KVPY Class-XI (SA) - 2018

x⁴ - x² + 2x - 1 = 0
(x²)² - (x -1)² = 0
⇒ (x² + x - 1) (x² - x + 1) = 0
(x² - x + 1) = 0 (This has imaginary roots.)
x² + x - 1 = 0 (This has two real roots.)
Hence, the number of real roots is 2.

S_m = m/2 [2a + (m - 1)d] = n ... (1)
S_n = n/2 [2a + (n - 1)d] = m ... (2)
By (1) and (2),
(m - n)a + (m - n){m + n - 1} d/2 = -(m - n)
⇒ 2a + (m + n - 1)d = -2 (m ≠ n)
⇒ S_{m + n} = m+n/2 [2a + (m + n - 1)d] = -(m + n)

Since both box and water are in a state of free fall (g = 0), weight is zero. So, water will not come out of the hole.

Using conservation of energy,
Increase in the kinetic energy of the alpha particle = Decrease in the electrostatic potential energy
ΔK = q (ΔV)
As we have charge on alpha particle = 2 times, the charge on electron,
( e = charge on electron)
q_α = 2q_e = 2e
Δ Ke = change of kinetic energy on accelerating an electron by 1 V
ΔK_α = 2ΔKe
1αV = 2eV

A lunar eclipse can occur on a new moon and full moon day.

Based on the principle of common ion effect, the dissociation of a weak electrolyte decreases in the presence of a common ion from a strong electrolyte.
dil. aq. solⁿ of NH₃ is NH₄OH(aq)
NH₄OH ⇌ NH₄⁺ + OH^–
NH₄Cl → NH₄⁺ + Cl^–
Due to common ion effect, equilibrium of NH₄OH shifts in backward direction, so [OH^–] ↓.
Decrease in the concentration of [OH^-] ions causes the [H⁺] to increase, and therefore, the pH decreases.

The maximum number of e^– in the shell with principal quantum number n is equal to 2n² (not n² + 2).

The energy of the electron in the 2s orbital decreases with the increase in the nuclear charge.

A filter paper soaked in KI turns brown when exposed to HNO₃ vapour due to liberation of iodine. The reaction is as follows:
6KI + 8HNO₃ → 6KNO₃ + 4H₂O + 2NO + 3I₂
Hence, X is KI.

Since oxygen has very little solubility in water, it does not dissolve in the blood plasma and needs a carrier molecule to be transported through the body.
Haemoglobin is the red pigment in the RBCs which combines with oxygen to form oxyhaemoglobin, and helps transport it to different parts of the body.

Molecules which cannot be directly used and synthesised by animal body, but may change primary process of a living system, are known as secondary metabolite.
Penicillin is an antibiotic obtained from Penicillium notatum, which is used for treatment of bacterial infection and provides artificial passive acquired immunity.

Lecithin is also called phospholipid molecule. It consists of two fatty acids, one glycerol, one phosphate and one choline molecule. It is also called phosphatidyl choline, which is an amphipathic molecule. It is a major membrane lipid constituting plasma membrane of the cell.

ψ_w of pure water is zero at standard temperature & atmospheric pressure because there is no solute in pure water. So, water potential (ψ_w) is zero. When some solute is dissolved in water, the water potential of pure water decreases.

An action potential is a rapid rise and subsequent fall in voltage or membrane potential across a cellular membrane with a characteristic pattern. Sufficient current is required to initiate a voltage response in a cell membrane. If the current is insufficient to depolarise the membrane to the threshold level, an action potential will not fire. Na⁺ channels open at the beginning of the action potential, causing depolarisation that stimulates neural conduction.

Juxtaglomerular cells of kidney produce erythropoietin hormone during oxygen deficiency. This hormone promotes red bone marrow to form RBCs (Erythropoiesis). Hemoglobin is the protein in red blood cells that helps blood carry oxygen throughout the body.

In plants, tendril is a modification of stem, leaves or petiole with a thread-like coiled shape. Functions of tendril are providing support, helping in climbing, providing attachment or cellular invasion by parasitic plant.
Examples:
1. Leaves modified into tendrils for climbing in pea
2. Stem tendrils, which develop from axillary bud in cucumber, pumpkins, watermelon and grapevines for climbing

The ribosome is a complex molecule made of ribosomal RNA molecules and proteins that form a factory for protein synthesis in cells. Ribosomes read the nucleotide sequence of a messenger RNA (mRNA) into a protein sequence, using the genetic code. They use transfer RNAs (tRNAs) to mediate this process of translation from the nucleotide language of RNA and DNA into the amino acid language of proteins.

Actin, myosin (contractile protein), troponin,and tropomyosin (regulatory protein) are essential for muscular contraction. Calcium is required by two proteins, troponin and tropomyosin, that regulate muscle contraction by blocking the binding of myosin to filamentous actin.

Trypsinogen is an inactive enzyme produced from pancreatic juice, which gets activated by the action of the enterokinase produced from intestinal gland in small intestine, into its active form trypsin.

Salamanders have soft and moist skin, covering their entire body and tail. These secretive and small creatures are principally active at night, and remain hidden under fallen logs and within moist leaf litter during the day. Specific species, such as the lungless salamanders, lack the primitive lungs that other amphibians have and breathe exclusively through their skin.

Prokaryotic cell, mature erythrocyte (animal), and mature sieve tube cell of phloem are examples of nucleus-less cell. In humans, mature red blood cells are flexible, oval, biconcave disks. They lack a cell nucleus and most organelles, in order to accommodate maximum space for hemoglobin. The biconcave shape allows RBCs to bend and flow smoothly through the body's capillaries. It also facilitates oxygen transport.

Salivary amylase is an enzyme in the mouth that initiates the digestion of carbohydrates in the form of starches by catalysing the hydrolysis of polysaccharides into disaccharides. However, the main site for carbohydrate digestion is the small intestine.

Eukaryotic cells contain a unique cluster of membrane vesicles, known as Golgi apparatus. It principally performs the function of packaging materials. Golgi apparatus is the important site of formation of glycoproteins and glycolipids. The Golgi apparatus receives proteins and lipids (fats) from the rough endoplasmic reticulum. It modifies some of them and sorts, concentrates and packs them into sealed droplets called vesicles.

Nastic movements are non-directional movements, which depend on external stimulus in plants.
These movements can be due to changes in turgor or changes in growth.
Examples:
(1) Folding of leaf of Mimosa pudica by touching
(2) Diurnal movement of leaves
(3) Response of insectivorous plant, such as Venus fly trap to their prey

O₂^2– contains 18 electrons that are the same as F₂.
So,
Bond order = 1
B₂ → σ1s² , σ *1s², σ 2s², σ *2s², π_2px¹ = π_2py¹
Bond order = 1/2 (N_b – N_a) = 1/2 (6 – 4) = 1
Thus, O₂^2– and B₂ have identical bond order.

Ethyl anions is less stable than the methyl anion because the methyl group exerts a +I effect, which increases e^– density on carbanion. So, the stability decreases.
Ethyl-free radical is more stable than the methyl-free radical because of the σ –p orbital conjugation (hyperconjugation) in the ethyl radical.

Genotype = 3ⁿ
n = Number of heterozygous pairs
= 3² = 3 × 3 = 9
Phenotype = 2ⁿ
n = Number of heterozygous pairs
= 2² = 2 × 2 = 4

[H⁺] = 10^–3 M
pH = –log₁₀ [H⁺]
= –log₁₀ [10^–3]
pH = 3
pH + pOH = 14
pOH = 14 – 3 = 11

Myopia or shortsightedness is corrected using a concave (curved inwards) lens, which is placed in front of a myopic eye, moving the image back to the retina and making it clearer.
Hyermetropia or longsightedness is corrected using a convex (outward facing) lens. This is placed in front of a hypermetropic eye, moving the image forward and focusing it correctly on the retina.

The part of the tendril in contact with the support does not grow as fast as the other parts of the plant. It coils around the support with the help of the plant hormone called auxin. This causes the tendril to circle around the object and thus, cling to it.
The unequal growth is due to more secretion of auxin in the other end and less secretion in the end in contact with the support.