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1 Basic Concepts of Chemistry

20.0 mg of a magnesium carbonate sample decomposes on heating to give carbon dioxide and 8.0 mg magnesium oxide. What will be the percentage purity of magnesium carbonate in the sample?
[Atomic weight of Mg=24]

1) 60
2) 80
3) 75
4) 96

Option 1

Option 2 Solution:

Solution

MgCO_3?→MgO+CO₂?

The molar masses of MgCO₃? and MgO are 24+12+48=84 g/mol and 24+16=40 g/mol respectively.

Thus, 8 g MgO will be obtained from (84 / 40)?×8=16.8 mg of MgCO₃??.

The percentage purity of magnesium carbonate in the sample is 16.8 ?×100/ 20=80%

Option 3

Option 4

2 Aldehydes Ketones and Carboxylic Acids

In Benzilic acid rearrangement,

1) Benzil is converted into Benzilic acid

2) Benzilic acid is converted into Benzil

3) Benzoin is converted into Benzilic acid

4) C₆H₅CHO is converted into Benzoin

Option 1 Solution:

Option 2

Option 3

Option 4

3 Differentiation

y=(c₁cosx + c₂sinx)e^−x+ c₃e^x

1) $$\displaystyle \frac{d^{3}y}{dx^{3}}-\frac{d^{2}y}{dx^{2}}+2y=0$$
2) $$\displaystyle \frac{d^{3}y}{dx^{3}}-y=0$$
3) $$\displaystyle \frac{d^{3}y}{dx^{3}}+\frac{d^{2}y}{dx^{2}}-2y=0$$
4) $$\displaystyle \frac{d^{2}y}{dx^{2}}-y=0$$

Option 1

Option 2

Option 3 Solution:

Solution

$$\displaystyle y=(c_1\cos x+c_2\sin x)e^{-x}+c_{3}e^{x}$$ ....... $$(i)$$
$$\Rightarrow \displaystyle \frac{dy}{dx} = -(c_1\cos x+c_2\sin x)e^{-x}+(-c_1\sin x+c_2\cos x)e^{-x}+c_3e^{x}$$
$$\Rightarrow \displaystyle \frac{dy}{dx} = -y+2c_3e^{x}+(-c_1\sin x+c_2\cos x)e^{-x}$$ ........ $$(ii)$$ [Using $$(i)$$]

Differentiating above equation,

$$\Rightarrow \displaystyle \frac{d^2y}{dx^2} = -\frac{dy}{dx}+2c_3e^x-(-c_1\sin x+c_2\cos x)e^{-x}+(-c_1\cos x-c_2\sin x)e^{-x}$$
Now using $$(i)$$ and $$(ii)$$
$$\displaystyle \frac{d^2y}{dx^2} =-2\frac{dy}{dx}+4c_3e^x-y+c_3e^x-y=-2\frac{dy}{dx}-2y+5c_{3}e^x $$ (iii)
Again differentiating,
$$\displaystyle \frac{d^3y}{dx^3} = -2\frac{d^2y}{dx^2}-2\frac{dy}{dx}+5c_{3}e^x$$ (iv)
Eliminating $$c_3$$ from (iii) and (iv) we get,
$$\displaystyle \frac{d^3 y}{dx^3}+\frac{d^2 y}{dx^2}-2y=0$$

Option 4

4 Trigonometric Equations

If 4nα = π, then the value of tanα. tan2α. tan3α. tan4α. ….. tan(2n – 2)α tan (2n – 1)α is

1) 0
2) 1
3)-1
4) 1/2

Option 1

Option 2 Solution:

Solution

2nα = π/2

(tanα tan(2n – 1)α) (tan 2α tan(2n – 2)α) ….. (tan (n – 1)α tan (n + 1)α).tan nα

(tanα tan(π/2 - α))(tan2αtan(π/2-2α))......tanπ/4 =1

Option 3

Option 4

5 Matrices

If A=[a_ij] is a square matrix of even order such that [a_ij]=i²−j², then

1) A is a skew-symmetric matrix and |A| = 0
2) A is skew-symmetric matrix and |A| is a square
3) A is symmetric matrix and |A| = 0
4) none of these

Option 1

Option 2 Solution:

Solution

Given $$\displaystyle \left [ a_{ij} \right ]=i^{2}-j^{2}$$
$$\Rightarrow a_{ii}=0$$ for all i.

Also, for $$i \ne j, a_{ij}=i^{2}-j^{2}=-(j^{2}-i^{2})=-a_{ji}$$
$$\Rightarrow a_{ij}=-a_{ji}$$

Hence, A is skew-symmetric matrix.
For order 2, $$A=\begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix}$$
$$|A|=9\ne 0$$

Hence, A is a skew-symmetric matrix and |A| is a square

Option 3

Option 4

6 Ray Optics

A ray of light is incident on the hypotenuse of a right-angled prism after travelling parallel to the base inside the prism. If μ is the refractive index of the material of the prism, the maximum value of the base angle for which light is totally reflected from the hypotenuse is

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTION

7 Chemical Thermodynamics

The value of standard electrode potential for the change will be

1) - 0.072 V
2) 0.385 V
3) 0.770 V
4) - 0.270 V

Option 1

Option 2

Option 3 Solution:

SOLUTIONS

Option 4

8 Complex Numbers

The area of the triangle with vertices z,izand−iz is ?

1) $$\large\frac{1}{2}|z|^2$$
2) $$\large\frac{1}{4}|z|^2$$
3) $$|z|^2$$
4) $$\large\frac{1}{8}|z|^2$$

Option 1

Option 2

Option 3 Solution:

SOLUTION

If z=x+iy, then the point represented by z is (x,y)

i2 = −1

$$|z|=\sqrt{x^2+y^2}$$

Area of the triangle with vertices (x₁,y₁),(x₂,y₂)and(x₃,y₃) is given by $$\large \frac{1}{2}\left|\begin {array}{ccc}x_1 & y_1 & 1\\x_2 & y_2 &1\\x_3 &y_3&1\end {array}\right|$$

Let z = x + iy

iz=i(x+iy)=xi+i²y=−y+xi

−iz=−i(x+iy)=−xi−i²y=y−xi

Then the vertices of the triangle are given by
A(z)=(x,y),B(iz)= (−y,x)andC(−z)=(y,−x)

Area of ΔABC = $$\large \frac{1}{2}\left|\begin {array}{ccc}x & y & 1\\-y & x &1\\y &-x&1\end {array}\right|$$

=1/2|1(xy−xy)−1(−x²−y²)+1(x²+y²)|

=x²+y²=|z|²

Option 4

9 Classification of Elements and Periodicity

The ions O²⁻, F⁻, Na⁺, Mg²⁺ and Al³⁺ are isoelecronic. Their ionic radii show:

1) a significant increase from O²⁻ to Al³⁺
2) a significant decrease from O²⁻ to Al³⁺
3) an increase from O²⁻ to F⁻ and then decrease from Na⁺ to Al³⁺
4) an decrease from O²⁻ to F⁻ and then increase from Na⁺ to Al³⁺

Option 1

Option 2 Solution:

SOLUTION

As all the given species are iso electronic, the ionic radii decreases with increasing atomic number as effective nuclear charge increases. Hence from O²⁻ to Al³⁺, the size decreases significantly

Option 3

Option 4

10 Kinematics

How many seconds does it take a metal ball to drop 200. m from rest?

1) 6.39 s
2) 4.28 s
3) 8.58 s
4) 7.29 s

Option 1 Solution:

Option 2

Option 3

Option 4

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