Algebra

PrepBank · পাতা ১৪ / ১৪ · ১,৩০১–১,৩৬৭ / ১,৩৮০

১,৩০১.

If x² + yz + zx + xy is divided by x + y, the result is-

ক
(x + z)
খ
(x - y)
গ
(x - z)
ঘ
(x + y)

ব্যাখ্যা

Question: If x² + yz + zx + xy is divided by x + y, the result is-

Solution:
x² + yz + zx + xy
= x² + xy + zx + yz
= x(x + y) + z(x + y)
= (x + y)(x + z)

∴ divided by (x + y) the result = (x + y)(x + z)/(x +y) = (x + z)

১,৩০২.

A worker was hired for 5 days. Each day, he was paid Tk. 20 more than what he was paid for the previous day of work. The total amount he was paid in the first 3 days of work equaled the total amount he was paid in the last 2 days. What was his starting pay?

ক
Tk. 60
খ
Tk. 80
গ
Tk. 90
ঘ
Tk. 100

ব্যাখ্যা

Question: A worker was hired for 5 days. Each day, he was paid Tk. 20 more than what he was paid for the previous day of work. The total amount he was paid in the first 3 days of work equaled the total amount he was paid in the last 2 days. What was his starting pay?

Solution:
Let
Starting payment was = x

Salary for the 1st 3 days:
x, x + 20, x + 40

Salary for the last 2 days:
x + 60, x + 80

Now
sum of the salary of the 1st 3 days = sum of the salary of the last 2 days
x + x + 20 + x + 40 = x + 60 + x + 80
⇒ 3x + 60 = 2x + 140
⇒ 3x - 2x = 140 - 60
⇒ x = 80

∴ The starting payment was Tk. 80

১,৩০৩.

In a group of 15, 7 have studied Latin, 8 have studied Greek, and 3 have not studied either. How many of these studied both Latin and Greek?

ক
ক) 0
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5
ঙ
ঙ) None of these

ব্যাখ্যা

Question: In a group of 15, 7 have studied Latin, 8 have studied Greek, and 3 have not studied either. How many of these studied both Latin and Greek?

Solution:
There are a group of 15
3 have not studied either.
∴ The number of student who either studied Latin or Greek = 15 - 3 = 12

Here,
The number of student studied Latin = n(L) = 7
The number of student studied Greek = n(G) = 8

Let,
The number of student studied both Latin and Greek = n(L ∩ G)

Now,
n(L ∪ G) = n(L) + n(G) - n(L ∩ G)
⇒ n(L ∩ G) = n(L) + n(G) - n(L ∪ G)
= 7 + 8 - 12
= 15 - 12
= 3

∴ 3 students of these studied both Latin and Greek.

১,৩০৪.

The product of the roots of the equation 2a² - 5a + m = 10 is - 3. Find the value of m.

ক
ক) 1
খ
খ) 2
গ
গ) 4
ঘ
ঘ) 8

ব্যাখ্যা

Question: The product of the roots of the equation 2a² - 5a + m = 10 is - 3. Find the value of m.

Solution:
Rearranging the given equation we have 2a² - 5a + (m - 10) = 0

We know that,
if ax² + bx + c = 0 is a quadratic equation, then the product of their roots = c/a

Given the product of the roots = - 3
⇒ (m - 10)/2 = - 3
⇒ (m - 10) = - 6
⇒ m = - 6 + 10
∴ m = 4

১,৩০৫.

Find an equation of the horizontal line containing the point (3, 2).

ক
x = 3
খ
y = 3
গ
y = 2
ঘ
x = 2

ব্যাখ্যা

Question: Find an equation of the horizontal line containing the point (3, 2).

Solution:
The equation of the horizontal line containing the point (3, 2) is y = 2.
A horizontal line has a constant y-value for all points on the line.
Since the line must pass through the point (3, 2), its y-coordinate must be 2.

১,৩০৬.

If the nth term of an arithmetic progression is 7n + 1, then what is the common difference?

ক
5
খ
7
গ
- 6
ঘ
3

ব্যাখ্যা

Question: If the n^th term of an arithmetic progression is 7n + 1, then what is the common difference?

Solution:
The nth term of an arithmetic progression is Tn = 7n + 1
n = 1 then, T₁ = 7 × 1 + 1 = 8
n = 2 then, T₂ = 7 × 2 + 1 = 15
n = 3 then, T₃ = 7 × 3 + 1 = 22
n = 4 then, T₄ = 7 × 4 + 1 = 29
............................

Common difference,
T₂ - T₁ = 15 - 8 = 7
T₄ - T₃ = 29 - 22 = 7

∴ The common difference is 7.

১,৩০৭.

Mr. Shihab moved 3/4th of his lawn in 5/4 hours. Mr. Anik makes twice as fast and finishes the remaining job. How many minutes did Mr. Anik work?

ক
25 min
খ
20 min
গ
12.5 min
ঘ
10 min

ব্যাখ্যা

Question: Mr. Shihab moved 3/4th of his lawn in 5/4 hours. Mr. Anik makes twice as fast and finishes the remaining job. How many minutes did Mr. Anik work?

Solution:
Mr. Shihab moved 3/4th of his lawn in 5/4 hours.
Mr. Shihab completed work in (5/4) × (4/3)
= 5/3 hours

Mr. Anik makes twice as fast
Mr. Anik takes time 5/6 hours
= (5/6) × 60 min
= 50 min to finish the job

Mr. Anik takes time to finish the remaining job = 50/4 min
= 12.5 min

১,৩০৮.

The sum of the first 16 terms of an Arithmetic Progression(AP) whose first term and third term are 5 and 15 respectively is-

ক
640
খ
720
গ
680
ঘ
600
ঙ
700

ব্যাখ্যা

Question: The sum of the first 16 terms of an Arithmetic Progression(AP) whose first term and third term are 5 and 15 respectively is-

Solution:
1^st term = 5
3^rd term =15
∴ 5 + d + d = 15
⇒ 2d = 10
∴ d = 5

16^th term = a + 15d
= 5 + 15 × 5
= 80

∴ The sum of the first 16 terms = (n/2)[2a + (n - 1)d]
= (16/2)[2 × 5 + (16 - 1)5]
= 8 × (10 + 75)
= 8 × 85
= 680

১,৩০৯.

If x + y = a and x - y = b, then 2xy =

ক
ক) (b² - a²)/2
খ
খ) (a² - b²)/2
গ
গ) (a - b)/2
ঘ
ঘ) ab/2

ব্যাখ্যা

দেওয়া আছে,
x + y = a
x- y = b

আমরা জানি
4xy = (x + y)² - (x - y)²
2xy = (a² - b²)/2

১,৩১০.

If x < 10 and 2x - 3y = 8 which of the following must be true?

ক
y < 4
খ
y < 6
গ
y > 5
ঘ
y = 5
ঙ
None

ব্যাখ্যা

Question: If x < 10 and 2x - 3y = 8 which of the following must be true?

Solution:
Given that,
x < 10 and 2x - 3y = 8
We need to determine which statement must be true from the options.

Now,
2x - 3y = 8
⇒ 3y = 2x - 8
⇒ y = (2x - 8)/3
Since x < 10, substitute into the expression for y,
⇒ y < {(2 × 10) - 8}/3
⇒ y < (20 - 8)/3
⇒ y < 12/3
∴ y = 4
∴ if x < 10, then y < 4.

So the statement that must be true is y < 4.

১,৩১১.

x - y = 4

and

, find the value of

ক
74
খ
94
গ
106
ঘ
110

ব্যাখ্যা

$Question: x - y = 4$ and , find the value of

Solution:
Given that,
$x - y = 4$
and

We know,

১,৩১২.

If x + y = 7, then the value of x³ + y³ + 21xy is?

ক
256
খ
343
গ
426
ঘ
386

ব্যাখ্যা

Question: If x + y = 7, then the value of x³ + y³ + 21xy is?

Solution:
x + y = 7
⇒ (x + y)³ = 7³
⇒ x³ + y³ + 3(x + y)xy = 343
⇒ x³ + y³ + 21xy = 343

১,৩১৩.

4x = 52, x² - 5² = ?

ক
81
খ
9
গ
144
ঘ
12

ব্যাখ্যা

Question: 4x = 52, x² - 5² = ?

Solution:
দেওয়া আছে
4x = 52
x = 52/4
x = 13

আবার
x² - 5² = 13² - 5²
= 169 - 25
= 144

১,৩১৪.

The roots of a quadratic equation ax² + bx + c = 0 will be real and unequal, if -

ক
b² - 2ac > 0
খ
b² - 4ac < 0
গ
b² - 4ac = 0
ঘ
b² - 4ac > 0

ব্যাখ্যা

Question: The roots of a quadratic equation ax² + bx + c = 0 will be real and unequal, if -

Solution:
The roots of a quadratic equation ax² + bx + c = 0 will be irrational and unequal if b² - 4ac < 0.
The roots of a quadratic equation ax² + bx + c = 0 will be real and unequal if b² - 4ac > 0.
The roots of a quadratic equation ax²+ bx + c = 0 will be real and equal, if b² - 4ac = 0.

১,৩১৫.

Find the simplified value of : (6 + 2x) (4 - 2x).

ক
ক) 24 + 12x - 4x
খ
খ) 24 + 20x - 4x
গ
গ) 24 - 4x - 4x²
ঘ
ঘ) 24 + 4x - 4x

ব্যাখ্যা

প্রশ্ন: Find the simplified value of : (6 + 2x) (4 - 2x).

সমাধান:

১,৩১৬.

Factor completely: x³ - 8.

ক
(x - 2)(x² + 2x + 4)
খ
(x + 2)(x² - 2x + 4)
গ
(x - 8)(x² - 2x + 4)
ঘ
(x + 8)(x² + 2x + 4)

ব্যাখ্যা

Question: Factor completely: x³ - 8.

Solution:
We know that,
a³ - b³ = (a - b)(a² + ab + b²)

∴ x³ - 8
= x³ - 2³
= (x - 2)(x² + x.2 + 2²)
= (x - 2)(x² + 2x + 4)

১,৩১৭.

If a + b = 4 and ab = - 12, then a - b equals to-

ক
± 8
খ
± 24
গ
± 2
ঘ
± 6

ব্যাখ্যা

Question: If a + b = 4 and ab = - 12, then a - b equals to-

Solution:
দেওয়া আছে
a + b = 4
ab = - 12

আমরা জানি
(a - b)² = (a + b)² - 4ab
⇒ (a - b)² = (4)² - 4 × (- 12)
⇒ (a - b)² = 16 + 48
⇒ (a - b)² = 64
⇒ a - b = ±√64
∴ a - b = ±8

১,৩১৮.

What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

ক
- 10
খ
- 1/10
গ
10
ঘ
1/10

ব্যাখ্যা

Question: What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

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,
20x - 2y = 6
2y = 20x - 6
y = 10x - 3
(1) নং এর সাথে তুলনা করে পাই,
m = 10

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - (1/10)

১,৩১৯.

If x + (1/x) = 4 then, x - (1/x) = ?

ক
± √10
খ
5
গ
3
ঘ
± 2√3

ব্যাখ্যা

Question: If x + (1/x) = 4 then, x - (1/x) = ?

Solution:
দেওয়া আছে, x + 1/x = 4

আমরা জানি,
(x - 1/x)² = (x + 1/x)² - 4 . x . 1/x
⇒ (x - 1/x)² = 4² - 4 [মান বসিয়ে]
⇒ (x - 1/x)² = 16 - 4
⇒ (x - 1/x)² = 12
⇒ x - (1/x) = ± √12
∴ x - (1/x) = ± 2√3

১,৩২০.

Solve the following equation: (x/3) - (1/12) = (1/6) + (x/4)

ক
- 2
খ
3
গ
4
ঘ
12

ব্যাখ্যা

Question: Solve the following equation: x/3 - 1/12 = 1/6 + x/4

Solution:
Given,
x/3 - 1/12 = 1/6 + x/4
⇒ x/3 - x/4 = 1/6 + 1/12
⇒ (4x - 3x)/12 = (2 + 1)/12
⇒ x/12 = 3/12
⇒ 12x = 3 × 12
∴ x = 3

১,৩২১.

ক
ক) 1
খ
খ) - 1
গ
গ) - 2
ঘ
ঘ) 2

ব্যাখ্যা

Question:

Solution:

১,৩২২.

Quadratic equation corresponding to the roots 2 + √5 and 2 - √5 is-

ক
x² - 4x - 1 = 0
খ
x² + 4x - 1 = 0
গ
x² - 4x + 1 = 0
ঘ
x² + 4x + 1 = 0

ব্যাখ্যা

Question: Quadratic equation corresponding to the roots 2 + √5 and 2 - √5 is-

Solution:
The quadratic equation is: x² - (Sum of roots)x + Product of roots = 0

Let the roots of the equation be A and B.
A = 2 + √5 and B = 2 - √5

∴ A + B = 2 + √5 + 2 - √5 = 4

∴ A × B = (2 + √5)(2 - √5) = 4 - 5 = - 1

Then equation is
x² - 4x - 1 = 0

১,৩২৩.

Find out the missing number in the following series: 256, 16, 4, ____?

ক
2
খ
4
গ
16
ঘ
None of these

ব্যাখ্যা

Question: Find out the missing number in the following series. 256, 16, 4, ____?

Solution:
√256 = 16
√16 = 4
√4 = 2

So, missing number is = √4 = 2

১,৩২৪.

What is the next term in the sequence 4, 9, 6,11, 8, 13, …….?

ক
ক) 18
খ
খ) 16
গ
গ) 10
ঘ
ঘ) 9

ব্যাখ্যা

4, 9, 6,11, 8, 13
Here in both in even and in odd position numbers increased by 2
4, 6, 8, 10
and, 9, 11, 13, 15

So, the next number in the series is 10

১,৩২৫.

A fraction become 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is-

ক
12/13
খ
5/12
গ
5/13
ঘ
6/13

ব্যাখ্যা

Question: A fraction become 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is-

Solution:
Let the fraction be x/y.

Given that, the fraction becomes 1/3 when 1 is subtracted from the numerator.
⇒ (x - 1)/y = 1/3
⇒ 3(x - 1) = y
⇒ 3x - 3 = y
⇒ 3x - y = 3 -------------(1)

Given that, the fraction becomes 1/4 when 8 is added to its denominator.
⇒ x / (y + 8) = 1/4
⇒ 4x = y + 8
⇒ 4x - y = 8 ------------ (2)

By solving equations (1) & (2) by the method of elimination, we get:
⇒ 4x - y - 3x + y = 8 - 3
⇒ x = 5

From equation (1):
⇒ 3x - y = 3
⇒ 3 × (5) - y = 3
⇒ 15 - y = 3
⇒ -y = 3 - 15
⇒ - y = -12
⇒ y = 12

Thus, the fraction, x/y = 5/12

১,৩২৬.

A school has a total of 270 students. There are 90 students taking Physics, 75 taking English, and 39 taking both. Approximately what percentage of the students is taking either Physics or English?

ক
32%
খ
36%
গ
47%
ঘ
51%

ব্যাখ্যা

Question: A school has a total of 270 students. There are 90 students taking Physics, 75 taking English, and 39 taking both. Approximately what percentage of the students is taking either Physics or English?

Solution:
Students taking physics n(A) = 90 (these 90 include those 39 that take both)
Students taking english n(B) = 75 (these 75 also include those 39)
Students taking both n(A ∩ B) = 39
Students taking either Physics or English n(A ∪ B) = ?

We know
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 90 + 75 - 39 = 126

Required percentage = (126/270) × 100
= 46.67 %
= 47% (approx.)

১,৩২৭.

If two non-zero positive integers p and q are such that p = 4q and p < 8, then q = ?

ক
ক) 1
খ
খ) 2
গ
গ) ±3
ঘ
ঘ) 5

ব্যাখ্যা

Given, p = 4q
if q = 1 then p = 4
if q = 2 then p = 8 not acceptable as p have to be less than 8
So, the correct answer is 1

১,৩২৮.

If A > B, B > C and C > D, then which of the following conclusions is definitely wrong?

ক
ক) A > C
খ
খ) D > A
গ
গ) A > D
ঘ
ঘ) B > D

ব্যাখ্যা

Given, A > B, B > C and C > D
That means, A > B > C > D
So, A > C, A > D and B > D is correct
But, D > A is wrong

১,৩২৯.

If x > y, z < y and w < x which of the following is always true?

ক
ক) z > w
খ
খ) y > w
গ
গ) z < x
ঘ
ঘ) y = w

ব্যাখ্যা

Question: If x > y, z < y and w < x which of the following is always true?

Solution:
দেওয়া আছে,
x > y, z < y
∴ x > y > z

∴ w < x অর্থাৎ x বড় w ছোট।
কিন্তু y ও z এর সাথে w এর কোন সম্পর্ক নেই।

∴ x বড় z ছোট।
∴ x > z অথবা, z < x এটাই সত্য।

১,৩৩০.

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

ক
ক) √5
খ
খ) √13
গ
গ) √7
ঘ
ঘ) 0

ব্যাখ্যা

আমরা জানি, (x - 1/x)² = (x + 1/x)2 - 4x × 1/x
(x - 1/x)² = 3² - 4
(x - 1/x)² = 9 - 4
∴ (x - 1/x) = √5

১,৩৩১.

Find the value of 17² - 4².

ক
272
খ
275
গ
271
ঘ
273

ব্যাখ্যা

Question: Find the value of 17² - 4².

Solution:
17² - 4²
= (17 + 4)(17 - 4)
= 21 × 13
= 273

১,৩৩২.

What is the sum of the following sequence: 5, 12, 19, 26, ... , 54?

ক
230
খ
236
গ
240
ঘ
254

ব্যাখ্যা

Question: What is the sum of the following sequence: 5, 12, 19, 26, ... , 54?

Solution:
এটি একটি সমান্তর ধারা (arithmetic series)।
প্রথম পদ, a = 5
সাধারণ অন্তর, d = 12 - 5 = 7
শেষ পদ= 54

আমরা জানি,
n তম পদ = a + (n - 1)d
⇒ 54 = 5 + (n - 1)7
⇒ 49 = 7(n - 1)
⇒ n - 1 = 7
⇒ n = 8

সমষ্টি, S_n = n/2{2a + (n - 1)d}
∴ S₈ = (8/2){2(5) + (8 - 1)7}
= 4{10 + (7 × 7)}
= 4{10 + 49}
= 4 × 59
= 236

অতএব, প্রদত্ত ধারাটির সমষ্টি হলো 236।

১,৩৩৩.

Find out the wrong number in the given series: 644, 328, 164, 84, 44, 24, 14.

ক
ক) 328
খ
খ) 164
গ
গ) 84
ঘ
ঘ) 44
ঙ
ঙ) 24

ব্যাখ্যা

644-320 = 324 ≠ 328
324-160 = 164
164-80 = 84
84-40 = 44
44-20 = 24
24-10 = 14

১,৩৩৪.

If one root of x² - (q + 2)x + 15 = 0 is 3, then the value of q is:

ক
12
খ
4
গ
6
ঘ
7

ব্যাখ্যা

Question: If one root of x² - (q + 2)x + 15 = 0 is 3, then the value of q is:

Solution:
Given equation: x² - (q + 2)x + 15 = 0

One root is x = 3. Substitute:
(3)²- (q + 2)(3) + 15 = 0
⇒ 9 - 3(q + 2) + 15 = 0
⇒ 9 - 3q - 6 + 15 = 0
⇒ 18 - 3q = 0
⇒ 3q = 18
∴ q = 6

১,৩৩৫.

31, 29, 24, 22, 17, ... What number should come next?

ক
15
খ
12
গ
13
ঘ
14
ঙ
None

ব্যাখ্যা

Question: 31, 29, 24, 22, 17, ... What number should come next?

Solution:
This is a simple alternating subtraction series, which subtracts 2, then 5.
31 - 2 = 29,
29 - 5 = 24,
24 - 2 = 22,
22 - 5 = 17,
17 - 2 = 15.

১,৩৩৬.

What is the sum of the numbers from 1 to 99?

ক
ক) 4950
খ
খ) 4650
গ
গ) 4750
ঘ
ঘ) 4850

ব্যাখ্যা

Question: What is the sum of the numbers from 1 to 99?

Solution:
the sum of the numbers from 1 to 99 is = n (n + 1)/2
= 99 (99 + 1)/2
= 99 × 100/2
= 99 × 50
= 4950

১,৩৩৭.

If ab + bc + ca = 31 and a² + b² + c² = 19 then, a + b + c = ?

ক
7
খ
9
গ
11
ঘ
13

ব্যাখ্যা

Solution: If ab + bc + ca = 31 and a² + b² + c² = 19 then, a + b + c = ?

Solution:
Given,
ab + bc + ca = 31
a² + b² + c² = 19

Now,
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
⇒ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (a + b + c)² = 19 + (2 × 31)
⇒ (a + b + c)² = 19 + 62
⇒ (a + b + c)² = 81
⇒ a + b + c = √81
∴ a + b + c = 9

১,৩৩৮.

Find the product of two consecutive numbers if three times the first number is 8 more than twice the second number.

ক
90
খ
111
গ
110
ঘ
120

ব্যাখ্যা

Question: Find the product of two consecutive numbers if three times the first number is 8 more than twice the second number.

Solution:
Let the numbers be a and a + 1.

According to the question:
3 × (first number) = 2 × (second number) + 8
⇒ 3a = 2(a + 1) + 8
⇒ 3a = 2a + 2 + 8
⇒ 3a = 2a + 10
⇒ 3a - 2a = 10
⇒ a = 10

∴ The numbers are 10 and 11.
Product = 10 × 11 = 110

১,৩৩৯.

The displacement of a particle S at time/is modelled by S = 10t - t². Find the displacement after 2 seconds.

ক
10m
খ
16m
গ
2m
ঘ
11m

ব্যাখ্যা

Question: The displacement of a particle S at time/is modelled by S = 10t - t². Find the displacement after 2 seconds.

Solution:
Given that,
S(t) = 10t - t²

We want S at t = 2 seconds.
Now,
S(2) = 10 × 2 - 2² = 20 - 4
∴ S(2) = 16

So the displacement after 2 seconds is 16.

১,৩৪০.

If B = {x : x ∈ N such that x² + 11x + 30 = 0} then B is ?

ক
Finite set
খ
Empty set
গ
Infinite set
ঘ
None of these

ব্যাখ্যা

Question: If B = {x : x ∈ N such that x² + 11x + 30 = 0} then B is ?

Solution:
Given that,
B = {x : x ∈ N such that x² + 11x + 30 = 0}
First let's solve the quadratic equation x² + 11x + 30 = 0
⇒ x² + 5x + 6x + 30 = 0
⇒ x(x + 5) + 6(x + 5) = 0
⇒ (x + 6)(x + 5) = 0
⇒ x = - 5 or - 6
Both solutions are negative numbers (- 5 and - 6). Natural numbers (N) are positive integers:

According to the definition of the given set x is a natural number but we know that neither x = - 5 nor x = - 6 is a natural number

So, the given set is an empty set i.e B = Ø

১,৩৪১.

If m > 0, y > 0, and x is m percent of 2y, then, in terms of y, m is what percent of x?

ক
5000/y
খ
y/200
গ
2y
ঘ
None of these

ব্যাখ্যা

Question: If m> 0, y> 0, and x is m percent of 2y, then, in terms of y, m is what percent of x?

Solution:
x = (m/100) × 2y
⇒ x/m = y/50
m/x = 50/y
⇒ m/x = (50/y) × 100% =5000/y%

১,৩৪২.

A sum of Tk. 312 was divided among 100 boys and girls in such a way that each boys gets Tk. 3.6 and each girl Tk. 2.4. The number of girls is :

ক
35
খ
40
গ
42
ঘ
45

ব্যাখ্যা

Question: A sum of Tk. 312 was divided among 100 boys and girls in such a way that each boys gets Tk. 3.6 and each girl Tk. 2.4. The number of girls is :

Solution:
ধরি,
বালিকার সংখ্যা = x জন
বালকের সংখ্যা = (100 - x) জন

1 জন বালক পায় 3.60 টাকা
(100 - x) জন বালক পায় = 3.60 (100 - x) টাকা

1 জন বালিকা পায় = 2.40 টাকা
x জন বালিকা পায় = 2.40x টাকা

প্রশ্নমতে,
3.60(100 - x) + 2.40x = 312
⇒ 360 - 3.60x + 2.40x = 312
⇒ 360 - 1.2x = 312
⇒ 360 - 312 = 1.2x
⇒ 1.2x = 48
⇒ x = 48/1.2
⇒ x = 40

১,৩৪৩.

The roots of the equation 9x² - bx + 81 = 0 will be equal, if the value of b is

ক
ক) ±9
খ
খ) ±18
গ
গ) ±27
ঘ
ঘ) ±54

ব্যাখ্যা

When roots are equal, b² = 4ac
⇒ b² = 4×9×81
⇒ b² = 36×81 = 6 × 6 × 9 × 9
∴ b = ± 54

১,৩৪৪.

For what value of k (k > 0) will the lines 3x + 4y = 1 and 2x + Ky = 7 be parallel?

ক
8/5
খ
3/4
গ
7/3
ঘ
8/3

ব্যাখ্যা

Question: For what value of k (k > 0) will the lines 3x + 4y = 1 and 2x + Ky = 7 be parallel?

Solution:
Two lines are parallel if their slopes are equal.
Rewrite both equations in slope-intercept form (y = mx + c).

Now first line,
3x + 4y = 1
⇒ 4y = - 3x + 1
⇒ y = (- 3/4)x + 1/4
∴ Slope m₁ = - 3/4

And second line,
2x + Ky = 7
⇒ Ky = - 2x + 7
⇒ y = (- 2/K)x + 7/K (since k > 0, k ≠ 0)
∴ Slope m₂ = - 2/k

For the lines to be parallel:
m₁ = m₂
⇒ - 3/4 = - 2/k
⇒ 3/4 = 2/k
⇒ 3k = 8
∴ k = 8/3

So the lines will be parallel when k = 8/3.

১,৩৪৫.

If x + 1/x = 6; find the value of 3x/(6x² - 5x + 6)?

ক
ক) 1/11
খ
খ) 3/32
গ
গ) 3/31
ঘ
ঘ) 5/27

ব্যাখ্যা

Question: If x + 1/x = 6; find the value of 3x/(6x² - 5x + 6)?

Solution:

১,৩৪৬.

If x²= 2.89, what is the value of x?

ক
ক) 0.7
খ
খ) 1.6
গ
গ) 1.7
ঘ
ঘ) 2.7

ব্যাখ্যা

Question: If x²= 2.89, what is the value of x?

Solution:
x² = 2.89
⇒ x = √2.89
= √(289/100)
= 17/10
= 1.7

১,৩৪৭.

If x + (1/x) = 2, then the value of x⁷ + (1/x⁵) =?

ক
2⁵
খ
2
গ
2¹²
ঘ
2⁷

ব্যাখ্যা

Question: If x + (1/x) = 2, then the value of x⁷ + (1/x⁵) =?

Solution:
x + (1/x) = 2
⇒ x² + 1 = 2x
⇒ x² - 2x + 1 = 0
⇒ (x - 1)² = 0
∴ x = 1

x⁷ + (1/x⁵)
= 1⁷ + 1/1⁵
= 1 + 1
= 2

১,৩৪৮.

If x + 1/x = 3 then x − 1/x =?

ক
ক) √5
খ
খ) √13
গ
গ) √7
ঘ
ঘ) 0

ব্যাখ্যা

x + 1/x = 3
Or, (x + 1/x)² = 9
Or, (x - 1/x)² + 4.x.1/x = 9
Or, (x - 1/x)² = 9 - 4 = 5
Or, (x - 1/x) = √5

১,৩৪৯.

If (5√5 × 5³)/5 ^{- 3/2} = 5^{a + 2}, then the value of a is-

ক
ক) 3
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

ব্যাখ্যা

Question: If (5√5 × 5³)/5 ^{- 3/2} = 5^{a + 2}, then the value of a is-

Solution:
Given that
(5√5 × 5³)/5^{- 3/2} = 5^{a + 2}
⇒ (5^{1 + (1/2) + 3})/5^{- 3/2} =5^{a + 2}
⇒ 5^{1 + (1/2) + 3 + (3/2)} = 5^{a + 2}
⇒ 1 + (1/2) + 3 + (3/2) = a + 2
⇒ (2 + 1 + 6 + 3)/2 = a + 2
⇒ 12/2 = a + 2
⇒ 6 = a + 2
a = 4

১,৩৫০.

Solve for y, √(2y - 3) = 5

ক
y = 28
খ
y = 5.5
গ
y = 4
ঘ
y = 14

ব্যাখ্যা

Question: Solve for y, √(2y - 3) = 5

Solution:
Given that,
⇒ √(2y - 3) = 5
⇒ (√(2y - 3))² = 5²⇒ 2y - 3 = 25
⇒ 2y = 25 + 3
⇒ 2y = 28
⇒ y = 28/2
∴ y = 14

১,৩৫১.

Quantity A = 4/100 and Quantity B = 0.012/3

ক
ক) Quantity B is greater
খ
খ) Quantity A equals Quantity B
গ
গ) Quantity A greater
ঘ
ঘ) Relationship indeterminate
ঙ
ঙ) None of these

ব্যাখ্যা

Question: Quantity A = 4/100 and Quantity B = 0.012/3

Solution:
Quantity A = 4/100

Quantity B = 0.012/3
= 12/(3 × 1000)
= 4/1000

∴ Quantity A > Quantity B

১,৩৫২.

What is the slope of a line containing the points (1, 13) and (-3, 6)?

ক
ক) 0.14
খ
খ) 0.57
গ
গ) 1.75
ঘ
ঘ) 1.83

ব্যাখ্যা

Here the points line are (1, 13) and (-3, 6)
∴ Slope, m = Y₁ - Y₂ / X₁ - X₂
= (13 - 6)/(1 + 3)
= 7/4
= 1.75

১,৩৫৩.

A sum of Tk.312 was divided among 100 boys and girls in such a way that the boy gets Tk..3.60 and each girl Tk.. 2.40 the number of girls is -

ক
35
খ
40
গ
55
ঘ
50

ব্যাখ্যা

Let x be the number of boys and y be the number of girls.
Given total number of boys and girls = 100

x + y = 100 --------- (i)
A boy gets Tk. 3.60 and a girl gets Tk. 2.40
The amount given to 100 boys and girls = Tk. 312

3.60x + 2.40y = 312 ---------- (ii)

Solving (i) and (ii)
3.60x + 3.60y = 360 ---------- Multiply (i) by 3.60
3.60x + 2.40y = 312 ------------ (ii)

Equation (i) - (ii)
1.20y = 48
⇒ y = 48/1.20
⇒ y = 40

Number of girls = 40.

১,৩৫৪.

What is the value of

ক
ক) 2√6
খ
খ) 2
গ
গ) 3√6
ঘ
ঘ) 1

ব্যাখ্যা

Question: What is the value of

Solution:
{√(2 × 2 × 6) + √(6 × 6 × 6)}/√(4 × 4 × 6 )
= (2√6 + 6√6)/4√6
= 8√6/4√6
= 2

১,৩৫৫.

If f(y) = y³ + ky² - 4y - 8 , then for what value of k will f(- 2) = 0?

ক
- 3
খ
1
গ
2
ঘ
- 4

ব্যাখ্যা

Question: If f(y) = y³ + ky² - 4y - 8 , then for what value of k will f(- 2) = 0 ?

Solution:
Given that,
f(y) = y³ + ky² - 4y - 8
f(- 2) = (- 2)³ + k(- 2)² - 4(- 2) - 8
= - 8 + 4k + 8 - 8
∴ f(- 2) = 4k - 8

Set f(- 2) = 0
⇒ 4k - 8 = 0
⇒ 4k = 8
⇒ k = 8/4
∴ k = 2

So the value of k is 2.

১,৩৫৬.

What is the solution of the equation 3/(y + 1) = 4/(y - 2)?

ক
ক) 10
খ
খ) - 10
গ
গ) 4/3
ঘ
ঘ) 3/4

ব্যাখ্যা

Question: What is the solution of the equation 3/(y + 1) = 4/(y - 2)?

Solution:
3/(y + 1) = 4/(y - 2)
⇒ 3/(y + 1) - 4/(y - 2) = 0
⇒ (3y - 6 - 4y - 4)/(y + 1) (y - 2) = 0
⇒ - y - 10 = 0
∴ y = - 10

১,৩৫৭.

If a² + b² = 61 and ab = 30, find (1/a) + (1/b).

ক
11/30
খ
2/5
গ
13/30
ঘ
1/2

ব্যাখ্যা

Question: If a² + b² = 61 and ab = 30, find (1/a) + (1/b).

Solution:
দেওয়া আছে,
ab = 30
a² + b² = 61
⇒ (a + b)² - 2ab = 61
⇒ (a + b)² - 60 = 61
⇒ (a + b)² = 121
∴ a + b = 11

এখন,
(1/a) + (1/b)
= (b + a)/ab
= 11/30

১,৩৫৮.

If a³ - b³ = 208 and a - b = 4, then ab = ?

ক
10
খ
12
গ
20
ঘ
26

ব্যাখ্যা

Question: If a³ - b³ = 208 and a - b = 4, then ab = ?

Solution:
Given,
a³ - b³ = 208
a - b = 4

We know,
a³ - b³ = (a - b)³ + 3ab(a - b)
⇒ 208 = 4³+ 3ab · 4
⇒ 208 = 64 + 12ab
⇒ 12ab = 208 - 64
⇒ 12ab = 144
∴ ab = 12

১,৩৫৯.

If x is doubled and y is tripled in the expression z = (4x/y), then the value of z is _____.

ক
Doubled
খ
Multiplied by 6
গ
Multiplied by a factor 2/3
ঘ
None

ব্যাখ্যা

Question: If x is doubled and y is tripled in the expression z = (4x/y), then the value of z is _____.

Solution:
x এর দ্বিগুণ = 2x
y এর দিগুণ = 3y

এখন
z = (4 × 2x/3y)
= 8x/3y
= (4x/y) × (2/3)

The value of z is multiplied by a factor 2/3.

১,৩৬০.

If a series looks like 3, 4.78, 6.56, 8.34, 10.12, ..........., then which of the following comes next in the sequence?

ক
7.12
খ
9.24
গ
11.9
ঘ
11.92

ব্যাখ্যা

⊗ 3 + 1.78 = 4.78

⊗ 4.78 + 1.78 = 6.56

⊗ 6.56 + 1.78 = 8.34

⊗ 8.34 + 1.78 = 10.12

⊗ 10.12 + 1.78 = 11.9

১,৩৬১.

Find the value of 3(p + 5) - 2(2p - 3) + p

ক
21
খ
25 - p
গ
18
ঘ
3p

ব্যাখ্যা

Question: Find the value of 3(p + 5) - 2(2p - 3) + p

Solution: Given that,
3(p + 5) - 2(2p - 3) + p
= 3p + 15 - 4p + 6 + p
= (3p - 4p + p) + (15 + 6)
= 0 + 21
= 21

১,৩৬২.

If x/y = 2/3, then (x - y)/x?

ক
- 1/2
খ
- 1/3
গ
1/3
ঘ
1/2

ব্যাখ্যা

Question: If x/y = 2/3, then (x - y)/x?

Solution:
(x - y)/x
= x/x - y/x
= 1- y/x
= 1- 3/2
= (2 - 3)/2
= - 1/2

১,৩৬৩.

In a class of 92 students, 40 are taking English, 24 are taking Arabic and 10 are taking both courses. How many students are not enrolled in either course?

ক
32
খ
35
গ
38
ঘ
41

ব্যাখ্যা

Question: In a class of 92 students, 40 are taking English, 24 are taking Arabic and 10 are taking both courses. How many students are not enrolled in either course?

Solution:
Total students = 92
Students taking English n(E) = 40
Students taking Arabic n(A) = 24
Students taking both English and Arabic = 10

We know,
n(E ∪ A) = n(E) + n(A) - n(E ∩ A)
n(E ∪ A) = 40 + 24 - 10 = 54

∴ Not enrolled = Total students - n(E ∪ A) = 92 - 54 = 38

১,৩৬৪.

(a² - b² - 2bc - c²)/(a² + b² + 2ab - c²) is equivalent to?

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

ব্যাখ্যা

Question: (a² - b² - 2bc - c²)/(a² + b² + 2ab - c²) is equivalent to?

Solution:
(a² - b² - 2bc - c²)/(a² + b² + 2ab - c²)
= {a² - (b² + 2bc + c²)}/{(a² + b² + 2ab) - c²}
= {a² - (b + c)²}/{(a + b)² - c²}
= (a + b + c)(a - b - c)/(a + b + c)(a + b - c)
= (a - b - c)/(a + b - c)

১,৩৬৫.

ক
16√3
খ
18√3
গ
9√3
ঘ
24√3

ব্যাখ্যা

Question:

Solution:

১,৩৬৬.

If x + y = 2a then the value of {a/(x - a)} + {a/(y - a)} is-

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 0

ব্যাখ্যা

Question: If x + y = 2a then the value of {a/(x - a)} + {a/(y - a)} is

Solution:
a/(x - a) + a/(y - a)
= a(y - a) + a(x - a)}/(x - a)(y - a)
= (ay - a² + ax - a²)/(x - a)(y - a)
= a(x +y) - 2a²/(x - a)(y - a)
= a. 2a - 2a²/(x - a)(y - a)
= 2a² - 2a²/(x - a)(y - a)
= 0/(x - a)(y - a)
= 0

১,৩৬৭.

What must be added to the expression (4a² + 9b²) so that the same is a perfect square?

ক
ক) 6ab
খ
খ) 12ab
গ
গ) 18ab
ঘ
ঘ) 24ab
ঙ
ঙ) 8ab

ব্যাখ্যা

(2a)² + (3b)²
= (2a + 3b)² - 2.2a.3b
= (2a + 3b)² - 12ab
So, we have to add 12ab to make the sum a perfect square