Algebra

PrepBank · পাতা ৪ / ১৪ · ৩০১–৪০০ / ১,৩৮০

৩০১.

If 9x² - px + 16 is a square number, then p/2 = ?

ক
9
খ
12
গ
16
ঘ
24

ব্যাখ্যা

Question: If 9x² - px + 16 is a square number, then p/2 = ?

Solution:
9x² - px + 16
= (3a)² - 2 · 3a · 4 + 4²
= (3a)² - 24a + 4²

Here, p= 24
∴ p/2 = 24/2 = 12

৩০২.

Which of the following is a common factor of both x³ + 7x²- x -7 and 2x⁴- x² - 1 polynomials?

ক
(x - 2)(x + 1)
খ
(x-1)(x - 2)
গ
(x+1)(x -1)
ঘ
(x + 1)(x + 2)
ঙ
(x - 2)(x - 3)

ব্যাখ্যা

ধরি, P(x) = x³ + 7x² -x -7
এবং Q(x) = 2x⁴ - x² - 1
এখন, P(-1) = (-1)³+ 7(-1)²-(-1) - 7 = 0
∴ (x + 1), P(x) এর উৎপদক।
Q(-1) = 2(-1)⁴-(-1)² - 1 = 0
∴ (x + 1), Q(x) এর উৎপদক।

আবার, P(1) = (1)³+7(1)²-1-7 = 0
∴ (x-1), P(x) এর উৎপদক।
Q(1) = 2(1)⁴-(1)²-1 = 0
∴ (x - 1), Q(x) এর উৎপদক।

∴ (x + 1) এবং (x - 1) উভয়েই x³ + 7x² - x - 7 এবং 2x4 - x² - 1 বহুপদীদ্বয়ের উৎপাদক।

৩০৩.

If x² + 1/x² = 34, then x +1/x =?

ক
3
খ
4
গ
5
ঘ
None of these

ব্যাখ্যা

Question: If x² + 1/x² = 34, then x +1/x =?

Solution:
x² + 1/x² = 34
⇒ (x + 1/x)² - 2.x.(1/x) = 34
⇒ (x + 1/x)² = 36
∴ (x +1/x) = ± 6

৩০৪.

ক
1
খ
1/2
গ
0
ঘ
∞

ব্যাখ্যা

Question:

Solution:

৩০৫.

If √a = √4 - √5 then the value of a² - 18a + 81 is?

ক
160
খ
120
গ
80
ঘ
81

ব্যাখ্যা

Question: If √a = √4 - √5 then the value of a² - 18a + 81 is?

Solution:
Given,
√a = √4 - √5
⇒ (√a)²= (√4 - √5)²
⇒ a = (√4)² - 2 . √4 .√5 + (√5)²
⇒ a = 4 - 2√20 + 5
⇒ a = 9 - 2√20
⇒ a - 9 = - 2√20
⇒ (a - 9)² = (- 2√20)²⇒ a² - 2 . a . 9 + 9² = 80
∴ a² - 18a + 81 = 80

৩০৬.

If 7x - 3y = 31 and 9x - 5y = 41, then (x, y) =?

ক
(- 7, - 3)
খ
(- 4, 3)
গ
(5, 1)
ঘ
(4, - 1)
ঙ
(- 5, 3)

ব্যাখ্যা

Question: If 7x - 3y = 31 and 9x - 5y = 41, then (x, y) =?

Solution:
7x - 3y = 31 ................(1)
9x - 5y = 41 ...............(2)

(1) × 5 - (2) × 3 ⇒
35x - 15y - 27x + 15y = 155 - 123
8x = 32
x = 32/8
x = 4

(1) ⇒
7x - 3y = 31
28 - 3y = 31
3y = - 3
y = - 1

(x, y) = (4, - 1)

৩০৭.

যদি a² - √11a + 1 = 0 হয়, তবে a² + a^{- 2} এর মান কত?

ক
3
খ
6
গ
7
ঘ
9
ঙ
0

ব্যাখ্যা

প্রশ্ন: যদি a² - √11a + 1 = 0 হয়, তবে a² + a^{- 2} এর মান কত?

সমাধান:
a² - √11a + 1= 0
⇒ a² + 1 = √11a
⇒ (a²/a) + (1/a) = √11a/a
⇒ a + (1/a) = √11

প্রদত্ত রাশি = a² + a^{- 2}
= a² + (1/a²)
= {a + (1/a)}²- 2 ⋅ a ⋅ (1/a)
= (√11)² - 2
= 11 - 2
= 9

৩০৮.

ক
ক) f (0) = ∝
খ
খ) f (1) = - 1
গ
গ) f (1) = 0
ঘ
ঘ) f (- 1) = - 1/2

ব্যাখ্যা

Solution:
f(- 1) = (- 1)² + {1/(- 1 - 1)} - 1
= 1 - 1/2 - 1
= - 1/2

৩০৯.

ক
1
খ
0
গ
1/2
ঘ
2/3

ব্যাখ্যা

Question:

Solution:

৩১০.

If 3x + (3/x) = 5, then x³ + (1/x³) = ?

ক
260/27
খ
85/27
গ
-(10/27)
ঘ
-(90/27)

ব্যাখ্যা

Question: If 3x + (3/x) = 5, then x³ + (1/x)³ = ?

Solution:
Given,
3x + (3/x) = 5
x + (1/x) = 5/3

∴ x³ + (1/x)³
{x + (1/x)}3 - 3.x.(1/x){x + (1/x)}
= (5/3)³ - 3 × (5/3)
= (125/27) - 5
= (125 - 135)/27
= - (10/27)

৩১১.

If then a/b = ?

ক
0.16
খ
0.018
গ
0.016
ঘ
0.044

ব্যাখ্যা

Question: If then a/b = ?

Solution:
√(0.04 × 0.4 × a) = 0.4 × 0.04 × √b
Or, {√(0.04 × 0.4 × a)}² = (0.4 × 0.04 × √b)²
Or, 0.04 × 0.4 × a = (0.4 × 0.04)² × b
Or, a = {(0.4 × 0.04)² × b}/(0.4 × 0.04)
Or, a = (0.4 × 0.04) × b
Or, a/b = 0.4 × 0.04
∴ a/b = 0.016

৩১২.

In a school, 30 students study Mathematics, 20 students study Science, and 12 students study both subjects. If 8 students study neither Mathematics nor Science, find the total number of students in the school.

ক
38
খ
46
গ
50
ঘ
62
ঙ
None of these

ব্যাখ্যা

Question: In a school, 30 students study Mathematics, 20 students study Science, and 12 students study both subjects. If 8 students study neither Mathematics nor Science, find the total number of students in the school.

Solution:
Given that,
Students studying Mathematics, ∣M∣ = 30
Students studying Science, ∣S∣ = 20
Students studying both, ∣M ∩ S∣ = 12
Students studying neither = 8

We know,
∣M ∪ S∣ = ∣M∣ + ∣S∣ - ∣M ∩ S∣
= 30 + 20 - 12
= 38

∴ Total students in the class = students who study Mathematics or Science + students who play neither
= ∣M ∪ S∣ + Neither
= 38 + 8
= 46

৩১৩.

Find the diagonal and trace of the matrix

ক
1, 3, 6 and trace 10
খ
1, 2, 4 and trace 7
গ
1, - 5, 9 and trace 5
ঘ
4, - 5, 6 and trace 5

ব্যাখ্যা

Question: Find the diagonal and trace of the matrix

Solution: The diagonal of a matrix consists of the elements from the upper left corner of the matrix to the lower right corner.

Or in other words, if a matrix is A, then the diagonal elements are a₁₁, a₂₂, and a₃₃.
Thus, the diagonal of A consists of the numbers 1, -5, and 9.
The trace of a matrix is the sum of the diagonal elements.

Thus,
Trace, tr = 1-5+9 = 5.

৩১৪.

ক
5
খ
7
গ
123
ঘ
125

ব্যাখ্যা

Question:

Solution:

৩১৫.

If for non zero x, x² - 4x - 1 = 0, then the value of x² + (1/x²) is?

ক
ক) 14
খ
খ) 16
গ
গ) 18
ঘ
ঘ) 20

ব্যাখ্যা

Question: If for non zero x, x² - 4x - 1 = 0, then the value of x² + (1/x²) is?

Solution:
x² − 4x − 1 = 0
⇒ x² − 1 = 4x
⇒ x − (1/x) = 4 (divide x both sides)
⇒ {x − (1/x)}² = 4²
⇒ x² + (1/x²) − 2.x.(1/x) = 16
∴ x² + (1/x²) = 18

৩১৬.

If a = b = 2c and abc = 256, then a = ?

ক
2
খ
4
গ
6
ঘ
8

ব্যাখ্যা

Question: If a = b = 2c and abc = 256, then a = ?

Solution:
abc = 256
⇒ (2c) (2c) c = 256
⇒ 4c³ = 256
⇒ c³ = 64
⇒ c = 4

∴ a = 2c = (2 × 4) = 8

৩১৭.

ক
ক) - 13
খ
খ) - 7
গ
গ) 5
ঘ
ঘ) 4

ব্যাখ্যা

Question:

Solution:

৩১৮.

How many line segments can be drawn inside a pentagon that connects non-adjacent vertices?

ক
9
খ
2
গ
3
ঘ
5

ব্যাখ্যা

Question: How many line segments can be drawn inside a pentagon that connects non-adjacent vertices?

Solution:
A pentagon has 5 sides. We obtain the diagonals by joining the vertices in pairs.
Total number of sides and diagonals,
= ⁵C₂
= 10
This includes its 5 sides also.

∴ Diagonals = 10 – 5 = 5

৩১৯.

Find the median of the following numbers: 11, 25, 15, 21, 12, 17, 18, 22, 27, 29, 16, 20.

ক
17
খ
19
গ
20
ঘ
22

ব্যাখ্যা

Question: Find the median of the following numbers: 11, 25, 15, 21, 12, 17, 18, 22, 27, 29, 16, 20.

Solution:
The numbers if arranged in ascending order will be: 11, 12, 15, 16, 17, 18, 20, 21, 22, 25, 27, 29
Here the number of data is even, i.e. n = 12

We know,
Median = {Value of sum of (12/2)^th and (12/2 + 1)^th terms}/2
= (value of sum of 6^th and 7^th terms)/2
= (18 + 20)/2
= 38/2
= 19

So, the median is 19.

৩২০.

The sum of first 12 terms of the series 2, 5, 8, 11, ... ... ...

ক
196
খ
206
গ
222
ঘ
225

ব্যাখ্যা

The sum of first 12 terms
= 12/2{2 × 2 + (12 - 1)3} [ 1st term, a = 2 and common difference, d = 5 - 2 = 3 ]
= 6(4 + 33)
= 6 × 37
= 222

৩২১.

If x + 7 > 2 and x - 3 < 5 the value of x must be between which of the following pairs of numbers?

ক
- 5 and 8
খ
- 2 and 8
গ
3 and 10
ঘ
- 3 and 4
ঙ
3 and 7

ব্যাখ্যা

Question: If x + 7 > 2 and x - 3 < 5 the value of x must be between which of the following pairs of numbers?

Solution:
x + 7 > 2
x > - 5

Next we simplify
x - 3 < 5
x < 8

We know that x is greater than - 5 and less than 8.

৩২২.

If x = √10 + 3 then find the value of x - 1/x

ক
ক) 2√10
খ
খ) 6
গ
গ) 2
ঘ
ঘ) 12

ব্যাখ্যা

Question: If x = √10 + 3 then find the value of x - 1/x

Solution:
Given that
x = √10 + 3
1/x = 1/(√10 + 3)
=(√10 - 3) /(√10 + 3)(√10 - 3)
= (√10 - 3)/{(√10)² - (3)²}
= (√10 - 3)/(10 - 9)
= (√10 - 3)

x - 1/x = √10 + 3 - (√10 - 3)
= √10 + 3 - √10 + 3
= 6

৩২৩.

A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, then the number of hens will be:

ক
21
খ
23
গ
27
ঘ
25
ঙ
26

ব্যাখ্যা

Let the number of hens be x and the number of cows be y.
Then, x + y = 48 .... (i)
and 2x + 4y = 140
x + 2y = 70 .... (ii)
Solving (i) and (ii) we get:
x = 26, y = 22.
The required answer = 26.

৩২৪.

What is the sum of the squares of the digits from 1 to 10?

ক
ক) 365
খ
খ) 375
গ
গ) 385
ঘ
ঘ) 390

ব্যাখ্যা

As we know
1² + 2² + 3² + ............... + n² = n(n + 1)(2n + 1)/ 6
1² + 2² + 3² + ............... + 10² = 10(10 + 1)(2 × 10 + 1)/6
= (10 × 11 × 21)/6
= 385

৩২৫.

If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 term is-

ক
200
খ
3200
গ
1600
ঘ
2800

ব্যাখ্যা

Question: If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 term is-

Solution:

৩২৬.

If a + (1/a) = √5 what is the value of a³ + (1/a³) = ?

ক
2√5
খ
√3
গ
3√5
ঘ
8√5

ব্যাখ্যা

Question: If a + (1/a) = √5 what is the value of a³ + (1/a³) = ?

Solution:
Given,
a + (1/a) = √5

We know,
a³ + (1/a³) = {a + (1/a)}³ - 3 . a . (1/a) . {a + (1/a)}
= (√5)³ - 3 × √5
= 5√5 - 3√5
= 2√5

৩২৭.

State the order of the matrix is-

ক
2 × 3
খ
6
গ
3 × 2
ঘ
9

ব্যাখ্যা

Question: State the order of the matrix is-

Solution:
ম্যাট্রিক্সের মাত্রা বা ক্রম(Order of Matrix): একটি ম্যাট্রিক্সের সারি ও কলামের সংখ্যা যথাক্রমে m ও n হলে, ঐ ম্যাট্রিক্সকে m × n ক্রমের বা আকারের ম্যাট্রিক্স বলা হয়।
অর্থাৎ ম্যাট্রিক্সের আকার বা মাত্রা বোঝাতে প্রথমে সারি এবং পরে কলাম উল্লেখ করা হয়।
প্রদত্ত ম্যাট্রিক্সটি একটি আয়তাকার ম্যাট্রিক্স কারণ এর সারি ও কলাম অসমান।
এখানে,
সারি m = 2 এবং কলাম n = 3
∴ প্রদত্ত ম্যাট্রিক্সটি একটি 2 × 3 আকারের ম্যাট্রিক্স।

৩২৮.

What is the slope of a line perpendicular to the line whose equation is 8x + 3y = 14?

ক
14/3
খ
- 8/3
গ
3/14
ঘ
3/8

ব্যাখ্যা

Question: What is the slope of a line perpendicular to the line whose equation is 8x + 3y = 14?

Solution:
সরল রেখার সাধারণ সমীকরণ,
y = mx + c ......(1) (এখানেm = ঢাল)

যদি কোনো রেখার ঢাল হয় m, তবে তার লম্ব (perpendicular) রেখার ঢাল হবে,
m' = - (1/m)

এখন,
8x + 3y = 14
3y = - 8x + 14
y = - (8/3)x + 14/3
(1) নং এর সাথে তুলনা করে পাই,
m = - (8/3)

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - {1/- (8/3)} = 3/8

৩২৯.

If x is an odd integer, then which of the following is true?

ক
5x - 2 is even
খ
5x² + 2 is odd
গ
5x² + 3 is odd
ঘ
None of these

ব্যাখ্যা

x is odd
⇒ x² is odd [square of an odd number is always odd]
⇒ 5x² is odd
⇒ (5x² + 2) is odd. [The sum of an odd number and even number is always odd]

৩৩০.

What is to be added to the expression 2x/y, so that the sum is a perfect square?

ক
(x² - y²)/y²
খ
(x² + y²)/y²
গ
(x² + y²)/x²
ঘ
(y² - x²)/y²

ব্যাখ্যা

Question: What is to be added to the expression 2x/y, so that the sum is a perfect square?

Solution:
We know,
a² + 2ab + b² = (a + b)²
⇒ 2ab = a² + 2ab + b² - (a² + b²)

∴ 2x/y = 2 . x/y . 1
= (x/y)² + 2 . x/y . 1 + 1² - {(x/y)² + 1²}
= {(x/y) + 1}² - {(x²/y²) + 1}
= {(x + y)/y}² - (x² + y²)/y²

∴ (x² + y²)/y² is be added to the expression 2x/y, the sum is a perfect square.

৩৩১.

ক
20/27
খ
27/20
গ
6/8
ঘ
8/6

ব্যাখ্যা

Question:

Solution:

৩৩২.

The equation x² - kx + 36 = 0 has two equal roots, then the value of k is-

ক
± √6
খ
± 6
গ
± 12
ঘ
± 2√2

ব্যাখ্যা

Question: The equation x² - kx + 36 = 0 has two equal roots, then the value of k is-

Solution:
Here
a = 1, b = - k and c = 36
Since the equation has two equal roots
: b² - 4ac = 0
⇒ (- k)² - 4 × 1 × 36 = 0
⇒ k² = 144
⇒ k = ± √(144)
k = ± 12

৩৩৩.

m, n, o are natural numbers. If m> n> o then which of the following is not true?

ক
mno > 0
খ
mn - o > 0
গ
n - mo > 0
ঘ
None

ব্যাখ্যা

Question: m, n, o are natural numbers. If m> n> o then which of the following is not true?

Solution: let, m = 4, n = 3, o = 2
a) mno = 4 × 3 × 2 = 24>0
b) mn - o = 4 × 3 - 2 = 12 - 2 = 10 > 0
c) n - mo = 3 - 4 × 2 = 3 - 8 = -5 < 0

৩৩৪.

ক
47
খ
49
গ
51
ঘ
45

ব্যাখ্যা

Question:

Solution:
Given that,
x - 1/x = - √5
⇒ (x - 1/x)² = (- √5)²
⇒ x² + 1/x² - 2 . x . (1/x) = 5
⇒ x² + 1/x² = 5 + 2
⇒ (x² + 1/x²)² = 7
⇒ (x²)² + (1/x²)² + 2 . x². (1/x²) = 49
⇒ x⁴ + 1/x⁴ = 49 - 2
∴ x⁴ + 1/x⁴ = 47

৩৩৫.

If then what is the value of (3 - 2x) + (3 - 2x)²?

ক
3
খ
2
গ
1
ঘ
0

ব্যাখ্যা

Question: If then what is the value of (3 - 2x) + (3 - 2x)²?

Solution:
√(3 - 2x) = 1
⇒ {√(3 - 2x)}² = 1²
⇒ 3 - 2x = 1

∴ (3 - 2x) + (3 - 2x)² = 1 + 1² = 2

৩৩৬.

In a class of 50 students, 20 study History, 25 study Geography, and 10 study both subjects. How many students study neither subject?

ক
12 students
খ
25 students
গ
15 students
ঘ
10 students

ব্যাখ্যা

Question: In a class of 50 students, 20 study History, 25 study Geography, and 10 study both subjects. How many students study neither subject?

Solution:
Given that,
Total students = 50
Study History = 20
Study Geography = 25
Study both = 10

Students studying at least one subject = H + G - Both
= 20 + 25 - 10
= 35

Therefore, Students who study neither subject = Total students - at least one subject
= 50 - 35 = 15

∴ 15 students study neither subject.

৩৩৭.

A system of equations is shown below:
X + 1= 6
X - m=5
X + p = 4
X - q = 3
What is the value of I+m+p+q?

ক
ক) 3
খ
খ) 2
গ
গ) 6
ঘ
ঘ) 5

ব্যাখ্যা

We can multiply equation 2 by -1 ad equation 4 by -1, and we have:
x + l = 6
-x + m =- 5
x + p = 4
-x + q = -3

Adding these equations together we have:
l + m + p + q = 2

৩৩৮.

If A = {1, 2, 3} and B = Ø, what is the value of (A U B)?

ক
{1, 2, 3}
খ
{1, 2, 3, Ø}
গ
{2, 3, Ø}
ঘ
Ø

ব্যাখ্যা

Question: If A = {1, 2, 3} and B = Ø, what is the value of (A U B)?

Solution:
দেওয়া আছে,
A = {1, 2, 3}
এবং B = Ø
যেকোনো সেট A এবং ফাঁকা সেট (Ø) এর সংযোগ (union) হলো A

∴ (A ∪ B) = {1, 2, 3} ∪ Ø
= A
= {1, 2, 3}

৩৩৯.

If x² + y² = 13 and xy = 6, then (x + y)³ is-

ক
125
খ
512
গ
729
ঘ
216

ব্যাখ্যা

Question: If x² + y² = 13 and xy = 6, then (x + y)³ is-

Solution:
Given that,
x² + y² = 13 and xy = 6
⇒ (x + y)² = x² + y² + 2xy
⇒ (x + y)² = 13 + (2 × 6)
⇒ (x + y)² = 13 + 12
⇒ (x + y)² = 25 = 5²
∴ x + y = 5

Now,
(x + y)³ = 5³ = 125

৩৪০.

The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have-

ক
ক) Unique solution
খ
খ) Exactly two solutions
গ
গ) Infinitely many solutions
ঘ
ঘ) No solution
ঙ
ঙ) Exactly three solutions

ব্যাখ্যা

Given,
9x + 3y + 12 = 0 and 18x + 6y + 26 = 0
a1/a2 = 9/18 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = 12/26 = 6/13
Since, a1/a2 = b1/b2 ≠ c1/c2

So, the pairs of equations are parallel and the lines never intersect each other at any point, therefore there is no possible solution.

৩৪১.

If p + q = 12 and pq = 32, then the numerical value of (p/q) + (q/p) will be?

ক
ক) 2.50
খ
খ) 6.50
গ
গ) 4.25
ঘ
ঘ) None of these

ব্যাখ্যা

Question: If p + q = 12 and pq = 32, then the numerical value of (p/q) + (q/p) will be?

Solution:
Given that,
p + q = 12
and pq = 32

Now,
p + q = 12
(p + q)² = 12²
p² + 2pq + q² = 144
p² + q² + 2 × 32 = 144
p² + q² = 144 - 64
p² + q² = 80

Again,
(p/q) + (q/p)
= (p² + q²)/pq
= 80/32
= 2.50

৩৪২.

Find the arithmetic mean of the set of data: 6, 1, 5, 8, and 10

ক
4
খ
5
গ
6
ঘ
7

ব্যাখ্যা

Question: Find the arithmetic mean of the set of data: 6, 1, 5, 8, and 10.

Solution:
In the given question, total number = 5

Arithmetic mean = (6 + 1 + 5 + 8 + 10)/5
= 30/5
= 6

৩৪৩.

What should be the value of "Q" so that the expression (49 - 28x + Qx²) becomes a perfect square?

ক
2
খ
4
গ
8
ঘ
16

ব্যাখ্যা

Question: What should be the value of "Q" so that the expression (49 - 28x + Qx²) becomes a perfect square?

Solution:
(49 - 28x + Qx²)
= (7)² - 2 × 7 × 2x + (2x)² + Qx² - (2x)²
= (7 - 2x)² + Qx² - 4x²

The expression becomes a perfect square if,
Qx² - 4x² = 0
⇒ Qx² = 4x²
∴ Q = 4

Thus, when Q = 4, the expression is (7 - 2x)², which is a perfect square.

৩৪৪.

If 1/2 is a root of the quadratic equation x² - mx - 5/4 = 0, then value of m is:

ক
ক) 2
খ
খ) -2
গ
গ) -3
ঘ
ঘ) 3
ঙ
ঙ) 1

ব্যাখ্যা

Given,
x = 1/2 as the root of equation x² - mx - 5/4 = 0.
(1/2)² – m(1/2) – 5/4 = 0
1/4 - m/2 - 5/4 = 0
m = -2

৩৪৫.

If x + 1/x = m, the value of x³ + 1/x³ is-

ক
ক) 3m³ + 6m
খ
খ) m² - 2m
গ
গ) m³ + 3m
ঘ
ঘ) m³ - 3m

ব্যাখ্যা

Given that
x + 1/x = m

x³ + 1/x³ = (x + 1/x)³ - 3 x. (1/x)(x + 1/x)
= m³ - 3m

৩৪৬.

The remainder is 3, when a number is divided by 5. If the square of this number is divided by 5, then what is the remainder?

ক
5
খ
4
গ
7
ঘ
1

ব্যাখ্যা

We know that,
Dividend = [(Divisor × Quotient)] + Remainder
It is given that the remainder is 3 when a number (dividend) is divided 5(divisor).
Dividend and quotient are unknown,

Hence assume dividend as X and quotient as Y.
X = (5Y) + 3
The square of this number is divided by 5, therefore
X² = (5Y + 3)²
X² = (25Y²+ 30Y + 9)

On dividing this equation,
we get,
= 25Y² + 30Y + 9
= 25Y² + 30Y + 5 + 4
= 5(5Y²+ 6Y + 1) + 4

The remainder is 4, because 9 is not exactly divisible by 5 and we get 4 as remainder.

৩৪৭.

If x is a number such that x² - 3x + 2 = 0 and x² - x - 2 = 0, what is the value of x?

ক
- 1
খ
- 2
গ
1
ঘ
2

ব্যাখ্যা

Question: If x is a number such that x² - 3x + 2 = 0 and x² - x - 2 = 0, what is the value of x?

Solution:
x² - 3x + 2 = x² - x - 2
⇒ - 3x + 2 = - x - 2
⇒ 2x = 4
∴ x = 2

৩৪৮.

If A = {3, 4, 5} and B = {4, 5, 6, 8} determine (A ∩ B)

ক
{ }
খ
{3, 4}
গ
{4, 5}
ঘ
{3, 5}

ব্যাখ্যা

Question: If A = {3, 4, 5} and B = {4, 5, 6, 8} determine (A ∩ B)

Solution:
Given,
A = {3, 4, 5}
and B = {4, 5, 6, 8}

∴ A ∩ B ={3, 4, 5} ∩ {4, 5, 6, 8}
= {4, 5}

The set to be determined is: {4, 5}

৩৪৯.

What is the slope of the line perpendicular to the line by y = -5x + 9?

ক
ক) 5
খ
খ) -5
গ
গ) 1/5
ঘ
ঘ) -1/5

ব্যাখ্যা

y = -5x + 9
⇒ y + 5x = 9 .....(i)
সুতরাং (i) নং রেখাটির লম্বরেখার সমীকরণ 5y - x = k
⇒ y = 1/5x + k
∴ লম্ব রেখাটির ঢাল = 1/5

৩৫০.

If - 2x + 5 < 19, then the value of x:

ক
x > - 7
খ
x < 7
গ
x < - 5
ঘ
x > 9

ব্যাখ্যা

Question: If - 2x + 5 < 19, then the value of x:

Solution:
Here,
- 2x + 5 < 19
⇒ - 2x + 5 - 5 < 19 - 5
⇒ - 2x < 14
⇒ 2x > - 14
⇒ x > - 14/2
∴ x > - 7

৩৫১.

If A = {2, 4, 6, 8} and B = {1, 2, 3, 4}, what is A \ B?

ক
{2, 4}
খ
{1, 3}
গ
{6, 8}
ঘ
{1, 2, 3, 4}
ঙ
None of these

ব্যাখ্যা

Question: If A = {2, 4, 6, 8} and B = {1, 2, 3, 4}, what is A \ B?

Solution:
A \ B = {2, 4, 6, 8}\{1, 2, 3, 4}
= {6, 8}

৩৫২.

If (5 - 2x) ≤ 13, then which one is correct?

ক
x ≥ - 2
খ
x ≤ - 4
গ
x ≥ - 4
ঘ
x ≤ - 2

ব্যাখ্যা

Question: If (5 - 2x) ≤ 13, then which one is correct?

Solution:

Given,
⇒ 5 - 2x ≤ 13
⇒ 5 - 2x - 5 ≤ 13 - 5
⇒ - 2x ≤ 8
∴ x ≥ - 4

৩৫৩.

If x² - 3x + 1 = 0, and x > 1, then what is the value of x - 1/x?

ক
√7
খ
√6
গ
√5
ঘ
√11

ব্যাখ্যা

Question: If x² - 3x + 1 = 0, and x > 1, then what is the value of x - 1/x?

Solution:
Given, x² - 3x + 1 = 0
⇒ (x²/x) - (3x/x) + (1/x) = 0 [উভয়পক্ষকে x দ্বারা ভাগ করে]
⇒ x - 3 + 1/x = 0
⇒ x + 1/x = 3

এখন, (x - 1/x)² = (x + 1/x)² - 4 . x . 1/x
⇒ (x - 1/x)² = (3)² - 4
⇒ (x - 1/x)² = 9 - 4
⇒ (x - 1/x)² = 5
∴ x - 1/x = ±√5

যেহেতু x > 1 দেওয়া আছে, তাই x - 1/x > 0
সুতরাং, x - 1/x = √5

৩৫৪.

If n(A ∪ B) = 61, n(A) = 30, n(B) = 54 then what is the value of n(A ∩ B)?

ক
ক) 22
খ
খ) 24
গ
গ) 23
ঘ
ঘ) 25

ব্যাখ্যা

Question: If n(A ∪ B) = 61, n(A) = 30, n(B) = 54 then what is the value of n(A ∩ B)?

Solution:
দেওয়া আছে,
n(A ∪ B) = 61
n(A) = 30
n(B) = 54

আমরা জানি,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
⇒ 61 = 30 + 54 - n(A ∩ B)
⇒ n(A ∩ B) = 84 - 61
∴ n(A ∩ B) = 23

৩৫৫.

What is the sum of two consecutive even numbers the difference of whose squares is 84?

ক
ক) 48
খ
খ) 58
গ
গ) 42
ঘ
ঘ) 46

ব্যাখ্যা

Let the numbers be x and x + 2.
Then, (x + 2)² - x² = 84
⇒ 4x + 4 = 84
⇒ 4x = 80
⇒ x = 20.
∴ The required sum
= x + (x + 2)
= 2x + 2
= 42

৩৫৬.

How many prime numbers are there between 90 and 100?

ক
ক) 2
খ
খ) 1
গ
গ) 3
ঘ
ঘ) Nill

ব্যাখ্যা

Question: How many prime numbers are there between 90 and 100?

Solution:
৯০ ও ১০০ এর মধ্যে মৌলিক সংখ্যা ১ টি - ৯৭।

৩৫৭.

Solve the inequality: 3(2x - 5) + 1 > 4(x - 3)

ক
x < 1
খ
x > 3
গ
x > 2
ঘ
x > 1

ব্যাখ্যা

Question: Solve the inequality: 3(2x - 5) + 1 > 4(x - 3)

Solution:
Given inequality,
3(2x - 5) + 1 > 4(x - 3)
⇒ 6x - 15 + 1 > 4x - 12
⇒ 6x - 14 > 4x - 12
⇒ 6x - 4x > −12 + 14
⇒ 2x > 2
∴ x > 1

So, the solution of the inequality is x > 1.

৩৫৮.

What value of x satisfies to equation x - 2 = 2 - x?

ক
ক) 2
খ
খ) 1
গ
গ) 0
ঘ
ঘ) - 2

ব্যাখ্যা

Question: What value of x satisfies to equation x - 2= 2 - x?

Solution:
x - 2 = 2 - x
⇒ x + x = 2 + 2
⇒ 2x = 4
∴ x = 2

৩৫৯.

If 2x = 3y = 10, then 12xy =?

ক
1200
খ
200
গ
120
ঘ
40

ব্যাখ্যা

Question: If 2x = 3y = 10, then 12xy =?

Solution:
2x = 3y = 10

12xy
= 2x × 6y
= 10 × 2 × 3y
= 20 × 10
= 200

৩৬০.

If A = {1, 4, 9, 16, 25}, the number of proper subsets of A is

ক
15
খ
16
গ
31
ঘ
32

ব্যাখ্যা

Question: If A = {1, 4, 9, 16, 25}, the number of proper subsets of A is

Solution:
দেওয়া আছে,
A = {1, 4, 9, 16, 25}

সেটের উপাদান সংখ্যা = 5

∴ প্রকৃত উপসেট সংখ্যা = 2ⁿ - 1
= 2⁵ - 1
= 32 - 1
= 31

৩৬১.

If C = {a, b, 1, 2} and D = {3, 4, 6, 8} then C union D is-

ক
{a, b, 1, 2, 3, 4, 6, 8}
খ
{a, b}
গ
{1, 2, 3, 4, 6, 8}
ঘ
{a, 1, 2, 3, 4, 6}

ব্যাখ্যা

Question: If C = {a, b, 1, 2} and D = {3, 4, 6, 8} then C union D is-

Solution:
C union D = C ∪ D = {a, b, 1, 2} ∪ {3, 4, 6, 8}
= {a, b, 1, 2, 3, 4, 6, 8}

৩৬২.

If the sum of an infinite geometric series is 20 and the common ratio r = 1/2, what is the first term?

ক
19
খ
13
গ
15
ঘ
10

ব্যাখ্যা

Question: If the sum of an infinite geometric series is 20 and the common ratio r = 1/2, what is the first term?

Solution:
Here,
r = 1/2
a = ?

We know that,
S_∞ = a/(1 - r)
⇒ 20 = a/(1 - 1/2)
⇒ 20 = a/(1/2)
⇒ 20 = 2a
∴ a = 10

৩৬৩.

If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is-

ক
30
খ
31
গ
32
ঘ
22

ব্যাখ্যা

প্রশ্ন: If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is-

সমাধান:
এখানে A সেটের উপাদান সংখ্যা ৫টি
A সেটের উপসেট সংখ্যা হবে ২৫ টি = ৩২টি

∴ A সেটের প্রকৃত উপসেট এর সংখ্যা (৩২ - ১) টি = ৩১টি

৩৬৪.

If 2x + 3y = 11 and xy = 5, then find the value of 8x³ + 27y³ =?

ক
471
খ
321
গ
288
ঘ
341

ব্যাখ্যা

Question: If 2x + 3y = 11 and xy = 5, then find the value of 8x³ + 27y³ = ?

Solution:
Given that,
2x + 3y = 11 and xy = 5

Now,
8x³ + 27y³
= (2x)³ + (3y)³
= (2x + 3y)³ - 3 × 2x × 3y (2x + 3y) ;[a³ + b³ = (a + b)³ - 3ab(a + b)]
= (2x + 3y)³ - 18xy(2x + 3y)
= 11³ - 18 × 5 × 11
= 1331 - 990
= 341
∴ The required answer is 341.

৩৬৫.

If x = 3 + 2√2, find the value of .

ক
2
খ
4
গ
2√2
ঘ
√2
ঙ
3√2

ব্যাখ্যা

Question: If x = 3 + 2√2, find the value of .

Solution:
Given,

৩৬৬.

Solution set of the inequality: x - 5 > 4x + 7 is

ক
(- ∞, - 4)
খ
[- ∞, - 4)
গ
(- ∞, - 4]
ঘ
[- ∞, - 4]

ব্যাখ্যা

Question: Solution set of the inequality: x - 5 > 4x + 7 Is

Solution:
x - 5 > 4x + 7
⇒ - 5 > 4x - x + 7
⇒ - 5 > 3x + 7
⇒ - 5 - 7 > 3x
⇒ - 12 > 3x
⇒ - 12/3 > 3x/3
⇒ - 4 > x
⇒ x < - 4

∴ নির্ণেয় সমাধান সেট: (- ∞, - 4)

৩৬৭.

In a group of 60 people 27 people like Coca Cola and 42 people like Pepsi Cola. Each person likes at least one of the two drinks. How many of these people like both Coca Cola and Pepsi Cola?

ক
ক) 7
খ
খ) 9
গ
গ) 11
ঘ
ঘ) 13

ব্যাখ্যা

সবাই দুইটির যেকোনো একটি পছন্দ করে,
∴ T = n(c) + n(p) - n(c∩p)
⇒ 60 = 27 + 42 - n(c∩p)
⇒ n(c∩p) = 69 - 60
⇒ n(c∩p) = 9

৩৬৮.

0, 2, 6, 8, 16, 30, 54,?

ক
88
খ
95
গ
100
ঘ
122

ব্যাখ্যা

Question: 0, 2, 6, 8, 16, 30, 54,?

Solution:
0 + 2 + 6 = 8
2 + 6 + 8 = 16
6 + 8 + 16 = 30
8 + 16 + 30 = 54

16 + 30 + 54 = 100

৩৬৯.

ক
16
খ
22
গ
34
ঘ
42

ব্যাখ্যা

Question:

Solution:

৩৭০.

In a class of 50 students, 18 students like Biology, 20 students like Chemistry, and 22 students like Physics. It is found that 4 students like both Biology and Chemistry, 5 students like both Biology and Physics, and 6 students like both Chemistry and Physics. If 3 students like none of these subjects, find the number of students who like all three subjects.

ক
2
খ
3
গ
6
ঘ
7

ব্যাখ্যা

Question: In a class of 50 students, 18 students like Biology, 20 students like Chemistry, and 22 students like Physics. It is found that 4 students like both Biology and Chemistry, 5 students like both Biology and Physics, and 6 students like both Chemistry and Physics. If 3 students like none of these subjects, find the number of students who like all three subjects.

Solution:
Let the number of students who like all three subjects = x.
Students liking at least one subject = Total students - Students liking none
= 50 - 3
= 47

According to question,
⇒ Biology + Chemistry + Physics - (Bio & Chem + Bio & Phys + Chem & Phys) + x = 47
⇒ 18 + 20 + 22 - (4 + 5 + 6) + x = 47
⇒ 60 - 15 + x = 47
⇒ 45 + x = 47
∴ x = 2

৩৭১.

When a positive integer X is divided by Y, the quotient is 11 and the remainder is 5. When X is divided by (Y + 3) the quotient is 9 and the remainder is 2. What is the value of X?

ক
137
খ
151
গ
163
ঘ
172
ঙ
None

ব্যাখ্যা

Question: When a positive integer X is divided by Y, the quotient is 11 and the remainder is 5. When X is divided by (Y + 3) the quotient is 9 and the remainder is 2. What is the value of X?

Solution:
Given that,
When X is divided by Y. Then we get,
X = 11Y + 5 .......(1)

And,
When X is divided by Y + 3. Then we get,
X = 9(Y + 3) + 2 = 9Y + 27 + 2 = 9Y + 29.......(2)

From (1) and (2), Then we get,
⇒ 11Y + 5 = 9Y + 29
⇒ 11Y - 9Y = 29 - 5
⇒ 2Y = 24
⇒ Y = 24/2
∴ Y = 12

From (1),
X = 11Y + 5 = (11 × 12) + 5 = 132 + 5 = 137

So the value of X is 137.

৩৭২.

A geometric series has its first term as 1 divided by square root of 2, and its common ratio is √2. Which term in the sequence is 16√2?

ক
8^th
খ
11^th
গ
10^th
ঘ
None of these

ব্যাখ্যা

Question: A geometric series has its first term as 1 divided by square root of 2, and its common ratio is √2. Which term in the sequence is 16√2?

Solution:
First term, a = 1/√2
Common ratio, r = √2

Let, the n-th term be = ar^{n - 1} = 16√2
⇒ (1/√2) (√2)^{n - 1} = 16√2
⇒ (√2)^{n - 1} = 32
⇒ (√2)^{n - 1} = (√2)¹⁰
⇒ n - 1 = 10
∴ n = 11

So the 11^th term is 16√2.

৩৭৩.

If a² + b² + c² = 138 and (ab + bc + ca) = 131, Then (a + b + c) =?

ক
15
খ
20
গ
25
ঘ
28

ব্যাখ্যা

Question: If a² + b² + c² = 138 and (ab + bc + ca) = 131, Then (a + b + c) =?

Solution:
a² + b² + c² = 138
(ab + bc + ca) = 131

Now
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (a + b + c)² = 138 + 2 × 131
⇒ (a + b + c)² = 400
⇒ (a + b + c)² = 20²
∴ a + b + c = 20

৩৭৪.

If x/(2x + y + z) = y/(x + 2y + z) = z/(x + y + 2z) = a, then find a if x + y + z ≠ 0

ক
ক) 1/2
খ
খ) 1/3
গ
গ) 1/4
ঘ
ঘ) 1/6

ব্যাখ্যা

Given that
x/(2x + y + z) = y/(x + 2y + z) = z/(x + y + 2z) = a

x/(2x + y + z) =a
x = a(2x + y + z)

y/(x + 2y + z) = a
y = a(x + 2y + z)

z/(x + y + 2z) = a
z = a(x + y + 2z) =

x + y + z = a(2x + y + z + x + 2y + z + x + y + 2z)
(x + y + z) = a(4x + 4y + 4z)
(x + y + z) = 4a(x + y + z)
4a = 1
a = 1/4

৩৭৫.

10 - [4 - {3 - (3 - 3 - 6)}] is equal to:

ক
ক) 5
খ
খ) 10
গ
গ) 15
ঘ
ঘ) 20

ব্যাখ্যা

Question: 10 - [4 - {3 - (3 - 3 - 6)}] is equal to:

Solution:
Given,
10 - [4 - {3 - (3 - 3 - 6)}]
= 10 - [4 - {3 - (- 6)}]
= 10 - [4 - {3 + 6}]
= 10 - (4 - 9)
= 10 - (- 5)
= 10 + 5
= 15

৩৭৬.

If (3 + √p) > 2√p, which of these statements cannot be false?

ক
p > 17
খ
p > 15
গ
p > 19
ঘ
p < 9
ঙ
p > 4

ব্যাখ্যা

Question: If (3 + √p) > 2√p, which of these statements cannot be false?

Solution:
(3 + √p) > 2√p
⇒ 3 > 2√p - √p
⇒ 3 > √p
⇒ √p < 3
⇒ p < 3²
∴ p < 9

৩৭৭.

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
∴ 6 + N এর মান ঋনাত্মক।

৩৭৮.

and

, find the value of

ব্যাখ্যা

Question:

and

, find the value of

Now, the given expression is,

৩৭৯.

If x - 2 = √3 then what is the value of x⁴ + (1/x)⁴?

ক
204
খ
198
গ
194
ঘ
144

ব্যাখ্যা

Question: If x - 2 = √3 then what is the value of x⁴ + (1/x)⁴?

Solution:
Given,
x - 2 = √3
⇒ x = 2 + √3

∴ (1/x) = 2 - √3

∴ x + (1/x) = 2 + √3 + 2 - √3 = 4

∴ x² + (1/x²) = {x + (1/x)}² - 2 . x . (1/x)
= (4)² - 2
= 16 - 2
= 14

∴ x⁴ + (1/x)⁴ = {x² + (1/x²)}² - 2. x². (1/x²)
= (14)² - 2
= 196 - 2
= 194

৩৮০.

ক
- 2
খ
0
গ
2
ঘ
4

ব্যাখ্যা

Question:

Solution:

৩৮১.

If x²b⁴ = ab^{- 1}, what is a in terms of b and x ?

ক
x²b³
খ
x²b^{- 3}
গ
x²b⁵
ঘ
x²b^{- 5}
ঙ
x²b⁶

ব্যাখ্যা

Question: If x²b⁴ = ab^{- 1}, what is a in terms of b and x ?

Solution:
x²b⁴ = ab^{- 1}
⇒ a/b = x²b⁴
⇒ a = x²b⁴.b
⇒ a = x²b^{4 + 1}
⇒ a = x²b⁵

৩৮২.

Solve the inequality 2 ≤ - 4 - 3x < 17

ক
- 7 > x ≥ - 2
খ
- 7 < x ≤ 2
গ
7 < x ≤ - 2
ঘ
- 7 < x ≤ - 2

ব্যাখ্যা

Question: Solve the inequality 2 ≤ - 4 - 3x < 17

Solution:
2 ≤ - 4 - 3x < 17
⇒ 2 + 4 ≤ - 4 - 3x + 4 < 17 + 4
⇒ 6 ≤ - 3x < 21
⇒ - 6 ≥ 3x > - 21
⇒ - 6/3 ≥ 3x/3 > - 21/3
⇒ - 2 ≥ x > - 7
∴ - 7 < x ≤ - 2

৩৮৩.

The value of (x - y)³ + (x + y)³ + 6x(x² - y²)

ক
ক) 4x³
খ
খ) 8x³
গ
গ) 6x³
ঘ
ঘ) 9x³

ব্যাখ্যা

Question: The value of (x - y)³ + (x + y)³ + 6x(x² - y²)

Solution:
(x - y)³ + (x + y)³ + 6x(x² - y²)
= (x - y)³ + (x + y)³ + 3.2x(x - y)(x + y)
Let
a = x - y
b = x + y
a + b = x - y + x + y = 2x

Given expression
= (x - y)³ + (x + y)³ + 3.2x(x - y)(x + y)
= a³ +b³ + 3(a + b)ab
= a³ + b³ + 3ab(a + b)
= (a + b)³
= (2x)³
= 8x³

৩৮৪.

Observe the following diagram and answer the question. Find the number of students who play any two of the three sports.

ক
9
খ
5
গ
11
ঘ
13
ঙ
None of the above

ব্যাখ্যা

Question: Observe the following diagram and answer the question. Find the number of students who play any two of the three sports.

Solution:
The shaded part represents the students who play any two of the three sports which is shown below:

Hence, the students who play any two of the three sports are 6 + 5 = 11.

৩৮৫.

Which of the following inequalities is an algebraic expression for the shaded part of the number line above?

ক
|x| ≤ 5
খ
|x - 2| ≤ 3
গ
|x - 1| ≤ 4
ঘ
|x + 1| ≤ 4
ঙ
None of these

ব্যাখ্যা

Question:

Which of the following inequalities is an algebraic expression for the shaded part of the number line above?

Solution:
From the number line it follows that - 5 ≤x ≤ 3
(A) |x| ≤ 5 ⇒ - 5 ≤ x ≤ 5. Discard.

(B) |x - 2| ≤ 3 ⇒ - 3 ≤ x - 2 ≤ 3 ⇒ add 2 to all parts: - 1 ≤ x ≤ 5. Discard.

(C) |x - 1| ≤ 4 ⇒ - 4 ≤ x - 1≤ 4 ⇒ add 1 to all parts: - 3 ≤ x ≤ 5. Discard.

(D) |x +1| ≤ 4 ⇒ - 4 ≤ x + 1 ≤ 4 ⇒ subtract 1 from all parts: - 5 ≤ x ≤ 3. OK.

৩৮৬.

If a + b + c = 5 and a² + b² + c² = 35, find the value of a³ + b³ + c³ - 3abc.

ক
255
খ
200
গ
352
ঘ
220

ব্যাখ্যা

Question: If a + b + c = 5 and a² + b² + c² = 35, find the value of a³ + b³ + c³ - 3abc.

Solution:
Given, a + b + c = 5 and a² + b² + c² = 35

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (5)² = 35 + 2(ab + bc + ca)
⇒ 25 = 35 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 25 - 35
⇒ 2(ab + bc + ca) = - 10
∴ ab + bc + ca = - 5

Now,
a³ + b³ + c³ - 3abc
= (a + b + c){a² + b² + c² - (ab + bc + ca)}
= 5 × {35 - (- 5)}
= 5 × 40
= 200

Therefore, the value is 200.

৩৮৭.

If x + (1/x) = 3, then x - (1/x) = ?

ক
√3
খ
3
গ
√5
ঘ
2

ব্যাখ্যা

Question: If x + (1/x) = 3, then x - (1/x) = ?

solution:
Given,
x + (1/x) = 3

We know,
{x - (1/x)}² = {x + (1/x)}² - 4 . x . 1/x
⇒ {x - (1/x)}² = 3² - 4
⇒ {x - (1/x)}² = 9 - 4
⇒ {x - (1/x)}² = 5
∴ x - (1/x) = √5

৩৮৮.

What is the sum of the series:
11² + 12² + 13² + ........ + 20²

ক
2465
খ
2845
গ
2485
ঘ
2495

ব্যাখ্যা

Question: What is the sum of the series:
11² + 12² + 13² + ........ + 20²

Solution:
11² + 12² + 13² + ........ + 20²
= (1² + 2² + 3² + .... + 20²) - (1² + 2² + 3² + .... + 10²)
= [{20(20 + 1)(2 × 20 + 1}/6] - [{10(10 + 1)(2 × 10 + 1)}/6]
= {(20 × 21 × 41)/6} - {(10 × 11 × 21)/6}
= 2870 - 385
= 2485

৩৮৯.

Karim can do a job in 15 minutes and his brother takes twice as long to do the same job. If they work together, how long it takes to complete the job?

ক
5
খ
7.5
গ
10
ঘ
12.5

ব্যাখ্যা

Question: Karim can do a job in 15 minutes and his brother takes twice as long to do the same job. If they work together, how long it takes to complete the job?

Solution:
করিমের সময় লাগে ১৫ মিনিট
তার ভাইয়ের সময় লাগে ৩০ মিনিট

১ মিনিটে তারা কাজ সম্পন্ন করে = (১/১৫) + (১/৩০)
= (২ + ১)/৩০
= ৩/৩০
= ১/১০ অংশ

∴ সম্পূর্ণ কাজ করতে সময় লাগে = ১০ মিনিট

৩৯০.

If x = √3 + √2, then find the value of x³ - (1/x)³ = ?

ক
18√2
খ
36
গ
22√2
ঘ
54
ঙ
36√3

ব্যাখ্যা

Question: If x = √3 + √2, then find the value of x³ - (1/x)³ = ?

Solution:
দেওয়া আছে, x = √3 + √2
সুতরাং, 1/x = 1/(√3 + √2)
= (√3 - √2)/{(√3 + √2)(√3 - √2)}
= (√3 - √2)/{(√3)² - (√2)²}
= (√3 - √2)/(3 - 2)
= √3 - √2

অতএব, x - (1/x) = (√3 + √2) - (√3 - √2)
= √3 + √2 - √3 + √2
= 2√2

আমরা জানি,
x³ - (1/x)³ = {x - (1/x)}³ + 3 . x . 1/x . {x - (1/x)}
= (2√2)³ + 3(2√2)
= (8 × 2√2) + 6√2
= 16√2 + 6√2
= 22√2

∴ নির্ণেয় মান হলো 22√2

৩৯১.

Two whole numbers whose sum is 64, cannot be in the ratio:

ক
ক) 5 : 3
খ
খ) 7 : 1
গ
গ) 3 : 4
ঘ
ঘ) 9 : 7

ব্যাখ্যা

Question: Two whole numbers whose sum is 64, cannot be in the ratio:

Solution:
5 : 3 হলে, সংখ্যা দুটি 5x ও 3x
5x + 3x = 64
⇒ 8x = 64
∴ x = 8
সংখ্যা দুটি 40, 24

7 : 1 হলে, সংখ্যা দুটি 7x ও x
7x + x = 64
⇒ 8x = 64
∴ x = 8
সংখ্যা দুটি 56, 8

3 : 4 হলে, সংখ্যা দুটি 3x ও 4x
3x + 4x = 64
⇒ 7x = 64
∴ x = 64/7
যেহেতু ৬৪, ৭ দ্বারা নিঃশেষে বিভাজ্য নয়, এক্ষেত্রে সংখ্যা দুটি পূর্ণসংখ্যা হবে না।

9 : 7 হলে, সংখ্যা দুটি 9x ও 7x
9x + 7x = 64
⇒ 16x = 64
∴ x = 4
সংখ্যা দুটি 36, 28

৩৯২.

If 5x + 1/3x = 4, then what is the value of 9x² + 1/25x² = ?

ক
ক) 118/22
খ
খ) 114/25
গ
গ) 117/27
ঘ
ঘ) 117/24

ব্যাখ্যা

প্রশ্ন : If 5x + 1/3x = 4, then what is the value of 9x² + 1/25x² = ?

সমাধান :

5x + 1/3x = 4

Multiply by 3/5
⇒ 3x + 1/5x = 12/5

Squaring on both sides
⇒ (3x + 1/5x)² = (12/5)²
⇒ 9x² + 1/25x² + 2 × 3x × (1/5x) = 144/25
⇒ 9x² + 1/25x² = 144/25 – 6/5
⇒ 9x² + 1/25x² = (144 – 30)/25

∴ The value is 114/25

৩৯৩.

In an AP, the ratio of the 2nd term to the 7th term is 1/3. If the 5th term is 11, what is the 15th term?

ক
ক) 33
খ
খ) 28
গ
গ) 31
ঘ
ঘ) 36

ব্যাখ্যা

Question: In an AP, the ratio of the 2nd term to the 7th term is 1/3. If the 5th term is 11, what is the 15th term?

Solution:
The 2nd term is a + d.
The 7th term is a + 6d

ATQ,
(a + d)/(a + 6d) = 1/3
⇒ 3a + 3d = a + 6d
∴ 2a = 3d
∴ a = (3d)/2

The 5th term is a + 4d = 11
⇒ (3d)/2 + 4d = 11
⇒ 3d + 8d = 22
⇒ 11d = 22
∴ d = 2

∴ a = (3 × 2)/2 = 3
∴ The 15th term is: a + 14d
= 3 + 14 × 2
= 3 + 28
= 31

৩৯৪.

If x + (1/x) = 3, then the value of (3x² - 4x + 3)/(x² - x + 1) is?

ক
3/4
খ
1/2
গ
5/2
ঘ
7/5

ব্যাখ্যা

Question: If x + (1/x) = 3, then the value of (3x² - 4x + 3)/(x² - x + 1) is?

Solution:
Given that,
x + (1/x) = 3
⇒ (x² + 1)/x = 3
∴ x² + 1 = 3x

Now,
(3x² - 4x + 3)/(x² - x + 1)
= (3x² + 3 - 4x)/(x² + 1 - x)
= {3(x² + 1) - 4x}/(x² + 1 - x)
= {(3 × 3x) - 4x}/(3x - x) [মান বসিয়ে]
= (9x - 4x)/2x
= 5x/2x
= 5/2

৩৯৫.

If 0 < x ≤ 1, then which one of the following is the maximum value of (x - 1)² + x ?

ক
ক) - 1
খ
খ) - 2
গ
গ) 0
ঘ
ঘ) 1

ব্যাখ্যা

0 < x ≤ 1, হলে x এর মান 0 থেকে বড় কিন্তু 1 এর চেয়ে ছোট বা সমান।

(x - 1)² + x এর সর্বোচ্চ মান বের করার জন্য x = 1 ধরে পাই,
(x - 1)² + x = (1 - 1)² + 1
= 0² + 1
= 0 + 1
= 1

৩৯৬.

If one root of x² - (p - 1)x + 10 = 0 is 5, then the value of P is-

ক
6
খ
7
গ
8
ঘ
10

ব্যাখ্যা

Question: If one root of x² - (p - 1)x + 10 = 0 is 5, then the value of P is-

Solution:
x² - (p - 1)x + 10 = 0
Putting x = 5
5²-(p - 1)5 + 10 = 0
⇒ 25 - 5p + 5 + 10 = 0
⇒ 40 - 5p = 0
⇒ 5p = 40
∴ p = 8

৩৯৭.

If a + b + c = 12 and a² + b² + c² = 56, then what is the value of ab + bc + ca ?

ক
26
খ
34
গ
42
ঘ
44

ব্যাখ্যা

Question: If a + b + c = 12 and a² + b² + c² = 56, then what is the value of ab + bc + ca ?

Solution:
দেওয়া আছে,
a + b + c = 12
a² + b² + c² = 56
আমরা জানি,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (12)² = 56 + 2(ab + bc + ca)
⇒ 144 = 56 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 144 - 56
⇒ 2(ab + bc + ca) = 88
⇒ ab + bc + ca = 88/2
∴ ab + bc + ca = 44

৩৯৮.

In a class 75% passed in English, 60% in Mathematics and 25% failed in both the subjects. What is the percentage who passed in both subjects?

ক
60%
খ
55%
গ
50%
ঘ
45%

ব্যাখ্যা

Question: In a class 75% passed in English, 60% in Mathematics and 25% failed in both the subjects. What is the percentage who passed in both subjects?

Solution:
75% passed in English then fail 25%
60% passed in Mathematics then fail 40%
Failed in both 25%

Total no. Of fail = (25 + 40 - 25) = 40%

Then passed is 60%

৩৯৯.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

Question:

Solution:

৪০০.

If 5 - 3x ≤ 14, then what is the value of x?

ক
(- ∞, - 3]
খ
[3, ∞)
গ
(- ∞, 3)
ঘ
[- 3, ∞)

ব্যাখ্যা

প্রশ্ন: If 5 - 3x ≤ 14, then what is the value of x?

Solution:
5 - 3x ≤ 14
⇒ - 3x ≤ 14 - 5
⇒ - 3x ≤ 9
⇒ 3x ≥ -9 [উভয় পক্ষকে -1 দ্বারা গুণ করলে]
⇒ x ≥ - 9/3
∴ x ≥ - 3

সমাধানটিকে ব্যবধি (interval) আকারে প্রকাশ করলে হয়: [- 3, ∞)
এখানে তৃতীয় বন্ধনী [ দ্বারা বোঝায় যে - 3 সমাধান সেটের অন্তর্ভুক্ত, এবং ∞ এর পাশে প্রথম বন্ধনী ) বোঝায় যে এটি অসীম পর্যন্ত বিস্তৃত।