KVPY SB/SX Mock Test - 4

sin x + sin y = 7/5 ------(1)

cos x + cos y = 1/5 ------(2)

By (1)² + (2)², we get

2 + 2 sin x sin y + 2 cos x cos y = 2

⇒ sin x sin y + cos x cos y = 0

cos (x - y) = 0

Therefore, x - y = 90^o

By (1) x (2), we get
sin x cos x + sin x cos y + sin y cos x + sin y cos y = 7/25

⇒ sin (90 + y) cos x + sin (x + y) + sin (x - 90) cos y = 7/25

⇒ cos y cos x + sin (x + y) - cos x cos y = 7/25

⇒ sin (x + y) = 7/25

f(x) = x³ + ax² + bx + c

f'(x) = 3x² + 2ax + b

D = 4a² - 4.3.b = 4(a² - 3b)

If a² < 2b

⇒ a² < 3b

⇒ D < 0

⇒ f'(x) = 0 has no real roots.

Hence, f(x) = 0 has 1 real and two imaginary roots.

n + 1, n + 2, ... n + 18
(1) False, if n = 19
(3) False if n = 15
16 to 33
20, 25, 30 → only three multiples of 5
(4) Number of odd integers in S_n = 9
Every third odd integer is the multiple of 3.
So, maximum prime numbers = 6

Diethyl ether is a good solvent for the formation of Grignard reagents because ether is aprotic in nature, while water, chloroform, ethyl acetate or alcohol would protonate and destroy the Grignard reagent. This is because the carbon atom in Grignard reagent is highly nucleophilic. This would form a hydrocarbon on reaction with water, alcohol etc.

Isoelectronic species have the same number of electrons. As ₂₅Mn²⁺ and ₂₆Fe³⁺ both have 23 electrons, both are an isoelectronic pair.

It is a high-spin complex as Cl^- is a weak field ligand. In [CoCl₄]^2-, the oxidation state of Co is +2, in which 3 unpaired electrons are present; this gives the spin-only magnetic moment equal to 3.83 BM.

Glycolysis is the metabolic pathway that converts glucose into two molecules of pyruvate. The process occurs in the cytoplasm of the cell.

Centrosome is an organelle near the nucleus of a cell which contains the centrioles (in animal cells) and from which, the spindle fibres develop in cell division.

A Ti or tumour inducing plasmid is a plasmid that often is a part of the genetic equipment that Agrobacterium tumefaciens and Agrobacterium rhizogenes (bacteria) used to transduce their genetic material to plants.

Fossils are more common in some kinds of sedimentary rocks than others. Fossils are most common in limestones. That is because most limestones consist partly or mostly of the shells of organisms.

Metaphase is the stage in which the chromosomes can be best visualised. Metaphase is a stage of mitosis or M phase of the cell cycle.

Silent mutations are mutations in DNA that do not significantly alter the phenotype of the organism in which they occur. Silent mutations can occur in non-coding regions (outside of genes or within introns), or they may occur within exons. These mutations do not cause any functional change in the protein product.

A conditioned reflex is an automatic response established by training to an ordinarily neutral stimulus. An unconditioned reflex is a reflex that is inborn or dependent on physiological maturation rather than on learning. Salivation in response to the aroma of food is an example of both.

Mules are a hybrid of two species – a female horse and a male donkey – so they end up with an odd number of chromosomes. A horse has 64 chromosomes and a donkey has 62. A mule inherits 63. Mules can be either male or female, but, because of the odd number of chromosomes, they can't reproduce.

Fruits produce ethylene on ripening. Ethylene serves as a ripening hormone. The fruit wrapped in paper ripens faster as ethylene produced is retained better.

Embryogenesis is the process by which the embryo forms and develops. Embryogenesis starts with the fertilisation of the egg cell (ovum) by a sperm cell (spermatozoon). Once fertilised, the ovum is referred to as a zygote, a single diploid cell. The zygote undergoes mitotic divisions with no significant growth (a process known as cleavage) and cellular differentiation, leading to development of a multicellular embryo.

The total length of double helix DNA in an organism can be calculated by multiplying the total number of base pairs with distance between two consecutive base pairs.
Let the total number of base pairs be x.
Length of DNA = Total number of base pairs X Distance between two consecutive base pairs
1 mm = x X 0.34 X 10^-9 m
By solving this, we get x = 2.9 X 10⁶ bp
This is equivalent to 3 million base pairs.

Base pairing is according to Chargaff rule. According to it, a hydrogen bonds with T by two H bonds. G is hydrogen bonded to C by 3 H bonds during DNA replication. When RNA synthesis occurs from DNA, T is replaced by U.

Some of the carbon dioxide is transported dissolved in the plasma. Some carbon dioxide is transported as carbaminohaemoglobin. However, most carbon dioxide is transported as bicarbonate.

Modern evolutionary theory consists of the concepts of Darwin modified by knowledge concerning the process of natural selection. This explains that adaptability has genetic basis, which proves fitness of an organism and nature selects organism for its fitness and allows to produce its progeny in large numbers.

Pachytene is the third stage of the prophase of meiosis, following zygotene, during which the paired chromosomes shorten and thicken, the two chromatids of each separate and exchange of segments between chromatids may occur.

A nonsense mutation is a point mutation in a sequence of DNA that results in a premature stop codon. Entry of T causes formation of stop codon.

According to the law of segregation, during gamete formation, the alleles for each gene segregate from each other so that each gamete carries only one allele for each gene. According to the law of independent assortment, genes for different traits can assort or segregate independently during the formation of gametes.

Blood thinners do not actually thin blood, but they can stop blood clots from forming or growing larger. Hence, they prevent cardiovascular accidents, which can cause stroke.

In Hardy-Weinberg equation (p² + 2pq + q² = 1), p² represents the frequency of the homozygous genotype AA, q² represents the frequency of the homozygous genotype aa and 2pq represents the frequency of the heterozygous genotype Aa.
Heterozygous population = 2pq = .48

K_max = 3 - 2.3 = 0.7 eV
Now from W.E. theorem:
If voltage difference is 1 V, then distance = 5 mm.
So, if voltage difference is 0.7 V, then d = 0.7 x 5 = 3.5 mm

N₁V₁ = N₂V₂

Mili eq. of hypo = 0.25 x 100

Mili eq. of hypo = 25

Eq. of hypo = 0.025

Mole of hypo = 0.025 x 1

Weight of hypo = 0.025 x 248 = 6.2 g

Each of the two alternative forms of a gene that arise by mutation and are found at the same place on a chromosome are called alleles. These exist in pairs and exhibit a dominant-recessive relationship.

It is estimated that marine plants produce between 70 and 80 percent of the oxygen in the atmosphere. Nearly all marine plants are single-celled, photosynthetic algae.

Let the red colour be represented by R and white by W (as both are dominant). Parental generation: RR X WWF1 generation will give roan colour ---RW

On self crossing F1: RW X RW, the genotypes formed will be: RR : RW : RW: RR.

This gives a genotypic ratio of 1 : 2 : 1.

An ORF is a continuous stretch of codons that contains a start codon (usually AUG) and a stop codon (usually UAA, UAG or UGA). An ATG codon within the ORF (not necessarily the first) may indicate where translation starts.

The condition given can be explained by Wobble hypothesis. The Wobble Hypothesis explains why multiple codons can code for a single amino acid. One tRNA molecule (with one amino acid attached) can recognise and bind to more than one codon, due to the less-precise base pairs that can arise between the 3rd base of the codon and the base at the 1st position on the anti-codon.

The total cross-sectional area of capillaries arising from an artery is much greater than that of the artery. This leads to lowering of the blood pressure when it reaches the capillary network.

Nocturnal animals have retinas that contain a high percentage of rods to increase sensitivity to low light conditions.

Differentiation of cells in the embryo is brought about by both internal cellular factors as well as extracellular factors, that act on the cell from the outside.

Individuals are predicted to behave more altruistically and less competitively toward their relatives, because they share a relatively high proportion of their genes (e.g., one-half for siblings and one-eighth for cousins). Consequently, by helping a relative reproduce, an individual passes its genes to the next generation, increasing their Darwinian fitness. Altruistic behaviour toward relatives may at some later time lead to increased competition between relatives, reducing or even completely removing the net selective advantage of altruism.