Test: CAT Quantitative Aptitude- 4 - CAT MCQ

The sum of the roots of a quadratic equation ax² + bx + c is - b / a
⇒  -(2m - 1)/m = 6/(3m + 1)
⇒ 6m² - m -1 = -6m
⇒ 6m² + 5m -1 = 0
⇒ m = 1/6 or -1
Thus, the sum of the values of m: 1/6 -1 = -5/6

When a right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm, then solid formed is a cone whose height, h = 8 cm.
The radius of the cone, r = 6 cm.
Slant height of the cone, l = 10 cm.
∴ Volume of the cone

Hence, the volume and surface area of the cone are 301.7 cm³.

n³ + 100 = (n + 10)(n² - 10n + 100) - 900

Therefore, if n + 10 divides n³ + 100, then it must also divide 900. Since we are looking for the largest n, n is maximized whenever n + 10 is, and since the largest divisor of 900 is 900, we must have:

n + 10 = 900 → n = 890

The largest n is therefore 890.

For a given set containing N elements, the number of non-zero subsets = 2^N - 1
Number of non-zero subsets in S = 2⁴⁰⁰ - 1

The product of the elements of a subset will be odd when all the elements of the subset are odd.

The odd elements in S are S_odd = (1, 3, 5 , ..., 399)
Number of non-zero subsets in S_odd = 2²⁰⁰-1

Number of subsets where product of its elements is even = (2⁴⁰⁰ - 1) -(2²⁰⁰-1) = 2⁴⁰⁰ - 2²⁰⁰

Hence, option 1.

The maximum area of trapezium will be obtained when the two sides are on the opposite sides of the diameter.

The height of the trapezium = OE + OF
OE = √OC² - √EC² = √3
OF = √OB² - √FB² = 2√2

∴ the area of the trapezium = (1 / 2) x (2√2 + √3) x (6 + 4) = 5(2√2 + √3)

Hence, option 2.

We know that,
Distance remaining = Total distance - Distance travelled
⇒ d(t) = 32 - 30t

⇒ 3M + 9 + M + 5 + 140= 180
⇒ 4M = 26
⇒ M = 6.5°

If we draw a figure according to the information provided in the question then we can note that:
In the question it is good that each girl is facing the center of the circle.

Radius of circle =10m, Diameter of circle =20m

We need to find the shortest distance between any two girls faces each other.

We can see from the image that oppositely standing girls will only face each other so the shortest distance between any two girls facing each other will be equal to the diameter of the circle ie 20 m.

The distances travelled for various time intervals are:


Hence, total distance in 2 hour = 
= 60 + 55 = 115 km

• Here, sell price = Rs 540, profit = Rs 90
• So, Cost price = 540 – 90 = Rs 450
• Now, new sell price = (450 + 200) × 90 / 100 [as, 10% discount on mark price]
• = 650 × 90 / 100 = Rs. 585
• So, profit = 585 – 450 = Rs. 135
• Always profit to be on C.P
• So, on C.P Rs. 450 profit is Rs. 135
• So, on C.P Rs. 100, profit is 135 × 100 / 450 = 30%
• So, profit percentage is 30%.

When the policeman arrives, the thief has traveled 6 meters.
So, the time taken by the policeman to catch the thief along the circular track = 6/1.5 = 4 seconds.

The total distance covered by the policeman = 12 meters.
Thus, the minimum distance is when the policeman takes the route via the center i.e. he travels a length of 2 x 5 = 10 meters.

The minimum time taken = 10/3 seconds.

The minimum distance covered by the thief = 1.5 x (10/3) = 5 meters

Hence, option 3.

 x = 4⁴ x 5⁶ x 6⁵ x 7¹⁰ x 8⁸
⇒ x = 2³⁷ x 3⁵ x 5⁶ x 7¹⁰ = (2 x 3) (2³⁶ x 3⁴ x 5⁶ x 7¹⁰)

Power of 2 can take values of 0, 2 , 4 , . . . , 36 = 19
Power of 3 can take values of 0, 2,4 = 3
Power of 5 can take values of 0, 2,4, 6 = 4
Power of 7 can take values of 0, 2,4, 6, 8, 10 = 6

Total number of factors = 19 x 3 x 4 x 6 = 1368 

If x is a prime number greater than 1, then x is not a factor of (x - 1)!

There are 29 primes up to 110. 4 is also one such number.

Hence, option 4.

Let the speed of the stream be v kmph.
Speed of boat in still water is 24 kmph.
Hence, the speed of boat upstream is (24 – v) kmph and the speed of boat down stream is (24 + v) kmph.

Putting v = 6 kmph:
Thus, the speed of boat in still water is 4 times as fast as the speed of the stream.

The minimum value of a quadratic expression ax² + bx + c (when a > 0) occurs at x = -b / 2a and its minimum value is -(b² - 4ac) / 4a

The minimum value of the equation is = [4(3)(-23) - (11)²]/12 = -397 / 12

Hence, option 4.

Area of each slab = 72 / 50 m² = 1.44 m²
Length of each slab =√1.44 =1.2m =120cm

Let the total population of the town be x.
The number of people who read XYZ = 0.37x - 8000
0.37x+0.37x - 8000 + 0.27x = x
⇒ x = 8,00,000

Hence, option 1.

x² + x - 2 < 0
⇒ (x - 1) (x + 2) < 0
⇒ (x - 1) < 0 and (x + 2) > 0 or (x - 1) > 0 and (x + 2) < 0
⇒ x < 1, x > -2 or x > 1, x < -2 which is not possible.

∴ -2 < x < 1
⇒ (x - 1) < 0 and (x + 2) > 0 

Hence, option 3.

When the minute hand travels 60 minutes, the hour hand travels 5 minutes.

The minute hand travels 55 minutes more than the hour hand in an hour.
In 54 minutes, the Minute hand travels = (54 x 55/ 60) = 49.5

Hence, option 3.

Let the number of hours worked by B be x.
The hours worked by A = (12 - x)
The manhours of work finished by A = (12 - x - 2) x 0.6 + 2

∵ (12 - x - 2) x 0.6 + 2 = x
⇒ x = 5

The total manhours of work done by them in a day = 2 * x = 2 * 5 = 10

Thus, the number of days required to finish the work = 120/10 =12 days

Hence, option 2.

Let the number of apples in the carton with the least number of apples be x and the second least number of apples be y.
 ∵ x +y + x + 100 = 270
⇒ 2x +y = 170...(i)

Also, the total amount spent on the two cartons with x and y apples = 4500 - 1900 = 2600
⇒ 20x + 25y = 2600
⇒ 4x + 5y = 520.. .(ii)

From (i) and (ii)
x = 55,y = 60

The lowest number of apples in a cartoon = 55.

Hence, option 3.

Let the length of the saree be ‘x’ cm.

x - (1/4) x - (2/5) x = 3850
⇒ (7/20) x = 3850
⇒ x = 11000 cm = 110 m