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1 Differentiation

f(x) = ?x−1?+?x−3? then f′(2)=

1) ?−2? .
2) 2
3) 0
4) 1
.

Option 1

Option 2

Option 3 Solution:

Solution

Hence f′(2)=0

Option 4

2 Thermal Properties Of Matter

Four rods with different radii r and length / are used to connect two heat reservoirs at different temperature. Which one will conduct most heat ?

1) $$r = 1 \, cm, l = 1 \, m $$
2) $$r = 1 \, cm , l = \frac{1}{2} m$$
3) $$r = 2 \, cm, l = 2 \, m $$
4) $$r = 2 \, cm , l = \frac{1}{2} m$$

Option 1

Option 2

Option 3

Option 4 Solution:

Solution:

$$H = \frac{KAd\theta}{L} \propto \frac{A}{L } = \frac{\pi r^{2}}{L} $$

$$ \left(\frac{A}{L}\right)_{1} = \frac{\pi\left(1\right)^{2}}{1} = 2\pi\,\, units \hspace10mm \left(\frac{A}{L}\right)_{2} = \frac{\pi\left(2\right)^{2}}{\frac{1}{2}} = 8\pi \,\, units.$$

$$ \left(\frac{A}{L}\right)_{3} = \frac{\pi\left(1\right)^{2}}{1} = \pi \,\, units \hspace10mm \left(\frac{A}{L}\right)_{_4 }= \frac{\pi\left(2\right)^{2}}{2} = 2\pi\,\, units$$

So rod (2) will conduct most heat per second.|

3 Matrices

Let A, B, C, D be (not necessarily square) real matrices such that A^T=BCD;B^T=CDA;C^T=DAB and D^T=ABC
for the matrix S=ABCD, consider the two statements.
I S³=S
II S²=S4

1) II is true but not I
2) I is true but not II
3) Both I and II are true
4) Both I and II are false

Option 1

Option 2 Solution:

Solution

$${ A }^{ T }=BCD$$
$${ B }^{ T }=CDA$$
$${ C }^{ T }=DAB$$
$${ D }^{ T }=ABC$$
$$S=ABCD$$
$${ S }^{ T }={ \left[ ABCD \right] }^{ T }$$
$$={ D }^{ T }{ C }^{ T }{ B }^{ T }{ A }^{ T }$$
$$=(ABC)(DAB)(CDA)(BCD)$$
$$=A(BCD)A(BCD)A(BCD)$$
$$=\left( A{ A }^{ T } \right) \left( A{ A }^{ T } \right) \left( A{ A }^{ T } \right) $$
$$={ \left( A{ A }^{ T } \right) }^{ 3 }$$
$$={ (ABCD) }^{ 3 }$$
$$={ S }^{ 3 }$$
$$\therefore { S }^{ 3 }={ S }^{ T }\Rightarrow \therefore $$ I is true
$${ S }^{ 2 }={ (ABCD) }^{ 2 }$$
$$={ (A{ A }^{ T } })^{ 2 }$$
$${ S }^{ 4 }={ (ABCD) }^{ 4 }$$
$$={ \left( A{ A }^{ T } \right) }^{ 4 }$$
$${ S }^{ 2 }\neq { S }^{ 4 }\Rightarrow \therefore $$ II is not true
$$\therefore $$ I is true but not II

Option 3

Option 4

4 Ray Optics

The apparent depth of a needle lying at the bottom of the tank, which is filled with water of refractive index $$1.33$$ to a height of $$12.5 \,cm$$ is measured by a microscope to be $$9.4\, cm$$. If water is replaced by a liquid of refractive index $$1.63$$ upto the same height. What distance would the microscope have to be moved to focus on the needle again?

1) $$1.73\,cm$$
2) $$2.13\,cm$$
3) $$1.5\,cm$$
4) $$2.9\,cm$$

Option 1 Solution:

Solution:

Apparent depth $$= \frac{\text{Real deapth}}{^{a}\mu_{l}}$$

Here, Real depth $$= 12.5\, cm$$ and

$$^{a}\mu_{l} = 1.63$$

$$\therefore $$ Apparent depth $$= \frac{12.5}{1.63} = 7.67\,cm$$

Now the microscope will have to shift from its initial position to focus $$9.4 \,cm$$ depth object to focus $$7.67\, cm$$ depth object.

Shift distance $$= 9.4 - 7.67 $$

$$= 1.73 \,cm$$

Option 2

Option 3

Option 4

5 Organic Compounds Containing Nitrogen

Identify C in the following reaction,

1) Benzamide
2) Benzoic acid
3) Chlorobenzene
4) Aniline

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTIONS

6 Some Basic Principles of Organic Chemistry

Dipole moment is shown by

1) cis-1, 2-dichloroethene
2) trans-1, 2-dichloroethene
3) trans-1 2-dichloro-2 pentene
4) Both (1) and (3)

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTIONS

C is – 1, 2 – dichloro and trans 1, 2- dichloro – 2 – pentene have dipole moment where as in trans – 1, 2 – dichloro ethene bond moments cancel out hence it has μ = OD.
The dipole moment is the vector sum of all the bond.

7 Probability

An aircraft has three engines A, B and C. The aircraft crashes if all three engines fail. The probabilities of failure are 0.03, 0.02 and 0.05 for engines A, B, and C respectively. What is the probability that the aircraft will not crash?

1) 0.00003
2) 0.00090
3) 0.99997
4) 0.90307

Option 1

Option 2

Option 3 Solution:

Solution

P(aircraft will crash)=0.03 × 0.02 × 0.05 = 0.00003
P(air craft will not crash)=1−0.00003 = 0.99997

Option 4

8 Experimental Skills

There is an error of ± 0.04 cm in the measurement of the diameter of a sphere. When the radius is 10 cm, the percentage error in the volume of the sphere is
1) ± 1.2
2) ± 1.0
3) ± 0.8
4) ± 0.6

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTION

9 Aldehydes Ketones and Carboxylic Acids

Which of the following compounds on reaction with C₂H₅MgBr and water will form 2 - methyl - 2 - butanol

1) CH₃COCH₃
2) CH₃CH₂CHO
3) CH₃COOH₃
4) CH₃CH₂OH

Option 1 Solution:

SOLUTION

A ketone always gives a tertiary alcohol. The alcohol mentioned in the question contains

5 carbon atoms. Grignard reagent mentioned has two carbon atoms. Hence the ketone to be selected should have three carbon atoms. Thus acetone is the right choice.

Option 2

Option 3

Option 4

10 Laws of Motion

A particle of mass 0.3 kg is subjected to a force F = -Kx with k = 15 N/m What will be its initial acceleration if it is released from a point 20 cm away from the origin

1) 5 m/s² 2) 10 m/s² 3) 3 m/s² 4) 15 m/s²

Option 1

Option 2 Solution:

SOLUTION:

Force on particle at 20 cm away F= Kx
F = 15 × 0.2 = 3 N

Option 3

Option 4

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