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Bank Math

মোট প্রশ্ন১৬,১২৪এই পাতা১০০প্রতি পাতা১০০
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Bank Math

PrepBank · পাতা ৬৮ / ১৬১ · ৬,৭০১–৬,৮০০ / ১৬,১২৪

৬,৭০১.
The following figure consists of twenty seven cubes. How many cubes will get closed on all sides by other cubes? 
  1. ক
    ক) 1
  2. খ
    খ) 2
  3. গ
    গ) 3
  4. ঘ
    ঘ) 6
ব্যাখ্যা
প্রশ্ন: The following figure consists of twenty seven cubes. How many cubes will get closed on all sides by other cubes? 


সমাধান: 
একমাত্র মাঝের ঘনক এর ৬ পাশেই ঘনক দ্বারা বেশিষ্ট থাকে।
বাকি গুলো কোন না কোন পাশে খোলা থাকে।

তাই, সঠিক উত্তর ১টি হবে।
৬,৭০২.
The average salary of 30 officers in a firm is Tk.120 and the average salary of laborers is Tk. 40. Find the total number of laborers if the average salary of the firm is Tk. 50.
  1. ক
    180
  2. খ
    210
  3. গ
    240
  4. ঘ
    420
  5. ঙ
    None of these
ব্যাখ্যা
Question: The average salary of 30 officers in a firm is Tk.120 and the average salary of laborers is Tk. 40. Find the total number of laborers if the average salary of the firm is Tk. 50.

Solution:
The sum of salary of officers will be = 30 × 120 = 3600
Let the number of labourers = X.
ATQ,
3600 + 40X = 50(30 + X)
⇒ 3600 + 40X = 1500 + 50X
⇒ 2100 = 10X
∴ X = 210
৬,৭০৩.
A jar contains 6 white balls, 4 red balls, and 2 black balls. Two balls are drawn one after the other without replacement. Find the probability that both balls drawn are white.
  1. ক
    1/4
  2. খ
    5/22
  3. গ
    1/6
  4. ঘ
    5/12
ব্যাখ্যা

Question: A jar contains 6 white balls, 4 red balls, and 2 black balls. Two balls are drawn one after the other without replacement. Find the probability that both balls drawn are white.

Solution:
A jar contains 6 white balls, 4 red balls and 2 black balls.
∴ Total balls = 6 + 4 + 2 = 12
Two balls are drawn successively without replacement.

Now, 
Probability first ball is white = 6/12 = 1/2
After drawing one white ball, 5 white balls remain and the total number of balls remaining = 11
Probability second ball is white = 5/11

Since the draws are dependent (without replacement), multiply the probabilities.
∴ P(both white) = (6/12) × (5/11)
= (1/2) × (5/11)
= 5/22

So the probability that both balls are white is 5/22.

৬,৭০৪.
The compound interest on Tk. 30000 at 7% per annum is Tk. 4347. The period (in years) is-
  1. ক
    3 years
  2. খ
    4 years
  3. গ
    1 years
  4. ঘ
    2 years
ব্যাখ্যা

Question: The compound interest on Tk. 30000 at 7% per annum is Tk. 4347. The period (in years) is-

Solution:
Given that,
Principal, P = Tk. 30000
Compound Interest, CI = Tk. 4347
Rate, r = 7% per annum
And, Amount, A = P + CI = 30000 + 4347 = 34347

We know,
A = P(1 + r/100)n
⇒ 34347 = 30000(1 + 7/100)n
⇒ (107/100)n = 34347/30000
⇒ (107/100)n = 11449/10000
⇒ (107/100)n = (107/100)2
∴ n = 2 

Hence, the period = 2 years.

৬,৭০৫.
If a + b = √3 and a = √2 + b, what is the value of 4ab?
  1. ক
    0
  2. খ
    - 1
  3. গ
    1
  4. ঘ
    - 3
ব্যাখ্যা
Question: If a + b = √3 and a = √2 + b, what is the value of 4ab?

Solution: 
given,
a + b = √3
a = √2 + b
a - b = √2

4ab = (a + b)2 - (a - b)2
= 3 - 2
= 1
৬,৭০৬.
35 men can make 70 toys in 8 days by working 8 hours a day. How many days will 56 men need to make 84 toys by working 12 hours every day?
  1. ক
    4 days
  2. খ
    5 days
  3. গ
    3 days
  4. ঘ
    6 days
ব্যাখ্যা
Question: 35 men can make 70 toys in 8 days by working 8 hours a day. How many days will 56 men need to make 84 toys by working 12 hours every day?

Solution:
Total man-hours for 70 toys:
35 × 8 × 8 = 2240 man-hours
So, Man-hours per toy = 2240/70 = 32

Total man-hours for 84 toys:
84 ×  32 = 2688 man-hours
Total man-hours per day for 56 men working 12 hours:
56 × 12 = 672 man-hours/day

∴ Time to complete work = 2688/672 = 4 days
৬,৭০৭.
The sum of principal and simple interest of a certain amount of money would be Tk. 460 after 3 years from now and Tk. 500 after 5 years from now. What is the total interest rate?
  1. ক
    ক) 5%
  2. খ
    খ) 12%
  3. গ
    গ) 15%
  4. ঘ
    ঘ) 20%
ব্যাখ্যা
Question: The sum of principal and simple interest of a certain amount of money would be Tk. 460 after 3 years from now and Tk. 500 after 5 years from now. What is the total interest rate?

Solution: 
ধরি,
মুনাফার হার = r
মূলধন = P

দেওয়া আছে,
৩ বছরে মুনাফা আসলে হয় = ৪৬০ টাকা
৫ বছরে মুনাফা আসলে হয় = ৫০০ টাকা

∴ ২ বছরে সরল মুনাফা = ৫০০ - ৪৬০ = ৪০ টাকা
তাহলে, ৩ বছরের সরল মুনাফা = (৪০/২)৩ = ৬০ টাকা

∴ মূলধন, P = ৪৬০ - ৬০ = ৪০০ টাকা

আমরা জানি,
I = Pnr
r = I/Pn
= 60/(400 × 3)
= 0.05 × 100%
= 5%
৬,৭০৮.
6 men can pack 12 boxes in 7 days by working for 7 hours a day. In how many days can 14 men pack 18 boxes if they work for 9 hours a day?
  1. ক
    3.5 days
  2. খ
    5 days
  3. গ
    7.5 days
  4. ঘ
    12 days
ব্যাখ্যা

If 'w1' work is done by 'm1' men by working for 'h1' hours per day in 'd1' days & 'w2' is work done by men 'm2' working for 'h2' hours per day in 'd2' days,
then,
m1d1h1/w1 = m2d2h2/w2

Since we need to find 'd2', we can rearrange the formula as
d2 = (m1d1h1w2)/(m2d2h2w1)
= (6 x 7 x 7 x 18)/(14 x 9 x 12)
= 3.5 days

৬,৭০৯.
When heated an iron bar expands 0.2%. If the increased length is 1 cm, what is the original length of the bar?
  1. ক
    ক) 500cm
  2. খ
    খ) 5cm
  3. গ
    গ) 0.97cm
  4. ঘ
    ঘ) 1.97cm
  5. ঙ
    ঙ) 0.95cm
ব্যাখ্যা
Question: When heated an iron bar expands 0.2%. If the increased length is 1 cm, what is the original length of the bar?

Solution:
Let
the original length of the bar = L cm

Increase in length  0.2% of L =  (0.2L)/100 cm
ATQ,
(0.2L)/100 = 1
⇒ 0.2L =  100
⇒ L = 100/0.2
⇒ L = 500 

∴ The original length is 500 cm.
৬,৭১০.
Find the ratio of purchase price to sell price if there is loss of 12.5%?
  1. ক
    ক) 7 : 8
  2. খ
    খ) 8 : 7
  3. গ
    গ) 2 : 25
  4. ঘ
    ঘ) 25 : 2
ব্যাখ্যা

ধরি, ক্রয়মূল্য 100 টাকা
12.5% ক্ষতিতে বিক্রয়মূল্য (100 - 12.5) = 87.5 টাকা
ক্রয়মূল্য : বিক্রয়মূল্য = 100 : 87.5 = 8 : 7

৬,৭১১.
If 4(x - 2/3) = 0, what is the value of x?
  1. ক
    ক) 2/3
  2. খ
    খ) -2/3
  3. গ
    গ) 8/3
  4. ঘ
    ঘ) -8/3
ব্যাখ্যা
দেয়া আছে,
4(x - 2/3) = 0
x - 2/3 = 0
x =  2/3
৬,৭১২.
A man has Tk. 640 in the denominations of one-taka notes, five-taka notes and ten-taka notes. The number of notes of each denomination is equal. Simplify this with equation and what is the total number of notes that he has?
  1. ক
    80
  2. খ
    100
  3. গ
    120
  4. ঘ
    140
ব্যাখ্যা
Question: A man has Tk. 640 in the denominations of one-taka notes, five-taka notes and ten-taka notes. The number of notes of each denomination is equal. Simplify this with equation and what is the total number of notes that he has?
 
Solution:
Let the number of notes of each denomination be x.

Then
x + 5x + 10x = 640
⇒ 16x = 640
∴ x = 40

Hence, the total number of notes = 3x = 120
৬,৭১৩.
p + q : q + r : r + p = 4 : 5 : 6 and p + q + r = 30, find q =?
  1. ক
    8
  2. খ
    6
  3. গ
    12
  4. ঘ
    4
ব্যাখ্যা
Question: p + q : q + r : r + p = 4 : 5 : 6 and p + q + r = 30, find q =?

Solution:
Given that,
p + q : q + r : r + p = 4 : 5 : 6
and p + q + r = 30 ......(1)

Let,
p + q = 4k, q + r = 5k, r + p = 6k

Now,
⇒ p + q + q + r + r + p = 4k + 5k + 6k
⇒ 2(p + q + r) = 15k
⇒ p + q + r = 15k/2
⇒ 15k/2 = 30 [from 1]
⇒ k = 60/15
∴ k = 4

And,
⇒ r + p = 6k
⇒ r + p = 6 × 4
∴ r + p = 24 .....(2)

From (1),
⇒ p + q + r = 30
⇒ q = 30 - (r + p)
⇒ q = 30 - 24
∴ q = 6
৬,৭১৪.
Tk. 700 is received among A,B and C so that A receives half as much as B and B half as much as C. Then C’s share is:
  1. ক
    ক) 200
  2. খ
    খ) 300
  3. গ
    গ) 400
  4. ঘ
    ঘ) 500
  5. ঙ
    ঙ) 600
ব্যাখ্যা

Let C = x
Then B = x/2
and A = x/4
A : B : C = 1 : 2 : 4
C's share = Tk. (4/7 × 700) = 400

৬,৭১৫.
If the radius of a sphere is increased by 20%, how much will the surface area be increased in percentage?
  1. ক
    ক) 21%
  2. খ
    খ) 24%
  3. গ
    গ) 44%
  4. ঘ
    ঘ) 40%
ব্যাখ্যা

Surface area of sphere = 4πr2 
If the new radius is 20% increased, then new surface area will be = 4π(1.2)2  = 5.76πr2 

Surface area Increased in percentage = (5.76πr2/4πr2 × 100) - 100 =  144 - 100 = 44%

৬,৭১৬.
If 16Pr - 1 : 15Pr - 1 = 16 : 7 then find r
  1. ক
    6
  2. খ
    12
  3. গ
    8
  4. ঘ
    10
ব্যাখ্যা
Question: If 16Pr - 1 : 15Pr - 1 = 16 : 7 then find r

Solution:
৬,৭১৭.
The value of  is-
  1. ক
    1.04
  2. খ
    0.96
  3. গ
    0.86
  4. ঘ
    0.69
  5. ঙ
    None of these
ব্যাখ্যা
Question: The value of is-

Solution:
৬,৭১৮.
5p - 3q = 42, 5p + 3q = 18. Given this system of equations. What is the value of ।p। + ।q।?
  1. ক
    ক) 2
  2. খ
    খ) 4
  3. গ
    গ) 6
  4. ঘ
    ঘ) 10
ব্যাখ্যা
দেয়া আছে,
5p - 3q = 42...........(1)
5p + 3q = 18...........(2)

(1) + (2)⇒
5p - 3q  + 5p + 3q  = 42 + 18 
বা, 10p = 60
p = 6 
(2) নং সমীকরণে p এর মান বসিয়ে পাই,
5 × 6 + 3q = 18
বা, 30 + 3q = 18
বা, 3q = 18 - 30 
বা, 3q = - 12
  q = - 4
এখন 
।p। + ।q। = ।6। + ।- 4। = 6 + 4 = 10
৬,৭১৯.
Father is aged three times more than his son Shafin. After 8 years, he would be two and a half times of Shafin's age. After further 8 years, how many times would he be of Shafin's age?
  1. ক
    2 times
  2. খ
    3 times
  3. গ
    1/2 times
  4. ঘ
    4 times
ব্যাখ্যা
Question: Father is aged three times more than his son Shafin. After 8 years, he would be two and a half times of Shafin's age. After further 8 years, how many times would he be of Shafin's age?

Solution:
Let, Shafin's present age be = a years.
Then, father's present age = (a + 3a) years
= 4a years

ATQ,
(4a + 8) = (5/2)(a + 8)
⇒ 8a + 16 = 5a + 40
⇒ 3a = 24
∴ a = 8

Hence, the required times = (4a + 16)/(a + 16) 
= 48/24
= 2
৬,৭২০.
- 2x + y - 3 = 0 এবং - 7y + 3x + 10 = 0 এর সমাধান কোনটি?
  1. ক
    x = 1, y = - 1
  2. খ
    x = - 1, y = 1
  3. গ
    x = - 1, y = 2
  4. ঘ
    x = - 1, y = - 1
  5. ঙ
    কোনটিই নয়
ব্যাখ্যা
প্রশ্ন: - 2x + y - 3 = 0 এবং - 7y + 3x + 10 = 0 এর সমাধান কোনটি?

সমাধান: 
- 2x + y - 3 = 0
- 2x + y = 3 ----------- (1)

- 7y + 3x + 10 = 0
3x - 7y = - 10 ----------- (2)

(1) নং কে 7 দ্বারা গুণ করে (2) নং এর সাথে যোগ করি-
- 14x + 7y = 21
3x - 7y = - 10 
- 11x = 11
∴ x = - 1

x এর মান (2) নং এ বসাই,
3 . (- 1) - 7y = - 10
- 3 - 7y = - 10
- 7y = - 7
∴ y = 1 

∴ নির্ণেয় সমাধান: (x, y) = (- 1, 1)
৬,৭২১.
The population of a town increased from 1,75,000 to 2,62,500 in a decade. The average percent increase of population per year is:
  1. ক
    8%
  2. খ
    3%
  3. গ
    5%
  4. ঘ
    10%
ব্যাখ্যা
Question: The population of a town increased from 1,75,000 to 2,62,500 in a decade. The average percent increase of population per year is-

Solution:
জনসংখ্যা বৃদ্ধি পায় ২৬২৫০০ - ১৭৫০০০ জন
= ৮৭৫০০ জন

১৭৫০০০ জনে ১০ বছরে বৃদ্ধি পায় ৮৭৫০০ জন
∴ ১৭৫০০০ জনে ১ বছরে বৃদ্ধি পায় ৮৭৫০০/১০ জন = ৮৭৫০ জন
∴ ১ জনে ১ বছরে বৃদ্ধি পায় = ৮৭৫০/১৭৫০০০ জন
∴ ১০০ জনে ১ বছরে বৃদ্ধি পায় = (৮৭৫০ × ১০০)/১৭৫০০০ জন
= ৫ জন
৬,৭২২.
Robi and Bobi together can paint a wall in 16 days. Robi can do it alone in 20 days. How many days would it take Bobi to do this work alone?
  1. ক
    48 days
  2. খ
    40 days
  3. গ
    24 days
  4. ঘ
    80 days
  5. ঙ
    96 days
ব্যাখ্যা

Question: Robi and Bobi together can paint a wall in 16 days. Robi can do it alone in 20 days. How many days would it take Bobi to do this work alone?

Solution:
Robi's 1 day's work 1/20
Robi's and Bobi's 1 day's work 1/16
∴ Bobi's 1 day's work = (1/16) - (1/20)
= (5 - 4)/80
= 1/80

1/80 part of the job done by Bobi in 1 day
∴ Full work done by Bobi in (80/1) days
= 80 days

৬,৭২৩.
What is the value of a if 9 - 12x + ax2 is an integer -
  1. ক
    ক) 2
  2. খ
    খ) 4
  3. গ
    গ) 6
  4. ঘ
    ঘ) 8
ব্যাখ্যা
Question: What is the value of a if 9 - 12x + ax2 is an integer -

Solution: 
9 - 12x + ax2
= 32 - 2 . 3 . √ax + (√ax)2
∴ 6√ax = 12x
⇒ √a = 2
⇒ (√a)2 = (2)2
∴ a = 4
৬,৭২৪.
A man travels 50 km at speed 25 km/hr and next 40 km at 20 km/hr and there after travel 90 km at 15 km/hr. His average speed is:
  1. ক
    ক) 14 km/hr.
  2. খ
    খ) 15 km/hr.
  3. গ
    গ) 18 km/hr.
  4. ঘ
    ঘ) 20 km/hr.
ব্যাখ্যা
Question: A man travels 50 km at speed 25 km/hr and next 40 km at 20 km/hr and there after travel 90 km at 15 km/hr. His average speed is:

Solution:
Here,
Total distance = 50 + 40 + 90 = 180 km.
Total time = 2 + 2 + 6 = 10 hr.

We know,
Average speed = Total distance/Total time
= 180/10
∴ Average speed = 18 km/hr.
৬,৭২৫.
Rakib can do a piece of work in 15 days, Rakib and Asif together can do in 12 days. If Asif does the work only for half a day daily then in how many days the work will be completed ?
  1. ক
    3/20 days
  2. খ
    20/3 days
  3. গ
    40/3 days
  4. ঘ
    3/40 days
ব্যাখ্যা

Question: Rakib can do a piece of work in 15 days, Rakib and Asif together can do in 12 days. If Asif does the work only for half a day daily then in how many days the work will be completed ?

Solution: 

Rakib's 1 day work = 1/15

Since, Rakib and Asif can together complete in 12 days
i.e. (Rakib + Asif)'s 1 day work = 1/12

Then,
Asif's 1 day work,
= (1/12) - (1/15) = (5 - 4)/60 = 1/60

If Asif Works only for half a day daily, then his 1 day work becomes (1/2) × (1/60)
  = (1/120)

Therefore, 1 day work of both Rakib and Asif,
=(1/15) + (1/120) = ( 8 + 1)/120 = 9/120

Hence, the work will be completed in 120/9 = 40/3 days

৬,৭২৬.
10 is how many times of (0.01)2?
  1. ক
    103
  2. খ
    104
  3. গ
    105
  4. ঘ
    107
ব্যাখ্যা
Question: 10 is how many times of (0.01)2?

Solution: 
10/(0.01)2
= 10/(1/100)2
= 10/(1/102)2
= 10/(1/104)
= 10 × 104
= 101 + 4
= 105
৬,৭২৭.
What will be the result if (4x + 20)/4 is subtracted from (x + 10)?
  1. ক
    5
  2. খ
    x
  3. গ
    x/2
  4. ঘ
    - x
  5. ঙ
    None of these
ব্যাখ্যা

Question: What will be the result if (4x + 20)/4 is subtracted from (x + 10)?

Solution:
Expression = (x + 10) - {(4x + 20)/4}
= (x + 10) - {4(x + 5)/4}
= (x + 10) - (x + 5)
= x + 10 - x - 5
= 5

৬,৭২৮.
A square and a circle have the same perimeter. The side length of the square is 11 cm. What is the area of the circle?
  1. ক
    154 square cm
  2. খ
    231 square cm
  3. গ
    77 square cm
  4. ঘ
    616 square cm
ব্যাখ্যা

Question: A square and a circle have the same perimeter. The side length of the square is 11 cm. What is the area of the circle?

Solution:
দেওয়া আছে,
বর্গক্ষেত্রের এক বাহুর দৈর্ঘ্য, a = 11 সে.মি.
∴ বর্গক্ষেত্রের পরিসীমা = 4 × a
= 4 × 11
= 44 সে.মি.

প্রশ্নমতে,
বৃত্তের পরিধি = বর্গক্ষেত্রের পরিসীমা
∴ 2πr = 44
⇒ 2 × (22/7) × r = 44
⇒ (44/7) × r = 44
⇒ r = 44 × (7/44)
∴ r = 7 সে.মি.

এখন,
বৃত্তের ক্ষেত্রফল = πr2
= (22/7) × 72
= (22/7) × 49
= 22 × 7
= 154 বর্গ সে.মি.

অতএব, বৃত্তের ক্ষেত্রফল = 154 বর্গ সে.মি.

৬,৭২৯.
What percent of 700 is 2.1?
  1. ক
    ক) 0.003
  2. খ
    খ) 0.3
  3. গ
    গ) 0.03
  4. ঘ
    ঘ) 3
ব্যাখ্যা
Question: What percent of 700 is 2.1?

Solution:
Let,
the percent is x.

ATQ,
x% of 700 = 2.1
⇒ (x/100) × 700 = 2.1
⇒ 7x = 2.1
⇒ x = 2.1/7
∴ x = 0.3
৬,৭৩০.
A ladder leans against a vertical wall making an angle of 60° with the ground. If the foot of the ladder is 4.6 m away from the base of the wall, find the length of the ladder.
  1. ক
    8 m
  2. খ
    9.2 m
  3. গ
    10.5 m
  4. ঘ
    12 m
ব্যাখ্যা

Question: A ladder leans against a vertical wall making an angle of 60° with the ground. If the foot of the ladder is 4.6 m away from the base of the wall, find the length of the ladder.

Solution:
 
Let AB be the wall and BC be the ladder.

 Then, ∠ACB = 60°
and AC = 4.6 m

We know,
cos∠ACB = AC/BC
⇒ cos60° = AC/BC
⇒ AC/BC= 1/2
⇒ BC = 2 × AC
⇒ BC = 2 × 4.6
∴ BC = 9.2 m

৬,৭৩১.
Milk and water are in the ratio of 4 : 5 in a mixture of 45 liters. To make the ratio equal how much milk should be added to the mixture ?
  1. ক
    5 liters
  2. খ
    9 liters
  3. গ
    10 liters
  4. ঘ
    15 liters
ব্যাখ্যা
Question: Milk and water are in the ratio of 4 : 5 in a mixture of 45 liters. To make the ratio equal how much milk should be added to the mixture ?

Solution: 
amount of milk
= (4/9)×45
= 20 litres
amount of water
= (5/9)×45
= 25 litres

the amount of milk to be added is
= (25-20)
= 5 litres.
৬,৭৩২.
The ratio of the length of a man and its shadow is 1 : √3. The angle of elevation of the sun is-
  1. ক
    15°
  2. খ
    45°
  3. গ
    30°
  4. ঘ
    60°
ব্যাখ্যা
Question: The ratio of the length of a man and its shadow is 1 : √3. The angle of elevation of the sun is-

Solution:

Let AB be a man and BC be its shadow
So that AB : BC = 1 : √3

Let, θ be the angle of elevation

∴tanθ = AB/BC = 1/√3 = tan30°

∴θ = 30°
৬,৭৩৩.
Puja invested 50000 for 2 years at 15% compound rate. What is the simple interest rate that she needs to put the same amount for the same time to ensure the same amount of return?
  1. ক
    18.126%
  2. খ
    16.125%
  3. গ
    16.5%
  4. ঘ
    12.126%
ব্যাখ্যা
Question: Puja invested 50000 for 2 years at 15% compound rate. What is the simple interest rate that she needs to put the same amount for the same time to ensure the same amount of return?

Solution: 
P = 50000
n = 2 years 
r = 15%

C = p(1 + r)n
= 50000(1 + 15/100)2
= 50000 ( 115/100)2
= 66125

return = 66125 - 50000 = 16125Tk

Let the simple interest rate is X%
we know, 
I = PnX/100
X = (100 × I)/ Pn
= (100 × 16125)/(50000 × 2)
= 16.125%
৬,৭৩৪.
  1. ক
    5
  2. খ
    -5
  3. গ
    0
  4. ঘ
    3
ব্যাখ্যা

Question:

Solution:

৬,৭৩৫.
What is the root cube of √4096?
  1. ক
    ক) 4
  2. খ
    খ) 8
  3. গ
    গ) 12
  4. ঘ
    ঘ) 24
ব্যাখ্যা
Question: What is the root cube of √4096?

Solution:
Here,
√4096 = 64

Now,
the root cube of 64 is = 3√64 = (43)1/3 = 4
৬,৭৩৬.
Evaluate the compound interest on Tk. 10101 for 3 years at the rate of 9% per annum compounded annually.
  1. ক
    Tk. 2980
  2. খ
    Tk. 30000
  3. গ
    Tk. 10101
  4. ঘ
    Tk. 33333
ব্যাখ্যা
Question: Evaluate the compound interest on Tk. 10101 for 3 years at the rate of 9% per annum compounded annually.

Solution:
A = P (1 + R/100)n
⇒ A = 10101(1 + 9/100)3 = 10101(109/100)3 = 13081.08

C.I. = A - P = 13081.08 - 10101 = 2980.08.
৬,৭৩৭.
What is the least number which when divided by 3, 5, 6, 8, 10, and 12 leaves a remainder 2 in each case, but when divided by 22 leaves no remainder?
  1. ক
    120
  2. খ
    242
  3. গ
    240
  4. ঘ
    132
  5. ঙ
    none of these
ব্যাখ্যা

Question: What is the least number which when divided by 3, 5, 6, 8, 10, and 12 leaves a remainder 2 in each case, but when divided by 22 leaves no remainder?

Solution:
The number, when divided by 3, 5, 6, 8, 10, and 12 leaves a remainder of 2. This means the number is 2 more than a multiple of their L.C.M.
LCM of 3, 5, 6, 8, 10, 12 = 120

The required number, N, must be in the form:
N = 120K + 2, where K is a positive integer.

N divisible by 22 ⇒ (120K + 2) ÷ 22 has remainder 0

120K + 2 = 22 × m ⇒ Check for smallest K

Try K = 2 ⇒ 120 × 2 + 2 = 242, divisible by 22

Hence, required number = 242

৬,৭৩৮.
If each side of a rectangle is increased by 10% the increase in the area of a rectangle will be ____ . 
  1. ক
    ক) 11%
  2. খ
    খ) 21%
  3. গ
    গ) 25%
  4. ঘ
    ঘ) 44%
ব্যাখ্যা
ধরি,
আয়তকারক্ষেত্রের দৈর্ঘ্য = x
আয়তকার ক্ষেত্রের প্রস্থ = y
∴ ক্ষেত্রফল = xy বর্গএকক

10% বৃদ্ধিতে নতুন দৈর্ঘ্য = x + x এর 10%
                                     = x + x এর 10/100
                                     = x + x/10 
                                     = 11x/10

10% বৃদ্ধিতে নতুন প্রস্থ  = y + y এর 10%
                                     = y + y এর 10/100
                                     = y + y/10 
                                     = 11y/10
∴ নতুন ক্ষেত্রফল = (11x/10) × (11y/10) বর্গএকক
                           = 121xy/100 বর্গএকক

ক্ষেত্রফল বৃদ্ধি = {(121xy/100) - xy} বর্গএকক
                      = 21xy/100 বর্গএকক

∴ ক্ষেত্রফল বৃদ্ধির হার = {(21xy × 100)/100 × xy}%
                                   = 21%
৬,৭৩৯.
The angles of a triangle are in the proportion of 1 : 2 : 3 and the length of the smallest side is 1 cm. what is the length of the longest side of the triangle?
  1. ক
    ক) 2 cm
  2. খ
    খ) 2.3 cm
  3. গ
    গ) 4 cm
  4. ঘ
    ঘ) 5 cm
ব্যাখ্যা
Question: The angles of a triangle are in the proportion of 1 : 2 : 3 and the length of the smallest side is 1 cm. what is the length of the longest side of the triangle?

Solution:
ত্রিভুজের তিনটি কোণের অনুপাত = 1 : 2 : 3

ধরি,
ত্রিভুজের তিনটি কোণ যথাক্রমে x, 2x, 3x

x + 2x + 3x = 180°
6x = 180°
x = 30°

ত্রিভুজের তিনটি কোণ যথাক্রমে = 30°, 60°, 90°
ত্রিভুজটি সমকোণী ত্রিভুজ। 


ΔABC এ 
cos60° = BC/AC
1/2 = 1/AC
AC = 2
৬,৭৪০.
A bag contains 5 red balls, 7 green balls, and 9 blue balls. One ball is picked at random. What is the probability that the ball is not green?
  1. ক
    1/7
  2. খ
    1/5
  3. গ
    2/3
  4. ঘ
    1/2
ব্যাখ্যা
Question: A bag contains 5 red balls, 7 green balls, and 9 blue balls. One ball is picked at random. What is the probability that the ball is not green?

Solution:
Total number of balls = 5 + 7 + 9
= 21

Number of non-green balls = 5 (red) + 9 (blue) = 14

∴ the probability of not getting a green ball = 14/21
= 2/3
৬,৭৪১.
An alloy contains zinc, copper and tin in the ratio 2 : 3 : 1 and another contains copper, tin and lead in the ratio 5 : 4 : 3. If equal weights of both alloys are melted together to form a third alloy, then the weight of copper and lead per kg in new alloy will be-
  1. ক
    1/4 and 1/6
  2. খ
    1/6 and 1/8
  3. গ
    1/8 and 1/4
  4. ঘ
    11/24 and 1/6
  5. ঙ
    11/24 and 1/8
ব্যাখ্যা

Question: An alloy contains zinc, copper and tin in the ratio 2 : 3 : 1 and another contains copper, tin and lead in the ratio 5 : 4 : 3. If equal weights of both alloys are melted together to form a third alloy, then the weight of copper and lead per kg in new alloy will be-

Solution:
Ratio of Zinc, Copper and Tin is given as, Z : C : T = 2 : 3 : 1 = 4 : 6 : 2
Now, let the first alloy be 12 kg (taken as 4 kg Zinc, 6 kg Copper and 2 Kg Tin)
Weight of second alloy = 12 kg as, C : T : L = 5 : 4 : 3 (taken as 5 kg Copper, 4 kg Tin and 3 Kg Lead)

Alloys are mixed together to form third alloy. Then the ratio of content in it,
Z : C : T : L = 4 : (6 + 5) : (2 + 4) : 3 = 4 : 11 : 6 : 3

 Weight of third alloy = 12 + 12 = 24 Kg.
∴ Weight of Copper = 11/24 
 And, weight of Lead = 3/24 
= 1/8

৬,৭৪২.
How many times in a day, the hands of a clock are straight?
  1. ক
    24 times
  2. খ
    44 times
  3. গ
    62 times
  4. ঘ
    22 times
ব্যাখ্যা
Question: How many times in a day, the hands of a clock are straight?

Solution:
The no of times the hands of a clock are straight = 22 times in 12 hours

∴ In a day or 24 hours, the hands of a clock are straight for 22×2 times.
= 44 times
৬,৭৪৩.
A train takes 8 seconds to cross a pole and 18 seconds to cross a platform of length 120m. What is the length of the train?
  1. ক
    80 m
  2. খ
    150 m
  3. গ
    200 m
  4. ঘ
    96 m
ব্যাখ্যা

Question: A train takes 8 seconds to cross a pole and 18 seconds to cross a platform of length 120m. What is the length of the train?

Solution:
Let the length of the train be L meters.

Now,
Time to cross a pole = 8 s
Distance covered = length of train = L
∴ Speed = L/8 m/s

Again, 
Platform length = 120 m
Distance covered = L + 120
Time taken = 18 s
∴ Speed = (L + 120)/18 m/s

ATQ, 
L/8 = (L + 120)/18
⇒ 18L = 8L + 960
⇒ 18L - 8L = 960
⇒ 10L = 960
⇒ L = 960/10
∴ L = 96 m

So the length of the train is 96 meters.

৬,৭৪৪.
What is the greatest number which divides 24, 28 and 34 and leaves the same remainder in each case?
  1. ক
    ক) 2
  2. খ
    খ) 1
  3. গ
    গ) 3
  4. ঘ
    ঘ) 4
ব্যাখ্যা

If the remainder is same in each case and remainder is not given,
HCF of the differences of the numbers is the required greatest number.
34 - 24 = 10
34 - 28 = 6
28 - 24 = 4
Hence, the greatest number which divides 24, 28 and 34 and gives the same remainder
= HCF of 10, 6, 4
= 2

৬,৭৪৫.
The average age of a group of 10 students is 15 years. When 5 more students join the group, the average age increase by 1 year. The average age of the new students is?
  1. ক
    16
  2. খ
    18
  3. গ
    20
  4. ঘ
    23
ব্যাখ্যা
Question: The average age of a group of 10 students is 15 years. When 5 more students join the group, the average age increase by 1 year. The average age of the new students is?

Solution:
Total age of 10 students = 10 × 15 = 150 years

Total age of 15 students = 15 × 16 = 240 years

Total age of 5 new students = 240 - 150 = 90 years

∴ Average age of 5 new students = 90/5 = 18 years
৬,৭৪৬.
Which one of the following is an irrational number?
  1. ক
    ক) (√32)/(√16)
  2. খ
    খ) π
  3. গ
    গ) √1000
  4. ঘ
    ঘ) All of them
ব্যাখ্যা
Question: Which one of the following is an irrational number?

• √32/√16 = (√16 × √2) / √16 = √2
• π
• √1000

Here, all of them are irrational numbers.
৬,৭৪৭.
By Selling 45 lemons for Tk. 40, a man loses 20%. How many should he sell for Tk..24, to gain 20% in the transaction?
  1. ক
    16
  2. খ
    18
  3. গ
    20
  4. ঘ
    21
ব্যাখ্যা
Question: By Selling 45 lemons for Tk. 40, a man loses 20%. How many should he sell for Tk..24, to gain 20% in the transaction?

Solution: 
SP per Lemons = 40/45 = Tk. 0.88.
Let, CP was Tk. X per Lemons.
SP = CP - 20% of CP
⇒ 0.88 = 0.8X
⇒ X = Tk. 1.11

To get 20% profit,
SP = 1.11 + 20% of 1.11 = Tk.1.33

Thus, he can sell = 24/1.33
= 18 Lemons in Tk. 24.
৬,৭৪৮.
The difference between two positive numbers is 8 and the difference of their squares is 160. What is the smallest number?
  1. ক
    6
  2. খ
    10
  3. গ
    12
  4. ঘ
    7
ব্যাখ্যা

Question: The difference between two positive numbers is 8 and the difference of their squares is 160. What is the smallest number?

Solution:
Let the numbers be x and (x + 8)

According to the question,
(x + 8)2 - x2 = 160
⇒ x2 + 16x + 64 - x2 = 160
⇒ 16x + 64 = 160
⇒ 16x = 96
⇒ x = 6

∴ The smallest number is = 6

৬,৭৪৯.
A is twice as fast as B and B is thrice as fast as C. The journey covered by C in 78 minutes will be covered by A in-
  1. ক
    ক) 13 min
  2. খ
    খ) 17 min
  3. গ
    গ) 7 min
  4. ঘ
    ঘ) 12 min
  5. ঙ
    ঙ) 13.5 min
ব্যাখ্যা

The ratio of speeds of A, B, C = 6 : 3 : 1
The ratio of time taken by A, B, C = 1/6 : 1/3 : 1
To simplify it, we will multiply it by the LCM of ratio of speeds given.
Hence, the ratio of time taken by A, B, C = 1 : 2 : 6

[Speed is inversely proportional to time, meaning if speed increases, time decreases. So, ratio of time would be reciprocal of the ratio of speed given. ]

Time taken by C to covered given distance = 78 = 6 × 13
The ratio of time of A and C = 1 : 6
Thus, time taken by A = 13 min.

Alternative method:
Let C's speed be x metres/min
Let time taken by A be y min
Then B's speed = 3x metres/min
And, A's speed = 6x metres/min
Ratio of speed of A and C = Ratio of time by C and A
6x : x = 78 : y
6x/x = 78/y
y = 13 min

৬,৭৫০.
What is the angle between the hour and minute hand of a clock when it is 10 : 20 pm by the clock?
  1. ক
    190°
  2. খ
    170°
  3. গ
    150°
  4. ঘ
    180°
ব্যাখ্যা
Question: What is the angle between the hour and minute hand of a clock when it is 10 : 20 pm by the clock?

Solution:
Angle = |(11M - 60H)/2|°
= |{(11 × 20) - (60 × 10)}/2|°
= |(220 - 600)/2|°
= |-380/2|°
= |-190|°
= 190° [যেহেতু এটি একটি প্রবদ্ধকোণ]

∴ The angle between the hour and minute hand = 360° - 190° = 170°
৬,৭৫১.
3 men, 4 women, and 6 children can complete a work in 7 days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?
  1. ক
    5
  2. খ
    6
  3. গ
    7
  4. ঘ
    8
ব্যাখ্যা
Question: 3 men, 4 women, and 6 children can complete a work in 7 days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

Solution:
Let 1 woman's 1 day work = x.
Then, 1 man's 1 day work = x/2 and 1 child's 1 day work  x/4.

ATQ,
(3x/2) + 4x + (6x/4) = 1/7
⇒ 28x/4 = 1/7
∴ x = 1/49

1 woman alone can complete the work in 49 days.
So, to complete the work in 7 days, number of women required = 49/7 = 7.
৬,৭৫২.
A man sells 4000 common shares of a Company x (each of par value Tk. 10), which pays a dividend of 40% at Tk. 30 per share. He invests the sale proceeds in ordinary shares of Company Y (each of par value Tk. 25) that pays a dividend of 15%. If the market value of Company Y is Tk. 15, find the number of shares of Company Y purchased by the man -
  1. ক
    ক) 16000
  2. খ
    খ) 20000
  3. গ
    গ) 12000
  4. ঘ
    ঘ) 8000
ব্যাখ্যা

Just too much information is given in the question to confuse. This is a straight and simple question
Market Value of Company X (his selling price) = Tk. 30
Total shares sold = 4000
The amount he gets = Tk. (4000 × 30)
He invests this amount in ordinary shares of Company Y
Market Value of Company Y(His purchasing price) = 15
Number of shares of company Y which he purchases = (4000 × 30)/15
= Tk. 8000.

৬,৭৫৩.
The volume of a right circular cylinder is 25π cubic units and its height is 4 units. What is the circumference of its base?
  1. ক
    5π
  2. খ
    10π
  3. গ
    20π
  4. ঘ
    10√2π
ব্যাখ্যা

Question: The volume of a right circular cylinder is 25π cubic units and its height is 4 units. What is the circumference of its base?

Solution:
আমরা জানি, একটি সিলিন্ডারের আয়তন = πr2h
যেখানে, r হলো ভূমির ব্যাসার্ধ এবং h হলো উচ্চতা।

প্রশ্নমতে,
πr2 × 4 = 25π
⇒ 4r2 = 25
⇒ r2 = 25/4
⇒ r = √(25/4)
⇒ r = 5/2 = 2.5 একক

সিলিন্ডারের ভূমির পরিধি = 2πr
= 2π × 2.5
= 5π একক

∴ সিলিন্ডারটির ভূমির পরিধি হলো 5π একক।

৬,৭৫৪.
Nisha is 15 years elder to Romi. If 5 years ago, Nisha was 3 times as old as Romi, then find Nisha’s present age.
  1. ক
    32.5 years
  2. খ
    27.5 years
  3. গ
    25 years
  4. ঘ
    24.9 years
  5. ঙ
    None of these
ব্যাখ্যা
Question: Nisha is 15 years elder to Romi. If 5 years ago, Nisha was 3 times as old as Romi, then find Nisha’s present age.

Solution:
Let age of Romi be y
Nisha is 15 years elder than Romi = (y + 15).
So Nisha's age 5 years ago = (y + 15 - 5).
Romi's age before 5 years = (y - 5)

5 years ago, Nisha is 3 times as old as Romi
(y + 15 - 5) = 3 (y - 5)
⇒ (y + 10) = (3y - 15)
⇒ 2y = 25
⇒ y = 12.5

Romi's age = 12.5 years
Nisha's age = (y + 15) = (12.5 + 15) = 27.5 years.
৬,৭৫৫.
x2 - (a + b)x + ab = 0; x = ?
  1. ক
    a, b
  2. খ
    2a
  3. গ
    b/2
  4. ঘ
    a/b
ব্যাখ্যা
Question: x2 - (a + b)x + ab = 0; x = ?

Solution:
Given that
x2 - (a + b)x + ab = 0
x2 - ax - bx + ab = 0
x(x - a) - b(x - a) = 0
(x - a)(x - b) = 0

হয় 
x - a = 0
x = a

অথবা
x - b = 0
x = b

নির্ণেয় সমধান x = a, b
৬,৭৫৬.
If x + 5 > 2 and x - 3 < 7, the value of x must be between which of the following pairs of numbers?
  1. ক
    - 3 and 10
  2. খ
    - 3 and 4
  3. গ
    2 and 7
  4. ঘ
    3 and 4
  5. ঙ
    3 and 10
ব্যাখ্যা
Question: If x + 5 > 2 and x - 3 < 7, the value of x must be between which of the following pairs of numbers?

Solution:
x + 5 > 2
x > - 3

Next we simplify
x - 3 < 7.
x < 10

We know that x is greater than - 3 and less than 10.
৬,৭৫৭.
If x and y are whole numbers such that xy = 64, then the value of (x - 2)y + 1 is = ?
  1. ক
    10
  2. খ
    216
  3. গ
    1000
  4. ঘ
    676
ব্যাখ্যা

Question: If x and y are whole numbers such that xy = 64, then the value of (x - 2)y + 1 is = ?

Solution:
দেওয়া আছে,
xy = 64
⇒ xy = 82
এখানে, x = 8 এবং y = 2

এখন,
(x - 2)y + 1 = (8 - 2)2 + 1  [মান বসিয়ে]
= 63
= 216

৬,৭৫৮.
The average of 60 numbers is 25. If two numbers 40 and 50 are discarded, find the average of the remaining numbers.
  1. ক
    20.17
  2. খ
    24.31
  3. গ
    25.17
  4. ঘ
    29.17
ব্যাখ্যা
Question: The average of 60 numbers is 25. If two numbers 40 and 50 are discarded, find the average of the remaining numbers.

Solution:
Given,
Average of 60 numbers = 25
Sum of 50 numbers = 25 × 60 = 1500
Sum of discarded numbers = 40 + 50 = 90
Sum of remaining numbers = 1500 - 90 = 1410
Now, total remaining numbers = 60 - 2 = 58

Average of remaining numbers = 1410/58 = 24.31
৬,৭৫৯.
28√x + 1426 = three-fourths of 2984, find x.
  1. ক
    ক) 659
  2. খ
    খ) 694
  3. গ
    গ) 841
  4. ঘ
    ঘ) 859
ব্যাখ্যা
প্রশ্ন: 28√x + 1426 = three-fourths of 2984, find x.

সমাধান:
28√x +1426 = (3/4) × 2984
⇒ 28√x +1426 = 2238
⇒ 28√x = 2238 - 1426
⇒ 28√x = 812
⇒ √x = 812/28
⇒ √x = 29
⇒ x = (29)2
∴ x = 841
৬,৭৬০.
In one hour, a boat goes 11 km/hr along the stream and 5 km/hr against the stream. The speed of the boat in still water (in km/hr) is-
  1. ক
    3 km/hr
  2. খ
    5 km/hr
  3. গ
    8 km/hr
  4. ঘ
    9 km/hr
ব্যাখ্যা
Question: In one hour, a boat goes 11 km/hr along the stream and 5 km/hr against the stream. The speed of the boat in still water (in km/hr) is-

Solution:
Speed in still water = (11 + 5)/2 kmph
= 16/2 kmph
= 8 kmph.
৬,৭৬১.
Rabbi's regular pay is 30 Taka per hour up to 40 hours. Overtime is twice the payment for regular time. If he was paid 1680 Taka, how many hours of overtime did Rabbi work?
  1. ক
    5 hours
  2. খ
    6 hours
  3. গ
    7 hours
  4. ঘ
    8 hours
ব্যাখ্যা
Question: Rabbi's regular pay is 30 Taka per hour up to 40 hours. Overtime is twice the payment for regular time. If he was paid 1680 Taka, how many hours of overtime did Rabbi work?

Solution:
40 ঘন্টার জন্য রাব্বির নিয়মিত বেতন = (30 × 40) = 1200 টাকা।
Overtime এর টাকার পরিমান = (1680 - 1200) টাকা = 480 টাকা
যেহেতু, Overtime এর প্রতিদিনের টাকার পরিমান নিয়মিত বেতন এর দ্বিগুন,
সেহেতু মোট overtime কাজ করার সময় = 480 ÷ (30×2) ঘন্টা = 8 ঘন্টা।
৬,৭৬২.
Which number is 25% less than 120?
  1. ক
    80
  2. খ
    86
  3. গ
    90
  4. ঘ
    100
ব্যাখ্যা
Question: Which number is 25% less than 120?

Solution:
Let the number be x.
x = 120 - 120 × 0.25
⇒ x = 120 - 30
∴  x = 90

So, the number is 90.
৬,৭৬৩.
A invested some money in 10% stock at 96. If B wants to invest in an equally good 12% stock, he must purchase a stock worth of -
  1. ক
    ক) Tk. 117.08
  2. খ
    খ) Tk. 109.70
  3. গ
    গ) Tk. 115.20
  4. ঘ
    ঘ) Tk. 105.09
ব্যাখ্যা
For an income of Tk. 10, investment = Tk. 96
For an income of 12, investment
= Tk. (96/10) × 12
= Tk. 115.20

Hence, He must purchase a stock worth of Tk. 115.20
৬,৭৬৪.
If 22x + 1 = 128 then, what is the value of x?
  1. ক
    ক) 7
  2. খ
    খ) 6
  3. গ
    গ) 3
  4. ঘ
    ঘ) 4
ব্যাখ্যা
Question: If 22x + 1 = 128 then, what is the value of x?

Solution:
22x + 1 = 128
⇒ 22x + 1 = 27
⇒ 2x + 1 = 7
⇒ 2x = 6
⇒ x = 3
৬,৭৬৫.
When the length of a rectangle grows by 25%, what percentage reduction in the width is required to maintain the same area?
  1. ক
    15%
  2. খ
    20%
  3. গ
    22%
  4. ঘ
    30%
ব্যাখ্যা
Question: When the length of a rectangle grows by 25%, what percentage reduction in the width is required to maintain the same area?

Solution:
We can let the original length = 4, and the original width = 10.
So the original area is 40.

When the length increases by 25% we have 4 × 1.25 = 5
Thus, for the area to remain 40, the width must be 8.

Since {(8 - 10)/10} × 100 = - 20,

we see that 8 is 20 percent less than 10. In other words, the width has to decrease by 20%.
৬,৭৬৬.
If 1 + sinθ = mcosθ than what is the value of cotθ?
  1. ক
     (m2 - 1)/2m
  2. খ
    2m/(m2 + 1)
  3. গ
    m/(m2 - 1)
  4. ঘ
    2m/(m2 - 1)
ব্যাখ্যা

Question:  If 1 + sinθ = mcosθ than what is the value of cotθ? 

Solution:
Given that,
1+ sinθ = m cos θ
⇒ (1 + sinθ)/cosθ = m
⇒ (1/cosθ) + (sinθ/cosθ) = m
∴ secθ + tanθ = m ...............(i)

We know,
(secθ + tanθ) (secθ - tanθ) = 1
⇒ m(secθ - tanθ) = 1
⇒ secθ - tanθ = 1/m .................(ii)

Now, (i) - (ii) ⇒
secθ + tanθ - (secθ - tanθ) = m - (1/m)
⇒ secθ + tanθ - secθ + tanθ = (m2 - 1)/m
⇒ 2tanθ = (m2 - 1)/m
⇒  tanθ = (m2 - 1)/2m
⇒  1/cotθ = 1/{(m2 - 1)/2m}
∴ cotθ = 2m/(m2 - 1)

৬,৭৬৭.
Ismael drives from City A to B at 40 km per hour and returns over the same road at 30 km per hour and spends 15 hours away from home including a one hour stop for lunch. What is the distance (in km) between City A and City B?
  1. ক
    ক) 108
  2. খ
    খ) 120
  3. গ
    গ) 140
  4. ঘ
    ঘ) 240
ব্যাখ্যা

Let, distance from A to B = x km
ATQ, 
x/40 + x/30 + 1 = 15
Or, (3x + 4x)/120 = 15 - 1
Or, 7x = 14 × 120
Or, x = 240
∴ distance = 120 km

৬,৭৬৮.
A 30-meter pole has fractured and bent over to make a 30° angle with the ground, remaining partially attached. How high from the base did it break?
  1. ক
    8 meters
  2. খ
    10 meters
  3. গ
    18√3 meters
  4. ঘ
    15 meters
ব্যাখ্যা

Question: A 30-meter pole has fractured and bent over to make a 30° angle with the ground, remaining partially attached. How high from the base did it break?

Solution:

ধরি,
খুটিটি x মিটার উচুতে ভেঙ্গেছিল।
∴ অপর ভাঙ্গা অংশের দৈর্ঘ্য = (30 - x) মিটার

এখন, 
sin θ = লম্ব/অতিভুজ
বা, sin θ = x/(30 - x)
বা, sin 30° = x/(30 - x)
বা, 1/2 = x/(30 - x)
বা, 2x = 30 - x
বা, 2x + x = 30
বা, 3x = 30
⇒ x = 10

∴ খুটিটি ভূমি থেকে 10 মিটার উচুতে ভেঙ্গেছিল।

৬,৭৬৯.
What is the greatest number of four digits which is divisible by 15, 25, 40 and 75 ?
  1. ক
    9400
  2. খ
    9700
  3. গ
    9200
  4. ঘ
    9600
  5. ঙ
    9900
ব্যাখ্যা

Greatest number of four digits = 9999
LCM of 15, 25, 40 and 75 = 600
9999 ÷ 600 = 16, remainder = 399
Hence, greatest number of four digits which is divisible by 15, 25, 40 and 75 = 9999 - 399 = 9600

৬,৭৭০.
If 30 men can do a piece of work in 20 hours, then in how many hours will 12 men do it?
  1. ক
    18 hours
  2. খ
    30 hours
  3. গ
    40 hours
  4. ঘ
    50 hours
  5. ঙ
    None of these
ব্যাখ্যা
Question: If 30 men can do a piece of work in 20 hours, then in how many hours will 12 men do it?

Solution:
As number of workers increase, the time required decreases. Hence, this is a problem related to indirect proportion.
Let the number of hours be x.
12 : 30 : : 20 : x
⇒ 12/30 = 20/x
⇒ x = (20 × 30)/12
∴ x = 50

12 men require 50 hours to complete the same work.
৬,৭৭১.
A batsman in his 12th innings makes a score of 120, and thereby increases his average by 5. The average score after 12th innings is-
  1. ক
    65
  2. খ
    70
  3. গ
    75
  4. ঘ
    80
  5. ঙ
    85
ব্যাখ্যা
Question: A batsman in his 12th innings makes a score of 120, and thereby increases his average by 5. The average score after 12th innings is-

Solution:
Let, average runs after 12 innings = x 
Average runs after 11 innings = x - 5

ATQ,
12x = (x - 5) × 11 + 120
⇒ 12x = 11x - 55 + 120
⇒ 12x - 11x = 65
∴ x = 65
৬,৭৭২.
In a cricket team of 11 players, the captain is 25 years old, while the wicketkeeper is 3 years older. When these two are excluded, the average age of the remaining players drops by one year compared to the entire team. Find the team's average age.
  1. ক
    18 years
  2. খ
    22 years
  3. গ
    28 years
  4. ঘ
    32 years
ব্যাখ্যা
Question: In a cricket team of 11 players, the captain is 25 years old, while the wicketkeeper is 3 years older. When these two are excluded, the average age of the remaining players drops by one year compared to the entire team. Find the team's average age.

Solution:
Let the average age of the whole team by x years.
⇒ 11x - (25 + 28) = 9(x -1)
⇒ 11x - 53 = 9x - 9 
⇒ 11x - 9x = 44
⇒ 2x = 44
⇒ x = 22.

So, the average age of the team is 22 years.
৬,৭৭৩.
Dhruvo has to divide his rectangular field into two parts from one corner to the other using fence. If the area of the field is 540m2 and the length of the field is 36m then what will be the length of the fence needed?
  1. ক
    32 m
  2. খ
    35 m
  3. গ
    39 m
  4. ঘ
    40 m
ব্যাখ্যা
Question: Dhruvo has to divide his rectangular field into two parts from one corner to the other using fence. If the area of the field is 540m2 and the length of the field is 36m then what will be the length of the fence needed?

Solution: 
Breadth of the field = 540/36 = 15 m

 the length of the fence = diagonal of the field 
= √(152 + 362)
= √1521
= 39 m
৬,৭৭৪.
In the figure, AD and BC are lines intersecting at O. What is the value of a?
  1. ক
    18
  2. খ
    15
  3. গ
    12
  4. ঘ
    None of the above
ব্যাখ্যা
Question: In the figure, AD and BC are lines intersecting at O. What is the value of a?

Solution:
Here,
y = 3x + 30 ..........(1)

5y/3 = 5x + 5a
⇒ y/3 = x + a
⇒ y = 3x + 3a ...........(2)

From (1) and (2) we get,
3x + 3a = 3x + 30
⇒ 3a = 30
∴ a = 10
৬,৭৭৫.
The cost of the paint is 36.50 Tk. per kg. If 1 kg of paint covers 16 square feet, how much will it cost to paint the outside of a cube having 8 feet on each side?
  1. ক
    678 Taka
  2. খ
    676 Taka
  3. গ
    786 Taka
  4. ঘ
    876 Taka
ব্যাখ্যা
Question: The cost of the paint is 36.50 Tk. per kg. If 1 kg of paint covers 16 square feet, how much will it cost to paint the outside of a cube having 8 feet on each side?

Solution:
Total surface area = 6 × 82 square ft.
∴ total paint needed = (6 × 82)/16 kg
= 24 kg

Total cost = (24 × 36.5) taka
= 876 Taka
৬,৭৭৬.
3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?
  1. ক
    ক) 9
  2. খ
    খ) 10
  3. গ
    গ) 11
  4. ঘ
    ঘ) 12
  5. ঙ
    ঙ) 15
ব্যাখ্যা

Let the required number of working hours per day be x.
More pumps , Less working hours per day (Indirect Proportion)
Less days, More working hours per day (Indirect Proportion)
Pumps 4 : 3 and Days 1 : 2 } :: 8:x
=> (4 × 1 × x) = (3 × 2 × 8)
=> x = 12

৬,৭৭৭.
If p is a positive fraction less than 1, then -
  1. ক
    ক) 1/p is less than 1
  2. খ
    খ) 1/p is a positive integer
  3. গ
    গ) p2 is less than p
  4. ঘ
    ঘ) 2/p - p is a positive number
ব্যাখ্যা

p < 1
⇒ 1/p > 1
⇒ 2/p > 2
2/p - p > 2 - p > 0
[∵ p < 1]
Hence, (2/p - p) is a positive number.

৬,৭৭৮.
A dog sees a cat 80 m away. The cat runs at a speed of 5 m/s while the dog chases it at a speed 2 m/s more than that of the cat. Before the dog is able to catch the cat, how much distance has it already run?
  1. ক
    ক) 50 m
  2. খ
    খ) 100 m
  3. গ
    গ) 130 m
  4. ঘ
    ঘ) 200 m
ব্যাখ্যা

Let distance travelled by cat before dog catches it be D
We know, time for which Dog and Cat ran is same
∴ T = T
∴ D/5 = (D + 80)/7 [D = S x T]
∴ D = 200 m

৬,৭৭৯.
In a certain sequence, the first term is 1, and each successive term is 1 more than the reciprocal of the term that immediately precedes it. What is the fifth term of the sequence?
  1. ক
    3/5
  2. খ
    5/8
  3. গ
    8/5
  4. ঘ
    5/3
ব্যাখ্যা
Question: In a certain sequence, the first term is 1, and each successive term is 1 more than the reciprocal of the term that immediately precedes it. What is the fifth term of the sequence?

Solution:
1st Term = 1
2nd Term = 2
3rd Term= 1/2 +1 = 3/2
4th Term = 2/3 +1 = 5/3
5th Term = 3/5 +1 = 8/5
৬,৭৮০.
A cistern of capacity 8000 litres measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The thickness of the bottom is:
  1. ক
    ক) 90 cm
  2. খ
    খ) 1 dm
  3. গ
    গ) 1.1 cm
  4. ঘ
    ঘ) 2 m
ব্যাখ্যা

Let the thickness of the bottom be x cm.
Then, [(330 - 10) x (260 - 10) x (110 - x)] = 8000 x 1000
320 x 250 x (110 - x) = 8000 x 1000
(110 - x) =8000 x 1000/320 x 250= 100
 x = 10 cm = 1 dm.

৬,৭৮১.
The difference between the length and breadth of a rectangle is 23 m. if its area is 2520 m2, then its perimeter is:
  1. ক
    206 m
  2. খ
    114 m
  3. গ
    103 m
  4. ঘ
    298 m
ব্যাখ্যা

Question: The difference between the length and breadth of a rectangle is 23 m. if its area is 2520 m2, then its perimeter is:

Solution:
Let,
The breadth is x m
The length is x + 23 m 

ATQ,
x(x + 23) = 2520
⇒ x2 + 23x - 2520 = 0
⇒ x2 + 63x - 40x - 2520 = 0
⇒ x(x + 63) - 40(x + 63) = 0
⇒ (x + 63)(x - 40) = 0
∴ x = - 63 Or x = 40 
We ignore the negative value.
so, x = 40

∴ The breadth is 40 m
∴ The length is 40 + 23 m = 63 m

∴ The perimeter is 2(63 + 40)m = 2 × 103 m = 206 m 

৬,৭৮২.
In covering a certain distance, the speed of A and B are in the ratio of 3 : 5. A takes 30 minutes more than B to reach the destination. The time taken by A to reach the destination is:
  1. ক
    ক) 75 minutes
  2. খ
    খ) 60 minutes
  3. গ
    গ) 55 minutes
  4. ঘ
    ঘ) 45 minutes
ব্যাখ্যা
A এবং B এর গতিবেগের অনুপাত =  3 : 5
A এর গতিবেগ =3x 
B এর গতিবেগ = 5x 

নির্দিষ্ট স্থানে B পৌঁছাতে সময় নেয় t মিনিট 
নির্দিষ্ট স্থানে A পৌঁছাতে সময় নেয় t + 30 মিনিট 

এখন,
3x(t + 30) = 5xt
3t + 90 = 5t 
5t - 3t = 90 
2t = 90 
t = 45

নির্দিষ্ট স্থানে A পৌঁছাতে সময় নেয় (45 + 30) মিনিট = 75 মিনিট
৬,৭৮৩.
If P = {x ∈ N : 2 < x ≤ 6 and x is natural number} and Q = {x ∈ N : x even number and x ≤ 8} then (P - Q) = ?
  1. ক
    ক) {4, 6}
  2. খ
    খ) {3, 5}
  3. গ
    গ) {2, 8}
  4. ঘ
    ঘ) None of the above
ব্যাখ্যা
Question: If P = {x ∈ N : 2 < x ≤ 6 and x is natural number} and Q = {x ∈ N : x even number and x ≤ 8} then (P - Q) = ?

Solution:
দেওয়া আছে,
P = {x ∈ N : 2 < x ≤ 6 এবং x স্বাভাবিক সংখ্যা}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x জোড় সংখ্যা এবং x ≤ 8}
∴ Q = {2, 4, 6, 8}

এখন,
(P - Q) = {3, 4, 5, 6} - {2, 4, 6, 8}
= {3, 5}
৬,৭৮৪.
If the area of the trapezium, whose parallel sides are 8 cm and 12 cm is 40 sq. cm, what will be the distance between the parallel sides?
  1. ক
    2 cm
  2. খ
    4 cm
  3. গ
    5 cm
  4. ঘ
    8 cm
ব্যাখ্যা
Question: If the area of the trapezium, whose parallel sides are 8 cm and 12 cm is 40 sq. cm, what will be the distance between the parallel sides?

Solution:
Parallel sides of a trapezium = 8 cm, and 12 cm
Area of trapezium = (1/2)(sum of the parallel sides) × distance between the parallel sides
40 = (1/2)(8 + 12) × distance 
⇒ 40 = 10 × distance
⇒ distance = 40/10 = 4 cm

So, the distance between the parallel lines of trapezium = 4 cm.
৬,৭৮৫.
If x/y = 5/3 then (3x - 2y)/(3x + 2y) =?
  1. ক
    ক) 2 : 7
  2. খ
    খ) 3 : 7
  3. গ
    গ) 4 : 7
  4. ঘ
    ঘ) 1 : 7
ব্যাখ্যা
Question: If x/y = 5/3 then (3x - 2y)/(3x + 2y) =?

Solution:
x : y = 5/3
⇒ x/y = 5/3
⇒ 3x/2y = (5 × 3)/(3 × 2) [Multiplying by 3/2]
⇒ 3x/2y = 15/6
⇒ (3x - 2y)/(3x + 2y) = (15 - 6)/(15 + 6)
⇒ (3x - 2y)/(3x + 2y) = 9/21 
⇒ (3x - 2y)/(3x + 2y) = 3/7
∴ (3x - 2y)/(3x + 2y) = 3 : 7
৬,৭৮৬.
If 4x2 - 6x + 1 = 0, then the value of 8x3 + 1/8x3 is-
  1. ক
    27
  2. খ
    18
  3. গ
    36
  4. ঘ
    9
ব্যাখ্যা

Question: If 4x2 - 6x + 1 = 0, then the value of 8x3 + 1/8x3 is- 

Solution:

৬,৭৮৭.
A can do a certain job in 12 days. B is 60% more efficient than A. How many days does B alone take to do the same job?
  1. ক
    ক) 7.2 days
  2. খ
    খ) 8 days
  3. গ
    গ) 7.5 days
  4. ঘ
    ঘ) 9 days
ব্যাখ্যা
Question: A can do a certain job in 12 days. B is 60% more efficient than A. How many days does B alone take to do the same job?

Solution:
A completes one job in 12 days.
Hence the fraction of the job completed by A in 1 day =1/12

Since B is 60% more efficient than A,
Hence the fraction of the total job completed by B on one day =1/12 + 1/12 × (60/100)
= 1/12 + 1/20
= (5 + 3)/60
= 8/60
= 2/15

B can finish the job in 15/2 = 7.5 days
৬,৭৮৮.
The sum of five consecutive multiples of 3 is 165. What is the largest number?
  1. ক
    45
  2. খ
    36
  3. গ
    27
  4. ঘ
    39
ব্যাখ্যা

Question: The sum of five consecutive multiples of 3 is 165. What is the largest number?

Solution:
ধরি, পাঁচটি ক্রমিক 3 এর গুণিতক যথাক্রমে x, (x + 3), (x + 6), (x + 9) এবং (x + 12).

প্রশ্নমতে,
x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 165
⇒ 5x + 30 = 165
⇒ 5x = 165 - 30
⇒ 5x = 135
⇒ x = 135/5
⇒ x = 27

∴ বৃহত্তম সংখ্যা = x + 12
= 27 + 12 = 39

৬,৭৮৯.
The ages of Moni and Roni are in the ratio 6 : 5 respectively. After 10 years, the ratio of their ages will be 8 : 7. What is the difference in their ages now?
  1. ক
    5 years
  2. খ
    8 years
  3. গ
    12 years
  4. ঘ
    4 years
ব্যাখ্যা
Question: The ages of Moni and Roni are in the ratio 6 : 5 respectively. After 10 years, the ratio of their ages will be 8 : 7. What is the difference in their ages now?

Solution:
Let the present ages of Moni and Roni be 6x and 5x respectively, where x is a common factor.

After 10 years, the ratio of their ages will be 8 : 7. then,
⇒ (6x + 10) : (5x + 10) = 8 : 7
⇒ 6x + 10/5x + 10 = 8/7
⇒ 7(6x + 10) = 8(5x + 10)
⇒ 42x + 70 = 40x + 80
⇒ 42x - 40x = 80 - 70
⇒ 2x = 10
⇒ x = 10/2
∴ x = 5

Now,
Moni's age = 6x = 6 × 5 = 30 years
Roni's age = 5x= 5 × 5 = 25 years

∴ The difference in their ages now = 30 − 25 = 5 years
৬,৭৯০.
One filling pipe P is three times faster than another filling pipe Q, if P can fill tank in 24 hours, then what is the time taken to completely fill the tank if both the pipes are opened together?
  1. ক
    ক) 12 hours
  2. খ
    খ) 16 hours
  3. গ
    গ) 18 hours
  4. ঘ
    ঘ) 14 hours
ব্যাখ্যা
Time taken by P = 24 hours
So, Q takes = 24 x 3 = 72 hours
Required time to fill the tank with both pipe = (24 x 72)/(24+72) = 18 hours
৬,৭৯১.
The average of five numbers is 32. The sum of the first three numbers is 75. If the fourth number is 28, what is the fifth number?
  1. ক
    50
  2. খ
    53
  3. গ
    57
  4. ঘ
    60
ব্যাখ্যা

Question: The average of five numbers is 32. The sum of the first three numbers is 75. If the fourth number is 28, what is the fifth number?

Answer:
Average of five numbers = 32
Total sum = 5 × 32
= 160

Sum of first three numbers = 75.
Fourth number = 28.

Sum of first four numbers = 75 + 28 = 103.

∴ Fifth number = total sum - sum of first four 
= 160 - 103
= 57

So the fifth number is 57.

৬,৭৯২.
A motorcycle at a speed of 60km/hr, it covers certain distance in 3 hours and the train can cover the same distance in 1 and half hours then the speed of train is -
  1. ক
    80km/hr
  2. খ
    120km/hr
  3. গ
    160km/hr
  4. ঘ
    50km/hr
ব্যাখ্যা

The Distance covered by the motorcycle with speed 60km/hr in 3 hours = (60 x 3)
= 180 km
Now,
Speed = Distance/Time
Since the train covers the same 180 km in 3/2 hours (we can write 1 and half hours as 3/2hours)
Then the speed of the train = 180/ (3/2)
= 180 × (2/3)
= 120 km/hr.
Hence, the train travelled at the speed of 120km/hr to cross 180km in 1 and half hour.

৬,৭৯৩.
The area of incircle of an equilateral triangle of side 42 cm is ___ cm2
  1. ক
    ক) 462
  2. খ
    খ) 452
  3. গ
    গ) 442
  4. ঘ
    ঘ) 432
ব্যাখ্যা

Radius of incircle
=a/2√3
=42/ 2√3
=7√3
Area of incircle
=22/7×49×3
=462 cm2

৬,৭৯৪.
At the beginning of each year, the price of item X is 10 percent higher than its price at the beginning of the previous year. During three consecutive years, if the price of item X is Tk. 80 at the beginning of the first year, what is its price at the beginning of the third year?
  1. ক
    Tk. 88
  2. খ
    Tk. 96
  3. গ
    Tk. 96.8
  4. ঘ
    Tk. 100
  5. ঙ
    Tk. 160
ব্যাখ্যা
Question: At the beginning of each year, the price of item X is 10 percent higher than its price at the beginning of the previous year. During three consecutive years, if the price of item X is Tk. 80 at the beginning of the first year, what is its price at the beginning of the third year?

Solution:
The price of item X is increasing 10% each year then the previous year.

At the beginning of the first year, the price of item X = Tk. 80

At the beginning of the second year, the price of item X = 80 × (1 + 10/100) = Tk. (80 × 1.1) = Tk. 88

At the beginning of the third year, the price of item X = 88 × (1 + 10/100) = Tk. (88 × 1.1) = Tk. 96.8
৬,৭৯৫.
If (a - b)2 = 4 and ab = 15, then what is the value of (a2 + b2)?
  1. ক
    16
  2. খ
    34
  3. গ
    22
  4. ঘ
    10
ব্যাখ্যা
Question: If (a - b)2 = 4 and ab = 15, then what is the value of (a2 + b2)?

Solution: 
a2 + b2
= (a - b)2 + 2ab 
= 4 + 2 × 15
= 4 + 30
= 34
৬,৭৯৬.
If a + b = 13 and a - b = 3 , then find the value of a2 + b2
  1. ক
    ক) 69
  2. খ
    খ) 79
  3. গ
    গ) 89
  4. ঘ
    ঘ) 91
  5. ঙ
    ঙ) 96
ব্যাখ্যা

a2 + b2
= 1/2{(a + b)2 + (a - b)2}
= 1/2(132 + 32)
= 1/2(169 + 9)
= 89

৬,৭৯৭.
The difference between two positive numbers is 4 and the difference of their squares is 96. The largest number is -
  1. ক
    ক) 10
  2. খ
    খ) 14
  3. গ
    গ) 20
  4. ঘ
    ঘ) 25
ব্যাখ্যা
Question: The difference between two positive numbers is 4 and the difference of their squares is 96. The largest number is - 

Solution: 
Let the numbers be X and (X + 4)

then,
(X + 4)2 - X2 = 96
X2 + 8X + 16 - X2 = 96
8X + 16 = 96
8X = 80
X = 10

hence, the largest number is = (10 + 4) = 14
৬,৭৯৮.
A pipe can fill a cistern in 20 hours. Once the cistern is half full, three additional identical pipes are opened. How long will it take to fill the cistern completely? 
  1. ক
    6 hours
  2. খ
    8 hours 30 minutes
  3. গ
    10 hours 30 minutes
  4. ঘ
    12 hours 30 minutes
ব্যাখ্যা

Question: A pipe can fill a cistern in 20 hours. Once the cistern is half full, three additional identical pipes are opened. How long will it take to fill the cistern completely?

Solution:

Work done by 1 pipe in 1 hour = 1/20
∴ Time to fill half the cistern with 1 pipe = (1/2) ÷ (1/20) = 10 hours

After cistern is half full,
three additional identical pipes are opened, 
total pipes = 4
Work done by 4 pipes in 1 hour = 4 × (1/20) = 1/5
Time to fill remaining half = (1/2) ÷ (1/5)
= 2.5 hours
= 2 hours 30 minutes

∴ Total time to fill cistern = 10 + 2.5
= 12.5 hours = 12 hours 30 minutes

৬,৭৯৯.
The area of a right-angled tringle is 20 time its base. What is its height? 
  1. ক
    ক) 40 cm
  2. খ
    খ) 20 cm
  3. গ
    গ) 30 cm
  4. ঘ
    ঘ) 10 cm
ব্যাখ্যা
Let the base and height of a right angled triangle be b, h respectively
Given area of a right angled triangle is 20times its base
⇒ (1/2)​b h= 20b
⇒ h = 40 cm
Height of a right angled triangle is 40 cm
৬,৮০০.
The price of salt has risen by 25%. If a family wants to keep their expenses on salt the same as earlier, the family will have to decrease its consumption of salt by
  1. ক
    25%
  2. খ
    30%
  3. গ
    15%
  4. ঘ
    10%
  5. ঙ
    None
ব্যাখ্যা
Question: The price of salt has risen by 25%. If a family wants to keep their expenses on salt the same as earlier, the family will have to decrease its consumption of salt by

Solution:
Let the initial expenses on salt was Tk. 100.

Now, the price of salt has risen 25%.

So, to buy the same amount of salt, they need to spend,
= (100 + 25% of 100) = Tk. 125.

But, They want to keep expenses on salt, so they have to cut Tk. 25 in the expenses to keep it to Tk. 100.

Now, % decrease in Consumption,
(25/125) × 100 = 20%