Match the following: Column I (A) In...

MathematicsScalar (Dot) ProductJEE Advanced 2015Difficult

Match the following:

List-I

(P)

R^{2}

, if the magnitude of the projection vector of the vector

α \hat{i} + β \hat{j}

3 \hat{i} + \hat{j}

3

and if

α = 2 + 3 β

, then possible value of

∣ α ∣

is/are

(Q)

Let

a

and

b

be real numbers such that the function

f (x) = {- 3 a x^{2} - 2, b x + a^{2}, x < 1 x \geq 1

if differentiable for all

x \in R

. Then possible value of

a

is (are)

(R)

Let

ω \neq = 1

be a complex cube root of unity. If

(3 - 3 ω + 2 ω^{2})^{4 n + 3} + (2 + 3 ω - 3 ω^{2})^{4 n + 3} + (- 3 + 2 ω + 3 ω^{2})^{4 n + 3} = 0

, then possible value (s) of

n

is (are)

(S)

Let the harmonic mean of two positive real numbers

a

and

b

be 4. If

q

is a positive real number such that

a, 5, q, b

is an arithmetic progression, then the value(s) of

∣ q - a ∣

is (are)

List-II

(1)

(2)

(3)

(4)

(5)

Answer:P → 2, P → 2, R → NaN, Q → 5

Visualized Solution (Hindi)

Multi-Concept Challenge Overview

The problem consists of four independent mathematical challenges.
Part A: Vector projections in $R^{2}$ .
Part B: Differentiability of piecewise functions.
Part C: Complex cube roots of unity and power series.
Part D: Harmonic Mean and Arithmetic Progressions.

Part A: Vector Projection Setup

Let $v = α \hat{i} + β \hat{j}$ and $u = 3 \hat{i} + \hat{j}$ .
Magnitude of projection of $v$ on $u$ is $\frac{∣ v \cdot u ∣}{∣ u ∣}$ .
Given: $\frac{∣ 3 α + β ∣}{( 3 ) ^{2} + 1 ^{2}} = 3$ .
Simplifying the denominator: $\frac{∣ 3 α + β ∣}{2} = 3 ⟹ ∣ 3 α + β ∣ = 23$ .

Part A: Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣ α ∣

Given relation: $α = 2 + 3 β ⟹ β = \frac{α - 2}{3}$ .
Substitute $β$ into $∣ 3 α + β ∣ = 23$ :
$3 α + \frac{α - 2}{3} = 23 ⟹ \frac{3 α + α - 2}{3} = 23$ .
$∣4 α - 2∣ = 6 ⟹ 4 α - 2 = 6$ or $4 α - 2 = - 6$ .
$α = 2$ or $α = - 1$ . Thus, $∣ α ∣ \in {1, 2}$ .

Part B: Continuity Condition

For $f (x)$ to be differentiable at $x = 1$ , it must be continuous at $x = 1$ .
$L H L = lim_{x \to 1^{-}} (- 3 a x^{2} - 2) = - 3 a - 2$ .
$R H L = lim_{x \to 1^{+}} (b x + a^{2}) = b + a^{2}$ .
Equating them: $- 3 a - 2 = b + a^{2} \dots (1)$

Part B: Differentiability and Solving

Differentiating: $f^{'} (x) = {- 6 a x, b, x < 1 x > 1$ .
Equating derivatives at $x = 1$ : $- 6 a (1) = b ⟹ b = - 6 a \dots (2)$ .
Substitute (2) into (1): $- 3 a - 2 = - 6 a + a^{2} ⟹ a^{2} - 3 a + 2 = 0$ .
$(a - 1) (a - 2) = 0 ⟹ a = 1, 2$ .

Part C: Complex Roots Insight

Let $A = 3 - 3 ω + 2 ω^{2}$ , $B = 2 + 3 ω - 3 ω^{2}$ , $C = - 3 + 2 ω + 3 ω^{2}$ .
Observe: $A + B + C = 2 (1 + ω + ω^{2}) = 0$ .
Also, $B = A ω$ and $C = A ω^{2}$ .
The equation becomes $A^{k} + (A ω)^{k} + (A ω^{2})^{k} = 0$ , where $k = 4 n + 3$ .

Part C: Power Condition for Zero

$A^{k} (1 + ω^{k} + ω^{2 k}) = 0 ⟹ 1 + ω^{k} + ω^{2 k} = 0$ .
This holds if $k$ is not a multiple of $3$ .
$k = 4 n + 3 \equiv n (mod 3)$ .
So, $n \neq \equiv 0 (mod 3)$ .
For $n \in {1, 2, 3, 4, 5}$ , possible values are $n = 1, 2, 4, 5$ .

Part D: Arithmetic Progression Relations

$a, 5, q, b$ are in AP. Let common difference be $d$ .
$d = 5 - a$ .
$q = 5 + d = 5 + (5 - a) = 10 - a$ .
$b = q + d = (10 - a) + (5 - a) = 15 - 2 a$ .

Part D: Harmonic Mean Substitution

$H M = \frac{2 ab}{a + b} = 4 ⟹ ab = 2 (a + b)$ .
Substitute $b = 15 - 2 a$ :
$a (15 - 2 a) = 2 (a + 15 - 2 a) ⟹ 15 a - 2 a^{2} = 2 (15 - a)$ .
$15 a - 2 a^{2} = 30 - 2 a ⟹ 2 a^{2} - 17 a + 30 = 0$ .

Part D: Final Calculation

Factorizing $2 a^{2} - 17 a + 30 = 0$ : $(2 a - 5) (a - 6) = 0$ .
Case 1: $a = \frac{5}{2} = 2.5$ . Then $q = 10 - 2.5 = 7.5$ . $∣ q - a ∣ = ∣7.5 - 2.5∣ = 5$ .
Case 2: $a = 6$ . Then $q = 10 - 6 = 4$ . $∣ q - a ∣ = ∣4 - 6∣ = 2$ .
Possible values for $∣ q - a ∣$ are $2$ and $5$ .

Final Matching and Conclusion

(A) $∣ α ∣ \in {1, 2} \to$ Match: 2 (Option 1 in Column 2)
(B) $a \in {1, 2} \to$ Match: 1, 2 (Options 0, 1 in Column 2)
(C) $n \in {1, 2, 4, 5} \to$ Match: 1, 2, 4, 5 (Options 0, 1, 3, 4 in Column 2)
(D) $∣ q - a ∣ \in {2, 5} \to$ Match: 2, 5 (Options 1, 4 in Column 2)

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Comprehension Passage

Let

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Question 1:

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(A)

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Which of the following values of $α$ satisfy the equation $(1 + α)^{2} (2 + α)^{2} (3 + α)^{2} (1 + 2 α)^{2} (2 + 2 α)^{2} (3 + 2 α)^{2} (1 + 3 α)^{2} (2 + 3 α)^{2} (3 + 3 α)^{2} = - 648 α$ ?

* Multiple Correct Options

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Match the following:

List-I

List-II

Visualized Solution (Hindi)

Multi-Concept Challenge Overview

Part A: Vector Projection Setup

Part A: Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣α∣

Part B: Continuity Condition

Part B: Differentiability and Solving

Part C: Complex Roots Insight

Part C: Power Condition for Zero

Part D: Arithmetic Progression Relations

Part D: Harmonic Mean Substitution

Part D: Final Calculation

Final Matching and Conclusion

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following values of α satisfy the equation ​(1+α)2(2+α)2(3+α)2​(1+2α)2(2+2α)2(3+2α)2​(1+3α)2(2+3α)2(3+3α)2​​=−648α?

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following values of α satisfy the equation ​(1+α)2(2+α)2(3+α)2​(1+2α)2(2+2α)2(3+2α)2​(1+3α)2(2+3α)2(3+3α)2​​=−648α?

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If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

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Which of the following values of $α$ satisfy the equation $(1 + α)^{2} (2 + α)^{2} (3 + α)^{2} (1 + 2 α)^{2} (2 + 2 α)^{2} (3 + 2 α)^{2} (1 + 3 α)^{2} (2 + 3 α)^{2} (3 + 3 α)^{2} = - 648 α$ ?

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