KVPY Class-XI (SA) - 05-11-2017

Let b = 10, c = 9 and d = 8
Sum of three sides > 4^th side
b + c + d > a
a < 27
Maximum possible length of the largest side = 26

x² - y² = 12345678 (x, y ∈ I⁺)
The R.H.S. is even, so x and y should be odd integers.
The difference of the squares of two odd integers is a multiple of 8, but the R.H.S. is not a multiple of 8.
So, S is the empty set.

72^x . 48^y = 6^xy
3^{2x + y}.2^{3x + 4y} = 2^xy.3^xy
Compare: 2x + y = xy ... (i)
and 3x + 4y = xy ... (ii)
From (i) and (ii), x = -3y
Put in (i).
-5y = -3y²
⇒ y = 5/3
So, x + y = -2y = - 10/3

43361 = p₁ × p₂
Numbers co-prime with 43361 = 42900
Numbers non co-prime with 43361 = 461
Let 3 × 5 = 15
3k = {3, 6, 9, 12, 15}
5k = {5, 10, 15}
Number of non co-prime with 15 = {3, 6, 9, 12, 15, 5, 10}
= 5 + 3 -1 = 7
Similarly, for 43361,
Number of non co-prime with 43361 = p₁ + p₂ - 1
p₁ + p₂ - 1 = 461
p₁ + p₂ = 462

x⁵ - 6x⁴ + 11x³ - 5x² - 3x + 2 = 0

(x - 1)(x - 2)(x³ - 3x + 1) = 0

The equation (x³ - 3x + 1) has non-integer roots.

∴ Required sum of roots = 3

Isotopes: Molecules having same number of protons
Number of Neutrons = Mass Number - Atomic Number
Isobars: Molecules having same number of nucleons

Eye lens produces upright down (inverted) image.
Iris controls the aperture in the eye.

Na is the most electropositive element in the list and has the least ionisation energy (IE) as it acquires an octet configuration after the loss of one electron. Moreover, it is an element of the period III, whereas others belong to the period II.

Within a period, the ionisation energy increases across the period, but the IE of N is more than that of O.
This is because, N has half filled stable configuration and hence the ionisation energy is high.
Hence, the correct order is:
Na < B < O < N

On moving down the group, the basic strength of the oxides increases and acidic strength decreases.
This is because the electronegativity of the central atom decreases.
Moreover, the oxidation state of phosphorus in P₂O₅ is +5, but that of other elements is +3 in the listed oxides.

Copper is extracted from its sulphide ore by first roasting followed by auto reduction.

S^-2 and Ca²⁺  have 18 electrons each; hence, they are isoelectronic.

The shell with n = 3 has subshells: 3s (accommodates 2 electrons), 3p (accommodates 6 electrons) and 3d (accommodates 10 electrons).
Hence, the third shell can accommodate total 18 electrons.

Atomic radius decreases across a period and increases down a group.

The major excretory product of birds is uric acid in the form of a white paste or powder and conserves as much water as possible.

Codon degeneracy means that several of the amino acids are coded by more than one codon. The degeneracy of the genetic code is what accounts for the existence of synonymous mutations.

Interphase refers to all stages of the cell cycle other than mitosis. During interphase, cellular organelles double in number, the DNA replicates, and protein synthesis occurs. DNA replicates in S phase of interphase.

Gene flow is the transfer of gene or allele from one population to another. Gene flow may change the frequency and/or the range of alleles in the populations.

The tobacco mosaic virus affects photosynthetic tissue of the host leading to distortion, blistering and necrosis.

Waste products diffuse out of placenta into maternal blood because it is permeable for ammonia and other nitrogenous wastes.

Nucleosome consists of DNA wrapped around histone during DNA packaging. It is not membrane-bound organelle. The nucleosomes are part of what is called chromatin, the DNA and proteins that make up the chromosomes.

Tyrosine with iodine forms thyroxine. Thyroxine (T4) contains four iodine atoms.

The maximum number of oxygen molecules that can bind to one molecule of haemoglobin is four, due to the presence of four heme molecules.

Smooth endoplasmic reticulum helps in the formation of lipid synthesis and metabolism, the production of steroid hormones, and detoxification.

ATP and NADPH are assimilatory power produced in light reaction, i.e. used in dark reaction.

Hypothalamus directly controls the production of glucocorticoids and androgens. By ACTH-RH, which acts on adenohypophysis for release of ACTH, ACTH stimulates the adrenal cortex for the production of glucocorticoids and androgens.

Acetaminophen is paracetamol not produced by plant or fungi; it is artificially formed.

Lamarck is best known for his theory of inheritance of acquired characteristics, first presented in 1801. If an organism changes during life in order to adapt to its environment, those changes are passed on to its offspring.

Total number of ways to choose three distinct numbers from A = ³⁰C₃
Multiple of 3 in set A = {3, 6, 9, ..., 30}
Multiple of 3 but not 9 in set A = {3, 6, 12, 15, ..., 30}
When choosing three distinct numbers and the product is not a multiple of 3 and 9, then number of ways = ²⁰C₃
When choosing three distinct numbers and the product is not a multiple of 9 but 3, then number of ways = (⁷C₁ × ²⁰C₂)
Number of ways in which one can choose three distinct numbers from A so that the product of the chosen numbers is divisible by 9
= ³⁰C₃ - ²⁰C₃ - (⁷C₁ × ²⁰C₂) = 1590

The nucleus does not divide and it contains 6 ng of DNA at the end of S phase as in this phase, DNA replication occurs.

In photosynthesis, photophosphorylation and in aerobic respiration, oxidative phosphorylation occurs that require proton gradient.

VC = TV + IRV + ERV

= (600 + 2500 + 800) ml

= 3900 ml

Honeybee and ant are haploid organisms, so they produce sperm without involving meiosis.