Class 12th
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New answer posted
a year agoContributor-Level 10
The equation of the plane passing through the line of intersection is:
(2x - 7y + 4z - 3) + λ (3x - 5y + 4z + 11) = 0.
The plane passes through the point (-2, 1, 3).
(2 (-2) - 7 (1) + 4 (3) - 3) + λ (3 (-2) - 5 (1) + 4 (3) + 11) = 0
(-4 - 7 + 12 - 3) + λ (-6 - 5 + 12 + 11) = 0
(-2) + λ (12) = 0 ⇒ 12λ = 2 ⇒ λ = 1/6.
Substitute λ back into the equation:
(2x - 7y + 4z - 3) + (1/6) (3x - 5y + 4z + 11) = 0
Multiply by 6:
6 (2x - 7y + 4z - 3) + (3x - 5y + 4z + 11) = 0
12x - 42y + 24z - 18 + 3x - 5y + 4z + 11 = 0
15x - 47y + 28z - 7 = 0.
This is the equation ax + by + cz - 7 = 0.
So, a=15, b=-47, c=28.
We need to find the value of 2a + b
New answer posted
a year agoContributor-Level 9
R_eq = 10 + (50 * 20) / (50 + 20) = 170/7 Ω
⇒ I = 170 / (170/7) = 7A
⇒ x = 10 * 7 = 70V ⇒ Voltage across 10Ω resistor
New answer posted
a year agoContributor-Level 10
A = [0, sin α], [sin α, 0]
A² = A ⋅ A = [0, sin α], [sin α, 0] ⋅ [0, sin α], [sin α, 0]
= [00 + sinαsinα, 0sinα + sinα0], [sinα0 + 0sinα, sinαsinα + 00]
= [sin²α, 0], [0, sin²α] = (sin²α)I
A² - (1/2)I = [sin²α, 0], [0, sin²α] - [1/2, 0], [0, 1/2]
= [sin²α - 1/2, 0], [0, sin²α - 1/2]
det (A² - (1/2)I) = (sin²α - 1/2)² - 0 = 0
sin²α - 1/2 = 0
sin²α = 1/2
sin α = ±1/√2
A possible value for α is π/4.
New answer posted
a year agoContributor-Level 10
Given r * a = r * b, which means r * a - r * b = 0 ⇒ r * (a - b) = 0.
This implies that vector r is parallel to vector (a - b).
So, r = λ (a - b) for some scalar λ.
a - b = (2i - 3j + 4k) - (7i + j - 6k) = -5i - 4j + 10k.
So, r = λ (-5i - 4j + 10k).
We are also given r ⋅ (i + 2j + k) = -3.
λ (-5i - 4j + 10k) ⋅ (i + 2j + k) = -3
λ (-51 - 42 + 10*1) = -3
λ (-5 - 8 + 10) = -3
λ (-3) = -3 ⇒ λ = 1.
So, r = 1 * (-5i - 4j + 10k) = -5i - 4j + 10k.
We need to find r ⋅ (2i - 3j + k).
(-5i - 4j + 10k) ⋅ (2i - 3j + k) = (-5) (2) + (-4) (-3) + (10) (1)
= -10 + 12 + 10 = 12.
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