A pendulum of length L oscillates with an angular amplitude of θ = 60° and time period T. Let T₀ = 2π√(L/g) be the time period for small angle of oscillations, where g is the acceleration due to gravity. If air resistance is negligibly small and the string remains straight, then which one of the following is correct?

The correct answer is A, The motion repeats after time T only once. This is the incorrect statement, because periodic motion repeats after every interval T, again and again.

The time period T of a pendulum is the time for one complete oscillation, that is, the smallest interval after which the bob returns to the same position moving in the same direction. Since the motion is periodic, it repeats after T, 2T, 3T and so on, without limit as long as no energy is lost. The formula T = 2π√(L/g) also assumes small swings; for large angles the period grows a little, so T stays constant only when the angular displacement is small.

B is correct: the period is by definition the least time after which motion repeats.

C is correct: if the motion repeats after T, it also repeats after any whole multiple nT.

D is correct: the period is independent of amplitude only for small angular displacements.

Exam tip: period = shortest repeat time; motion also repeats at every nT; isochronism holds only for small swings.

The correct answer is A, Mitochondria, nucleus, chloroplast. These are the three places in a eukaryotic cell where DNA is found.

The nucleus holds nearly all of a cell's DNA, packed with proteins into chromosomes. Mitochondria and chloroplasts each carry a small circular DNA of their own, together with 70S ribosomes, which lets them make some of their own proteins and divide independently. This supports the idea that both organelles descend from bacteria that once lived inside early cells. Every organelle in this group therefore contains DNA, which no other option can claim.

B is wrong: Golgi bodies package and send out proteins and have no DNA.

C is wrong: the plasma membrane is a lipid-protein layer with no DNA.

D is wrong: ribosomes are made of RNA and protein; they contain no DNA.

Exam tip: DNA in a eukaryotic cell: nucleus, mitochondria, chloroplasts; ribosomes contain RNA, not DNA.

The correct answer is A, Ribosome. A ribosome contains RNA, but it has no genes of its own; it reads messenger RNA sent from the DNA of the nucleus or an organelle.

Ribosomes are made of ribosomal RNA and proteins and are the sites of protein synthesis. Their rRNA is structural and catalytic: it helps join amino acids, but it does not carry a code for any protein. The instructions come from mRNA, which is copied from DNA. The nucleus holds the cell's main DNA, while mitochondria and chloroplasts have small circular DNA molecules that code for some of their own proteins. Ribosomes have no membrane and no DNA, so they cannot direct their own protein making.

B is wrong: the nucleus contains the chromosomes, which carry most of the cell's protein-coding genes.

C is wrong: mitochondria have their own DNA that codes for some proteins of the respiratory chain.

D is wrong: chloroplasts have their own DNA that codes for proteins such as part of the enzyme RuBisCO.

Exam tip: ribosome = protein factory without its own genes; nucleus, mitochondria and chloroplasts carry DNA.

The correct answer is A, 1 only. A chemical reaction produces only internal forces, which cannot change the motion of the centre of mass but can change internal kinetic energy.

By Newton's third law, internal forces come in equal and opposite pairs, so their sum is zero. The centre of mass of a system therefore accelerates only under an external force; with none acting, it keeps moving at the same velocity v whatever happens inside. Internal forces can, however, do work on the particles. A reaction that releases energy, such as an explosion, makes the fragments fly apart faster, raising their kinetic energy relative to the centre of mass. A shell that bursts in mid-air is the classic example: the pieces scatter, yet the centre of mass follows the original path.

Statement 2 is wrong: internal forces can and do change the kinetic energy of the parts relative to the centre of mass.

So B and C, which accept Statement 2, are wrong, and D is wrong because Statement 1 is correct.

Exam tip: internal forces never change total momentum or centre-of-mass velocity, but they can change kinetic energy.

The correct answer is A, Density of water at 4°C is 1000 kg/m³. Water is densest at 4 °C, where one cubic metre has a mass of almost exactly 1000 kg.

Most liquids grow steadily denser as they cool, but water behaves oddly below 4 °C: it expands again as it cools towards 0 °C. This is called the anomalous expansion of water, and it makes 4 °C the point of maximum density, about 1000 kg/m³ or 1 g/cm³. Because of it, in winter the densest water at 4 °C sinks to the bottom of lakes while ice forms on top, so fish survive beneath the frozen surface.

B is wrong: at 0 °C liquid water is slightly less dense, about 999.8 kg/m³, because it has expanded below 4 °C.

C is wrong: 100 kg/m³ is a tenth of water's real density, far too low.

D is wrong: 10 kg/m³ is only a hundredth of the true value of about 1000 kg/m³ at 4 °C.

Exam tip: water is densest at 4 °C (1000 kg/m³ = 1 g/cm³); ice is lighter, about 917 kg/m³, so it floats.