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Class 12th Determinants

The number of θ $\in (0, 4 π)$ for which the system of linear equations

3 (sin 3θ) x – y + z = 2

3 (cos 2θ) x + 4y +3z = 3

6x + 7y + 7z = 9, has no solution, is:

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Vishal Baghel

Contributor-Level 10

$Δ = | \begin{matrix} 3 s i n 3 θ & - 1 & 1 \\ 3 c o s 2 θ & 4 & 3 \\ 6 & 7 & 7 \end{matrix} |$

= 3 sin3θ (28 – 21) + (21 cos 2θ - 18) + 1 (21 cos 2θ - 24)

$Δ = 21 s i n 3 θ + 42 c o s 2 θ - 42$

for no solution

sin 3θ + 2 cos 2θ = 2

$θ = π, 2 π, 3 π, \frac{π}{6}, \frac{5 π}{6}, \frac{13 π}{6}, \frac{17 π}{6}$

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Class 12th Sequence and Series

For n $\in$ N, let S_n = ${z \in C : | z - 3 + 2 i | = \frac{n}{4}}$ and T_n = ${z \in C : | z - 2 + 3 i = \frac{1}{n}} .$ Then the number of elements in the set ${n \in N : S_{n} \cap T_{n} = ?}$ is:

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Vishal Baghel

Contributor-Level 10

$S_{n} = {z \in C : | z - 3 + 2 i | = \frac{n}{4}}$

represents a circle with centre C1 (3, 2) and radius

$r_{1} = \frac{n}{4}$

Similarly Tn represents circle with centre C2 (2, 3) and radius

$r_{1} = \frac{1}{n}$

$\sqrt[]{2} < | \frac{n}{4} - \frac{1}{n} |$

n take infinite values.

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Class 12th Complex Numbers and Quadratic Equations

If a, b, $γ$ , $δ$ , are the roots of the equation x⁴ + x³ + x² + x + 1 = 0, then $α^{2021} + β^{2021} + γ^{2021} + δ^{2021}$ is equal to:

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Vishal Baghel

Contributor-Level 10

From properties of nth root of unity

$1^{2021} + α^{2021} + β^{2021} + γ^{2021} + δ^{2021} = 0$

$\Rightarrow α^{2021} + β^{2021} + γ^{2021} + δ^{2021} = - 1$

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Class 12th Maths

The total number of functions, f : {1,2,3,4} -> {1,2,3,4,5,6} such that f(1) + f(2) = f(3), is equal to:

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Vishal Baghel

Contributor-Level 10

Case 1 : If f (3) = 3 then f (1) and f (2) take 1 or 2

No. of ways = 2.6 = 12

Case 2 : If f (3) = 5 then f (1) and f (2) take 2 or 3

OR 1 and 4

No. of ways = 2.6.2 = 24

Similarly for all other cases

Total no. of ways 90.

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Class 12th Chemistry

Given below are two statements:

Statement I: On heating with KHSO₄, glycerol is dehydrated and acrolein is formed.

Statement II: Acrolein has fruity odour and can be used to test glycerol's presence.

Choose the correct option.

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Vishal Baghel

Contributor-Level 10

$G l y c e r o l \to_{Δ}^{K H S O_{4}} \underset{(P u n g e s t, s u f f o c a t i n g o d o u r)}{A c r o l e i n}$

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Class 12th Chemistry

Drugs used to bind to receptors inhibiting its natural function and blocking a message are called:

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Vishal Baghel

Contributor-Level 10

Based upon properties of drugs.

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Class 12th Biomolecules

During the denaturation of proteins, which of these structure will remain intact?

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Vishal Baghel

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During denature of protein, 1° structure remains the same, whereas 2° and 3° structure get destroyed.

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Class 12th Chemistry

Melamine polymer is formed by the condensation of:

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Vishal Baghel

Contributor-Level 10

Melamine + formaldehyde -> Melamine formaldehyde polymer

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Class 12th Chemistry

An organic compound 'A' on reaction with NH₃ followed by heating gives compound B. Which on further strong heating gives compound C(C₈H₅NO₂). Compound C on sequential reaction with ethanolic KOH, alkyl chloride and hydrolysis with alkali gives a primary amine. The compound A is: