Q.A protozoan reproduces by binary fission. What will be the number of protozoans in its population after six generations?

The key concept here is continuous growth, where the rate of increase is proportional to the current amount. This is the classic model for continuously compounded interest, population growth, or radioactive decay (but with a positive rate). The differential equation is $\frac{dP}{dt} = rP$, whose solution is an exponential function. The intuition: the larger the principal, the faster it grows, leading to a "snowball" effect.

We are told the principal increases continuously at 4% per annum. That "4%" is the instantaneous rate $r = 0.04$ (not an annual percentage yield after compounding). So we use the continuous compounding formula.

1. Set up the model Let $P(t)$ be the principal after $t$ years. The continuous growth rate is $r = \frac{4}{100} = 0.04$. The differential equation is

The solution (by separation of variables or known formula) is

where $P_0 = 1000$ is the initial principal.

2. State the tripling condition We want the time $t$ such that the principal triples:

Substitute the expression:

Cancel $P_0$ (since it's nonzero):

3. Solve for $t$ Take the natural logarithm of both sides:

Using $\ln(e^x) = x$, we get:

So

Step 1 — Core assumption of exponential growth

When resources in a habitat are unlimited, each species has the potential to realise its full biotic potential, and the population grows exponentially — described by the equation $\frac{dN}{dt} = rN$, producing a J-shaped curve.

Step 2 — Checking each statement

• (A) "Resources are limited" — false for the exponential model; unlimited resources is precisely what allows unchecked exponential growth. This is the assumption of the logistic, not exponential, model.
• (B) "Population grows in a geometric fashion" — true; this is the defining character of exponential/J-shaped growth. …

We are given a first-order differential equation: the rate of change of $f$ is proportional to $f$ itself. This is the hallmark of exponential functions — the only functions whose derivative is a constant multiple of the original function. The constant here is $2$, so we expect $f(x)=Ce^{2x}$. The initial condition $f(0)=e^3$ will determine $C$.

Let’s work through it step by step.

1. Separate variables The equation is $\frac{df}{dx}=2f$. We can rewrite it as

provided $f \neq 0$ (which will be true here since the initial value is positive). This isolates $f$ on one side and $x$ on the other.

2. Integrate both sides

The left side gives $\log|f|$, and the right side gives $2x + C$ (where $C$ is the constant of integration). So

3. Solve for $f$ Exponentiate both sides:

Since $e^{C}$ is a positive constant, we can write $f(x) = \pm e^{C} e^{2x}$. Let $K = \pm e^{C}$, an arbitrary constant. Then

4. Apply the initial condition We are given $f(0) = e^3$. Substituting $x=0$:

So $K = e^3$, and the solution is

$f(x) = e^3 e^{2x} = e^{2x+3}.$ …

The key idea is recognizing that the equation $f'(x)=f(x)$ is the classic differential equation whose only solutions (up to a constant factor) are exponential functions. This is because the derivative being equal to the function itself means the function grows at a rate proportional to its current value — the hallmark of exponential growth.

Why this works:

If you differentiate $e^x$, you get $e^x$ back. So any function of the form $f(x)=Ce^x$ will satisfy $f'(x)=Ce^x = f(x)$. The constant $C$ is then determined by the given condition $f(1)=2$.

1. Set up the general solution The differential equation $f'(x)=f(x)$ has the general solution

where $C$ is a constant. This is because the derivative of $Ce^x$ is $Ce^x$, matching the original function.

2. Use the initial condition We know $f(1)=2$. Substitute $x=1$ into $f(x)=Ce^x$:

3. Find $f(3)$ Now plug $x=3$ into the function: $f(3) = \frac{2}{e} \cdot e^3 = 2e^{2}.$ …

Step 1 — What an ECG records

An electrocardiogram is a graphical record of the electrical activity of the heart during one cardiac cycle, picked up by electrodes on the body surface. A standard ECG has three components:

Step 2 — Test each option

• (A) "P is the contraction of only left atria" — WRONG. The impulse from the SA node spreads over both atria; the P wave therefore reflects the excitation of the right and left atria together. Restricting it to the left atrium alone misstates the ECG. ✓ this is the mis-interpretation the question asks for …

Step 1 — Benign vs malignant tumours

Cancer cells lose the contact inhibition that normally stops a cell dividing once it touches its neighbours, so they go on dividing to form a mass — a tumour. Tumours are of two kinds:

• Benign tumours normally remain confined to their original location, do not spread to other parts of the body, and cause little damage.
• Malignant tumours are a mass of proliferating cells (neoplastic/tumour cells) that grow very rapidly, invade and damage the surrounding normal tissue, and starve the normal cells by competing for nutrients.

Step 2 — The defining terror: metastasis

NCERT states it plainly: "the most feared property of malignant tumours is METASTASIS."

In metastasis:

1. Cells sloughed off from the primary tumour enter the bloodstream.
2. They are carried to distant sites in the body.
3. Wherever they lodge, they start a NEW tumour — a secondary growth.

Step 3 — Why metastasis, and not the other three, is the most feared

All four options are genuine features of malignancy, but they differ in consequence: …

The key here is understanding what each term means in the context of tumor biology. The question describes two specific behaviors: rapid division and movement to distant parts of the body to cause new tumors. The second part — the spread — is the defining feature being tested.

Let’s break down the options:

1. Metastasis – This is the process by which cancer cells break away from the original (primary) tumor, travel through the bloodstream or lymphatic system, and form new tumors (secondary tumors) in other organs. This matches the description exactly. The word itself comes from Greek meta (beyond) + stasis (standing), meaning "displacement."

2. Metagenesis – This refers to alternation of generations in some organisms (e.g., jellyfish), where a sexual phase alternates with an asexual phase. It has nothing to do with cancer spread.

3. Teratogenesis – This is the process by which birth defects (congenital malformations) are produced, often due to exposure to harmful substances (teratogens) during pregnancy. Not related to tumor spread.

4. Mitosis – This is the normal process of cell division that produces two identical daughter cells. While cancer cells do divide rapidly via mitosis, mitosis itself is not a property unique to malignant tumors — it happens in all dividing cells. The question specifically asks about the property of moving to distant parts and causing new tumors, not just rapid division. …