilibria are

$N O (g) + 0.5 O_{2} (g) \to K_{1} N O_{2} (g)$ and
$2 N O_{2} (g) \to K_{2} 2 N O (g) + O_{2} (g)$

Q. The angle between the curves

\frac{x^{2}}{a^{2} + k_{1}} + \frac{y^{2}}{b^{2} + k_{1}} = 1

\frac{x^{2}}{a^{2} + k_{2}} + \frac{y^{2}}{b^{2} + k_{2}} = 1

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants $K_{1}, K_{2} a n d K_{3}$ as shown in Fig. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant K is given by

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