Class 12th
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New answer posted
2 months agoContributor-Level 10
To find the values of a and b, we evaluate the left-hand limit (LHL) and right-hand limit (RHL) at x=0 and equate them.
LHL: lim (x→0) sin (a+3)/2 * x / x = (a+3)/2. The full limit evaluates to (a+3)/2. So, (a+3)/2 = b.
This gives the relation a - 2b + 3 = 0 — (I)
RHL: lim (x→0) [√ (x+bx³) - √x] / (bx²).
Rationalize the numerator:
lim (x→0) [ (x+bx³) - x] / [bx² (√ (x+bx³) + √x)]
= lim (x→0) bx³ / [bx² (√x (√ (1+bx²) + √x)]
= lim (x→0) x / [√x (√ (1+bx²) + 1)] = lim (x→0) √x / (√ (1+bx²) + 1) = 0/2 = 0.
So, b = 0.
Substituting b=0 into equation (I): a - 2 (0) + 3 = 0 ⇒ a = -3.
New answer posted
2 months agoContributor-Level 10
Let X1, ., X2n be the first 2n observations and Y1, ., Yn be the last n observations.
Given:
ΣXi / 2n = 6 => ΣXi = 12n (i)
ΣYi / n = 3 => ΣYi = 3n (ii)
Combined mean: (ΣXi + ΣYi) / 3n = 5 => ΣXi + ΣYi = 15n. This is consistent with (i) and (ii).
Combined variance: (ΣXi^2 + ΣYi^2) / 3n - (mean)^2 = 4
(ΣXi^2 + ΣYi^2) / 3n - 5^2 = 4
ΣXi^2 + ΣYi^2 = (4 + 25) * 3n = 87n.
New observations are Xi + 1 and Yi - 1.
New mean = (Σ (Xi + 1) + Σ (Yi - 1) / 3n = (ΣXi + 2n + ΣYi - n) / 3n = (15n + n) / 3n = 16n / 3n = 16/3.
New variance k:
k = (Σ (Xi + 1)^2 + Σ (Yi - 1)^2) / 3n - (new mean)^2
k = (Σ (Xi^2 + 2Xi + 1) + Σ (Yi^2 - 2Yi + 1) / 3
New answer posted
2 months agoContributor-Level 10
The differential equation is rearranged to dt/dx - xt = -e? ²/², where t = 1/ (y+1).
This is a linear first-order differential equation. The integrating factor (I.F.) is e^ (∫-x dx) = e? ²/².
The solution is t * (I.F.) = ∫ Q (x) * (I.F.) dx + c.
t * e? ²/² = ∫ -e? ²/² * e? ²/² dx + c = ∫ -1 dx = -x + c.
Substituting t = 1/ (y+1) back: e? ²/² / (y+1) = -x + c.
Using the initial condition y (2) = 0:
e? ²/ (0+1) = -2 + c ⇒ c = e? ² + 2.
The solution is e? ²/² / (y+1) = 2 + e? ² - x.
New answer posted
2 months agoContributor-Level 10
Given f (x) = ax^2 + bx + c.
f (-1) = a - b + c = 2
f' (x) = 2ax + b, so f' (-1) = -2a + b = 1
f' (x) = 2a, so f' (-1) = 2a = 1/2
From 2a = 1/2, we get a = 1/4.
Substituting a into -2a + b = 1: -2 (1/4) + b = 1 => -1/2 + b = 1 => b = 3/2.
Substituting a and b into a - b + c = 2: 1/4 - 3/2 + c = 2 => -5/4 + c = 2 => c = 13/4.
So, f (x) = (1/4)x^2 + (3/2)x + 13/4 = (1/4) (x^2 + 6x + 13).
We need to find f (1):
f (1) = (1/4) (1^2 + 6 (1) + 13) = (1/4) (1 + 6 + 13) = (1/4) (20) = 5.
New answer posted
2 months agoContributor-Level 10
PES University conducts PESSAT for admission into BBA programme. The minimum eligibility for the programme is passing Class 12 with atleast 50% of aggregate marks. Scoring 60% in Class 12 makes the candidate eligible for admission into BBA. However, the candidate must qualify PESSAT and the subsequent personal interview round to get a confirmed seat in the course at the university.
New answer posted
2 months agoContributor-Level 9
CuSO? 5H? O is blue vitriol which is represented as [Cu (H? O)? ]. H? O.SO? Here, secondary valencies are 4 H? O and 1H? O molecule outside coordination sphere is involved in hydrogen bonding.
New answer posted
2 months agoContributor-Level 10
Given vectors a? and b? such that |a? | = |b? | and a? ⋅ b? = 0 (they are orthogonal).
The problem implies |a? |=|b? |=1.
Let c? = a? + b? + a? x b?
To find the magnitude of c? , we calculate |c? |²:
|c? |² = c? ⋅ c? = (a? + b? + a? x b? ) ⋅ (a? + b? + a? x b? ).
This expands to |a? |² + |b? |² + |a? x b? |² because all other dot products are zero (e.g., a? ⋅ b? = 0, a? ⋅ (a? x b? ) = 0).
|a? x b? |² = (|a? |b? |sin (90°)² = |a? |²|b? |².
So, |c? |² = |a? |² + |b? |² + |a? |²|b? |² = 1² + 1² + 1²*1² = 3.
∴ |c? | = √3.
To find the angle θ between c? and a? , we compute their dot product:
c? ⋅ a? = (a? + b? + a
New answer posted
2 months agoContributor-Level 10
Given functions f (x) = (x-2)/ (x-3) and g (x) = 2x-3.
First, find the inverse functions f? ¹ (x) and g? ¹ (x).
For f? ¹ (x): y = (x-2)/ (x-3) ⇒ y (x-3) = x-2 ⇒ xy - 3y = x-2 ⇒ xy-x = 3y-2 ⇒ x (y-1) = 3y-2 ⇒ x = (3y-2)/ (y-1). So, f? ¹ (y) = (3y-2)/ (y-1).
For g? ¹ (x): y = 2x-3 ⇒ y+3 = 2x ⇒ x = (y+3)/2. So, g? ¹ (y) = (y+3)/2.
We are given f? ¹ (x) + g? ¹ (x) = 13/2.
(3x-2)/ (x-1) + (x+3)/2 = 13/2.
Multiply by 2 (x-1): 2 (3x-2) + (x+3) (x-1) = 13 (x-1).
6x - 4 + x² + 2x - 3 = 13x - 13.
x² + 8x - 7 = 13x - 13.
x² - 5x + 6 = 0.
(x-2) (x-3) = 0.
The possible values of x are 2 and 3. Note that x=3 is not in the domain of t
New answer posted
2 months agoContributor-Level 9
Match the following:
Antifertility drug → Norethindrone
Antibiotic → Salvarsan
Tranquilizer → Meprobamate
Artificial Sweetener → Alitame
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