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Concept:

When A² + B² + C² = 0, it implies A = B = C = 0 (since the squares of real numbers are non-negative).

Substitute the values of A, B, and C for determinant calculation into the matrix

Calculation:

$\begin{vmatrix} 1& cos0& cos0\ cos0&1&cos0 \ cos0&cos0&1 \end{vmatrix}$

Since, Cos0 =1

Thus Matrix becomes 

$\begin{vmatrix} 1& 1& 1\ 1&1&1 \ 1&1&1 \end{vmatrix}$

Now determinant = 1[(1×1 - 1×1)] - 1[(1×1 - 1×1)] + 1[(1×1 - 1×1)]

= = 1(0) - 1(0) + 1(0) = 0

∴ The value of the determinant is 0.

Hence, the correct answer is Option 2.

Calculation:

Given,

Let ω be a non-real cube root of unity, so $\omega^{3}=1$ and $1+\omega+\omega^{2}=0$.

Consider the determinant

$\Delta(x)= \begin{vmatrix} x+1 & \omega & \omega^{2}\ \omega & x+\omega^{2} & 1\ \omega^{2} & 1 & x+\omega \end{vmatrix}=0.$

Step 1 — Column operation:  Replace the first column by $C_{1}-C_{2}$:

$\Delta(x)= \begin{vmatrix} x^{2}-1 & ,2!k!+!1, & 1\ k-1 & k+2 & 1\ 0 & 3 & 1 \end{vmatrix}.$

Step 2 — Expansion along the third row:

$\Delta(x)= -3! \begin{vmatrix} k^{2}-1 & 1\ k-1 & 1 \end{vmatrix} + \begin{vmatrix} k^{2}-1 & 2k+1\ k-1 & k+2 \end{vmatrix},$

which simplifies to

$\Delta(x)=x(x^{2}-1)-x\bigl(\omega+\omega^{2}\bigr) =x(x^{2}-1)+x =x^{3}.$

Step 3 — Equate to zero:

$\Delta(x)=0 ;\Longrightarrow; x^{3}=0 ;\Longrightarrow; x=0.$

∴ The root of the equation is  $x = 0$.

Hence, the correct answer is Option 1.

Concept:

The key concept involves simplifying complex numbers in the polar form. We use the argument (angle) of the complex number and De Moivre's theorem, which states:

  (r(cosθ + i sinθ))ⁿ = rⁿ(cos(nθ) + i sin(nθ))

Calculation:

⇒ $\left( \frac{\sqrt{3} + i}{\sqrt{3} - i} \right)^3$

Multiply the numerator and the denominator by the conjugate of the denominator

⇒ $\frac{\sqrt{3} + i}{\sqrt{3} - i} \times \frac{\sqrt{3} + i}{\sqrt{3} + i}$

⇒ $\frac{(\sqrt{3} + i)^2}{(\sqrt{3} - i)(\sqrt{3} + i)}$

 $\frac{2 + 2\sqrt{3}i}{4} = \frac{1 + \sqrt{3}i}{2}$

Convert to polar form. The modulus r is 

$r = \sqrt{\left( \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$

The argument θ is 

$\theta = \tan^{-1}\left( \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

So the polar form of the complex number is

$1 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$

To cube the complex number, we use De Moivre’s Theorem

In our case, r = 0 so,

$\left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)^3 = \cos \pi + i \sin \pi = -1 + 0i = -1$

Thus, the result of cubing the complex number is $\boxed{-1}$

Hence, the correct answer is Option 1.

Calculation:

Given,

The equation is $x^2 - x + 1 = 0$

We need to find the value of the following expression:

$\left( x - \frac{1}{x} \right)^2 + \left( x - \frac{1}{x} \right)^4 + \left( x - \frac{1}{x} \right)^8$

The equation $x^2 - x + 1 = 0$ is solved as follows:

$x = \frac{1 \pm \sqrt{-3}}{2} = e^{i \pi / 3} \quad \text{or} \quad x = e^{-i \pi / 3}$

Now, substitute the value of $x$ into the expression $x - \frac{1}{x}$:

$x - \frac{1}{x} = i\sqrt{3}$

Evaluate the powers of $x - \frac{1}{x}$

Now, let's evaluate each term in the expression:

$\left( x - \frac{1}{x} \right)^2 = (i \sqrt{3})^2 = -3$

$\left( x - \frac{1}{x} \right)^4 = (-3)^2 = 9$

$\left( x - \frac{1}{x} \right)^8 = 9^2 = 81$

Now, sum the values:

$-3 + 9 + 81 = 87$

∴ The value of the expression is 87.

Hence, the correct answer is Option 3.

Concept:

If all consonants are treated as a single unit, the problem reduces to arranging the unit (CPTL) and vowels (A, I, A).

Within the consonant unit, the consonants themselves can be permuted.

When letters repeat, the arrangements are divided by the factorial of the repeated letters.

Calculation:

⇒ Treat consonants CPTL as a single unit.

⇒ Total "super units" = {CPTL, A, I, A} = 4 units.

⇒ Arrangements of super units = 4! / 2! (since A repeats).

⇒ Arrangements of super units = 24 / 2 = 12. ...(1)

⇒ Internal arrangements of consonants CPTL = 4!.

⇒ Internal arrangements = 4 × 3 × 2 × 1 = 24. ...(2)

⇒ Total arrangements = (Arrangements of super units) × (Internal arrangements of consonants).

⇒ Total arrangements = 12 × 24.

⇒ Total arrangements = 288.

∴ The total number of 7-letter words = 288.

Hence, the correct answer is Option 3.

Concept:

1. The amplitude (or argument) of a complex number $z = r(\cos\theta + i\sin\theta)$ is given by $\text{amp}(z) = \theta$.
2. The conjugate of $z$, denoted as $\overline{z}$, has an amplitude $\text{amp}(\overline{z}) = -\theta$, because conjugating a complex number reflects it across the real axis in the Argand plane.

Formula Used:

$\text{amp}(z) + \text{amp}(\overline{z}) = \theta + (-\theta) = 0$.

Calculation:

$z = r(\cos\theta + i\sin\theta)$

$\overline{z} = r(\cos\theta - i\sin\theta)$

$\text{amp}(z) + \text{amp}(\overline{z}) = \theta + (-\theta) = 0$

Conclusion:

$\therefore \text{amp}(z) + \text{amp}(\overline{z}) = 0$.

Hence, the correct answer is Option 1.

Calculation:

Given,

Number of diagonals, D = 20 

Formula for the number of diagonals in a polygon is:

$D = \frac{n(n - 3)}{2}$

Substitute the value of D = 20  in the formula:

⇒ $20 = \frac{n(n - 3)}{2}$

Multiply both sides by 2:

⇒ $40 = n(n - 3)$

⇒ $n^2 - 3n - 40 = 0$

We now solve this quadratic equation:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

⇒ $n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-40)}}{2(1)}$

⇒ $n = \frac{3 \pm 13}{2}$

The two solutions are:

⇒ $n = \frac{3 + 13}{2} = 8$

or

⇒ $n = \frac{3 - 13}{2} = -5$

Since the number of sides cannot be negative, we choose n = 8.

∴ The polygon has 8 sides.

Hence, the correct answer is Option 3.

Calculation:

Number of vowels = 2 (E, I). Positions of vowels are fixed.

⇒ Arrangements of vowels = 2! = 2 × 1 = 2

Number of consonants = 3 (D, L, H). Positions of consonants are fixed.

⇒ Arrangements of consonants = 3! = 3 × 2 × 1 = 6

Total arrangements = Arrangements of vowels × Arrangements of consonants.

⇒ Total arrangements = 2 × 6 = 12

∴ The total number of arrangements is 12.

Hence, the correct answer is Option 3.

Concept:

The number of positive integer solutions for the equation x + y + z = n is given by the combination formula:

Number of solutions = C(n - 1, k - 1), where k is the number of variables.

Formula Used:

C(n, r) = n! / [r! × (n - r)!]

Calculation:

Here, n = 5 and k = 3.

The number of solutions is:

⇒ C(5 - 1, 3 - 1) = C(4, 2)

⇒ C(4, 2) = 4! / [2! × (4 - 2)!]

⇒ C(4, 2) = 24 / [2 × 2]

⇒ C(4, 2) = 6

The correct answer is Option 3.

Concept:

A term in the binomial expansion of (a + b)ⁿ is given by T_k+1 = C(n, k) × a^n-k × b^k.

For a term to be rational, the exponents of both √3 and 5^1/4 must be integers.

Formula Used:

In (√3)^n-k, n-k must be even for it to be rational.

In (5^1/4)^k, k must be a multiple of 4 for it to be rational.

Calculation:

Let n = 12:

⇒ For √3^n-k to be rational, n-k must be even.

⇒ Since n = 12, k must also be even.

⇒ For (5^1/4)^k to be rational, k must be a multiple of 4.

⇒ The values of k that satisfy both conditions (k is even and a multiple of 4) are:

⇒ k = 0, 4, 8, and 12.

⇒ These correspond to 4 rational terms in the expansion.

Hence, the Correct answer is Option 3.