Bank Math

PrepBank · পাতা ১৪০ / ১৬১ · ১৩,৯০১–১৪,০০০ / ১৬,১২৪

১৩,৯০১.

The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half-yearly is -

ক
ক) 6.07%
খ
খ) 6.08%
গ
গ) 6.09%
ঘ
ঘ) 6.05%

ব্যাখ্যা

Compound interest for 1 year.
2{3%(P)} + 3%{3%(P)}
= 6%(P) + 0.09%(P)
= 6.09%(P)
That is an effective interest rate for 1 year = 6.09%.

১৩,৯০২.

At what annual simple interest rate will Tk. 13,000 become Tk. 18,850 in 5 years?

ক
12%
খ
11%
গ
10%
ঘ
9%

ব্যাখ্যা

Question: At what annual simple interest rate will Tk. 13,000 become Tk. 18,850 in 5 years?

Solution:
Given,
P = Tk. 13,000
A = Tk. 18,850
∴ I = A - P = Tk. (18,850 - 13,000) = Tk. 5850
n = 5 years

We know,
I = Prn
⇒ r = I/Pn
= 5850/(13000 × 5)
= {(5850 × 100)/(13000 × 5)}%
= 9%

১৩,৯০৩.

Two ships, Alpha and Beta, start towards each other from two ports, 160 km apart. The speeds of Ship Alpha and Ship Beta in still water are 16 km/h and 24 km/h respectively. If Ship Alpha proceeds downstream and Ship Beta proceeds upstream, they will meet after how many hours?

ক
4 hours
খ
5 hours
গ
6 hours
ঘ
6.5 hours

ব্যাখ্যা

Question: Two ships, Alpha and Beta, start towards each other from two ports, 160 km apart. The speeds of Ship Alpha and Ship Beta in still water are 16 km/h and 24 km/h respectively. If Ship Alpha proceeds downstream and Ship Beta proceeds upstream, they will meet after how many hours?

সমাধান:
ধরি, স্রোতের গতিবেগ হলো x কিমি/ঘন্টা।
ধরি, জাহাজ দুটি t ঘন্টা পর মিলিত হবে।

জাহাজ Alpha (স্রোতের অনুকূলে) এর গতিবেগ = (16 + x) কিমি/ঘন্টা।
জাহাজ Beta (স্রোতের প্রতিকূলে) এর গতিবেগ = (24 - x) কিমি/ঘন্টা।

তারা একে অপরের দিকে আসছে, তাই তাদের আপেক্ষিক গতিবেগ হলো তাদের গতির যোগফল।
আপেক্ষিক গতিবেগ = (16 + x) + (24 - x) কিমি/ঘন্টা
= (16 + 24 + x - x) কিমি/ঘন্টা
= 40 কিমি/ঘন্টা।

প্রশ্নমতে,
দূরত্ব = আপেক্ষিক গতিবেগ × সময়।
160 = 40 × t
⇒ t = 160/40
⇒ t = 4 ঘন্টা।
∴ জাহাজ দুটি 4 ঘন্টা পর মিলিত হবে।

১৩,৯০৪.

An observer who is 1.8 meters tall is standing 20 meters away from a tower. If the angle of elevation from his eye to the top of the tower is 45°, what is the height of the tower?

ক
20 meters
খ
21.8 meters
গ
22.8 meters
ঘ
24 meters

ব্যাখ্যা

Question: An observer who is 1.8 meters tall is standing 20 meters away from a tower. If the angle of elevation from his eye to the top of the tower is 45°, what is the height of the tower?

Solution:

পর্যবেক্ষকের উচ্চতা, CD = 1.8 মিটার
এখানে, CD = EB
টাওয়ারের উচ্চতা = AB

এখন,
tan∠C = AE/CE
⇒ tan45° = AE/20
⇒ 1 = AE/20
∴ AE = 20

∴ AB = AE + BE
= 20 + 1.8
= 21.8 m

∴ টাওয়ারটির উচ্চতা 21.8 meters.

১৩,৯০৫.

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.

ক
ক) Tk. 16000
খ
খ) Tk. 18000
গ
গ) Tk. 12000
ঘ
ঘ) Tk. 8000

ব্যাখ্যা

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) = Tk. 15

Number of shares of company Y which he purchases = (4000 × 30)/15
= Tk. 8000.

১৩,৯০৬.

A train 300 meters long passes a pole in 20 seconds. How long will it take to pass a platform that is 450 meters long?

ক
30 seconds
খ
45 seconds
গ
50 seconds
ঘ
60 seconds

ব্যাখ্যা

Question: A train 300 meters long passes a pole in 20 seconds. How long will it take to pass a platform that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 300/20
= 15 m/s

Total distance to pass the platform = Length of train + Length of platform
= 300 m + 450 m
= 750 m

∴ Required time = Distance/Speed
= 750/15
= 50 seconds

∴ The train will take 50 seconds to pass the platform.

১৩,৯০৭.

Find the greatest number which on dividing 1567 and 1979, leaves remainders 3 and 7 respectively.

ক
45
খ
86
গ
68
ঘ
58

ব্যাখ্যা

Question: Find the greatest number which on dividing 1567 and 1979, leaves remainders 3 and 7 respectively.

Solution:
1567 - 3 = 1564
1979 - 7 = 1972

বৃহত্তম সংখ্যাটি হবে 1564 এবং 1972 এর গসাগু।

ইউক্লিডীয় পদ্ধতিতে গসাগু বের করি,

∴ গসাগু = 68

∴ নির্ণেয় বৃহত্তম সংখ্যা = 68

১৩,৯০৮.

The angle between the minute hand and the hour hand of a clock when the time is 8 : 30 is:

ক
80°
খ
75°
গ
60°
ঘ
105°

ব্যাখ্যা

Question: The angle between the minute hand and the hour hand of a clock when the time is 8 : 30 is:

Solution:
= │(11m - 60h)/2│°
= │(11 × 30 - 60 × 8)/2│°
= │(330 - 480)/2│°
= │- 150/2│°
= 75°

১৩,৯০৯.

Sunrise : Sunset-

ক
ক) Dawn : Twilight
খ
খ) Noon : Midnight
গ
গ) Morning : Night
ঘ
ঘ) Energetic : Lazy

ব্যাখ্যা

At sunrise morning begins, similarly at sunset night begins.

১৩,৯১০.

A rectangular field will be fenced on three side leaving one side of 15 feet uncovered. If the area of the field 450 square feet. How many feet of fencing is required?

ক
90 feet
খ
72 feet
গ
80 feet
ঘ
75 feet
ঙ
65 feet

ব্যাখ্যা

Question: A rectangular field will be fenced on three side leaving one side of 15 feet uncovered. If the area of the field 450 square feet. How many feet of fencing is required?

Solution:
আয়তাকার মাঠের এক পাশের দৈর্ঘ্য = 15 ফুট
আয়তাকার মাঠের ক্ষেত্রফল = 450 বর্গ ফুট

আয়তাকার মাঠের অন্য পাশের দৈর্ঘ্য = 450/15 = 30 ফুট

∴ মোট বেড়ার দৈর্ঘ্য = (30 + 15 + 30) = 75 ফুট

১৩,৯১১.

The side of a square is increased by 10%, by what percent will the area be increased?

ক
16%
খ
21%
গ
32%
ঘ
100%

ব্যাখ্যা

Question: The side of a square is increased by 10%, by what percent will the area be increased?

Solution:
আমরা জানি,
বর্গক্ষেত্রের ক্ষেত্রফল = (বাহু × বাহু) বর্গ একক

ধরি,
বর্গক্ষেত্রের এক বাহুর দৈর্ঘ্য = 10 মিটার
∴ বর্গক্ষেত্রের ক্ষেত্রফল = (10 × 10) বর্গমিটার
= 100 বর্গমিটার

আবার,
10% বৃদ্ধিতে বর্গক্ষেত্রের বাহুর সংখ্যা = 10 + (10 এর 10%) মিটার
= 10 + 1 = 11 মিটার

∴ 10% বৃদ্ধিতে বর্গক্ষেত্রের ক্ষেত্রফল = (11 × 11) = 121 বর্গমিটার

∴ বর্গক্ষেত্রের ক্ষেত্রফল বৃদ্ধি পেয়েছে = (121 - 100) = 21 বর্গমিটার

∴ বর্গক্ষেত্রের ক্ষেত্রফল বৃদ্ধি পাবে = 21%

১৩,৯১২.

X can complete a task in 12 days and Y in 18 days. If they work together for 3 days, what fraction of work is left?

ক
3/10
খ
1/6
গ
5/12
ঘ
7/12

ব্যাখ্যা

Question: X can complete a task in 12 days and Y in 18 days. If they work together for 3 days, what fraction of work is left?

Solution:
The work rate of X is 1/12 (since X can complete the work in 12 days).
The work rate of Y is 1/18 (since Y can complete the work in 18 days).
In 3 days they complete = 3{(1/12) + (1/18)}
= 3 × (5/36)
= 5/12

Then the fraction of work that is left is = 1 - (5/12)
= (12 - 5)/12
= 7/12

১৩,৯১৩.

The average age of A, B, C, D and E is 48 years. The average age of A and B is 40 years and the average of C and D is 50 years. Age of E is :

ক
55 years
খ
60 years
গ
65 years
ঘ
70 years

ব্যাখ্যা

Question: The average age of A, B, C, D and E is 48 years. The average age of A and B is 40 years and the average of C and D is 50 years. Age of E is :

Solution:
A + B + C + D + E = 48 × 5 = 240
A + B = 40 × 2 = 80
C + D = 50 × 2 = 100
Therefore,
E = (A + B + C + D + E) - (A + B + C + D)
E = 240 - 80 - 100
E = 60 years

১৩,৯১৪.

It is possible for which of the following groups of crops to grow together in field 2-

ক
ক) F, G, and H
খ
খ) F, H, and I
গ
গ) G, H, and J
ঘ
ঘ) G, I, and J

ব্যাখ্যা

Question: It is possible for which of the following groups of crops to grow together in field 2-

Solution:
G, H, I, J ফিল্ড ২ তে উৎপাদন করা যেতে পারে।
J, H একই ফিল্ডে উৎপাদন করা যায় না।

অতএব একসাথে ফিল্ড ২ এ উৎপাদন করা যাবে GHI অথবা GIJ.

১৩,৯১৫.

The average of nine numbers is 60, that of the first five numbers is 55 and the next three is 65. The ninth number is 10 less than the tenth number. Then, tenth number is-

ক
80
খ
76
গ
82
ঘ
78

ব্যাখ্যা

Question: The average of nine numbers is 60, that of the first five numbers is 55 and the next three is 65. The ninth number is 10 less than the tenth number. Then, tenth number is-

Solution:
Average of nine numbers = 60
Average of first five numbers = 55 and
average of next three numbers = 65
Tenth number = Ninth number + 10

The sum of nine numbers = 60 × 9 = 540
The sum of the first five numbers = 55 × 5 = 275
The sum of the next three numbers = 65 × 3 = 195
Ninth number = (540 - 275 - 195) = (540 - 470) = 70

∴ Tenth number = 70 + 10 = 80

১৩,৯১৬.

A pipe can fill a tank in x hours, and another can empty it in y hours. In how many hours do they together fill it in (y > x)?

ক
xy/(y - x) hours
খ
xy/(x - y) hours
গ
(x - y)/xy hours
ঘ
(y - x)/xy hours
ঙ
None of the above

ব্যাখ্যা

Question: A pipe can fill a tank in x hours, and another can empty it in y hours. In how many hours do they together fill it in (y > x)?

Solution:
Pipe fills the tank in x hours
∴ filling rate = 1/x tank/hour
Pipe empties the tank in y hours
∴ emptying rate = 1/y tank/hour
We are told y > x, so the filling pipe is faster than the emptying pipe

When both pipes are open, the net rate = (1/x) - (1/y)

∴ Time to fill the tank = Total work/Net rate
= 1/[(1/x) - (1/y)]
= 1/[(y - x)/xy]
= xy/(y - x) hours

১৩,৯১৭.

A pole 120 m long breaks at a point and the broken part bends such that it makes an angle of 30° with the ground (without getting separated). The length of the broken part is:

ক
65 meters
খ
70 meters
গ
80 meters
ঘ
100 meters

ব্যাখ্যা

Question: A pole 120 m long breaks at a point and the broken part bends such that it makes an angle of 30° with the ground (without getting separated). The length of the broken part is:

Solution:

খুঁটির মোট দৈর্ঘ্য = 120 মিটার
ধরি,ভাঙা অংশটির দৈর্ঘ্য = x মিটার
∴ অবশিষ্ট অংশটির দৈর্ঘ্য = (120 - x) মিটার
মই ভূমির সাথে কোণ তৈরি করে, θ = 30°

আমরা জানি,
sinθ = লম্ব/অতিভুজ
⇒ sin 30° =(120 - x)/x
⇒ 1/2 = (120 - x)/x
⇒ x = 2(120 - x)
⇒ x = 240 - 2x
⇒ 3x = 240
∴ x = 80 মিটার

অতএব, খুঁটির ভাঙা অংশটির দৈর্ঘ্য = 80 মিটার।

১৩,৯১৮.

1/4, 1/2, 3/4 এর গড় কোনটি?

ক
5/4
খ
2/3
গ
1/2
ঘ
3/4

ব্যাখ্যা

প্রশ্ন: 1/4, 1/2, 3/4 এর গড় কোনটি?

সমাধান:
1/4, 1/2, 3/4 এর যোগফল = (2 + 4 + 6)/8
= 12/8
= 3/2

1/4, 1/2, 3/4 এর গড় = (3/2)/3
= (3/2) × (1/3)
= 1/2

১৩,৯১৯.

The 5th and 12th term of an arithmetic progression are -21 and 21 respectively. What is the 21th term?

ক
145
খ
165
গ
135
ঘ
75

ব্যাখ্যা

Let first term be 'a' and common difference be 'd'
5th term = a + 4d = - 21
12th term = a + 11d = 21
Therefore, d = 6 and a = - 45
21th term = - 45 + 20 × 6 = - 45 + 120 = 75

১৩,৯২০.

If the radius of a right cylinder is tripled and the height is reduced by 40%, then what would the percentage change in volume?

ক
320%
খ
440%
গ
460%
ঘ
540%

ব্যাখ্যা

Question: If the radius of a right cylinder is tripled and the height is reduced by 40%, then what would the percentage change in volume?

Solution:
We know,
Volume of a cylinder, V = πr²h .....(1)

Given that,
New radius = 3r
New height = h - 40% of h
= (1 - 0.40)h
= 0.6h

∴ New volume, V' = π(3r)²(0.6h)
= π × 9r² × 0.6h
= π × 5.4 × r²h
= 5.4 × (πr²h)
= 5.4 × V ; [From 1]
So, new volume = 5.4 times the original volume

∴ Percentage change in volume = {(V' - V)/V} × 100%
= {(5.4V - V)/V} × 100%
= (4.4V/V) × 100%
= 4.4 × 100%
= 440% increase

So the volume increases by 440%.

১৩,৯২১.

At what rate of compound interest per annum will a sum of Tk. 1800 become Tk. 3042 in two years?

ক
15%
খ
21%
গ
12%
ঘ
30%

ব্যাখ্যা

Question: At what rate of compound interest per annum will a sum of Tk. 1800 become Tk. 3042 in two years?

Solution:
Given that,
Principal,
Amount after 2 years,
Time, years
Rate of interest per annum,

Now substituting the values,
C = P{1 + (r/100)}ⁿ
⇒ 3042 = 1800 × {1 + (r/100)}²
⇒ {1 + (r/100)}² = 3042/1800
⇒ {1 + (r/100)}² = 169/100
⇒ 1 + (r/100) = 13/10 ;[Taking the square root on both sides]
⇒ r/100 = (13/10) - 1
⇒ r/100 = (13 - 10)/10
⇒ r/100 = 3/10
⇒ r = (3 × 100)/10
∴ r = 30

Therefore, the rate of interest is 30%.

১৩,৯২২.

What is the maximum number of half-point bottles of cream that can be filled with a 4- gallon can of cream? (2 pt. = 1 qt. and 4 qt. = 1 gal)

ক
ক) 16
খ
খ) 24
গ
গ) 30
ঘ
ঘ) 64
ঙ
ঙ) None of these

ব্যাখ্যা

Question: What is the maximum number of half-point bottles of cream that can be filled with a 4- gallon can of cream? (2 pt. = 1 qt. and 4 qt. = 1 gal)

Solution:
The given information:
1 gallon (gal) = 4 quarts (qt)
1 quart (qt) = 2 pints (pt)
1 pint (pt) = 2 half-pints

∴1 gallon = 4 quarts = 8 pints =16 half-pints.

∴ 4 gallons = 4 × 16 half-pints = 64 half-pints

Therefore, the maximum number of half-pint bottles of cream that can be filled with a 4-gallon can of cream is 64 half-pints.

১৩,৯২৩.

Two small circular parks of diameter 24 meter and 10 meter are to be replaced by a bigger circular park. What would be the radius of this new park, if the new park has to occupy the same space as the two small parks?

ক
ক) 15 meter
খ
খ) 14 meter
গ
গ) 10 meter
ঘ
ঘ) 13 meter

ব্যাখ্যা

Question: Two small circular parks of diameter 24 meter and 10 meter are to be replaced by a bigger circular park. What would be the radius of this new park, if the new park has to occupy the same space as the two small parks?

Solution:
Let the radius of the new park be R m
Then,
πR²=π × 12² + π×5²
⇒πR² = 169π
⇒R² = 169
⇒R² =13²
R =13

১৩,৯২৪.

A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways the committee can be formed if it should have 1 man and 2 women?

ক
25 ways
খ
28 ways
গ
30 ways
ঘ
33 ways

ব্যাখ্যা

Question: A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways can the committee be formed if it should have 1 man and 2 women?

Solution:
We can select 1 man from 5 men in
= ⁵C₁ ways
= 5 ways

Selecting 2 women from 4 women in
= ⁴C₂
= 6

∴ The committee can be formed
= 5 × 6
= 30 ways

১৩,৯২৫.

City B is 5 miles east of city A. City C is 10 miles southeast of city B. Which of the following is the closest to the distance from city A to City C?

ক
11 miles
খ
12 miles
গ
13 miles
ঘ
14 miles

ব্যাখ্যা

Question: City B is 5 miles east of city A. City C is 10 miles southeast of city B. Which of the following is the closest to the distance from city A to City C?

Solution:

BD এবং DC দুটো সমান যেহেতু বিপরীত কোন দুইটাই 45°
অর্থাৎ, BDC ত্রিভুজ থেকে আমরা পাই,
BC² = BD² + DC²⇒ 10²= x²+ x²⇒ 2x² = 100
⇒ x² = 50
∴ x = 5√2

অনুরূপে, ADC থেকে পাই,
AC² = AD² + DC²⇒ AC² = (5 + x)²+ x²
= 5² + 2 · 5 · x + x² + x²
= 25 + 10 · 5√2 + (5√2)² + (5√2)²
= 25 + 50√2 + 50 + 50
= 125 + 50√2
= 125 + 70.71
= 195.71
∴ AC = √195.71 = 13.99
≈ 14 miles
অর্থাৎ, A থেকে C এর নিকটবর্তী দূরত্ব 14 মাইল।

১৩,৯২৬.

In how many ways can 5 different colored flags be arranged on a pole?

ক
32
খ
25
গ
120
ঘ
60

ব্যাখ্যা

Question: In how many ways can 5 different colored flags be arranged on a pole?

Solution:
Total ways = 5!
= 120

১৩,৯২৭.

Two pipes, A and B, can fill a cistern together in 3 hours. If each pipe were opened separately, pipe B would take 8 hours more than pipe A to fill the cistern. How long would it take pipe A to fill the cistern on its own?

ক
5 hours
খ
6 hours
গ
7 hours
ঘ
4 hours

ব্যাখ্যা

Question: Two pipes, A and B, can fill a cistern together in 3 hours. If each pipe were opened separately, pipe B would take 8 hours more than pipe A to fill the cistern. How long would it take pipe A to fill the cistern on its own?

Solution: Let the time taken by A alone be x hours.
Then time taken by B alone = x + 8 hours.
Rate of A = 1/x cistern/hour. Rate of B = 1/(x+8) cistern/hour.
Combined rate = 1/x + 1/(x+8) = 1/3 (since together they fill in 3 hours).

Now,
1/x + 1/(x+8) = 1/3
⇒ (x+8 + x) / [x(x+8)] = 1/3
⇒ (2x + 8) / [x(x+8)] = 1/3
⇒ 3(2x + 8) = x(x+8) [Cross multiply]
⇒ 6x + 24 = x² + 8x
⇒ x² + 2x - 24 = 0
⇒ (x + 6)(x - 4) = 0
So, x = 4 (positive value).

(Other root is negative and discarded.)
Therefore, A will take 4 hours alone.

১৩,৯২৮.

ক
2
খ
1
গ
3
ঘ
4

ব্যাখ্যা

Question:

Solution:

১৩,৯২৯.

The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is :

ক
ক) 35
খ
খ) 40
গ
গ) 50
ঘ
ঘ) None

ব্যাখ্যা

3 year ago the age was = 27×3 = 81
Currently sum of their age is = 81 + 9 = 90
5 year ago = 20×2 = 40
Now, the sum is = 40 + 10 = 50

So, Present age of the husband = 90 - 50 = 40

১৩,৯৩০.

The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the ratio of its diameter to its height.

ক
3 : 7
খ
7 : 3
গ
6 : 7
ঘ
7 : 6
ঙ
None of these

ব্যাখ্যা

Question: The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the ratio of its diameter to its height.

Solution:
(πr²h)/(2πrh) = 924/264
⇒ r = (924/264) × 2
∴ r = 7

And,
2πrh = 264
⇒ h = (264/2) × (7/22) × (1/7)
∴ h = 6

∴ Required ratio = (2r)/h
= 14/6
= 7 : 3.

১৩,৯৩১.

A shopkeeper marks his umbrella at 25% above the cost price. During rainy season sale, he offers a discount of 12% on marked price. Find his profit percentage.

ক
12%
খ
13%
গ
10%
ঘ
11%

ব্যাখ্যা

Question: A shopkeeper marks his umbrella at 25% above the cost price. During rainy season sale, he offers a discount of 12% on marked price. Find his profit percentage.

Solution:
Let,
Cost Price (CP) = tk 100
Marked Price (MP) = CP + 25% of CP
= 100 + (100 × 25/100)
= 100 + 25
= tk 125
Selling Price (SP) = MP - 12% of MP
= 125 - (125 × 12/100)
= 125 - 15
= tk 110

Profit = SP - CP
= 110 - 100
= tk 10

Profit percentage = (Profit × 100)/CP
= (10 × 100)/100
= 10%

১৩,৯৩২.

A certain sum of money becomes 2.5 times itself in 10 years at simple interest. What is the rate of interest per annum?

ক
7.5%
খ
10%
গ
12.5%
ঘ
15%

ব্যাখ্যা

Question: A certain sum of money becomes 2.5 times itself in 10 years at simple interest. What is the rate of interest per annum?

Solution:
Let the principal amount be P.
The sum becomes 2.5 times itself in 10 years.

So, A = 2.5P
∴ Simple Interest, I = 2.5P - P = 1.5P

We know, I = Pnr/100
1.5P = P × 10 × r/100
10r = 150
∴ r = 15%

১৩,৯৩৩.

A retail appliance store priced a video recorder at 20 percent above the wholesale cost of Tk. 200. If a store employee applied the 10 percent employee discount to the retail price to buy the recorder, how much did the employee pay for the recorder?

ক
ক) Tk. 198
খ
খ) Tk. 216
গ
গ) Tk. 220
ঘ
ঘ) Tk. 230

ব্যাখ্যা

20% of 200 = 40
At 20 percent above the wholesale cost of Tk. 200
So, the total price = 200 + 40 = Tk. 240
At 10 percent employee discount to the retail price to buy the recorder
So, employee paid for the recorder = tk. (240 - 10% of 240) = 240 - 24 = Tk. 216

১৩,৯৩৪.

When a certain number is divided by 7, the remainder is 0, if the remainder is not 0 when the number is divided by 14, then the remainder must be -

ক
ক) 7
খ
খ) 5
গ
গ) 3
ঘ
ঘ) 8

ব্যাখ্যা

Numbers can be divided by 7 are: 7, 14, 21, 28, 35
Among them, 14, 28 are divided by 14 with no remainder
But when 7, 21, 35 these numbers are divided by 14, the remainder is 7

১৩,৯৩৫.

If sinA + sin²A = 1 then, cos²A + cos⁴A =?

ক
- 1
খ
0
গ
1
ঘ
2

ব্যাখ্যা

Question: If sinA + sin²A = 1 then, cos²A + cos⁴A =?

Solution:
sinA + sin²A = 1
বা, sinA = 1 - sin²A
বা, sinA = cos²A

cos²A + cos⁴A
= cos²A + (cos²A)²
= cos²A + sin²A
= 1

১৩,৯৩৬.

The product of three consecutive even integers is 480. Find their sum.

ক
24
খ
14
গ
28
ঘ
32

ব্যাখ্যা

Question: The product of three consecutive even integers is 480. Find their sum.

Solution:
Let the three consecutive even integers are,
x - 2, x, x + 2

Now their product is,
⇒ (x - 2)(x)(x + 2) = 480
⇒ x(x² - 4) = 480
⇒ x³ - 4x = 480

Now try even integer values,
x = 6 than (6)³ - (4 × 6) = 216 - 24 = 192 not valid
x = 8 than (8)3 - (4 × 8) = 512 - 32 = 480 is valid
So,
x = 8
Then the three even integers are,
x - 2 = 6, x = 8, x + 2 = 10

Now their sum is = 6 + 8 + 10 = 24

১৩,৯৩৭.

The number of four-digit telephone numbers having at least one of their digits repeated is-

ক
9000
খ
10000
গ
3240
ঘ
4960
ঙ
None of these

ব্যাখ্যা

Question: The number of four-digit telephone numbers having at least one of their digits repeated is-

Solution:
The number of four-digit telephone numbers which can be formed using the digits of 0, 1, 2,...., 9 is 10⁴ = 10000The number of four digit telephone numbers which have none of their digits repeated is ¹⁰P₄ = 5040

Hence the required number =10000 - 5040 = 4960

১৩,৯৩৮.

(0.1 × 0.1 + 0.02)(0.2 × 0.2 + 0.01)^{- 1}is equal to-

ক
ক) 5/2
খ
খ) 2/5
গ
গ) 5/3
ঘ
ঘ) 3/5

ব্যাখ্যা

(0.1 × 0.1 + 0.02)(0.2 × 0.2 + 0.01)^{- 1}
= (0.1 × 0.1 + 0.02)/(0.2 × 0.2 + 0.01)
= (0.01 + 0.02)/(0.04 + 0.01)
= 0.03/0.05
= 3/5

১৩,৯৩৯.

A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3 : 5?

ক
4 litres, 8 litres
খ
6 litres, 6 litres
গ
5 litres, 7 litres
ঘ
7 litres, 5 litres

ব্যাখ্যা

Question: A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3 : 5?

Solution:
Let the cost of 1 litre milk be Tk. 1

Milk in 1 litre mix. in 1st can = 3/4 litre, C.P. of 1 litre mix. in 1st can Tk. 3/4
Milk in 1 litre mix. in 2nd can = 1/2 litre, C.P. of 1 litre mix. in 2nd can Tk. 1/2
Milk in 1 litre of final mix. = 5/8 litre, Mean price = Tk. 5/8

By the rule of alligation, we have:

⇒ Quantity of 2nd can : Quantity of 1st can = (3/4 - 5/8) : (5/8 - 1/2) = (6 - 5)/8 : (5 - 4)/8 = 1/8 : 1/8 = 1 : 1

∴ Quantity of mixure taken from each can = 12 × (1/2) = 6 litres

১৩,৯৪০.

Find the smallest fraction from the following:

ক
7/20
খ
9/50
গ
3/25
ঘ
7/125

ব্যাখ্যা

7/20 = 35/100
9/50 = 18/100
3/25 = 12/100
7/125 = 56/1000 = 5.6/100

১৩,৯৪১.

ক
2^7/8
খ
2^15/16
গ
2^31/32
ঘ
2^63/64
ঙ
2^3/4

ব্যাখ্যা

Number of 2 in square root is 5
2⁵ = 32
2⁵ - 1 = 31
Required solution:
2^{(32 - 1)/(32)}
= 2^31/32
---------------------------------------
Alternative way:
√ 2 [ √ 2 { √ 2 (√ 2 (√2 ) ) } ]
= √ 2 [ √2 { √2 ( √ ( 2 × 2^1/2) ) } ]
= √ 2 [ √2 { √2 ( √ ( 2^3/2) } ]
= √ 2 [ √2 { √ ( 2 × 2^3/4) } ]
= √ 2 [ √2 { √ 2^7/4} ]
= √ 2 [ √ {2 × 2^7/8} ]
= √ 2 × √ 2^15/8
= √ (2 × 2^15/16)
= √ 2^31/16
= 2^31/32

১৩,৯৪২.

If a man gains 10% by selling a watch for Taka 660, find the cost price.

ক
Taka 600
খ
Taka 620
গ
Taka 650
ঘ
Taka 670

ব্যাখ্যা

Question: If a man gains 10% by selling a watch for Taka 660, find the cost price.

Answer: Selling Price = 110% of Cost Price
→ Cost Price = 660 ÷ 1.1 = 600

Another Method,
With 10% gain,
Selling Price is 110 Taka when Cost Price is 100 Taka
Selling Price is 1 Taka when Cost Price is 100/110 Taka
Selling Price is 660 Taka when Cost Price is (100×660)/110 Taka = 600 Taka

১৩,৯৪৩.

An employee's annual salary was increased Tk. 25,000. If her new annual salary now equals Tk. 90,000, what was the percent increase?

ক
ক) 45%
খ
খ) 42%
গ
গ) 20%
ঘ
ঘ) None of these

ব্যাখ্যা

New annual salary = Tk. 90,000
Salary increase = Tk. 25,000.

Original salary
= Tk. (90,000 - 25,000).
= 65,000

Percentage Increase
= (25,000/ 65,000) × 100%
= 38.46%

১৩,৯৪৪.

For the past n days, the average (arithmetic mean) daily production at a company was 50 units. If today's production of 90 units raises the average to 55 units per day, what is the value of n?

ক
11
খ
12
গ
9
ঘ
7

ব্যাখ্যা

Question: For the past n days, the average (arithmetic mean) daily production at a company was 50 units. If today's production of 90 units raises the average to 55 units per day, what is the value of n?

Solution:
total production in n days is = 50n

ATQ,
50n + 90 = (n + 1)55
or, 50n + 90 = 55n + 55
or, 55n - 50n = 90 - 55
or, 5n = 35
∴ n = 7

১৩,৯৪৫.

A certain number of men can complete a piece of work in 180 days. If there are 30 men less, it will take 20 days more for the work to be completed. How many men were there originally?

ক
150
খ
165
গ
180
ঘ
300

ব্যাখ্যা

Question: A certain number of men can complete a piece of work in 180 days. If there are 30 men less, it will take 20 days more for the work to be completed. How many men were there originally?

Solution:
Let there be x men originally.
They were to complete the work in 180 days but as the number of persons is reduced to x - 30.
∴ Work takes 20 more days.
So the equation is
180x = (x - 30)200
⇒ 180x = 200x - 6000
⇒ 20x = 6000
⇒ x = 300.

১৩,৯৪৬.

A car covers the first 30km in 15 minutes and the second 30km in 10 minutes. Between the two actions, the car stopped for 5 minutes for fuel. What is the average speed of the car?

ক
100 km/hr
খ
120 km/hr
গ
140 km/hr
ঘ
90 km/hr

ব্যাখ্যা

Question: A car covers the first 30km in 15 minutes and the second 30km in 10 minutes. Between the two actions, the car stopped for 5 minutes for fuel. What is the average speed of the car?

Solution:
total distance = 30 + 30 = 60km
total time = 30 minutes = 1/2 hour

average = 60/(1/2) = 120km/hr.

১৩,৯৪৭.

In each expression below, N represents a negative integer. Which expression could have a negative value?

ক
ক) N²
খ
খ) 6 - N
গ
গ) -N
ঘ
ঘ) 6 + N

ব্যাখ্যা

যেহেতু,
N ঋনাত্মক পূর্ণসংখ্যা।
ধরি, N = -7

∴ (N)² = (-7)² = 49
6 - N = 6 - (-7) = 13
- N = -(-7) = 7
6 + N = 6 + (-7) = -1

তবে অপশন D তে N এর মান -6 বা তার কম হলে এটার মানও ধণাত্মক হতো। এজন্যই প্রশ্নে 'Could have a negative value' টার্ম ব্যবহার করা হয়েছে

∴ 6 + N এর মান ঋনাত্মক হতে পারে।

১৩,৯৪৮.

If y% of x = 29, then x = ?

ক
ক) 2900/y
খ
খ) 29x/y
গ
গ) 29y/x
ঘ
ঘ) 29xy

ব্যাখ্যা

y% of x = 29
Or, y/100 × x = 29
So, x = 29×100/y = 2900/y

১৩,৯৪৯.

If x = 4y = z and xyz = 128, then x = ?

ক
ক) 2
খ
খ) 4
গ
গ) 6
ঘ
ঘ) 8

ব্যাখ্যা

Question: If x = 4y = z and xyz = 128, then x = ?

Solution:
xyz = 128
⇒ (4y) y (4y) = 128
⇒ 16y³ = 128
⇒ y³ = 8
⇒ y³ = 2³
∴ y = 2

∴ x = 4 × 2
= 8

১৩,৯৫০.

A person rides a bicycle round a circular path of radius 50m. The radius of the wheel of the bicycle is 50 cm. The bicycle comes to the starting point for the first time of 1 hour. What is the number of revolution of the wheel in 15 minutes?

ক
15 times
খ
20 times
গ
25 times
ঘ
30 times

ব্যাখ্যা

Question: A person rides a bicycle round a circular path of radius 50m. The radius of the wheel of the bicycle is 50 cm. The bicycle comes to the starting point for the first time of 1 hour. What is the number of revolution of the wheel in 15 minutes?

Solution:
দেওয়া আছে,
বৃত্তের ব্যাসার্ধ = ৫০ মিটার
এবং চাকার ব্যাসার্ধ = ৫০ সে.মি. = ০.৫ মিটার

∴ চাকার পরিধি = ২πr = ২π × ০.৫ = π মিটার
সুতরাং, একবার ঘুরলে π মিটার যায়
আবার, বৃত্তের পরিধি = ২π × ৫০ = ১০০π মিটার
চাকাটি ১০০π মিটার অতিক্রম করলে ঘুরে = ১০০ বার

প্রশ্নমতে,
৬০ মিনিটে ঘুরে = ১০০ বার
১৫ মিনিটে ঘুরে = (১৫ × ১০০)/৬০ = ২৫ বার

১৩,৯৫১.

In the sequence (1/√2), 1, √2............... which term is 8√2?

ক
6
খ
9
গ
11
ঘ
12

ব্যাখ্যা

Question: In the sequence (1/√2), 1, √2............... which term is 8√2?

Solution:
দেওয়া আছে,
অনুক্রমটির প্রথম পদ, a = 1/√2
সাধারন অনুপাত, r = 1/(1/√2) = √2
n তম পদ = ar^{n - 1}

প্রশ্নমতে,
ar^{n - 1} = 8√2
⇒ (1/√2) × (√2)^{n - 1} = 8√2
⇒ (√2)^{n - 1} = 8√2 × √2
⇒ (√2)^{n - 1} = 16
⇒ (√2)^{n - 1} = (√2)⁸
⇒ n - 1 = 8
⇒ n = 8 + 1 = 9

অর্থাৎ অনুক্রমটির 9-তম পদ হলো 8√2

১৩,৯৫২.

A train, 150 m long, passes a pole in 15 seconds and another train of the same length travelling in the opposite direction in 12 seconds. The speed of the second train is-

ক
54 km/hr
খ
45 km/hr
গ
48 km/hr
ঘ
52 km/hr

ব্যাখ্যা

Question: A train, 150 m long, passes a pole in 15 seconds and another train of the same length travelling in the opposite direction in 12 seconds. The speed of the second train is-

Solution:
Given that,
Length of both trains = 150 m
First train passes a pole in 15 s
First train passes the second train (opposite direction) in 12 s

Now,
Speed of the first train,
= 150/15
= 10 m/s
Time taken by trains to cross each other = 12 sec
And, relative speed of two trains :
= (150 + 150)/12
= 25 m/s

∴ Speed of the second train is
= (25 - 10) × 18/5
= 15 × 18/5
= 54 km/hr

So the speed of the second train is 54 km/hr.

১৩,৯৫৩.

If p × q = p + q +p/q, then the value of 8 × 2 is?

ক
6
খ
10
গ
14
ঘ
16

ব্যাখ্যা

Question: If p × q = p + q +p/q, then the value of 8 × 2 is?

Solution:
দেওয়া আছে,
p × q = p + q +p/q

∴ 8 × 2 = 8 + 2 + 8/2
= 10 + 4
= 14

১৩,৯৫৪.

How many ways can the word QUESTION be arranged with 2 letters each time?

ক
24
খ
56
গ
74
ঘ
96

ব্যাখ্যা

Question: How many ways can the word QUESTION be arranged with 2 letters each time?

Solution:
'QUESTION' শব্দটিতে মোট 8 টি ভিন্ন ভিন্ন বর্ণ আছে (Q, U, E, S, T, I, O, N)

8টি ভিন্ন বর্ণ থেকে প্রতিবার 2টি বর্ণ নিয়ে সাজাতে হবে।

n টি ভিন্ন বস্তু থেকে r টি বস্তু নিয়ে সাজানোর উপায় = ⁿP_r = n!/(n - r)!

এখানে n = 8, r = 2
∴ ⁸P₂ = 8!/(8 - 2)!
= 8!/6!
= (8 × 7 × 6!)/(6!)
= 8 × 7
= 56

∴ 'QUESTION' শব্দের বর্ণগুলো থেকে প্রতিবার 2টি বর্ণ নিয়ে মোট 56 উপায়ে সাজানো যায়।

১৩,৯৫৫.

In a regiment the ratio between the number of officers to soldiers was 3 : 31 before. In a battle 6 officers and 22 soldiers were killed and the ratio become 1 : 13, the number of officers in the regiment before battle was?

ক
18
খ
21
গ
24
ঘ
27

ব্যাখ্যা

Question: In a regiment the ratio between the number of officers to soldiers was 3 : 31 before. In a battle 6 officers and 22 soldiers were killed and the ratio become 1 : 13, the number of officers in the regiment before battle was?

Solution:
Let, the number of officers and soldiers = 3x, 31x

ATQ,
(3x - 6)/(31x - 22) = 1/13
⇒ 39x - 78 = 31x - 22
⇒ 39x - 31x = - 22 + 78
⇒ 8x = 56
∴ x = 7

So, number of officer before = 3x
= 3 × 7
= 21

১৩,৯৫৬.

A boat covers a certain distance downstream in 1 hour, while it comes back in 3/2 hours. If the speed of the stream be 3 km/hr, what is the speed of the boat in still water?

ক
ক) 16km/hr
খ
খ) 14km/hr
গ
গ) 15km/hr
ঘ
ঘ) 12km/hr

ব্যাখ্যা

Question: A boat covers a certain distance downstream in 1 hour, while it comes back in 3/2 hours. If the speed of the stream be 3 km/hr, what is the speed of the boat in still water?

Solution:
Let the speed of the boat in still water be x km/hr
Then
Speed downstream = (x + 3) km/hr
Speed upstream = (x - 3) km/hr

ATQ,
(x + 3) × 1 = (x - 3) × (3/2)
⇒ 2x + 6 = 3x - 9
∴ x = 15km/hr

১৩,৯৫৭.

Man hardly believes a liar, ____ ?

ক
does he
খ
doesn't he
গ
are they
ঘ
aren't they

ব্যাখ্যা

• শূন্যস্থানে সঠিক উত্তর হবে - does he?
- Complete sentence: Man hardly believes a liar, does he?

• Tag question করার নিয়ম -
- সাধারণত Tag question ব্যবহৃত হয় পূর্বে উল্লেখিত কোন উক্তি সত্য না মিথ্যা তা নিশ্চিত হওয়ার জন্য।
- Statement positive হলে tag question টা negative হবে।
- আবার statement negative হলে tag question positive হবে।
- Subject ও Tense অনুসারে auxiliary verb দ্বারা tag question তৈরি হয়।

• Tag question এর নিয়মানুযায়ী বাক্যটি Negative হওয়ায় Tag question Negative হবে।
- Hardly = কদাচিৎ/ না বললেই চলে।
- Hardly, seldom, rarely ইত্যাদি থাকলে সাধারণত Present Indefinite Tense হয়।
- আর hardly নিজেই Negative অর্থ প্রকাশ করায়, এর সাথে কোন নাবোধক শব্দ বসে না।

• বাক্যটি Present indefinite tense হওয়ায় Tag question এর ক্ষেত্রে does he হবে।

১৩,৯৫৮.

If 20% of A = 30% of B = 1/6 of C, then A : B : C = ?

ক
ক) 15 : 10 : 21
খ
খ) 15 : 10 : 18
গ
গ) 15 : 8 : 12
ঘ
ঘ) 15 : 10 : 24

ব্যাখ্যা

Question: If 20% of A = 30% of B = 1/6 of c, then A : B : C = ?

Solution:
20% of A = 30% of B = 1/6 of c
(1/5)A = (3/10)B = (1/6)C
A/5 = 3B/10 = C/6

∴ A/5 = 3B/10
A/B = 15/10

and,
A/5 = C/6
A/C = 5/6 = 15/18

hence,
A : B : C = 15 : 10 : 18

১৩,৯৫৯.

A pipe can fill a cistern in just 4 hours. It took 8 hours to fill the tank after attaching an outgoing pipe to it at the same time. In how much time alone the outgoing pipe can empty the cistern?

ক
8 hours
খ
10 hours
গ
12 hours
ঘ
6 hours

ব্যাখ্যা

Question: A pipe can fill a cistern in just 4 hours. It took 8 hours to fill the tank after attaching an outgoing pipe to it at the same time. In how much time alone the outgoing pipe can empty the cistern?

Solution:
Let,
the outgoing pipe can empty the cistern in X hours.
total fill-up in one hour
= 1/4 - 1/X
= X - 4/4X

ATQ,
4X/ X - 4 = 8
4X = 8X - 32
X = 8 hours

১৩,৯৬০.

If 20 workers can complete a road construction in 45 days by working 6 hours a day, how many workers are required to complete the same construction in 30 days by working 9 hours a day?

ক
15 workers
খ
20 workers
গ
30 workers
ঘ
25 workers

ব্যাখ্যা

Question: If 20 workers can complete a road construction in 45 days by working 6 hours a day, how many workers are required to complete the same construction in 30 days by working 9 hours a day?

Solution:
দৈনিক ৬ ঘণ্টা কাজ করে ৪৫ দিনে একটি কাজ সম্পন্ন করে ২০ জন
দৈনিক ১ ঘণ্টা কাজ করে ৪৫ দিনে একটি কাজ সম্পন্ন করে ২০ × ৬ জন
দৈনিক ১ ঘণ্টা কাজ করে ১ দিনে একটি কাজ সম্পন্ন করে ১২০ × ৪৫ জন
দৈনিক ৯ ঘণ্টা কাজ করে ১ দিনে একটি কাজ সম্পন্ন করে (১২০ × ৪৫)/৯ জন
দৈনিক ৯ ঘণ্টা কাজ করে ৩০ দিনে একটি কাজ সম্পন্ন করে (১২০ × ৪৫)/(৯ × ৩০) জন
= ২০ জন

১৩,৯৬১.

A sum of Tk. 1550 was lent partly at 5% and partly at 8% p.a. simple interest. The total interest received after 3 years was Tk. 300. The ratio of the money lent at 5% to that lent at 8% is:

ক
ক) 16 : 9
খ
খ) 16 : 15
গ
গ) 31 : 15
ঘ
ঘ) 17 : 15

ব্যাখ্যা

Question: A sum of Tk. 1550 was lent partly at 5% and partly at 8% p.a. simple interest. The total interest received after 3 years was Tk. 300. The ratio of the money lent at 5% to that lent at 8% is:

Solution:
Let
the sum lent at 5% be Tk. x and that lent 8% be Tk. (1550 - x).

ATQ,
(x × 5 × 3)/100 + {(1550 - x) × 8 × 3}/100 = 300
⇒ 15x + 37200 - 24x = 30000
⇒ - 9x = 30000 - 37200
⇒ - 9x = - 7200
⇒ x = 7200/9
∴ x = 800

∴ Required ratio = 800 : 750
= 16 : 15

১৩,৯৬২.

5% merchandise of a trader was stolen another 5% was damaged during transportation. At what profit (in %) should he sell the rest of the goods so that he can make an overall profit of 20%?

ক
25%
খ
30%
গ
20%
ঘ
None of the above

ব্যাখ্যা

Question: 5% merchandise of a trader was stolen another 5% was damaged during transportation. At what profit (in %) should he sell the rest of the goods so that he can make an overall profit of 20%?

Solution:
ধরি,
১০০ টাকার পণ্য ছিল।
তাহলে, বিক্রয়যোগ্য পণ্য = ১০০ - ৫ - ৫ = ৯০ টাকার
এখন,
২০% লাভ করতে হলে ৯০ টাকার পণ্য বিক্রয় করতে হবে ১২০ টাকা।
অর্থাৎ লাভ করতে হবে = ১২০ - ৯০ = ৩০ টাকা

৯০ টাকায় লাভ করতে হবে = ৩০ টাকা
১ টাকায় লাভ করতে হবে = ৩০/৯০ টাকা
∴ ১০০ টাকায় লাভ করতে হবে = (৩০ × ১০০)/৯০ টাকা
= ৩৩.৩৩ টাকা বা ৩৩.৩৩%

১৩,৯৬৩.

If cosecθ + cotθ = 3, Find the value of cotθ.

ক
4/3
খ
4
গ
3/4
ঘ
1/3

ব্যাখ্যা

Question: If cosecθ + cotθ = 3, Find the value of cotθ.

Solution:
given,
cosecθ + cotθ = 3.......(i)

cosec²θ - cot²θ = 1
or, (cosecθ + cotθ)(cosecθ - cotθ) = 1
or, cosecθ - cotθ = 1/3.......(ii)

subtracting (i) from (ii) we get,
2cotθ = 8/3
cotθ = 4/3

১৩,৯৬৪.

The average of A and B is 30, and the average of B and C is 20. What is the value of (A - C)/2?

ক
5
খ
10
গ
15
ঘ
20
ঙ
None

ব্যাখ্যা

প্রশ্ন: The average of A and B is 30, and the average of B and C is 20. What is the value of (A - C)/2?

সমাধান:
১ম শর্তমতে,
(A + B)/2 = 30
⇒ A + B = 60 ...... (1)

২য় শর্তমতে,
(B + C)/2 = 20
⇒ B + C = 40 ...... (2)

(1) নং - (2) নং ⇒
A + B - B - C = 60 - 40
⇒ A - C = 20
⇒ (A - C)/2 = 20/2
∴ A - C = 10

১৩,৯৬৫.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
ক) 22 hours
খ
খ) 35 hours
গ
গ) 45 hours
ঘ
ঘ) 52 hours

ব্যাখ্যা

Suppose pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.
∴ 1/x + 2/x + 4/x = 1/5
⇒ 7/x = 1/5
⇒ x = 35 hours

১৩,৯৬৬.

In a 567 liters mixture of milk and water, the ratio of milk to water 7 : 2. To get a new mixture containing milk and water in the ratio 7: 3, the amount of water to be added is-

ক
81 liters
খ
63 liters
গ
49 liters
ঘ
27 liters
ঙ
21 liters

ব্যাখ্যা

Question: In a 567 liters mixture of milk and water, the ratio of milk to water 7 : 2. To get a new mixture containing milk and water in the ratio 7: 3, the amount of water to be added is-

Solution:
Quantity of milk in 567 liter of mixture = (7 × 567/9) litres
= 441 litres

Quantity of water = (567 - 441) litres
= 126 litres

Let,
x liter of water be added to become ratio = 7 : 3

According to the question,
7/3 = 441/(126 + x)
⇒ 7(126 + x) = 441 × 3
⇒ 882 + 7x = 441 × 3
⇒ 7x = 1323 - 882
⇒ 7x = 441
⇒ x = 441/7
∴ x = 63

Therefore, 63 liters of water is to be added.

১৩,৯৬৭.

The present age of john and marry are in the ratio of 6 : 4. Five years ago their ages were ratio of 5 : 3. How old is john now?

ক
ক) 42
খ
খ) 36
গ
গ) 30
ঘ
ঘ) 24

ব্যাখ্যা

Let, John's present age = 6x,
and, Mary’s present age = 4x
Therefore,
(6x - 5)/(4x - 5) = 5/3
Or, 20x - 25 = 18x - 15
Or, x = 5
∴ John’s age = 6 × 5 = 30 years

১৩,৯৬৮.

The mean of 8 numbers is 22. Five of them are 20, 18, 24, 16, and 26. Find the mean of the remaining three numbers.

ক
20
খ
22
গ
23
ঘ
24

ব্যাখ্যা

Question: The mean of 8 numbers is 22. Five of them are 20, 18, 24, 16, and 26. Find the mean of the remaining three numbers.

Solution:
Mean of 8 numbers = 22
Total sum of 8 numbers,
= 8 × 22
= 176

Sum of the given 5 numbers,
= 20 + 18 + 24 + 16 + 26
= 104

Sum of the remaining 3 numbers:
= 176 - 104
= 72

∴ Mean of the remaining 3 numbers,
= 72/3
= 24

১৩,৯৬৯.

Three-fifth of one-third of four-ninth of a number is 16. What is one-fourth of that number?

ক
72
খ
45
গ
86
ঘ
55

ব্যাখ্যা

Question: Three-fifth of one-third of four-ninth of a number is 16. What is one-fourth of that number?

Solution:
Let
the number is A

ATQ,
A × (4/9) × (1/3) × (3/5) = 16
⇒ 12A/135 = 16
⇒ 12A = 16 × 135
⇒ A = (16 × 135)/12
∴ A = 180

∴ One-fourth of that number is = 180/4 = 45

১৩,৯৭০.

If A is a zero matrix, then A + B = ?

ক
Zero matrix
খ
Matrix A
গ
Matrix B
ঘ
Identity matrix

ব্যাখ্যা

Question: If A is a zero matrix, then A + B = ?

Solution:
যদি A একটি শূন্য ম্যাট্রিক্স (zero matrix) হয়, তবে এর সব উপাদান শূন্য।
ম্যাট্রিক্স যোগ করার সময় প্রতিটি অবস্থানের উপাদানগুলি যথাক্রমে যোগ করা হয়।

তাই A + B মানে প্রতিটি অবস্থানে A-এর উপাদান এবং B-এর উপাদান যোগ করা।
যেহেতু A-এর সব উপাদান শূন্য, প্রতিটি অবস্থানে যোগফল শুধু B-এর উপাদানই থাকবে। তাই A + B = B.

এটি একটি মৌলিক বৈশিষ্ট্য যা শূন্য ম্যাট্রিক্সের সাথে যে কোনো ম্যাট্রিক্স যোগ করলে মূল ম্যাট্রিক্স অপরিবর্তিত থাকে।
সুতরাং সঠিক উত্তর হলো Matrix B।

- উত্তর: খ) Matrix B

১৩,৯৭১.

The twice of sum of the ages of a father and his son is 8 times the age of the son. If the average age of the father and the son is 30 year, what is father's age?

ক
ক) 42 years
খ
খ) 45 years
গ
গ) 36 years
ঘ
ঘ) 38 years

ব্যাখ্যা

Let father's age be x year and son's age be y year.
According to question,
2(x+y) = 8y _______(I)
and (x+y)/2 = 30
=> x+y = 60 year_______(II)
From equation (I) and (II)
8y = 120
y = 15 year,
Hence x = 45 year.

১৩,৯৭২.

If the population of a village went from 2105 to 1705, then the percent decrease in population is closest to

ক
ক) 19
খ
খ) 21
গ
গ) 22
ঘ
ঘ) 23

ব্যাখ্যা

Question: If the population of a village went from 2105 to 1705, then the percent decrease in population is closest to-

Solution:
শতকরা জনসংখ্যা হ্রাস পাবে = (২১০৫ - ১৭০৫) × ১০০%/২১০৫
= ৪০০ × ১০০%/২১০৫
= ১৯%

১৩,৯৭৩.

1/log₂24 + 1/log₃24 + 1/log₄24 = ?

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 24

ব্যাখ্যা

প্রশ্ন: 1/log₂24 + 1/log₃24 + 1/log₄24 = ?

সমাধান:
আমরা জানি,
1/log_ba= log_ab

প্রদত্তরাশি,
1/log₂24 + 1/log₃24 + 1/log₄24
= log₂₄2 + log₂₄3 + log₂₄4
= log₂₄(2 × 3 × 4)
= log₂₄24
= 1

১৩,৯৭৪.

Ravi asks Akash, what's your age and Akash replied 20 years hence my age will be 10 times of my age 10 years back. What is Akash’s present age?

ক
ক) 13 years
খ
খ) 11.3 years
গ
গ) 13.3 years
ঘ
ঘ) 14.3 years

ব্যাখ্যা

Let
Akash’s present age be x years
Akash’s age before 10 years = (x – 10)
Akash’s age after 20 years = (x + 20)
According to question
(x + 20) = 10(x – 10)
⇒ x + 20 = 10x – 100
⇒ 9x = 120
⇒ x = 13.3 years

∴ Present age of Akash is 13.3 years.

১৩,৯৭৫.

How many hours will it take a man to travel M miles at the rate of H miles per hour?

ক
ক) HM
খ
খ) H/M
গ
গ) M/H
ঘ
ঘ) H/2M

ব্যাখ্যা

question: How many hours will it take a man to travel M miles at the rate of H miles per hour?

solution:
given,
Distance = M miles
Speed = H miles per hour
Time = ?

we know,
Distance = Speed × Time
Time = Distance/Speed
= M/H

১৩,৯৭৬.

Of two numbers, 4 times the smaller one is less than 3 times the larger one by 5. If the sum of the number is larger than 6 times their difference by 6, find the two numbers.

ক
ক) 54, 41
খ
খ) 58, 43
গ
গ) 59, 41
ঘ
ঘ) 59, 43

ব্যাখ্যা

Let the number be x and y, such that x > y.
then, 3x - 4y = 5 ..........(i)
And (x + y) - 6 (x - y) = 6 ⇔ -5x + 7y = 6 ........(ii)
Solving (i) and (ii), we get : x = 59 and y = 43
Answer : 59, 43.

১৩,৯৭৭.

Pavel travels 96 km at a speed of 16 km/hr using a bike, 124 km at 31 km/h by car and another 105 km at 7 km/h in horse cart. Find his average speed for the entire distance travelled.

ক
20 km/h
খ
12 km/h
গ
13 km/h
ঘ
16 km/h

ব্যাখ্যা

Question: Pavel travels 96 km at a speed of 16 km/hr using a bike, 124 km at 31 km/h by car and another 105 km at 7 km/h in horse cart. Find his average speed for the entire distance travelled.

Solution:
We know,
Average speed = Total distance/Total time

Total distance = 96 + 124 + 105 = 325 km

Now,
Bike: 96 km at 16 km/h ∴ Time = 96/16 = 6 hours
Car: 124 km at 31 km/h ∴ Time = 124/31 = 4 hours
Horse cart: 105 km at 7 km/h ∴ Time = 105/7 = 15 hours

∴ Total time = 6 + 4 + 15 = 25 hours

∴ Average speed = Total distance/Total time
= 325/25
= 13 km/h

So Pavel's average speed for the entire journey is 13 km/h.

১৩,৯৭৮.

A wire can be bent in the form of a circle of radius 42 cm. If it is bent in the form of a square, then its area will be -

ক
ক) 3660 cm²
খ
খ) 4356 cm²
গ
গ) 5236 cm²
ঘ
ঘ) 5660 cm²

ব্যাখ্যা

Question: A wire can be bent in the form of a circle of radius 42 cm. If it is bent in the form of a square, then its area will be -

Solution:
দেয়া আছে,
বৃত্তের ব্যাসার্ধ r = 42 cm

বৃত্তের পরিধি = 2πr
= 2 × (22/7) × 42
= 2 × 22 × 6
= 264 cm

বর্গের এক বাহুর দৈর্ঘ্য = 264/4 = 66 cm

বর্গের ক্ষেত্রফল = (66)² cm² = 4356 cm²

১৩,৯৭৯.

A reduction of 10% in the price of tea enables a dealer to purchase 25 kg more tea for Tk 22500. What is the reduced price per kg of tea?

ক
81 tk
খ
90 Tk
গ
93 Tk
ঘ
97 Tk
ঙ
None

ব্যাখ্যা

Question: A reduction of 10% in the price of tea enables a dealer to purchase 25 kg more tea for Tk 22500. What is the reduced price per kg of tea?

Solution:
Let 10% of 22500 = 2250
Now,
25 kg = 2250
⇒ 1kg = 2250/25
∴ 1kg = 90

১৩,৯৮০.

A parallelogram has a base of 30m and height is 10m long. Then its area is-

ক
ক) 220 m²
খ
খ) 250 m²
গ
গ) 300 m²
ঘ
ঘ) 350 m²

ব্যাখ্যা

Question: A parallelogram has a base of 30m and height is 10m long. Then its area is-

Solution:
area = base × height
= 30 × 10
= 300 m²

১৩,৯৮১.

It was Friday on January 1, 2016. What was the day of the week on Jan 1, 2017?

ক
Saturday
খ
Friday
গ
Tuesday
ঘ
Sunday

ব্যাখ্যা

Question: It was Friday on January 1, 2016. What was the day of the week on Jan 1, 2017?

Solution:
• অধিবর্ষ নির্ণয়:
2016 সালটি 4 দ্বারা বিভাজ্য (2016 ÷ 4 = 504), তাই এটি একটি অধিবর্ষ (Leap Year)।

• অতিরিক্ত দিন নির্ণয়:
একটি অধিবর্ষে 366 দিন থাকে। 366 ÷ 7 = 52 সপ্তাহ এবং 2 দিন অতিরিক্ত (Odd Days) হয়।

এখন,
1লা জানুয়ারি 2016 দিনটি ছিল Friday।
যেহেতু 2016 সালটি অধিবর্ষ, তাই 1লা জানুয়ারি 2017 দিনটি হবে Friday এর থেকে 2 দিন বেশি।
⇒ Friday + 1 দিন = Saturday
⇒ Saturday + 1 দিন = Sunday

∴ 1লা জানুয়ারি 2017 দিনটি হবে Sunday।

১৩,৯৮২.

A dog takes 3 leaps for every 5 leaps of a hare. If one leap of a dog is equal to 3 leap of the hare then find the ratio of the speed of the dog to that of the hare is :

ক
ক) 8 : 5
খ
খ) 9 : 7
গ
গ) 9 : 5
ঘ
ঘ) 8 : 7

ব্যাখ্যা

Given that,
A dog takes 3 leaps for every 5 leaps of a hare
Therefore,
Dog : Hare = 3 : 5
One leap of a dog is equal to 3 leap of the hare
Therefore,
1 leap of dog = 3 leap of hare
Now the ratio becomes,
Dog : Hare = 3 (3) : 5
Dog : Hare = 9 : 5
Thus the ratio of the speed of the dog to that of the hare is 9 : 5

১৩,৯৮৩.

If 3 sides of a triangle are 6 cm, 8 cm, and 10 cm, then the altitude of the triangle, using the largest side as its base, will be -

ক
4.8 cm
খ
4.4 cm
গ
6 cm
ঘ
8 cm

ব্যাখ্যা

Question: If 3 sides of a triangle are 6 cm, 8 cm, and 10 cm, then the altitude of the triangle, using the largest side as its base, will be -

Solution:
Semi perimeter of the triangle is, S = (6 + 8 + 10)/2
= 12 cm

Area of the triangle is = √{s(s - a)(s - b)(s - c)}
= √{12(12 - 6) (12 - 8) (12 - 10)
= √(12 × 6 × 4 × 2)
= √576
= 24 sq. cm

Area of the triangle = (1/2) × base x height
⇒ 24 = (1/2) × b × h
⇒ b × h = 48
⇒ h = 48/b
⇒ h = 48/10
∴ h = 4.8 cm

১৩,৯৮৪.

A outlet pipe can empty a cistern in 8 hours. In what time will it empty 1/2 part of the cistern?

ক
ক) 2 hours
খ
খ) 3 hours
গ
গ) 4 hours
ঘ
ঘ) 5 hours

ব্যাখ্যা

Question: A outlet pipe can empty a cistern in 8 hours. In what time will it empty 1/2 part of the cistern?

Solution:
The outlet pipe empties one complete cistern in 8 hours.
Time taken to empty (1/2) × 8 = 4 hours

১৩,৯৮৫.

The length of a room is 5.5 m and the width is 3.75 m. Find the cost of paving the floor with slabs at the rate of Tk. 800 per square metre.

ক
11500 Tk
খ
16200 Tk
গ
12500 Tk
ঘ
16500 Tk

ব্যাখ্যা

Question: The length of a room is 5.5 m and the width is 3.75 m. Find the cost of paving the floor with slabs at the rate of Tk. 800 per square metre.
(একটি ঘরের দৈর্ঘ্য ৫.৫ মিটার এবং প্রস্থ ৩.৭৫ মিটার। প্রতি বর্গমিটার ৮০০ টাকা হারে মেঝেতে স্ল্যাব বসানোর খরচ কত হবে?)

Solution:
ফ্লোরের ক্ষেত্রফল
= (5.5 × 3.75) m²
= 20.625 m²

∴ মেঝে বাঁধানোর খরচ
= (800 × 20.625) Tk
=16500 Tk

১৩,৯৮৬.

If A and B are in the ratio 3:4 and B and C are in the ratio 12:13, then A and C will be in the ratio.

ক
3:13
খ
9:13
গ
13:9
ঘ
4:13

ব্যাখ্যা

Question: If A and B are in the ratio 3:4 and B and C are in the ratio 12:13, then A and C will be in the ratio.

Solution:
A and B are in the ratio 3:4 and B and C are in the ratio 12:13.

Another way,
Given, A and B are in the ratio = 3:4 -------(1)
B and C are in the ratio = 12:13. --------(2)

To combine, make B's value the same (LCM of 4 and 12 is 12).

Thus, multiplying (1) by 3, we get,
A:B = 9:12
and B:C = 12:13
Equating two ratios, we can achieve
→ A:C = 9:13

১৩,৯৮৭.

If (7^a)(7^b) = 7^c/7^d, what is d in terms of a b and c?

ক
ক) c/ab
খ
খ) c - a - b
গ
গ) a + b - c
ঘ
ঘ) c - ab

ব্যাখ্যা

(7^a)(7^b) = 7^c/7^d
Or, 7^{a + b} = 7^c-d
Or, a + b = c - d
Or, d = c - a - b

১৩,৯৮৮.

A group of men decided to do a job in 5 days. But since 10 men dropped out every day, the job completed at the end of the 7th day. How many men were there at the beginning?

ক
145
খ
115
গ
105
ঘ
90

ব্যাখ্যা

Question: A group of men decided to do a job in 5 days. But since 10 men dropped out every day, the job completed at the end of the 7th day. How many men were there at the beginning?

Solution:
Let X be the initial number of men then,
According to the question,
5X = X + (X - 10) + (X - 20) + (X - 30) + (X - 40) + (X - 50) + (X - 60)
or, 5X = 7X - 210
or, 2X = 210
or, X = 210/2
or, X = 105

১৩,৯৮৯.

(256)^0.16 × (256)^0.09 = ?

ক
0.2
খ
4
গ
1.6
ঘ
16

ব্যাখ্যা

Question: (256)^0.16 × (256)^0.09 = ?

Solution:
(256)^0.16 × (256)^0.09
= 256^{(0.16 + 0.09)}
= 256^0.25
= 256^(25/100)
= 256^1/4
= (4⁴)^1/4
= (4)^4/4
= 4¹
= 4

১৩,৯৯০.

Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

ক
9 minutes
খ
11 minutes
গ
12 minutes
ঘ
15 minutes

ব্যাখ্যা

Question: Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

Solution:
Given that,
Pipe A fills the tank in 24 minutes.
Pipe B fills the tank in 36 minutes.
Total time to fill the tank = 18 minutes.
Now,
LCM of 24 and 36 = 72 (Total capacity of the tank).
Efficiency of pipe A = 72/24 = 3 units/minute.
Efficiency of pipe B = 72/36 = 2 units/minute.

Let,
pipe B be turned off after x minutes.
Pipe A works for 18 minutes.
Pipe B works for x minutes.
Work done by A in 18 minutes = 3 × 18 = 54 units.
Work done by B in x minutes = 2x = 2x units.

Total work done = 54 + 2x = 72
⇒ 2x = 72 - 54
⇒ 2x = 18
⇒ x = 18/2
∴ x = 9

∴ Pipe B should be turned off after 9 minutes.

১৩,৯৯১.

ক
11
খ
47
গ
63
ঘ
31

ব্যাখ্যা

Question:

Solution:

১৩,৯৯২.

In a certain code 'TABLE' is written as WDEOH. How would 'SKY' be written in that code?

ক
BNY
খ
NYB
গ
TBN
ঘ
VNB

ব্যাখ্যা

Question: In a certain code 'TABLE' is written as WDEOH. How would 'SKY' be written in that code?

Solution:

TABLE is written as WDEOH
Here each letter is replaced by the third letter after it.
এখানে প্রতিটি বর্ণ তার পরের তৃতীয় বর্ণ দ্বারা প্রতিস্থাপিত হচ্ছে।

∴ SKY is written as VNB.

১৩,৯৯৩.

A boy divided the numbers 7654, 8506 and 9997 by a certain largest number and he gets the same remainder in each case. What is the common remainder?

ক
ক) 156
খ
খ) 211
গ
গ) 231
ঘ
ঘ) 199

ব্যাখ্যা

If the remainder is same in each case and remainder is not given,
HCF of the differences of the numbers is the required largest number.

9997 - 7654 = 2343
9997 - 8506 = 1491
8506 - 7654 = 852
Hence, the greatest number which divides 7654, 8506 and 9997 and leaves same remainder
= HCF of 2343, 1491, 852
= 213
Now we need to find out the common remainder.
Take any of the given numbers from 7654, 8506 and 9997, say 7654
7654/213 = 35,
remainder = 199.

১৩,৯৯৪.

If a - b = 6 and a²+b² = 116, then what is the value of ab?

ক
ক) 80
খ
খ) 60
গ
গ) 40
ঘ
ঘ) 20

ব্যাখ্যা

(a - b)² = a² - 2ab + b²
⇒ 6² = (a² + b²) - 2ab
⇒ 6² = 116 - 2ab
⇒ 36 = 116 - 2ab
⇒ 2ab = 116 - 36
⇒ 2ab = 80
⇒ ab = 40.

১৩,৯৯৫.

36 is divided into two parts such that 5 times of first part added to 3 times the second part makes 142. What are both parts?

ক
ক) 16, 18
খ
খ) 15, 17
গ
গ) 17, 19
ঘ
ঘ) 20, 22

ব্যাখ্যা

Let, first number be x and second number be y
Here,
x + y = 36 .... (i)
and, 5x + 3y = 142 ...(ii)

by multiplying the first equation by 3
3x + 3y = 108
5x + 3y = 142

by subtraction the second equation from the first
(3–5)x + (3–3)y = 108–142
Or, -2x = -34
Or, x = 17

by substituting x in the first equation
17 + y = 36
Or, y = 19

১৩,৯৯৬.

If x + y + z = 0, then, x³ + y³ + z³ + 3xyz is equal to -

ক
ক) 0
খ
খ) 6xyz
গ
গ) 12xyz
ঘ
ঘ) xyz

ব্যাখ্যা

As we know,
a³ + b³ + c³ - 3abc = a² + b² + c² - ab - bc - ca(a + b + c)
when (a + b + c) = 0
Then a³ + b³ + c³ - 3abc = 0
When x + y + z = 0
⇒ x³ + y³ + z³ = 3xyz
⇒ x³ + y³ + z³ + 3xyz = 3xyz +3xyz
= 6xyz.

১৩,৯৯৭.

A fruit shop has 12 types of fruits. You don’t like Mango and Papaya. How many ways can you select 5 different fruits from the ones you like?

ক
252
খ
320
গ
424
ঘ
180

ব্যাখ্যা

Question: A fruit shop has 12 types of fruits. You don’t like Mango and Papaya. How many ways can you select 5 different fruits from the ones you like?

Solution:
Given that,
Total fruits = 12
Fruits you don’t like = 2
∴ Fruits you can choose = 12 - 2 = 10
Number of fruits to choose = 5

∴ Number of ways = ¹⁰C₅ = 10!/5!(10 - 5)!
= (10 × 9 × 8 × 7 × 6 × 5!)/(5! × 5!)
= (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2)
= 252

So, there are 252 ways to select 5 different fruits from the ones you like.

১৩,৯৯৮.

If sum of two consecutive numbers is multiplied by 5, then the result is 105. What are the numbers?

ক
15, 16
খ
20, 21
গ
8, 9
ঘ
10, 11

ব্যাখ্যা

Question: If sum of two consecutive numbers is multiplied by 5, then the result is 105. What are the numbers?

Solution:
ধরি
ধারাবাহিক দুটি সংখ্যা,
x এবং x + 1

প্রশ্নমতে,
⇒ {x + (x + 1)} × 5 = 105
⇒ 2x + 1 = 105/5
⇒ 2x + 1 = 21
⇒ 2x = 21 - 1
⇒ x = 20/2
∴ x = 10

∴ সংখ্যাগুলো, x = 10 এবং x + 1 = 11

১৩,৯৯৯.

In how many years will Tk. 20 amounts to Tk. 24.20 at 10% per annum compound interest?

ক
2.5 years
খ
3.5 years
গ
2 years
ঘ
1.5 years

ব্যাখ্যা

Question: In how many years will Tk. 20 amounts to Tk. 24.20 at 10% per annum compound interest?

Solution:
Principal (P) = Tk. 20
Amount (A) = Tk 24.20
Rate (r) = 10% per annum
t = time in years

A = P(1 + r/100)^t
24.20 = 20(1 + 10/100)^t
⇒ 24.20 = 20(1.1^)t
⇒ 24.20/20 = (1.1)^t
⇒ 1.21 = (1.1)^t
⇒ (1.1)² = (1.1)^t
∴ t = 2

১৪,০০০.

125 gallons of a mixture contains 20% water. What amount of additional water should be added such that water content be raised to 25%?

ক
ক) 15/2 gallons
খ
খ) 13/2gallons
গ
গ) 9/2 gallons
ঘ
ঘ) 8(1/3) gallons

ব্যাখ্যা

In the original 125 gallons of mixture, 20% is water.
Hence, no. of gallons of other materials in the mixture: 125 x 80% = 100 gallons
In the new mixture, water makes up 25%, thus 75% is other materials.
As no. of gallons of others is unchanged, 100 gallons = 75% in the new mixture volume.
The total volume of the new mixture is : 100 / 75% = 100/ 0.75 = 133.33 gallons.
∴ Required additional amount of water = 133.3 – 125 = 8.33 = 8(1/3) gallons