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1 Matrices

The matrix $$X$$ for which $$\begin{bmatrix} 1&-4 \\ 3 & -2\end{bmatrix} X = \begin{bmatrix} -16 & -6\\ 7 & 2\end{bmatrix}$$ is

1) $$\begin{bmatrix} -2& 4\\ -3 & 1\end{bmatrix}$$
2) $$\displaystyle \begin{bmatrix} - \cfrac{1}{5} & \cfrac{2}{5} \\ - \cfrac{3}{10} & \cfrac{1}{5}\end{bmatrix}$$
3) $$\begin{bmatrix} -16& -6\\ 7 & 2\end{bmatrix}$$
4) $$\displaystyle \begin{bmatrix} 6& 2 \\ \cfrac{11}{2} & 2\end{bmatrix}$$

Option 1

Option 2

Option 3

Option 4 Solution:

Solution

Given, $$\begin{bmatrix} 1&-4 \\ 3 & -2\end{bmatrix} X = \begin{bmatrix} -16 & -6\\ 7 & 2\end{bmatrix}$$

Let $$A=\begin{bmatrix} 1 & -4 \\ 3 & -2 \end{bmatrix}$$ , $$B=\begin{bmatrix} -16 & -6 \\ 7 & 2 \end{bmatrix}$$

So, $$AX=B$$
$$\Rightarrow X=A^{-1}B$$

Here, $$|A|=10$$
Now, $$adj A= C^{T}={\begin{bmatrix} -2 & -3 \\ 4 & 1 \end{bmatrix}}^{T}$$

$$\Rightarrow adj A=\begin{bmatrix} -2 & 4 \\ -3 & 1 \end{bmatrix}$$

$$A^{-1}=\dfrac{1}{10}\begin{bmatrix} -2 & 4 \\ -3 & 1 \end{bmatrix}$$

$$\Rightarrow A^{-1}B=\dfrac{1}{10}\begin{bmatrix} 60 & 20 \\ 55 & 20 \end{bmatrix}$$

Hence, $$X=\begin{bmatrix} 6 & 2 \\ \dfrac{11}{2} & 2 \end{bmatrix}$$

2 Chemistry in Everyday Life

Commonly used antiseptic 'Dettol' is a mixture of

1) o-chlorophenoxylenol + terpineol

2) o-cresol + terpineol

3) phenol + terpineol

4) chloroxylenol + terpineol

Option 1

Option 2

Option 3

Option 4 Solution:

Solution

One of the most known antiseptics, Dettol is a mixture of terpineol and chloroxylenol. These ingredients make dettol an effective antiseptic-disinfectant that kills bacteria.

3 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

4 Oscillations

A simple pendulum is attached to the roof of a lift. Its time period of oscillation, when the lift is stationary is 0.5 s. Its frequency of oscillation when the lift falls freely will be

1) infinite
2) 2 Hertz
3) zero
4) 20 Hz

Option 1

Option 2

Option 3 Solution:

Solution

When lift falls with acceleration g, net effective force acting on pendulum is 0 (mg downwards due to gravity and mg upwards due to pseudo force.). Thus net acceleration will be zero.
Now, frequency, $$f=1/T=1/2\pi \sqrt{g_{eff}/l}=0$$ (because $$g_{eff}=0$$)

Option 4

5 Vector Algebra

A vector perpendicular to the plane containing the points A(1, -1, 2), B(2, 0, -1) and C(0, 2, 1) is

Option 1

Option 2 Solution:

Solution

Option 3

Option 4

6 Magnetic Effects of Current

Yellow light of wavelength 6000Å produces fringes of width 0.8mm in Youngs double slit experiment. If the source is replaced by another monochromatic source of wavelength 7500Å and the separation between the slits is doubled then the fringe width becomes

1) 0.1mm
2) 0.5mm
3) 4.3mm
4) 1mm

Option 1

Option 2 Solution:

Solution:

Fringe width in first case, $$\beta_{1} = \frac{D\lambda_{1}}{d}\quad...\left(i\right) $$

Fringe width in second case, $$\beta_{2}=\frac{D\lambda_{2}}{2d} \quad...\left(ii\right) $$

Divide equation $$\left(ii\right)$$ by $$\left(i\right)$$,

$$ \therefore \frac{\beta_{2}}{\beta_{1}}= \frac{D\lambda_{2}/ 2d}{D\lambda_{1} /d} = \frac{1}{2} \cdot\frac{\lambda_{2}}{\lambda_{1}} $$ or

$$\beta_{2} = \frac{1}{2}\cdot\frac{\lambda_{2}}{\lambda_{1}} \cdot\beta_{1} $$

$$\therefore \beta_{2} = \frac{1}{2}\times\frac{ 7500 Å}{6000 Å} \times 0.8 mm $$

$$= 0.5 mm $$

Option 3

Option 4

7 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

8 Chemistry in Everyday Life

An example of a psychedelic agent is

1) LSD

2) TNT

3) DNA

4) DDT

Option 1 Solution:

Option 2

Option 3

Option 4

9 Hydrocarbons

Which of the following compounds will not undergo Friedel -Craft's reaction easily?

1) Cumene
2) Xylene
3) Nitrobenzene
4) Toluene

Option 1

Option 2

Option 3 Solution:

SOLUTION

Nitro group being electron withdrawing, deactivates the benzene nucleus to such an extent so that it becomes incapable to give Friedel -Crafts reaction.
Nitrobenzene because of its unreactivity towards Friedel-Craft's reaction is used as a solvent for this reaction.

Option 4

10 Work Power Energy

The work done on a particle of mass m by a force

(K being a constant of appropriate dimension), when the particle is taken from the point (a, 0) to the point (0, a) along a circular path of radius a about the origin in the x-y plane is …

1) 2kπ / a
2) kπ / a
3) kπ / 2a
4) zero

Option 1

Option 2

Option 3

Option 4 Solution:

SOLUTION:−

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