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1 Biomolecules

Starting with three different amino acid molecules, how many different tripeptide molecules could be formed ?

1) 12
2) 9
3) 3
4) 6

Option 1

Option 2

Option 3

Option 4 Solution:

2 Semiconductors and Electronic Devices

An n-p-n transistor of current gain 80 in common emitter mode gives emitter current equal to 8.1 mA. The base current is

1) 0.1 $$\mu$$A
2) 0.1 mA
3) 0.01 mA
4) 0.01 $$\mu$$A

Option 1

Option 2 Solution:

Solution:

$$\beta=80$$

$$l_{c}=8.1mA$$

$$\left(\frac{l_{c}}{l_{b}}\right) =80\left(l_{e}=l_{b}+l_{c}\right)$$

$$\left(\frac{l_{e-l_b}}{l_{b}}\right) =80$$

Option 3

Option 4

3 Electric Charges And Fields

Consider a thin spherical shell of radius R with its centre at the origin , carrying uniform positive surface charge density. The variation of the magnitude of the electric field |E(r)| and the electric potential V(r) with the distance r from the centre, is best represented by which graph ?

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTION

4 General Principles & Process of Isolation of Metals

What produce when a mixture of Cu₂O and Cu₂S is heated?

1) O₂
2) SO₂
3) H₂S
4) None of these

Option 1

Option 2 Solution:

SOLUTION

It undergoes auto reduction forming copper and sulphur dioxide.
$$2{ Cu }_{ 2 }O+{ Cu }_{ 2 }S\xrightarrow { heat } 6Cu+S{ O }_{ 2 }$$

Option 3

Option 4

5 Titrations

For the reaction between KMnO₄? and H₂O₂?, the number of electrons transferred per mole of H₂?O₂? is:

1) 1

2) 2

3) 3

4) 4

Option 1

Option 2 Solution:

Solution

3H₂?O₂?+2KMnO₄?−>3O₂?+2MnO₂?+2KOH+2H₂?O

Number of electron transfer per mole ofH₂O₂? is 2.

Option 3

Option 4

6 Probability

In a given race, the odds in favour of four horses A,B,C & D are 1:3,1:4,1:5 and 1:6 respectively. Assuming that a dead heat is impossible, find the chance that one of them wins the race

1) 101/420

2) 101/210

3) 319/420

4) 111/420

Option 1

Option 2

Option 3 Solution:

Solution

Given, Odds in favour of four horses $$A,B,C,D$$ are $$1:3,1:4,1:5,1:6.$$
The probability of winning the horse A, $$P(A)=\dfrac{1}{4}$$

The probability of winning the horse B, $$P(B)=\dfrac{1}{5}$$

The probability of winning the horse C, $$P(C)=\dfrac{1}{6}$$

The probability of winning the horse D, $$P(D)=\dfrac{1}{7}$$
The probability that one of the horse winning the race $$=(P(A)+P(B)+P(C)+P(D))$$
$$=(\displaystyle\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7})$$

$$=\displaystyle\frac{(210+168+140+120)}{840}=\frac{319}{420}$$

Option 4

7 Laws of Motion

In the following figure, calculate the value of force exerted on the 1.5 kg block by the 2.5 kg block.Given that all surfaces are frictionless.

1)25.0 N

2)28.2 N

3) 27.6 N

4)19.1 N

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTION:

Force on 1.5 kg block due to 2.5 kg block=gravitational force on 2.5 g block along inclined plane
force=mgsin50=2.5 ×10 × 0.766=19.15 N

8 Differentiation

Let f be a non-negative function defined on the interval [0,1] .If $$\int_0^x \sqrt{1-(f'(t))^2} dt = \int _0^x f(t) dt, 0 \leq x \leq 2,$$ and $$f(0) =0$$ then,

1) $$f\left(\frac{1}{2} \right) < \frac{1}{2} $$ and $$f\left( \frac{1}{3} \right) > \frac{1}{3}$$
2) $$f\left(\frac{1}{2} \right) > \frac{1}{2} $$ and $$f\left( \frac{1}{3} \right) > \frac{1}{3}$$
3) $$f\left(\frac{1}{2} \right) < \frac{1}{2} $$ and $$f\left( \frac{1}{3} \right) < \frac{1}{3}$$
4) $$f\left(\frac{1}{2} \right) > \frac{1}{2} $$ and $$f\left( \frac{1}{3} \right) < \frac{1}{3}$$

Option 1

Option 2

Option 3 Solution:

Solution

Differentiating using Libnitz formula, we get

$$\sqrt{1-(f^{'}(x))^{2}}=f(x)$$

$$\sqrt{1-(f(x))^{2}}=f^{'}(x)=\cfrac{d(f(x))}{dx}$$

$$dx=\cfrac{d(f(x))}{\sqrt{1-(f(x))^{2}}}$$

Now, on integarting we get

$$x+C=\sin^{-1}(f(x))$$

Now for x to be real, we can say that

$$0\le f(x)<1 \, or\, f(x)<\cfrac{1}{2} and f(x)<\cfrac{1}{3}$$

Option 4

9 Complex Numbers

If m₁ , m₂ , m₃ and m₄ respectively denote the moduli of the complex numbers 1 + 4i, 3 + i,1- i and 2 - 3i , then the correct one, among the following is
1) m₁ < m₂ < m₃ < m₄
2) m₄ < m₃ < m₂ < m₁
3) m₃ < m₂ < m₄ < m₁
4) m₃ < m₁ < m₂ < m₄

Option 1

Option 2

Option 3 Solution:

SOLUTION

∴ m₁ = |1 + 4i|;m₂ = |3 + i|;m₃ = |1 — i|;
m₄ = |2 — 3i|

∴ m₁ = √1+6 = √17 ; m₂ =√9+1 = √10

m₃ =√1+1 =√2; m₄=√9+4 = √13
∴ m₃ < m₂ < m₄ < m₁
Here m₃ will be the smallest

Option 4

10 Fundamentals of Mathematics

Ten ants are on the real line. At time t=0, the k^th ant starts at the point k² and travelling at uniform speed, reaches the point (11−k)₂ at time t=1. The number of distinct time at which at least two ants are at the same location is?

1) 45

2) 11

3) 17

4) 9

Option 1

Option 2

Option 3 Solution:

Solution

Velocity of $$ k^{th} $$ ant $$ = \left ( 11-k \right )^{2} - k^{2} = 121 - 22k $$

Position of $$ k^{th} $$ ant at time $$ t : x_{k} = k^{2} + t\left ( 121-22k \right ) $$

Let $$ p^{th} $$ ant and $$ k^{th} $$ ant ($$ k $$ and $$ p $$ are distinct) be at the same position at time $$ t $$.

$$ \Rightarrow k^{2} + t\left ( 121 - 22k \right ) = p^{2} + t\left ( 121-22p \right ) $$

$$ \Rightarrow \left ( k^{2}-p^{2} \right ) -22t\left ( k-p \right ) = 0 $$

$$ \Rightarrow k+p = 22t $$

$$ k $$ and $$ p $$ both range from $$ 1 $$ to $$ 10 $$ and are distinct, so $$ (k+p) $$ attains each value in the set $$ \left \{ 3,4,5...18,19 \right \} $$.

Hence, there are 17 distinct times at which at least two ants are at the same location.

Option 4

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