Question 1 |

Equal sized circular regions are shaded in a square
sheet of paper of 1 cm side length. Two cases,
case M and case N, are considered as shown in the
figures below. In the case M, four circles are shaded
in the square sheet and in the case N, nine circles
are shaded in the square sheet as shown.

What is the ratio of the areas of unshaded regions of case M to that of case N?

What is the ratio of the areas of unshaded regions of case M to that of case N?

2:3 | |

1:1 | |

3:2 | |

2:1 |

Question 1 Explanation:

2d=1\Rightarrow d=\frac{1}{2}

Area of circle =\frac{\pi d^2}{4}=\frac{\pi}{16}

Total circle area = 4 \times \frac{\pi}{16}=\frac{\pi}{4}

3d=1\Rightarrow d=\frac{1}{3}

Area of circle =\frac{\pi d^2}{4}=\frac{\pi}{36}

Total circle area = 9 \times \frac{\pi}{36}=\frac{\pi}{4}

\frac{\text{Unshaded area M}}{\text{Unshaded area N}}=\frac{1^2-\frac{\pi}{4}}{1-\frac{\pi}{4}}=\frac{1}{1}=1:1

Question 2 |

Four cities P, Q, R and S are connected through oneway routes as shown in the figure. The travel time
between any two connected cities is one hour. The
boxes beside each city name describe the starting
time of first train of the day and their frequency of
operation. For example, from city P, the first trains
of the day start at 8 AM with a frequency of 90
minutes to each of R and S. A person does not spend
additional time at any city other than the waiting
time for the next connecting train. If the person starts from R at 7 AM and is required
to visit S and return to R, what is the minimum time
required?

6 hours 30 minutes | |

3 hours 45 minutes | |

4 hours 30 minutes | |

5 hours 15 minutes |

Question 2 Explanation:

R at 7 am

1 hour journey

Reached Q at 8 am

At Q buses available timings are 5 am, 7am, 9 am ..

Person started at Q at 9 am

1 hour journey reached P at 10 am.

Buses timings at P are 8 am, 9:30am, 11 am, 12:30 pm...

Person started at P at 11 am

1 hour journey

Reached S at 12 noon.

Buses timings at S are

8am, 8:45 am, 9:30 am, 10:15 am, 11am, 11:45 am, 12:30pm ....

Person started at S at 12:30 pm

1 hour journey

Reached R at 1:30 pm

Minimum Total time = 1:30 pm - 7 am

= 6 hrs 30 min

1 hour journey

Reached Q at 8 am

At Q buses available timings are 5 am, 7am, 9 am ..

Person started at Q at 9 am

1 hour journey reached P at 10 am.

Buses timings at P are 8 am, 9:30am, 11 am, 12:30 pm...

Person started at P at 11 am

1 hour journey

Reached S at 12 noon.

Buses timings at S are

8am, 8:45 am, 9:30 am, 10:15 am, 11am, 11:45 am, 12:30pm ....

Person started at S at 12:30 pm

1 hour journey

Reached R at 1:30 pm

Minimum Total time = 1:30 pm - 7 am

= 6 hrs 30 min

Question 3 |

Consider the following functions for non-zero
positive integers, p and q

f(p,q)=\underbrace{p \times p \times p \times ...p}_{q \;\;times} =p^q;\;\; f(p,1)=p

g(p,q)=p^{p^{p^{\vdots ^{q \;\;times}}}};\;\; g(p,1)=p

Which one of the following options is correct based on the above?

f(p,q)=\underbrace{p \times p \times p \times ...p}_{q \;\;times} =p^q;\;\; f(p,1)=p

g(p,q)=p^{p^{p^{\vdots ^{q \;\;times}}}};\;\; g(p,1)=p

Which one of the following options is correct based on the above?

f(2,2)=g(2,2) | |

f(g(2,2),2) \lt f(2,g(2,2)) | |

g(2,1) \neq f(2,1) | |

f(3,2) \gt g(3,2) |

Question 3 Explanation:

f(2,2) = 2 x 2 = 4

g(2,2) = 4

f(2,2) = g(2,2)

g(2,2) = 4

f(2,2) = g(2,2)

Question 4 |

For the past m days, the average daily production at
a company was 100 units per day.

If today's production of 180 units changes the average to 110 units per day, what is the value of m?

If today's production of 180 units changes the average to 110 units per day, what is the value of m?

18 | |

10 | |

7 | |

5 |

Question 4 Explanation:

Total production = Average x No. of days

P= 100 m

and

P + 180 = 110 (m+1)

Solving, m = 7

P= 100 m

and

P + 180 = 110 (m+1)

Solving, m = 7

Question 5 |

Which one of the groups given below can be assembled to get the shape that is shown above using each piece only once without overlapping with each other?

(rotation and translation operations may be used).

A | |

B | |

C | |

D |

Question 5 Explanation:

Priority: Shape, position and size.

As per size point of view parallelogram is not suitable.

As per size point of view parallelogram is not suitable.

Question 6 |

If f(x)=2 \ln\sqrt{e^x} what is the area bounded by f(x)
for the interval [0,2] on the x- axis?

\frac{1}{2} | |

1 | |

2 | |

4 |

Question 6 Explanation:

\begin{aligned}
f(x)&=2 \ln ( \sqrt{e^x})\\
&=2 \ln (e^{x/2})=2 \log _e e^{x/2}\\
f(x)&=2\left ( \frac{x}{2} \right )=x
\end{aligned}

Area =(1/2) x 2 x 2=2

Area =(1/2) x 2 x 2=2

Question 7 |

Which one of the following is a representation (not
to scale and in bold) of all values of x satisfying the
inequality 2-5x\leq -\frac{6x-5}{3} on the real number
line?

A | |

B | |

C | |

D |

Question 7 Explanation:

\begin{aligned}
2-5x \leq -\frac{6x-5}{3}\\
1-9x\leq 0\\
\Rightarrow 1\leq 9x\Rightarrow x\geq \frac{1}{9}
\end{aligned}

Question 8 |

In a 12-hour clock that runs correctly, how many
times do the second, minute, and hour hands of the
clock coincide, in a 12-hour duration from 3 PM in
a day to 3 AM the next day?

11 | |

12 | |

144 | |

2 |

Question 8 Explanation:

3 pm - 3am = 12 hrs

Coincide is

11 times in 12 hours

Coincide is

11 times in 12 hours

Question 9 |

An equilateral triangle, a square and a circle have
equal areas.

What is the ratio of the perimeters of the equilateral triangle to square to circle?

What is the ratio of the perimeters of the equilateral triangle to square to circle?

3\sqrt{3}: 2: \sqrt{\pi} | |

\sqrt{3\sqrt{3}}: 2: \sqrt{\pi} | |

\sqrt{3\sqrt{3}}: 4: 2\sqrt{\pi} | |

\sqrt{3\sqrt{3}}: 2: 2\sqrt{\pi} |

Question 9 Explanation:

\begin{aligned} \frac{\sqrt{3}}{4}a^2 &=s^2 &=\pi r^2&=k\\ a&=\sqrt{\frac{4k}{\sqrt{3}}}&s=\sqrt{k} &\;\;\; r=\sqrt{\frac{\pi}{k}}\\ &\text{Perimeters:}\\ 3a&:&4a&:&2 \pi ^2\\ 3\frac{2\sqrt{k}}{3^{1/4}}&:&4\sqrt{k}&:&2 \pi \frac{\sqrt{k}}{\sqrt{\pi}}\\ 3^{3/4}&:&2&:&\sqrt{\pi} \end{aligned}

Question 10 |

A block with a trapezoidal cross-section is placed over a block with rectangular cross section as shown above.

Which one of the following is the correct drawing of the view of the 3D object as viewed in the direction indicated by an arrow in the above figure?

A | |

B | |

C | |

D |

Question 10 Explanation:

View from arrow direction is

There are 10 questions to complete.

Answer to 113 question is option b plz correct it

Thank You Hania,

We have updated the answer.

for question 22 the options dosent has the right option it shall 80/81 rather than 81/80

mistake in question 25 , as per solution the data for april second row (second row last column) should be 70 but here its given as 170

ques no. 65 ans should be “c”…..

We have updated the Answer as (C).