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Maths Matrices

If A = $(\begin{matrix} \frac{1}{\sqrt[]{5}} & \frac{2}{\sqrt[]{5}} \\ \frac{- 2}{\sqrt[]{5}} & \frac{1}{\sqrt[]{5}} \end{matrix}),$ B = $(\begin{matrix} 1 & 0 \\ i & 1 \end{matrix}), i = \sqrt[]{- 1},$ and Q = A^TBA, then the inverse of the matrix AQ²⁰²¹A^T is equal to

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Vishal Baghel

Contributor-Level 10

AA^T = A^TA = l

$A {(A^{T} B A)}^{2021} A^{T}$

= $B^{2021} = {(I + P)}^{2021} = l + 2021 P = [\begin{matrix} 1 & 0 \\ 2021 i & 1 \end{matrix}]$

$\Rightarrow {(B^{2021})}^{- 1} = [\begin{matrix} 1 & 0 \\ - 2021 i & 1 \end{matrix}]$

where P = $[\begin{matrix} 0 & 0 \\ i & 0 \end{matrix}]$

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New answer posted

3 months ago

Maths Binomial Theorem

If ²⁰C_r is the co-efficient of x^r in the expansion of (1 + x)²⁰, then the value of $\sum_{r = 0}^{20} r^{2}$ ²⁰C_r is equal to

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Vishal Baghel

Contributor-Level 10

Kindly consider the following figure

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Maths Class 11th

The sum of the series $\frac{1}{x + 1} + \frac{2}{x^{2} + 1} + \frac{2^{2}}{x^{4} + 1} + . . . . + \frac{2^{100}}{x^{2^{100}} + 1} w h e n x = 2 i s :$

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Vishal Baghel

Contributor-Level 10

$S - \frac{1}{x - 1} = \frac{- 2^{101}}{{(x^{2^{100}})}^{2} - 1}$

For x = 2, S = 1 $\frac{2^{101}}{4^{101} - 1}$

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3 months ago

Maths Class 12th

A plane P contains the line x + 2y + 3z + 1 = 0 x – y – z – 6, and is perpendicular to the plane -2x + y + z + 8 = 0. Then which of the following points lies on P?

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Vishal Baghel

Contributor-Level 10

$⊥^{r} p l a n e s$

${\vec{n}}_{1}, {\vec{n}}_{2} = 0 \Rightarrow λ = \frac{3}{4}$

Where required plane P is

$(1 + λ) x + (2 - λ) y + (3 - λ) z + 1 - 6 λ = 0$

$\Rightarrow 2 x + y + 2 z - 5 = 0$

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New answer posted

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Maths Matrices

Let $θ \in (0, \frac{π}{2}) .$ $\frac{}{}$ If the system of linear equations.

$(1 + c o s^{2} θ) x + s i n^{2} θ y + 4 s i n 3 θ z = 0$

$c o s^{2} θ x + (1 + s i n^{2} θ) y + 4 s i n 3 θ z = 0$

$c o s^{2} θ x + s i n^{2} θ y + (1 + 4 s i n 3 θ) z = 0$

has a non trivial solution, then the value of is:

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Vishal Baghel

Contributor-Level 10

$Δ = 0$

sin 3 θ= $\frac{}{}$ $\frac{1}{2}$

but $0 < θ < \frac{π}{2}$ $\frac{}{}$

θ= 70°

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3 months ago

Maths Class 11th

The equation arg $(\frac{z - 1}{z + 1}) = \frac{π}{4}$ $\frac{}{} \frac{}{}$ represents a circle with:

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Vishal Baghel

Contributor-Level 10

Centre (0, 1)

Radius = $\sqrt[]{2}$

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3 months ago

Maths Class 12th

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set:

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Vishal Baghel

Contributor-Level 10

89 – k + k + 98 – k + x = 100

=> $87 \leq k \leq 89$

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3 months ago

Maths Functions

Let f(x) = $c o s (2 t a n^{- 1} s i n (c o t^{- 1} \sqrt[]{\frac{1 - x}{x}})), 0 < x < 1 . T h e n :$

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Vishal Baghel

Contributor-Level 10

$f (x) = \frac{1 - x}{1 + x}, 0 < x < 1$ (on simplification)

$f' (x) = \frac{- 2}{{(1 + x)}^{2}}$

${(1 - x)}^{2} f' (x) = - 2 {(f (x))}^{2}$

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3 months ago

Maths Class 11th

The sum of solutions of the equation $\frac{c o s x}{1 + s i n x} = | t a n 2 x |, x \in (- \frac{π}{2}, \frac{π}{2}) - {\frac{π}{4}, - \frac{π}{4}}$ $\frac{}{} \frac{}{} \frac{}{} \frac{}{} \frac{}{}$ is:

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Vishal Baghel

Contributor-Level 10

Given equation

$\Rightarrow \frac{c o s x}{1 + s i n x} = \frac{2 | s i n x | c o s x}{| c o s^{2} x - s i n^{2} x |}$

$\Rightarrow | 1 - 2 s i n^{2} x | = 2 | s i n x | (1 + s i n x)$

Put sin x = t

then equation,

$| 1 - 2 t^{2} | = 2 | t | (1 + t), t \in (- 1, 1) - {\pm \frac{1}{\sqrt[]{2}}}$

Case I : For $\geq \frac{1}{\sqrt[]{2}}$ $\frac{}{}$

the equation has no solution

Case II : For $0 \leq t < \frac{1}{\sqrt[]{2}}$ $\frac{}{}$

Equation t = $\frac{\sqrt[]{5} - 1}{4} \Rightarrow x = 18 °$ $\frac{}{}$

Case III : $F o r \frac{- 1}{\sqrt[]{2}} < t < 0$ $\frac{}{}$

Equation t = $\frac{- 1}{2} \Rightarrow x = - 30 °$ $\frac{}{}$

Case IV : For $T < - \frac{1}{\sqrt[]{2}}$ $\frac{}{}$

Equation t = $\frac{\sqrt[]{5} + 1}{4} \Rightarrow$ $\frac{}{}$ x = 54°

Sum of solutions = 18° - 30° - 54° = 66°

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New answer posted

3 months ago

Maths Class 11th

The number of three digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is___________.

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Vishal Baghel

Contributor-Level 10

Total number = 20 + 8 + 24 = 52

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