KVPY (SA) Mock Test - 5

2^x - x - 1 = 0
2^x = x + 1
At x = 0 ⇒ 2^o = 1 and x + 1 = 0 + 1 = 1
At x = 1 ⇒ 2¹ = 2 and 1 + 1 = 2

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
If a³ + b³ + c³ = 3abc, then either a + b + c = 0 or (a² + b² + c² - ab - bc - ca) = 0
If a + b + c = 0, then a + b = -c
Now, if (a² + b² + c² - ab - bc - ca) = 0, then (1/2) (2a² + 2b² + 2c² - 2ab - 2bc - 2ca) = 0
Or, (1/2)((a - b)² + (b - c)² + (c - a)²) = 0, which is only possible if a = b = c
As in the answer options, we only have a + b = -c, so it is correct.

Let one of the roots be P, so the other will be 2P.
Sum of the roots = P + 2P = 3P = - b/a
P = - b/3a
And product of roots P × 2P = 2P² = c/a
2 (-b/3a)² = c/a
2 × b²/9a² = c/a
So, 2b² = 9ac

Suppose n is odd.

Then, for n = 1, k is between 2 and 400.

Suppose n is odd.

Then, for n = 1, k is between 2 and 400.

Then, sum S₁ = 397/2 (3+399) = 200 × 397, which is not divisible by 9

So, n is also not of the form 3k + 1 and 3k + 2.

So, n is an even number.

Range of the projectile is the same if the angle of projection with horizontal is either θ or (90°- θ). Here, angles of projection are 30° and 60° (= 90° - 30°), so the range of two projectiles will be the same.

If the volume of two blocks is the same, then the buoyant force acting on the two blocks will be the same irrespective of their densities. So, the buoyant force required to overcome the gravitational force will be equal, i.e. F₁ = F₂. 

If a torch welder is working without his safety glasses, his retina is most likely to get affected.

If we take any charge from one point to another inside a charged spherical shell then no work will be done, this means that inside a spherical charge the potential at all points is the same and its value is equal to that on the surface, that is 10 V in this case.

Since there is no mass change and also no change in the distance between centres of both, the radius would be unchanged.

The IUPAC name of acetaldehyde (CH₃CHO) is ethanal.

The tendency to lose electrons increases as we go down a group in the case of alkali metals. So, Fr has the highest tendency to lose electrons.

In H-F, fluorine has very high electronegaivity, while hydrogen is very low, therefore there is large charge separation and so the dipole moment increases, however, H-Cl has more dipole moment than H-F, bcoz the H-F bond length is very small as compared to H-Cl, while the difference in charge is not much.

In the reductive cleavage, the ozonides are treated with a reducing agent such as Zn dust and water to form aldehydes or ketones or a mixture of both depending upon the structure of the alkene.

Since the enthalpy of elements in their standard state is taken to be zero, the heat of formation of a compound (ΔH_f) may be positive or negative

Besides urination, another important activity in which water is substantially lost from the body is expiration. Water is lost in sweat, in expired air, and in urine. Under normal conditions, the amount of water lost in the faeces is relatively small.

The cell body contains the necessary structures for keeping the neuron functional. It includes the nucleus, mitochondria, and other organelles.

Human males are termed as hemizygous for any X-chromosome genes, meaning that there are only half (hemi) as many alleles as normally present for a diploid individual. As a result, if males inherit X-linked recessive trait from their mothers, they will show the recessive phenotype.

DNA condensation takes place during the M phase and RNA synthesis stops.

The bacteria retain blue colour, which is due to low lipid contents in their cell wall.

Involucre - Bract
Ray florets - Launaea
Centrifugal - Cymose
Pyhllode - Acacia

It secretes insulin. Insulin causes cells in the liver, skeletal muscle and fat tissue to take up glucose from the blood and thus reduces the blood glucose level.

Haemodialysis, commonly called kidney dialysis or simply dialysis, is a process of purifying the blood of a person whose kidneys are not working normally.

Bile salts emulsify fats and solubilise them. This aids in digestion and absorption of fats.

Roots show movements towards water in soil to receive water. Therefore, roots are positively hydrotropic and not negatively hydrotropic.

A lacteal is a lymphatic capillary that absorbs dietary fats in the villi of the small intestine. Triglycerides are emulsified by bile and hydrolysed by the enzyme lipase, resulting in a mixture of fatty acids, di- and monoglycerides.

Insulin is a hormone produced by pancreas that helps in regulating blood sugar level. Diabetes is caused when the pancreas do not produce enough insulin.

Oxidative phosphorylation is the process in which ATP is formed as a result of the transfer of electrons from NADH or FADH₂ to O₂ by a series of electron carriers. This process, which takes place in mitochondria, is the major source of ATP in aerobic organisms.
As oxisomes are present in the inner mitochondrial space of the mitochondrion, they are associated with oxidative phosphorylation.

f(0) = p and f(1) = p - 3
Since f(0) and f(1) have opposite signs, so if p < 0, then p - 3 > 0, i.e. p > 3, which is not possible.
Or, p > 0 and p - 3 < 0
⇒ p < 3
⇒ 0 < p < 3

One root is i - 1 & other root should be - i - 1 as coefficients are rational.
Sum of roots = - 2. & Product = - i ^₂ + (- 1)² = 2
Equation reqd is x² + 2x + 2 = 0

Distance between centre of mass and inertial axes is the largest. So, torque will be high and kinetic energy will be high.

According to Lewis, acids are those species which can accept lone pair of electrons. BF₃ has incomplete octet; so it acts as Lewis acid.

Hybridisations of the carbon atom in case of ethane is sp³ and each hybrid orbital has 25% s-character.
Hybridisations of the carbon atom in case of ethene is sp² and each hybrid orbital has 33% s-character.
Hybridisations of the carbon atom in case of ethyne is sp and each hybrid orbital has 50% s-character.

Wings of kiwi, splint bone of horse and hind limbs of python are vestigial organs.

It must be recessive because both parents are normal, but one or both parents must have passed the disease allele to their daughter. Since one or both parents are heterozygous and are not affected by the Tay Sachs allele, the disease must be recessive.

A makes a double bond with T and C forms a triple bond with G.

People with cerebral palsy have various motor problems, such as spasticity, paralysis and even seizures.

Proctitis is the inflammation of the lining of the rectum.