GRE PAPER- I
1. DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. Two dice are thrown together Column A Column B The probability of getting a total of 7 The probability of getting a total more than 7 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given Column A : n(S) = 62 = 36 n(E) = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)} ∴ required probability = Column B : n (S) = 62 = 36. E = {(2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (3, 5), (4, 4), (4, 5), (5, 3), (5, 4), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
15
Solution. (b)
∴ Required probability = 36 12 2.
=
5
= 0.42.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. One litre of water was mixed to 3 litres of sugar solution containing 4% of sugar. Column A Column B % of sugar in solution now 2 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
Solution. (a) It's indirect proportion ⇒ 3 : 4 : : x : 4 ⇒ 4x = 12 ⇒ x = 3%. 3. DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. Column A Column B The perimeter of a rectangle whose area is 48 The perimeter of a rectangle whose area is 64 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
Solution. (d) For rectangle, Area = length × breadth and Perimeter = 2 (length + breadth) Hence, we can't calculate the perimeter if only area is given. 4. DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer.
B
90° O 4 A
O is the centre.
Column A Column B The area of shaded region. 4π (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
1 πr 2 − 1 1 1 × r × r = π (4) 2 − × 4 × 4 = 4π − 8 sq. unit 2 4 2
Solution. (b) Area of shaded region = 4 5.
A lizard comes out of a hole at one corner on the floor of a room, runs 5 ft along the foot of a wall, then climbs vertically up that wall (height of wall = 12 ft) and having reached the ceiling, crawls 9 ft along it at right angles to the wall. How far is the lizard from the hole? (a) 10 5 ft (e) 12 ft (b) 5 10 ft (c) 13 ft (d) 15 ft
Solution.
(b)
F E
D
C
A
B
The lizard crawls along AB = 5 ft, BC = 12 ft and CE = 9 ft AE = Diagonal of a 5 ft × 12 ft × 9 ft cuboid
AE = 25 + 144 + 81 = 5 10 ft
6.
A letter lock contains four rings and each ring contains five letters. If the lock opens in only one arrangement of four letters, how many unsuccessful events are possible? (a) 625 (b) 1024 (c) 624 (d) 1023 (e) 824 (c) Each ring contains five letters. Therefore, we have for each ring five different ways of bringing a letter to the opening position. The number of ways in which the four rings can combine = 5 × 5 × 5 × 5 = 625. But, of these attempts to open the lock, only one will be successful. Hence, the possible number of unsuccessful events = 625 – 1 = 624.
x y x2 2 = 3 , then 3 ÷ y2 = ?
Solution.
7.
If
(a) (e)
4 9 16 27
(b)
4 3
(c)
16 9
(d)
4 27
Solution.
x2 1 1 ⎛ x ⎞ 1⎛2⎞ 4 × = ⎜ ⎟ = ⎜ ⎟ = 3 y2 3 ⎜ y ⎟ 3⎝ 3⎠ 27 ⎝ ⎠ (d)
2
2
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. Column A Column B (a) (b) (c) (d) –16 quantity in column A is greater quantity in column B is greater both quantities are equal the relationship cannot be determined from the information given
( −2 )
( − 2 ) ( −2 )
( −2)( −2)
( −2)
Solution. (a) 9.
Column A :
⎡⎛ 1 ⎞ 2 ⎤ = ⎢⎜ − ⎟ ⎥ ⎢⎝ 2 ⎠ ⎥ ⎣ ⎦
−2
⎛ 1⎞ = ⎜− ⎟ ⎝ 4⎠
−2
= (−4) 2 = 16
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. 5 D C
3
A
Column A Area of shaded region (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given Solution. (c) Column A : Area of rectangle = 5 × 3 = 15 sq. units
1 × DC × AD = 15 1 sq . units × 5× 3 = 2 2 15 2
B E ABCD is a rectangle. Column B area of ABCD
Area of unshaded region = 2
∴ Area of shaded region = Area of rectangle –
15 −
Column B :
= Area of ABCD = 15 sq. unit.
∴ 1 2
15 15 sq . units = 2 2
(area of ABCD) =
15 2
sq. units.
10.
A motorcyclist travels for 10 hours, the first half at 21 km/h and the other half at 24 km/h. Find the distance travelled. (a) 225 km (b) 224 km (c) 200 km (d) 324 km (e) 350 km
Solution.
(b) Let the total distance = 2x
∴ x x 8x + 7 x 168×10 + = 10 ⇒ = 10 ⇒ 15x = 168×10 ⇒ x = ⇒ x = 112 km 21 24 168 15
∴ Total distance = 2 × 112 = 224 km. 11. DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. x% of 50 is 80. Column A Column B 80% of x 120 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
x × 50 = 80 ⇒ x = 160 100 (a) Given : 80
Solution.
Column A : 80% of x = 80% of 160 = 100 12.
×160 = 128.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. 2x – 1 + 2x + 1 = 1280 Column A Column B x 8 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
1280× 2 ⎛1 ⎞ ⇒ 2 x ⎜ + 2 ⎟ = 1280 ⇒ 2 x = = 512 ⇒ 2 x = 29 ⇒ x = 9. 5 ⎝2 ⎠ = 1280
Solution. (a) 13.
Given :
2x – 1
+
2x + 1
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer.
A 4 B
Column A (a) (b) (c) (d)
4 C
4
Column B
quantity in column A is greater quantity in column B is greater both quantities are equal the relationship cannot be determined from the information given
Solution. (c) Since all the sides of triangle are equal, therefore, triangle is an equilateral triangle. ⇒ All the angles are equal. Directions for questions 14 and 15 : The following bar-diagram shows the no. of students enrolled for their graduation in the different disciplines in a certain college. Study it carefully and answer the questions that follow.
No. of students
450 400 350 300 250 200 150 100 50 0
360
390
400
345
315
325
300
300
260
280
280
280
Arts Science Commerce
1994
1995 Years
1996
300
1997
335
1998
14.
What is the percentage difference between the Arts students of 1995 and 1997 ? (a) 0.27% (b) 0.12% (c) 2% (d) 1.23% (e) 2.4%
⎛ 300 360 ⎞ − ⎜ ⎟ × 100% ⎝ 860 1040 ⎠ ⎛ 15 9 ⎞ ⎜ − ⎟ × 100% ⎝ 43 26 ⎠
Solution.
(a) Required percentage difference = = (0.3488 – 0.3461) × 100% = 0.27%.
=
15.
What is the percentage increase in the no. of students from 1995 to 1998 ? (a) 28.5 (b) 30.4 (c) 24.5 (d) 32.5 (e) None of these (d) Total no. of students in 1995 = 300 + 280 + 280 = 860 and that in 1998 = 390 + 400 + 350 =1140 Required percentage increase =
1140 − 860 ×100 = 32.5 860 %
Solution.
16.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. x + 2y = 4 y – 2x = 2 Column A Column B y 2 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
Solution. (c) Given equations are x + 2y = 4 ....(1) y – 2x = 2 ....(2) Multiplying (1) by 2 and then add in equation (2), we get 5y = 10 ⇒ y = 2.
17.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer.
x= 11 13
Column A
Column B
350
x +1 x −1
x
(a) (b) (c) (d)
quantity in column A is greater quantity in column B is greater both quantities are equal the relationship cannot be determined from the information given
x= 11 x + 1 11 + 13 −24 ⇒ = = = −12. 13 x − 1 11 − 13 2
Solution. (b) 18.
(using componendo and dividendo)
If x and y are negative, then which of the following statements is/are always true? I. x + y is positive II. xy is positive III. x – y is positive. (a) I only (b) II only (c) III only (d) I and III only (e) II and III only (b) x < 0, y < 0 (x + y) < 0, xy > 0 and x – y may be + ve or – ve II is always true.
Solution.
19.
Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio must these metals be mixed so that the mixture may be 15 times as heavy as water? (a) 2 : 3 (b) 3 : 2 (c) 1 : 3 (d) 2 : 1 (e) None of these (b) Let the weight of water = x units ⇒ Weight of gold = 19x units and weight of copper = 9x units Also assume that gold and copper are added in the ratio g : c ⇒ 19x . g + 9x . c = 15x . (g + c) or 19g – 15g = 15c – 9c or 4g = 6c
g 6 3 = = or c 4 2
Solution.
Directions for questions 20 and 21 : Study the following pie-diagrams carefully and answer the questions that follow :
PERECENTAGE COMPOSITION OF HUMAN BODY Skin 1/10 Muscles 1/3 Bones 1/6 Water 70%
Proteins 16% Other dry elements 14%
Hormones and enzymes
Solution.
In the human body, what part is made of neither bones nor skin ?
1
(a) 40 (e) None of these
(b)
3 80
(c)
2 5
(d)
5 7
Solution. 21.
(d)
⎛ 1 1 ⎞ 11 1− ⎜ + ⎟ = Part of the body made of neither bones nor skin = ⎝ 6 10 ⎠ 15 .
What is the ratio of the distribution of proteins in the muscles to that of the distribution of proteins in the bones? (a) 1 : 18 (b) 1 : 2 (c) 2 : 1 (d) 18 : 1 (e) None of these
1 3 =6=2 1 3 1 16% of 6 . (c) Required ratio = 16% of
Solution. 22.
In a particular dam project, certain barriers have to be raised to prevent the flow of water. But in spite of raising a barrier, it might happen that the water force can break the barrier. The probability of a barrier failing is 20 per cent. Then, what should be the minimum number of barriers to be raised so that the overall probability of the project failing is less than 0.1 per cent? (a) 5 (b) 7 (c) 10 (d) 11 (e) 6 (a) Probability of each barrier failing = 20% = 0.2 With two barriers, probability of failure of both = 0.2 × 0.2 = 0.04 Proceeding in this manner, we have (0.2)5 = 0.00032 as the probability of failure of all 5 barriers, which is less than 0.1% (=0.001) Hence, minimum number of barriers = 5.
Solution.
23.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. A
30° D E B CE bisects the angle C.
Column A (a) (b) (c) (d) Column B
C
quantity in column A is greater quantity in column B is greater both quantities are equal the relationship cannot be determined from the information given
∠C = 90° – 30° = 60°
∴∠BCE = ∠DCE = 30° Column A : ∠CED = 90° – 30° = 60°
Solution. (c)
24.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. Distribution of students studying different discipline in a University.
Arts 34% Science Engg. 18%
Total no. of students = 8000 Column A Column B Number of students in Engg. 1000 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given Solution. (a) Column A : % of students in Engineering = 100 – (34 + 33 + 18) = 100 – 85 = 15%.
15 × 8000 = 1200.
∴ Number of students in Engineering = 15% of 8000 = 100 25.
DIRECTIONS: The question consists of two quantities. One in Column A and one in Column B, you are to compare the two quantities and then give the answer. Column A Column B LCM of 2, 3, 4, 5 LCM of 2, 3, 4, 5, 6 (a) quantity in column A is greater (b) quantity in column B is greater (c) both quantities are equal (d) the relationship cannot be determined from the information given
2 2, 3, 4, 5 1, 3, 2, 5
Solution. (c) Column A :
Column B :
∴ LCM = 2 × 3 × 2 × 5 = 60. 2 2, 3, 4, 5, 6 3 1, 3, 2, 5, 3 1, 1, 2, 5, 1
∴ LCM = 2 × 3 × 2 × = 60.
26.
In the following figure, CE and DE are equal chords of a circle with centre O. Arc AB is a quarter-circle. Then the ratio of the area of triangle CED to the area of triangle AOB is:
E
C
O
D
A
(a)
2 :1
B
(b)
3 :1
(c) 4 : 1
(d) 3 : 1
(e) 2 : 1 Solution. (e) Here, ∠CED = 90º (angle subtended by semicircle)
2 ∴ Area of ΔCED 2 Now, CD2 = CE2 + ED2 ⇒ CD2 = 2CE2 ⇒ (2.OA)2 = 2.(CE)2 ⇒ CE2 = 2.OA2 ...(1) ∠AOB =
=
1
CE × DE =
1
(CE ) 2
(Q CE = DE )
(CD = diameter = 2. radius = 2.OA)
Also,
1 × 360º = 90º 4
( arc AB is a quarter circle)
(Q OA = OB = radius of circle)
1 1 ∴ Area of ΔOAB = (OA ) . (OB) = OA 2 2 2 1 ⎛1⎞ 2 = . ⎜ ⎟CE 2 ⎝2⎠
[By (1)]
27.
Area of ΔCED 1 1 ⎛1⎞ = (CE ) 2 ÷ ⎜ ⎟CE 2 = 2 : 1 Area of ΔOBAB 2 2 ⎝2⎠ 1 1 25 1 A rope is 2 m long. How many pieces each of 2 m long can be cut from it ? ∴
(a) 15 (e) 14
(b) 16
(c) 17
(d) 18
Solution. 28.
51 2 = 51 = 17 3 3 (c) No. of pieces = 2
A train with 90 km/h crosses a bridge in 36 seconds. Another train, 100 metres shorter, crosses the same bridge at 45 km/h. Find the time taken by the second train to cross the bridge. (a) 68 s (b) 62 s (c) 72 s (d) 64 s (e) 78 s (d) Let, length of first train = L1 and Length of second train = L2 then L1 – L2 = 100 meters Speed of first train = V1 = 90 km/h = 25 m/s Speed of second train = V2 = 45 km/h = m/s Time taken by first train to cross a bridge t1 = 36 s Time taken by second train to cross a bridge t2 = ? Here, two trains are crossing the same bridge and length of the bridge is not known. Using the formula for two trains crossing the same object,
25
Solution.
V1× t1 – L1= V2 × t2 – L2 25 × 36 – L1= 2 × t2 – L2
25
100 = 25 × 36 – 2 × t2 t2 = 64 s.