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Language Development Milestones:
| Age | Expected Language |
|---|---|
| 2 months | Coos, social smile |
| 6 months | Babbles (ba, da, ma) |
| 9 months | Mama/dada (non-specific) |
| 12 months | 1–3 meaningful words (mama, dada specific) |
| 18 months | Minimum 10–20 words; points to body parts |
| 24 months | 2-word phrases; 50+ words |
| 3 years | 3-word sentences; strangers understand ~75% |
Red flag at 18 months: No single words = speech delay — requires formal evaluation.
Critical distinction:
- This child maintains eye contact, points, plays peek-a-boo = social development intact → less likely autism
- Speech delay without social communication deficits suggests: hearing loss, expressive language delay, bilingual exposure, or oral-motor issues
First step: Formal hearing test (audiometry) — hearing loss is the most common correctable cause of speech delay. Concurrent speech-language evaluation.
Reassurance without evaluation at 18 months is inappropriate — early intervention improves outcomes significantly.
Which is a strong electrolyte in aqueous solution?
Strong electrolytes dissociate completely. KI is an ionic salt → strong electrolyte. Acetic acid, NH₄OH, H₂CO₃ are weak electrolytes.
Two fair six-faced dice $A$ and $B$ are rolled together. Define events:
$E_1$: die $A$ shows 4, $\quad E_2$: die $B$ shows 2, $\quad E_3$: sum of both dice is odd.
Which of the following statements is NOT true?
Total outcomes $= 36$.
$P(E_1)=\frac{6}{36}=\frac{1}{6}$, $P(E_2)=\frac{1}{6}$, $P(E_3)=\frac{18}{36}=\frac{1}{2}$
$E_1$ and $E_2$: $P(E_1\cap E_2)=\frac{1}{36}=\frac{1}{6}\cdot\frac{1}{6}$ ✓ Independent.
$E_1$ and $E_3$: For sum odd with die A showing 4 (even), die B must be odd: 3 outcomes. $P(E_1\cap E_3)=\frac{3}{36}=\frac{1}{12}=\frac{1}{6}\cdot\frac{1}{2}$ ✓ Independent.
$E_2$ and $E_3$: For sum odd with die B showing 2 (even), die A must be odd: 3 outcomes. $P(E_2\cap E_3)=\frac{3}{36}=\frac{1}{12}=\frac{1}{6}\cdot\frac{1}{2}$ ✓ Independent.
$E_1\cap E_2\cap E_3$: Die A=4, Die B=2, sum=6 (even) — impossible! $P(E_1\cap E_2\cap E_3)=0$.
But $P(E_1)\cdot P(E_2)\cdot P(E_3)=\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{2}=\frac{1}{72}\neq 0$.
So $E_1,E_2,E_3$ are not mutually independent (as a triple), even though pairwise independent.
- (I) Siderite - (II) Kaolinite - (III) Malachite - (IV) Calamine
- (a) Zinc - (b) Copper - (c) Iron - (d) Aluminium
- Siderite: $FeCO_3$ (Iron)
- Kaolinite: $Al_2Si_2O_5(OH)_4$ (Aluminium)
- Malachite: $Cu_2CO_3(OH)_2$ (Copper)
- Calamine: $ZnCO_3$ (Zinc)
The solution of the differential equation $\dfrac{dy}{dx} = (x-y)^2$, subject to $y(1)=1$, is:
Let $v = x - y$, so $\frac{dv}{dx} = 1 - \frac{dy}{dx} = 1 - v^2$.
Separate: $\frac{dv}{1-v^2} = dx \Rightarrow \frac{1}{2}\ln\left|\frac{1+v}{1-v}\right| = x + C$
i.e. $\ln\left|\frac{1+x-y}{1-x+y}\right| = 2x + C'$
At $(1,1)$: $v=0$, $\ln(1)=2+C' \Rightarrow C'=-2$
Final: $-\ln\left|\dfrac{1-x+y}{1+x-y}\right| = 2(x-1)$ which matches option B.
Asim is now three times as old as Irfan. After 10 years, Asim will be twice as old as Irfan. Asim’s at this time is:
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