Class 12th
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New answer posted
4 months agoContributor-Level 10
In the ammonolysis reaction, HCl is produced as a byproduct. To neutralize this acidic impurity, the mixture is treated with NaOH.
New answer posted
4 months agoContributor-Level 10
P' (x) = a (x-1) (x+1) = a (x²-1).
P (x) = ∫ P' (x) dx = a (x³/3 - x) + b.
Given P (-3) = 0 ⇒ a (-9+3) + b = 0 ⇒ b = 6a.
Given ∫ P (x)dx = 18. Assuming the integration is over a symmetric interval like [-c, c] and using the fact that a (x³/3-x) is an odd function, ∫ (a (x³/3 - x)dx = 0. Then ∫ b dx = 18. If the interval is [-1, 1], this would be b (1 - (-1) = 2b = 18, so b=9.
With b=9, we find a = b/6 = 9/6 = 3/2.
So, P (x) = 3/2 (x³/3 - x) + 9 = x³/2 - 3x/2 + 9.
The sum of all coefficients is 1/2 - 3/2 + 9 = -1 + 9 = 8.
New answer posted
4 months agoContributor-Level 10
The equation of a plane is determined by a point it passes through and a normal vector. The plane passes through (1, -6, -5). Its normal vector (a, b, c) is perpendicular to two other vectors, derived from the given equations:
4a - 3b + 7c = 0
3a + 4b + 2c = 0
The direction of the normal vector (a, b, c) is found by the cross product of (4, -3, 7) and (3, 4, 2):
a = (-3) (2) - 7 (4) = -34.
b = 7 (3) - 4 (2) = 13.
c = 4 (4) - (-3) (3) = 25.
So the plane equation is -34 (x-1) + 13 (y+6) + 25 (z+5) = 0.
The point (1, -1, α) lies on this plane:
-34 (1-1) + 13 (-1+6) + 25 (α+5) = 0.
0 + 13 (5) + 25α + 125 = 0.
65 + 25α + 125 = 0 ⇒ 25α = -190
New answer posted
4 months agoContributor-Level 10
The vector PQ is given by Q - P = (-3-1, 5-3, 2-a) = (-4, 2, 2-a).
This vector is collinear with 2i - j + k = (2, -1, 1).
This means their components are proportional: -4/2 = 2/ (-1) = (2-a)/1.
From -2 = 2-a, we find a=4.
The midpoint M of PQ is (-3+1)/2, (5+3)/2, (2+4)/2) = (-1, 4, 3).
M lies on the plane 2x - y + z - b = 0.
Substitute the coordinates of M: 2 (-1) - 4 + 3 - b = 0 ⇒ -2 - 4 + 3 = b ⇒ b = -3.
New answer posted
4 months agoContributor-Level 10
The functional equation f (x+y) = f (x)f (y) implies f (x) = a? for some constant a.
Then f' (x) = a? ln (a).
Given f' (0) = 3, we have a? ln (a) = 3 ⇒ ln (a) = 3 ⇒ a = e³.
So, f (x) = e³?
We need to evaluate the limit: lim (x→0) (f (x)-1)/x = lim (x→0) (e³? -1)/x.
Using the standard limit lim (u→0) (e? -1)/u = 1, we can write:
lim (x→0) 3 * (e³? -1)/ (3x) = 3 * 1 = 3.
New answer posted
4 months agoContributor-Level 10
Given the matrix P = [2, -1], [5, -3].
The characteristic equation is det (P - λI) = 0, which is (2-λ) (-3-λ) - (-1) (5) = 0.
This simplifies to λ² + λ - 1 = 0.
By the Cayley-Hamilton theorem, the matrix P satisfies this equation: P² + P - I = 0, so P² = I - P.
To find P³: P³ = P * P² = P (I-P) = P - P² = P - (I-P) = 2P - I.
The problem asks for N=6, likely related to a higher power P? Continuing the pattern:
P? = 2P² - P = 2 (I-P) - P = 2I - 3P.
P? = 2P - 3P² = 2P - 3 (I-P) = 5P - 3I.
P? = 5P² - 3P = 5 (I-P) - 3P = 5I - 8P.
The solution N=6 must relate to a different question not fully transcribed, for example, if P^N = 5I - 8P.
New answer posted
4 months agoContributor-Level 10
This is a binomial probability problem with n=5. Let p be the probability of success and q be the probability of failure.
Given P (X=1) =? C? p¹q? = 5pq? = 0.4096 — (I)
Given P (X=2) =? C? p²q³ = 10p²q³ = 0.2048 — (II)
Divide (I) by (II): (5pq? ) / (10p²q³) = 0.4096 / 0.2048 = 2.
(1/2) * (q/p) = 2 ⇒ q/p = 4 ⇒ q = 4p.
Using p+q=1, we have p+4p=1 ⇒ 5p=1 ⇒ p=1/5.
And q = 4/5.
We need to find P (X=3) =? C? p³q².
P (X=3) = 10 * (1/5)³ * (4/5)² = 10 * (1/125) * (16/25) = 160 / 3125 = 32 / 625.
New answer posted
4 months agoContributor-Level 9
The following depicts a chemical reaction leading to a Major Product Y.
Mercury has low boiling point so is refined by distillation method.
Copper refining is done through electrolytic refining.
Silicon is refined by zone refining method.
Nickel is refined by vapour phase refining.
New question posted
4 months agoTaking an Exam? Selecting a College?
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