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Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$ for all $x\in\mathbb{R}$. Find $f(2)$.
Let $f'(1)=a$, $f''(2)=b$, $f'''(3)=c$. Then $f(x)=x^3+ax^2+bx+c$.
$f'(x)=3x^2+2ax+b \Rightarrow f'(1)=3+2a+b=a \Rightarrow a+b=-3$ ...(i)
$f''(x)=6x+2a \Rightarrow f''(2)=12+2a=b$ ...(ii)
$f'''(x)=6 \Rightarrow f'''(3)=6=c$ ...(iii)
From (ii): $b=12+2a$. Sub in (i): $a+12+2a=-3\Rightarrow3a=-15\Rightarrow a=-5$, $b=2$, $c=6$.
$f(2)=8+4(-5)+2(2)+6=8-20+4+6=\mathbf{-2}$
Wait: $f(2)=8-20+4+6=-2$. So correct index is 2 (option C: $-2$).
This is a classic presentation of Neonatal Respiratory Distress Syndrome (RDS), also called Hyaline Membrane Disease:
- Prematurity (36 weeks — late preterm)
- Grunting, retractions, nasal flaring, tachypnea
- Ground glass appearance + air bronchograms on CXR = pathognomonic of RDS
Pathophysiology: Deficiency of surfactant (produced by Type II pneumocytes from ~24 weeks, mature by ~35 weeks). Surfactant reduces alveolar surface tension. Without it:
\[\text{Surface tension} \uparrow \Rightarrow \text{Alveolar collapse} \Rightarrow \text{V/Q mismatch} \Rightarrow \text{Hypoxia}\]
- Treatment: Exogenous surfactant (beractant, poractant alfa) via endotracheal tube + respiratory support (CPAP/mechanical ventilation)
- Prevention: Antenatal corticosteroids (betamethasone 12 mg IM × 2 doses, 24 hrs apart) if delivery expected before 34 weeks — accelerates fetal lung maturity
Which list shows entries that should be credited to a partner's current account?
Option A is correct.
Credits to current account increase the partner's entitlement:
- Interest on partner's loan — reward for lending to the firm
- Partner's salary — appropriation in favour of the partner
- Share of profits — partner's portion of residual profit
Interest on drawings is debited (charged to) the partner.
Which financial statement directly represents and reflects the accounting equation ($\text{Assets} = \text{Liabilities} + \text{Owner's Equity}$)?
The Balance Sheet (also called the Statement of Financial Position) is a snapshot of a business at a point in time, structured as: $\text{Assets} = \text{Liabilities} + \text{Owner's Equity}$. This is a direct representation of the accounting equation. The income statement shows revenues and expenses; the cash flow statement shows cash movements; and the statement of changes in equity shows movements in equity accounts.
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