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

MathematicsEquation of a PlaneJEE Advanced 2006Difficult

Match the following :

List-I

(P)

Two rays

x + y = ∣ a ∣

and

a x - y = 1

intersects each other in the first quadrant in the interval

a \in (a_{0}, \infty)

, the value of

a_{0}

(Q)

Point

(α, β, γ)

lies on the plane

x + y + z = 2

. Let

a = α \hat{i} + β \hat{j} + γ \hat{k}

\hat{k} \times (\hat{k} \times a) = 0

, then

γ =

(R)

\int_{0}^{1} (1 - y^{2}) d y + \int_{1}^{0} (y^{2} - 1) d y

(S)

sin A sin B sin C + cos A cos B = 1

, then the value of

sin C =

List-II

(1)

(2)

4/3

(3)

\int_{0}^{1} 1 - x d x + \int_{- 1}^{0} 1 + x d x

(4)

Answer:P → 4, Q → 1, Q → 3, S → 4

Visualized Solution (Hindi)

Solving for Intersection in Part (A)

Given equations: $x + y = ∣ a ∣$ and $a x - y = 1$
Adding the equations: $(a + 1) x = 1 + ∣ a ∣ ⟹ x = \frac{1 + ∣ a ∣}{a + 1}$
Substituting $x$ to find $y$ : $y = ∣ a ∣ - \frac{1 + ∣ a ∣}{a + 1} = \frac{a ∣ a ∣ + ∣ a ∣ - 1 - ∣ a ∣}{a + 1} = \frac{a ∣ a ∣ - 1}{a + 1}$

Applying First Quadrant Constraints

For first quadrant: $x > 0$ and $y > 0$
From $x > 0$ : $\frac{1 + ∣ a ∣}{a + 1} > 0 ⟹ a + 1 > 0 ⟹ a > - 1$
From $y > 0$ : $\frac{a ∣ a ∣ - 1}{a + 1} > 0 ⟹ a ∣ a ∣ - 1 > 0$
For $a > 0$ , $a^{2} - 1 > 0 ⟹ a > 1$ . Thus $a_{0} = 1$ .

Vector Triple Product in Part (B)

Given: $\hat{k} \times (\hat{k} \times a) = 0$
Using identity: $(\hat{k} \cdot a) \hat{k} - (\hat{k} \cdot \hat{k}) a = 0$
Substitute $a = α \hat{i} + β \hat{j} + γ \hat{k}$ : $γ \hat{k} - (α \hat{i} + β \hat{j} + γ \hat{k}) = 0$
Result: $- α \hat{i} - β \hat{j} = 0 ⟹ α = 0, β = 0$

Finding Gamma on the Plane

Point $(α, β, γ)$ lies on $x + y + z = 2$
Substitute $α = 0, β = 0$ : $0 + 0 + γ = 2$
Final value: $γ = 2$

Evaluating Integrals in Part (C)

Expression: $\int_{0}^{1} (1 - y^{2}) d y + \int_{1}^{0} (y^{2} - 1) d y$
Property: $\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
Simplified: $2 \int_{0}^{1} (1 - y^{2}) d y = 2 [y - \frac{y ^{3}}{3}]_{0}^{1} = 2 (1 - \frac{1}{3}) = \frac{4}{3}$

Matching with Option (r)

Option (r): $\int_{0}^{1} 1 - x d x + \int_{- 1}^{0} 1 + x d x$
Let $1 - x = t$ and $1 + x = u$ : $\int_{1}^{0} t (- d t) + \int_{0}^{1} u d u$
Calculation: $\int_{0}^{1} t^{1/2} d t + \int_{0}^{1} u^{1/2} d u = \frac{2}{3} + \frac{2}{3} = \frac{4}{3}$

Trigonometric Analysis in Part (D)

Equation: $sin A sin B sin C + cos A cos B = 1$
Rearrange: $sin A sin B sin C = 1 - cos A cos B$
Since $sin C \leq 1$ : $sin A sin B \geq 1 - cos A cos B$
Identity: $cos A cos B + sin A sin B \geq 1 ⟹ cos (A - B) \geq 1$

Final Conclusion for Part (D)

Since $cos (A - B) \leq 1$ , then $cos (A - B) = 1 ⟹ A = B$
Substitute back: $sin^{2} A sin C + cos^{2} A = 1$
$sin^{2} A sin C = 1 - cos^{2} A = sin^{2} A$
Result: $sin C = 1$

Summary of Matches

Final Matches:
(A) $\to$ (s) : $a_{0} = 1$
(B) $\to$ (p) : $γ = 2$
(C) $\to$ (q, r) : Value $= 4/3$
(D) $\to$ (s) : $sin C = 1$

00:00 / 00:00

Conceptually Similar Problems

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(3)

(4)

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Let

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Comprehension Passage

Let

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[a b c] 187923777 = [000] \dots (E)

Question 1:

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

(A)

(B)

(C)

(D)

Answer:D

Question 2:

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

(A)

-2

(B)

(C)

(D)

-3

Answer:A

Question 3:

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(B)

(C)

6/7

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\infty

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* Multiple Correct Options

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(B)

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(C)

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(D)

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Answer:B, D

Match the following :

List-I

List-II

Visualized Solution (Hindi)

Solving for Intersection in Part (A)

Applying First Quadrant Constraints

Vector Triple Product in Part (B)

Finding Gamma on the Plane

Evaluating Integrals in Part (C)

Matching with Option (r)

Trigonometric Analysis in Part (D)

Final Conclusion for Part (D)

Summary of Matches

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.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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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