Question 2: The income of Amala is 20% more than that of Bimala and 20% less than that of Kamala. If Kamala's income goes down by 4% and Bimala's goes up by 10%, then the percentage by which Kamala's income would exceed Bimala's is nearest to
31
28
32
29
Let the income of Bimla be Rs. 100
Amla’s income = 1.2*100 = Rs. 120
Also, Amla’s income = $\frac{4}{5}\times$ Kamla’s income
Therefore, Kamla’s income = $\frac{5}{4}\times 120$=Rs. 150
Kamala's new income with 4% decrease = 0.96×150 =Rs. 144
Bimla’s new income with 10% increase = 1.1×100= Rs. 110
Question 3:
In a class, 60% of the students are girls and the rest are boys. There are 30 more girls than boys. If 68% of the students, including 30 boys, pass an examination, the percentage of the girls who do not pass is
Let number of girls and boys be 3x and 2x respectively.
Given,
3x-2x=30
Or x = 30. So the number of students = 3x+2x=5x =150.
Also, number of girls (3x) and boys (2x) = 90 and 60 respectively.
Number of students pass the exam = 68% of 150=102. Number of girls pass the exam = 102-30 = 72 .
Therefore, number of girls who fail in the exam = 90-72=18
Hence, percentage of girls fail the exam = $\frac{18}{90}\times 100=20%$
Question 4: On selling a pen at 5% loss and a book at 15% gain, Karim gains Rs. 7. If he sells the pen at 5% gain and the book at 10% gain, he gains Rs. 13. What is the cost price of the book in Rupees?
80
85
95
100
Let the cost prices of a pen and a books be x and y respectively.
Question 6:
Let S be the set of all points (x, y) in the x-y plane such that | x | + | y | ≤ 2 and | x | ≥ 1. Then, the area, in square units, of the region represented by S equals
The below figure shows the graph of both the expressions.
The required area = $4 \times$ the area of red triangle.
Area of red triangle =$\frac{1}{2}\times 1\times 1=\frac{1}{2}$ sq. units
Therefore, the required area = $4\times \frac{1}{2}=2$
Question 7: Ramesh and Gautam are among 22 students who write an examination. Ramesh scores 82.5. The average score of the 21 students other than Gautam is 62. The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh. The score of Gautam is
49
48
51
53
Let Gautam’s score be g. Also, let x be the average of all 22 students so that the total scores of all 22 students =22x.
Question 8: At their usual efficiency levels, A and B together finish a task in 12 days. If A had worked half as efficiently as she usually does, and B had worked thrice as efficiently as he usually does, the task would have been completed in 9 days. How many days would A take to finish the task if she works alone at her usual efficiency?
24
18
12
36
Let the work be LCM of (9 and 12) = 36 units.
Let the amount of work done in one day with their normal efficiencies by A and B be x and y units respectively.
Therefore, (x+y)×12=36
Or x+y =3 … (1)
Similarly,
(x/2+3y)×9=36
Or x/2 +3y = 4 … (2)
Solving (1) and (2) for x, we get x =2 units
Hence, A alone would take 36/x = 36/2 = 18 days to complete the work with her normal efficiency.
Question 9: If $a_{1}, a_{2}, \ldots$ are in A.P., then, $\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\cdots+\frac{1}{\sqrt{a_{n}}+\sqrt{a_{n+1}}}$ is equal to
$\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n-1}}}$
$\frac{n}{\sqrt{a_{1}}+\sqrt{a_{n+1}}}$
$\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}$
$\frac{n}{\sqrt{a_{1}}-\sqrt{a_{n+1}}}$
Shortcut:
For such questions, we can take value of n =1. The right option must give the first term i.e. $\frac{1}{\sqrt{{{a}_{1}}}+\sqrt{{{a}_{2}}}}$
Question 10: In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is
Question 11:
With rectangular axes of coordinates, the number of paths from (1,1) to (8,10) via (4,6), where each step from any point (x, y) is either to (x, y+1) or to (x+1, y), is
Let A, B and C represent the coordinates (1, 1), (8,10) and (4,6) respectively.
The number of ways to go from A to B via C = (The number of ways to go from A to C)×(The number of ways to go from C to B)
Question 12: Amala, Bina, and Gouri invest money in the ratio 3 : 4 : 5 in fixed deposits having respective annual interest rates in the ratio 6 : 5 : 4. What is their total interest income (in Rs) after a year, if Bina's interest income exceeds Amala's by Rs 250?
6350
7250
7000
6000
Ratio of their incomes = 3:4:5
Ratio of their interests = 6:5:4
Therefore, the ratio of their interest income = $\left( 3\times 6 \right):\left( 4\times 5 \right):\left( 5\times 4 \right)=18:20:20$
Let the interest incomes of Amala, Bina, and Gouri be 18x, 20x, and 20x respectively.
Given,
Bina's interest income exceeds Amala's by Rs 250
Therefore, 20x-18x=2x=250
Or x = 125.
Total interest incomes = 18x+20x+20x=58x = 58×125 =7250
Question 13: A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, 58 members can play both football and tennis, 25 can play both cricket and tennis, while 63 can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is
45
38
32
43
Let the number of members playing all three games be x.
Given, that all the members play at least one of these three games, hence the union of these three sets = 256.
Therefore,
256 =144+123+132)-(58+25+63)+x
Or x = 3.
Fitting the numbers in the venn diagram, we get
Clearly, the number of members playing only tennis = 43.
Question 14:
Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is
Clearly, the triangle will be right angled triangle and the hypotenuse would be the diameter of the circumcircle.
Getting the coordinates by substituting x and y with 0 alternatively in the equation 3x+5y-45=0, we have
Question 15:
Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?
Let one machine completes 1 unit of work per day.
Given, two machines can finish the job in 13 days
Therefore, the work of 2×1×13 = 26 units.
Also, let on man completes m units of work per day.
From the given condition:
3m+8×1 = 2(8m+3×1)
Or m = $\frac{2}{13}units$
Let it require ‘x’ number of men to complete the work in 13 days.
Question 16: Two cars travel the same distance starting at 10:00 am and 11:00 am, respectively, on the same day. They reach their common destination at the same point of time. If the first car travelled for at least 6 hours, then the highest possible value of the percentage by which the speed of the second car could exceed that of the first car is
Question 19: The wheels of bicycles A and B have radii 30 cm and 40 cm, respectively. While traveling a certain distance, each wheel of A required 5000 more revolutions than each wheel of B. If bicycle B traveled this distance in 45 minutes, then its speed, in km per hour, was
$18 \pi$
$12 \pi$
$16 \pi$
$14 \pi$
The distance travelled by bicycle A in one revolution = $2\pi {{r}_{a}}=2\pi \times 30=60\pi \ cm$
The distance travelled by bicycle B in one revolution = $2\pi {{r}_{b}}=2\pi \times 40=80\pi \ cm$
Let B makes ‘n’ revolutions to cover the distance. Then, A would make (n+5000) to cover the same distance.
Question 20: AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to
7.8
8.5
9.1
9.3
Refer to the figure below:
$\angle APB=\angle AQB={{90}^{0}}$ {angle in a semicircle is a right angle}
Question 21:
For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals
Question 22: A chemist mixes two liquids 1 and 2. One litre of liquid 1 weighs 1 kg and one litre of liquid 2 weighs 800 gm. If half litre of the mixture weighs 480 gm, then the percentage of liquid 1 in the mixture, in terms of volume, is
Question 24:
Consider a function $f$ satisfying $f(x+y)=f(x) f(y)$ where x, y are positive integers, and $f(1)=2$. If $f(a+1)+f(a+2)+\ldots+f(a+n)=16\left(2^{n}-1\right)$ then a is equal to
Question 26: Let $x$ and $y$ be positive real numbers such that $\log _{5}(x+y)+\log _{5}(x-y)=3,$ and $\log _{2} y-\log _{2} x=1-\log _{2} 3 .$ Then xy equals
Question 27: One can use three different transports which move at 10, 20, and 30 kmph, respectively. To reach from A to B, Amal took each mode of transport 1/3 of his total journey time, while Bimal took each mode of transport 1/3 of the total distance. The percentage by which Bimal’s travel time exceeds Amal’s travel time is nearest to
21
22
20
19
Let the total journey time taken by Amal be 3t hours.
Therefore, the total distance = 10t + 20t + 30t = 60t kms
Bimal took each mode of transport 1/3 of the total distance.
Therefore, total time taken by him = $\frac{20t}{10}+\frac{20t}{20}+\frac{20t}{30}=\frac{11t}{3}\ hours$
The percentage by which Bimal’s travel time exceeds Amal’s travel time $\left( \frac{\left( 11/3 \right)t-3t}{3t} \right)\times 100=22.22%$
Question 28: If the rectangular faces of a brick have their diagonals in the ratio $3: 2 \sqrt{3}: \sqrt{15},$ then the ratio of the length of the shortest edge of the brick to that of its longest edge is
$\sqrt{3}: 2$
$2: \sqrt{5}$
$1: \sqrt{3}$
$\sqrt{2}: \sqrt{3}$
Let the edges of the brick be a, b, and c such that $a<b<c$
Question 29: If the population of a town is p in the beginning of any year then it becomes 3+2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be
$(997) 2^{14}+3$
$(1003)^{15}+6$
$(1003) 2^{15}-3$
$(997)^{15}-3$
We can transform each of the options for ‘n’ years.
Question 30: A person invested a total amount of Rs 15 lakh. A part of it was invested in a fixed deposit earning 6% annual interest, and the remaining amount was invested in two other deposits in the ratio 2 : 1, earning annual interest at the rates of 4% and 3%, respectively. If the total annual interest income is Rs 76000 then the amount (in Rs lakh) invested in the fixed deposit was
Let the amount invested in fixed deposit be x lakhs.
Question 31: Meena scores 40% in an examination and after review, even though her score is increased by 50%, she fails by 35 marks. If her post-review score is increased by 20%, she will have 7 marks more than the passing score. The percentage score needed for passing the examination is
70
60
75
80
Let the total score of the exam be 100x.
Meena’s before review = 40% of 100x = 40x.
Her score after review = 40x + 50%of 40x = 60x.
Passing marks = 60x+35. .. (1)
Her score after increasing it by 20% of post revie score = 60x+20% of 60x = 72x
Passing marks = 72x-7. … (2)
Equating (1) and (2)
72x-7 = 60x + 35
$\Rightarrow x=3.5$
Hence, Total Marks = 100x= 350 and passing marks = 60x+35 = 245
Question 32:
In a race of three horses, the first beat the second by 11 metres and the third by 90 metres. If the second beat the third by 80 metres, what was the length, in metres, of the racecourse?
Let the first, second, and third horses be A, B and C respectively. Also, let the length of the racecourse be x.
Therefore,
$\frac{\left( x-11 \right)}{\left( x \right)}=\frac{\left( x-90 \right)}{\left( x-80 \right)}$
Question 34: The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157:3, then the sum of the two numbers is