Q.In a bank, principal increases continuously at the rate of $r\%$ per year. Find the value of $r$ if Rs 100 double itself in 10 years $(\log_e 2 = 0.6931)$.

Concept and Intuition

The logistic growth model describes how a population grows rapidly when resources are abundant (population size N much smaller than carrying capacity K) but growth slows and eventually plateaus as N approaches K, due to increasing resource limitation, captured by the term $\left[\dfrac{K-N}{K}\right]$. The constant r in this equation, called the intrinsic rate of natural increase, represents the population's inherent capacity to grow per individual per unit time in the absence of any limiting factor — it is determined by the species' biology (birth rate minus death rate under ideal conditions) and differs from K, which represents the maximum population size the environment can sustain, not a rate at all.

Step-by-Step Solution

1. Identify the variables in the given equation: N = population size, K = carrying capacity, r = the proportionality constant governing growth rate.
2. Recall that K is explicitly the environment's carrying capacity — this rules out option (C), since K, not r, represents carrying capacity. …

Concept and Intuition

Population ecology distinguishes the rate of change (differential form) from the closed-form solution describing population size at any time (integral form). Exponential growth assumes unlimited resources, so the rate of increase is proportional to the current population size.

Step-by-Step Solution

1. Start from the differential (instantaneous rate) form: $\dfrac{dN}{dt} = rN$, where $r = b-d$ (birth rate minus death rate).
2. Separate variables: $\dfrac{dN}{N} = r\,dt$.
3. Integrate both sides: $\log N = rt + C$.
4. Apply the condition $N=N_0$ at $t=0$: $C=\log N_0$.
5. Exponentiate: $N_t = N_0 e^{rt}$ — this is the integral (solved) form. …

Concept and Intuition

Whenever you see $\left(1+\dfrac{f(n)}{n}\right)^n$ with $f(n)\to L$, the limit is $e^{L}$. Here we must first simplify $b^{1/n}-1$ using the exponential expansion, since $b^{1/n}=e^{(\log b)/n}\approx 1+\dfrac{\log b}{n}$ for large $n$.

Step-by-Step Solution

1. Write $b^{1/n} = e^{\frac{\log b}{n}} = 1+\dfrac{\log b}{n}+O(1/n^2)$.
2. Then $a+b^{1/n}-1 = a + \dfrac{\log b}{n}+O(1/n^2)$.
3. So $\dfrac{a+b^{1/n}-1}{a} = 1+\dfrac{\log b}{an}+O(1/n^2)$. …

Concept and Intuition

Plant growth can follow two mathematical patterns:

• Arithmetic growth: after mitotic division, only one daughter cell continues to divide, the other differentiates — the growth increment per unit time stays constant, giving a linear increase, plotted as a straight line: $L_t = L_0 + rt$ (length at time t = initial length + rate × time). A classic example is a root elongating at a constant rate.
• Geometric growth: initially both daughter cells retain the capacity to divide, so growth accelerates — first a lag phase, then exponential (log) growth, described by $W_1 = W_0\, e^{rt}$ (final size = initial size × e^(rate × time)). This is the pattern of "unlimited" exponential growth seen early in most growing organisms/populations before resources become limiting.

Step-by-Step Solution

1. Identify arithmetic growth's linear form: $L_t = L_0 + rt$.
2. Identify geometric growth's exponential form: $W_1 = W_0\, e^{rt}$. …

Concept and Intuition

• Arithmetic growth: after mitosis, if only one daughter cell keeps dividing while the other differentiates, growth proceeds at a constant rate — root elongation at a constant rate is the classic example.
• Geometric growth: initial growth is slow (lag phase), then rapid/exponential; but as resources become limiting, the rate declines and growth plateaus (stationary phase) — plotted over time, this lag → exponential → plateau progression traces out the characteristic sigmoid (S-shaped) curve typical of organisms growing in a natural, resource-limited environment.
• Turgidity and extension growth: plant cells elongate mainly by taking up water, which raises internal turgor pressure against the cell wall, stretching it — turgor-driven extension is the dominant mode of plant cell growth.
• Cytokinins: these are synthesised predominantly in regions of active cell division — root apices, developing shoot buds, young leaves, and developing seeds/fruits — not in old, mature/senescing tissue.

Step-by-Step Solution

1. A: root elongation as arithmetic growth — correct. …

Concept and Intuition

Growth in plants is not measured by a single universal metric — the appropriate parameter depends on the organ's geometry and growth pattern. A structure that elongates in one direction (like a pollen tube, which grows only at its tip) is best measured by length. A structure that expands in a flat plane (like a dorsiventral/bifacial leaf, which has distinct upper and lower surfaces) is better measured by surface area, since length alone wouldn't capture its lateral expansion. Cell division rates (as in the maize root meristem) and cell enlargement magnitudes (as in watermelon fruit cells) are used to illustrate just how extreme and rapid plant growth can be at the cellular level.

Step-by-Step Solution

1. (i) Root apical meristems are highly active in cell division; a commonly cited figure is that a single maize root apical meristem can generate more than 17,500 new cells every hour — a standard NCERT-cited statistic. TRUE.
2. (ii) Cell enlargement can be dramatic; watermelon cells are cited as increasing in size up to 3,50,000-fold, illustrating that growth by cell enlargement can far exceed growth by cell division in magnitude. TRUE.
3. (iii) A pollen tube grows only from its tip in one direction toward the ovule, so its growth is naturally quantified by the increase in its length. TRUE. …

Concept and Intuition

Growth curves (bacterial growth curves, and analogously the "grand period of growth" in plants) typically show four phases: lag (adjustment, minimal growth), exponential/log (rapid, geometric increase as resources are abundant and unrestricted), stationary (growth rate equals death/limiting-factor rate, net growth plateaus), and senescence/decline (resources exhausted, growth rate falls, population/organism ages and declines). By definition, the steepest slope on the growth curve — i.e., the maximum rate of increase — occurs during the exponential phase, before nutrient depletion, waste accumulation, or space constraints begin to slow things down.

Step-by-Step Solution

1. Recall the four standard phases of a growth curve: lag, exponential, stationary, senescence.
2. Lag phase: growth is minimal as cells/organisms adapt — not the maximum rate.
3. Exponential phase: growth proceeds at its fastest, unimpeded rate — this IS the maximum. …

Concept and Intuition

Jagadish Chandra Bose pioneered instruments to demonstrate that plants respond to stimuli much like animals do, and needed a device sensitive enough to magnify and record the very slow, minute elongation growth of plant tissues.

Step-by-Step Solution

1. The Crescograph was designed by J.C. Bose specifically to magnify and record plant growth movements to an extraordinary degree of precision — down to about a millionth of a millimetre.
2. This instrument was central to Bose's demonstrations of plant responsiveness (growth response to stimuli) and remains the standard textbook answer to this fact. …

Concept and Intuition

Eichhornia crassipes, the "terror of Bengal," is a textbook example of an invasive alien species — introduced into India from its native South America for its attractive flowers, it spread uncontrollably, forms dense mats on standing water, drains dissolved oxygen (harming aquatic life), and propagates rapidly via vegetative fragmentation.

Step-by-Step Solution

1. (A) invasive weed of standing water bodies — TRUE.
2. (B) drains oxygen from water — TRUE, a well-documented ecological effect.
3. (C) "indigenous to India" — FALSE; it is an introduced species from South America. …