CAT Model Paper 3 Questions and Answers with Explanation Part 4

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Q: 19. A person started from his house one morning in a hurry. While leaving instead of looking at his wall clock, he looked at the mirror image on the mirror hanging on the opposite wall. The clock has no numbers on its dial and hence he mixed up the time. He reached the station in 10 minutes, saw a clock on the platform and realized his mistake as well as the fact that the time on the clock at home is exactly 1 hour behind that on the platform clock. What is the time on the platform clock?

(A) 5: 35

(B) 6: 25

(C) 6: 35

(D) 5: 25

Ans: C

Solution:

  • As the time on mirror clock is exactly 1 hour behind the time on platform clock, so both them must have their minutes hand on same position. As he took 10 min to reach station that mean the difference in platform clock and original clock is exactly 10 min. Now we will take the help of the options.
  • According to 5: 35, platform time is 5: 35, so original clock time is 5: 25, so mirror clock time must be 6: 35 (mirror vision of 5: 25) , but it is not 1 hour behind 5: 35, so this option is not correct.
  • According to 6: 25, platform time is 6: 25, so original clock time is 6: 15, so mirror clock time must be 5: 45 (mirror vision of 6: 15) , but it is not 1 hour behind 5: 35, so this option is not correct.
  • According to 6: 35, platform time is 6: 35, so original clock time is 6: 25, so mirror clock time must be 5: 35 (mirror vision of 6: 25) , it is 1 hour behind 6: 35, so this option is correct.

6: 35

Q: 20. ABCD is a trapezium such that CD is the diameter of a circle which touches the side AB at M. Further the circle passes through the midpoint of the diagonals, i.e.. , the diagonals AC and BD are bisected at the points they are cut by the circle. Calculate angle BDA.

(A)

(B)

(C)

(D) Cannot be determined.

Ans: B

Solution

Consider the following figure. CQ is joined and CP is dropped perpendicular to AB.

Q_20_Find the Angle BDA

We observe that Triangle DQC is a right-angled triangle as DC is the diameter of a circle.

Considering triangles DQC and BQC, we observe that CQ is common. Angle CQD = Angle

and (by question) . Hence the triangles CQD and CQB are congruent. Hence , where r is the radius of the circle. DC is parallel to AM and angle CPM is . Hence . Similarly angle OMP and angle are equal to . Hence . Triangle CPB is right angled at P. CP = r and hypotenuse . Hence sine (angle CBP) . This implies angle . Now PBQ = alternate angle QDC and angle QDC angle (As the triangles CDQ and CBQ are congruent) .

Hence angle Angle as given by option () .

Q: 21. Let be an equilateral triangle with centroid at O. Let be a line parallel to the base passing through the centroid cutting the sides and at A and B respectively. Let P be any arbitrary point on . Perpendiculars x and y are dropped on the sides and respectively from the point P. If the altitude of the triangle is h, which among the following best describes the relation between and h?

(A)

(B)

(C)

(D)

Ans: D

Solution:

Q_21_Equilateral Triangle with Centroid at O

Consider the figure given above. The perpendiculars to the triangle from the point P are x and y. We further drop another perpendicular from p to XY and call it z. The altitude of triangle XYZ is h. If P is joined to X, Y, and Z respectively, the triangle XYZ is partitioned off into three triangles which are PXZ, PXY and PZY respectively. Let each side of the XYZ have length = a units.

Hence area triangle XYZ = . Hence (Since AB passes through the centroid, ) . Hence the correct answer is given by option () .

Q: 22. If , and , find q in terms of a.

(A)

(B)

(C)

(D)

Ans: A

Solution

Given that

Or, ; or simplifying, we get, or,

Again as given by option () .

Directions for questions 23 - 25: Answer the questions based on the given data.

Alia Boxwala, a graduate from a premier B School in India, currently a resident of Kolkata, envisaged a unique business opportunity with zero investment. She started lunch delivery service for office goers across the city near to the well-connected Sovabazar- Tollygunj metro route. For these, she appointed some undergraduates, who would carry the lunch boxes from the respective residence or caterer in morning and deliver the same to the respective individual by 1 pm. Entire service is decentralized during pick-up and delivery time. The most attractive part was the

standard pricing. Irrespective of distance, or weight of the boxes, one has to pay ₹ 600 per month. However, an additional charge of ₹ 30 per month will be levied if the distance between pick-up and delivery metro stations crosses 10 km. The model works as follows:

Q_23_to_25_Table of Model Works
9: 30 am- 10: 15 am10: 30 am- 11: 00 am11: 00 am- 12: 00 am12: 00 am- 12: 15 pm12: 30 – 1 pm
Pick-upMeeting at the nearby metro stationJourney timeHand-over to the respective delivery boysDelivery Time

She appointed separate manpower for (i) pick-up, (ii) travelling by metro with the boxes and (iii) delivery. Each pick-up and delivery boy was supposed to carry out not more than 30 orders. She appointed 36 additional manpower for travelling with the boxes. 18 among them would start travelling from Sovabazar to Tollygunj, with 2 of them getting down at each destination station collect/handover lunch boxes from the pickup/slash delivery boys, while the other 18 who would travel from Tollygunj to Sovabazar are performing the same functions. Given below are the details for the financial year 2012 - 2013. Assume that order status is constant across all 12 months.

Order Status for a month

Q_23_to_25_1_Table of Order Status for a Month
ToFromSova BazarGirish ParkChandni ChawkEspla-nadePark StreetMaidanRabindra SarovarJatin Das ParkKalighatTolly-Gunj
Sovabazar21510151112
Girish park51518
Chandni Chawk13714131213
Esplanade251015
Park Street1221118151317
Maidan15171413
Rabindra Sarovar1012111761316
Jatindas Park12161011181412411
Kalighat1212147
Tollygunj1119141713122

She had decided to pull order from a place only if the number of orders from or to that place is at least 10, except for delivery within the same location. For delivery within the same location, she managed without any additional manpower, since the pick-up boys get free after hand-over the boxes at the stations and can manage delivery to the same station after that. Each worker gets a fixed salary of ₹ 3000 per month. Over and above this they would also get an additional commission, which is certain percentage of the revenue earned, as decided by Alia at the end of every month. They are supposed to travel at their own cost.

The distances of the locations (calculated from Sovabazar) are as follows:

Calculated from Sovabazar

Q: 23. Assuming Alia did not appoint any extra manpower, what is the employee strength of her organization?

(A)

(B)

(C)

(D) None of these

Ans: A

Solution:

Manpower can be calculated from the following table:

Q_23_Table of Calculated the Manpower
ToFromSova BazarGirish ParkChandni ChawkEspla-nadePark StreetMaidanRabindra SarovarJatin Das ParkKalighatTolly-GunjTotalManpower
Sovabazar21510151112653
Girish park51518382
Handni Chaw13714131213723
Esplanade251015502
Park Street1221118151317974
Maidan15171413592
Rabindra Sarovar1012111761316853
Jatindas Park121610111814124111084
Kalighat1212147454
Tollygunj1119141713122883
Total58114537777705352575670728
Manpower242333222225

Number of manpower = Quotient of , if it is not divisible by 30.

For calculating number of delivery boys, the entries like Sova-Sova, Giri-Giri etc. will not be counted.

Thus, total manpower boys for travelling .

Option A

Q: 24. If Alia decides to spend 10 % of revenue earned on commission, then what is her profit at the end of the financial year (assuming no additional order and no additional cost) ?

(A) ₹ 12.3 lacs

(B) ₹ 19.4 lacs

(C) ₹ 13.9 lacs

(D) None of these

Ans: C

Solution:

Total number of orders , within which are Tolly - Sova (11) + Tolly - Giri (19) + Sova - Tolly (12) + Sova-Kali (11) . Hence, total revenue . Fixed salary = . Exp on commission of revenue, i.e.. , ₹ 510948. Thus profit ₹ 1394532 ₹ 13.9 lacs.

₹ 13.9 lacs

Q: 25. Institute of Sigma, India (ISI) , offers prestigious Sigma Three Award to those services that has less than 3 errors in 1 lac of deliveries within the same financial year. On 7th May, due to heavy rain, Suraj, a delivery boy failed to deliver an order on time. Assuming no such error before this one, how many more errors can Alia afford to achieve Sigma Three Award (assume 25 working days each month) ?

(A)

(B)

(C)

(D)

Ans: D

Solution:

Total number of deliveries to be executed in the year . Thus, number of affordable errors . Thus 5 more errors are acceptable.