KVPY Class-XI (SA) - 06-11-2016

16⁵5¹⁶
= 16 × 16⁴ × 5¹⁶
= 16 × 10¹⁶
It is an 18 digit number.

t² = at + b ; a, b ∈ I⁺
t³ = at² + bt
= a(at + b) + bt
= a² t + bt + ab
⇒ t³ = (a² + b)t + ab , check possibility for a, b ∈ I⁺ from options.
(1) a² + b = 4
ab = 3 possible for (a = 1, b = 3)
(2) a² + b = 8
ab = 5 not possible
(3) a² + b = 10
ab = 3 possible (a = 3, b = 1)
(4) a² + b = 6
ab = 5 possible (a = 1, b = 5)

(1 + a + b)² = 3 (1 + a² + b² )
1 + a² + b² + 2a + 2ab + 2b = 3 + 3a² + 3b²
1 ^. a + a ^. b + 1 ^. b = 1² + a² + b²
So, 1 = a = b is the only possible solution pair.

Let the sides be a, a and h.
So, 4a + 4h + 4a = 2(a² + ah + ah)
2a + 2h + 2a = a² + 2ah
⇒ a² - 4a = 2h - 2ah
⇒ (a² - 1) + 1 - 4(a - 1) - 4 = 2h(1 - a)
⇒ (a - 1)(a + 1) - 4(a - 1) - 3 = 2h(1 - a)
⇒ 2h = 3/a - 1 + 4 - (a + 1)
So, a = 2 and h = 2 are the only integral solution (a and h are positive integers).
Sum of all edges = (4 × 2) + (4 × 2) + (4 × 2) = 24

n = ab
ab = a² + b³
⇒ 10a + b = a² + b³
⇒ a(10 - a) + b(1 - b)(1 + b) = 0
⇒ a(10 - a) = (b - 1)(b) (b + 1)
If b = 2; a(10 - a) = 6 ⇒ No value of 'a'
If b = 3; a(10 - a) = 24 ⇒ a ∈ {4, 6}
{The numbers are 43 and 63.}
b = 4; a(10 - a) = 60 ⇒ No value of a
b = 5; a(10 - a) = 120 ⇒ No need to check further
∴ The numbers are 43 and 63.

f(xy) = f(x) + f(y)
⇒ f(x) = log_ax
So, f(12) = 24
⇒ log_a12 = 24
⇒ 12 = a²⁴ and f(8) = 15
⇒ log_a8 = 15
⇒ 3log_a2 = 15
⇒ log_a2 = 5
So, f(48) = log_a48 = log_a12 + log_a4
= log_a12 + log_a2²
= 24 + 2 ^. 5
= 34

Since the voltage production is based upon A.C. supply and this voltage is D.C, which is constant, no flux will change in secondary and no voltage will be induced.

High density is not the reason for its uses in clinical thermometers.

2LiOH + CO₂ → Li₂CO₃ + H₂O

Number of moles of Li = 1/24
Number of moles of CO₂ used = 1/48
Mass of CO₂ = 1/48 × 44 = 0.916 g

As we move from left to right in a period atomic, radius decrease due to increase in Z_eff. So, the largest radius is of lithium.

The distance between 2 nucleotides/nitrogen bases is 3.4 angstroms and human DNA containing 6.6 x 10⁹ bp multiplied by this distance gives a length of 2.2 meters

Vaccine of Diphtheria, Pertussis and Tetanus (DPT) consists of
(i) Toxoids of Diptheria and Tetanus
(ii) Heat killed cells of Pertussis

Lipase = Enzyme (Lipid digesting)
Amylase = Enzyme (Starch digesting)
Trypsin = Enzyme (Endopeptidase)
Bilirubin = Bile pigment

The bicarbonate buffer system is an acid-base mechanism involving the balance of carbonic acid, bicarbonate ion and carbon dioxide in order to maintain pH in the blood. The pH of avian blood is maintained by bicarbonate ion.

Podocytes are cells of squamous epithelium of bowman's capsule of nephrons.

According to Chargaffs rule, the molar concentration of purines is equal to molar concentration of pyrimidines.
A + G = T + C
So, if Adenine is 20%, then T is also 20% because A always pairs with T, total of 40%, which we then subtract from 100%, leaving us with 60%. Because cytosine and guanine are complimentary base pairs, we now have the answer to the amount of guanine in this DNA sequence: 30%
Hence, G is 30% and C is also 30%.

Pellagra can be cured by - Niacin
- Nicotinamide
- Tryptophan
Tryptophan is a precursor of Niacin. All three can prevent pellagra.

In all living organisms, there are 64 codons and 3 codons are stop or termination codons, i.e. UAA, UAG and UGA which do not code for any amino acids.

Order of Silkworm ⇒ Lepidoptera
(Bombax mori)

Relaxin is a hormone secreted by the placenta that causes the cervix to dilate and prepares the uterus for the action of oxytocin during labour.

The connecting links between reptiles and mammals are:
(1) Platypus
(2) Echidna

The genetic makeup of Turner's syndrome is 44 + XO. So, there are a total of 45 chromosomes only.

Chipko movement was headed by social activist Sunder Lal Bahuguna in Uttarakhand to save trees from felling.

The primary carbon skeletons used for gluconeogenesis are derived from pyruvate, lactate, glycerol, and the amino acids alanine and glutamine. Lysine cannot be converted to glucose.

Causative agent of Syphilis is Treponema pallidum.

a⁵ - a³ + a = 2; a ∈ R⁺
Let f(a) = a⁵ - a³ + a - 2; {Note f'(a) > 0 ∀ a ∈ R⁺}
For a⁶ = 3,
a = 3^1/6 = 1.2 {use calculator}
We get f(1.2) < 0
At a = 4^1/6,
f(4^1/6) > 0
So, one root in a⁶ ∈ (3, 4)

nx² - 7√nx + n = 0, where n is a positive integer.

D = 49n - 4n²

= n(49 - 4n)

D ≠ 0 for any n ∈ I⁺. So, the roots are distinct.

For roots to be real, D ≥ 0

So, n ≤ 49/9

So, n can be {1, 2, 3, ...,12}

Clearly, the product of the roots is 1.

We need a 3-digit number which is divisible by 4 and 5 both.
i.e. Their last two digits are: 00, 20, 40, 60 and 80.
Now, ending with 00 are:
{100, 200, 900}
If a digit repeats other than '0', then they are:
{220, 440, 660, 880}
But 220 can be permuted according to the condition as {220, 202}.
So, there are '8' other favorable cases.
If the number has no digit repeated, like 320:
320 can be permuted in 4 ways.
{302, 230, 320, 203}
So, such numbers are: 8 x 4 x 4 = 128
Total favourable cases = 9 + 8 + 128 = 145
So, required probability = 145/900 = 29/180 

The partial pressure of carbon dioxide in outside air (dry air at sea level) is 0.3 mm Hg.

Neurogenic heart → King crab

Bronchial heart → Shark

Pulmonary heart → Human

(i) Hydrolase catalyses the hydrolysis of ester, ether, peptide, glycosidic, C-C, C-halide or P-N bonds.
(ii) Lyase catalyses the breaking of chemical groups other than hydrolysis.
(iii) Isomerase catalyses the inter-conversion of optical, geometric or positional isomers.
(iv) Ligase catalyses the linking together of two compounds.