Algebra

জোড় স্থানগুলোতে সর্বদা 3 স্থির রেখে প্রথম সংখ্যা থেকে তৃতীয়, পঞ্চম, সপ্তম সংখ্যায় যথাক্রমে
5; 5 × 2 = 10; 10 × 2 = 20 ......... এভাবে বাড়বে ।
তাই, 4 + 5 = 9
9 + (5 × 2)
= 9 + 10
= 19
19 + (10 × 2)
= 19 + 20
= 39 হবে ।
ANswer: 39.

Question: Kalam earns Tk. 7.50 per hour on days other than Friday and twice the rate on Friday. Last week he worked a total of 60 hours, including 8 hours on Friday. What is his earnings for the week?

Solution: 
During the week, Kalam worked a total of 60 - 8 = 52 hours at a rate of Tk. 7.50 per hour.
On Friday, he worked 8 hours at a rate of Tk. 7.50 × 2 = Tk. 15.00 per hour.

Therefore, his total earnings for the week were (52 × 7.50 + 8 × 15) = Tk. 510

Let us assume that he answered x question correctly. Marks scored by him in x question = 2x
Then, wrong answer would be = 60 – x
Marks lost by him in (60 – x) questions = (60 – x)×1
ATQ,
2x – (60 – x) = 39
Or, 3x = 99
∴ x = 33

Question: If x + (2/x) = 4, what is the value of x³ + (8/x³)?

Solution: 
Here, x + (2/x) = 4

Now, 
x³ + (8/x³)
= (x)³ + (2/x)³
= {(x + (2/x)}³ - 3 . x . 2/x {x + (2/x)}
= 4³ - 3 . 2 . 4
= 64 - 24
= 40

Question: Find the domain of f(m) = 1/(4m + 3).

Solution:
দেওয়া আছে,
f(m) = 1/(4m + 3)

আমরা জানি,
একটি ভগ্নাংশের হর(denominator) শূন্য হতে পারবে না।
অর্থাৎ,
4m + 3 ≠ ০
or, 4m ≠ - 3
or, m ≠ - (3/4)

∴ f(m) এর ডোমেইন = R - {- 3/4}

Question: For a geometric sequence, the first term a = 5 and the common ratio r = 3. What is the sum of the first 4 terms?

Solution:
প্রদত্ত গুণোত্তর ধারাটির, প্রথম পদ, a = 5
সাধারণ অনুপাত, r = 3
পদের সংখ্যা, n = 4
যেহেতু r = 3 > 1,

∴ n সংখ্যক পদের সমষ্টির সূত্র:
S_n = a(rⁿ - 1)/(r - 1)
∴ S₄ = 5(3⁴ - 1)/(3 - 1)
= 5(81 - 1)/2
= 5 × 80/2
= 5 × 40
= 200

অতএব, প্রথম 4টি পদের সমষ্টি হলো 200।

Question:


Solution:

শেষের মিনিটে সে slip করবে না, অর্থাৎ 54 মিটার উঠার পর 6 মিটার উঠে গেলে সে আর নিচে নামবে না।
তাই এখন দেখি এই 54 মিটার সে কত মিনিটে উঠে।
প্রতি 2 মিনিটে বানরটি উঠে = 6 - 3 = 3 মিটার।
অর্থাৎ 3 মিটার উঠে = 2 মিনিটে
∴ 54 মিটার উঠে = (2 × 54)/3
= 36 মিনিটে 
∴ খুটি বেয়ে উঠতে সময় লাগবে = 36 + 1 = 37 মিনিট

Question: Find out the missing number : 52, 64, 76, ____ , 100

Solution:
52 + 12 = 64
64 + 12 = 76
76 + 12 = 88
88 + 12 = 100

Let the denominator of the required fraction be X.
Given that,
the numerator of a fraction is 3 less than its denominator;
then its numerator becomes X-3.
Then the required fraction (X-3)/X ...(1)
If we add 10 to the numerator(X-3), original fraction (X-3)/X is increased by 1(3/7) (i.e., 10/7).
i.e.,
(X -3)/X + 10/7 = (X - 3 + 10)/X
(X - 3)/X + 10/7 = (X + 7)/X
(X + 7)/X - (X - 3)/X = 10/7
10/X = 10/7
X = 7.
Therefore, (X-3)/X = 4/7.
Hence the required fraction is 4/7.

Question: What is the nature of the roots of the equation 9x² + 12x + 4 = 0?

Solution:
Given that, 
9x² + 12x + 4 = 0  
Here,  
a = coefficient of x² = 9  
b = coefficient of x = 12  
c = constant term = 4  

Discriminant = b² - 4ac  
= (12)² - 4 × 9 × 4  
= 144 - 144  
= 0
When the discriminant = 0, the roots are real and equal.

Therefore, the roots are real and equal.

Note: 
- If b² - 4ac > 0 and a perfect square ⇒ roots are real, unequal and rational  
- If b² - 4ac > 0 but not a perfect square ⇒ roots are real, unequal and irrational  
- If b² - 4ac = 0 ⇒ roots are real and equal  
- If b² - 4ac < 0 ⇒ no real roots (complex roots)

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

প্রশ্ন: If 5x - (5/x) = 15, then what is the value of x³ - (1/x)³?

সমাধান:
দেওয়া আছে,
5x - 5/x = 15
⇒ (5x - 5/x)/5 = 15/5
∴ x - 1/x = 3

এখন,
x³ - (1/x)³
= (x - 1/x)³ + 3 . x . (1/x)(x - 1/x)
= (x - 1/x)³ + 3(x - 1/x)
= 3³ + 3 × 3
= 27 + 9
= 36

দেয়া আছে, 3x + 4y = 12 এবং kx + 12y = 30 
যদি, 3/k = 4/12 হয় তাহলে সমীকরণ জোটটির অসংখ্য সমাধান থাকবে।
এখন, 3/k = 4/12
⇒ 4k = 12 × 3
⇒ k = 36/4
⇒ k = 9

a b c d e f g h i j k l m n o p q r s t u v w x y z

Every polynomial of the form ax³ + bx + c with a, b > 0 has exactly one real roots.
Hence, 2x³ + 8x - 7 or 2x³ + 8x + (-7) has one real root.

Question: In a survey of 1,000 consumers it is found that 720 consumers liked product A and 450 liked product B. What is the least number that must have liked both the products?

Solution:
Given that,
Total consumers = 1000
Consumers who like product A = 720
Consumers who like product B = 450

We know, 
n(A U B) = n(A) + n(B) - n(A ∩ B)
⇒ 1000 = 720 + 450 - n(A ∩ B)  
⇒ 1000 = 1170 - n(A ∩ B)
⇒ n(A ∩ B) = 1170 - 1000
∴ n(A ∩ B) = 170

So 170 consumers like both the products A and B.

Question: In a geometric sequence, the third term is 16 and the sixth term is 128. What is the first term?

Solution:
Let the first term of the geometric sequence be a
and the common ratio be r.
Third term = 16
∴ ar² = 16 ....... (1)
Again,
Sixth term = 128
∴ ar⁵ = 128 ....... (2)

Now, divide equation (2) by equation (1),
ar⁵/ar² = 128/16
⇒ r³ = 8
⇒ r³ = 2³
∴ r = 2
Substitute the value of r into equation (1).
a(2)² = 16
⇒ 4a = 16
∴ a = 4

Therefore, the first term of the geometric sequence is 4.

Question: If 7 - 2x ≤ 15, then what is the value of x?

Solution:
Given inequality:
7 - 2x ≤ 15

Subtract 7 from both sides:
-2x ≤ 8

Divide both sides by -2 (and reverse the inequality sign):
x ≥ -4

So, the solution set is x ∈ [-4, ∞) 

Given,
x - 1/x = √3
⇒ (x - 1/x)² = (√3)²
⇒ (x + 1/x)² - 4.x.(1/x) = 3
⇒ (x + 1/x)² = 3 + 4 = 7
∴ x + 1/x = √7

The general equation of a sphere is: (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) represents the center of the sphere
As, here x² + y² + z² = 4
or, (x - 0)² + (y - 0)² + (z - 0)² = 2²,
So, it's an equation of a sphere where (0, 0, 0) represents the center of the sphere and '2' is it's radius

Question: A particle moves such that its displacement is described by S = 10t - t², what is its 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.

Question: If
​
​Solution:
​

x - 2y = 4
এখানে অপশন থেকে দেখলে উত্তর বের করা সহজ হবে
অপশন a, b, d থেকে মান বসালে সমাধান 4 হবে না
অপশন c থেকে x = 4 এবং Y = 0 বসালে সমীকরণের সমাধান হবে 4

(2a - b)/(2a + b) + 2/9
= ({2a - b)/a}/{(2a + b)/a} + 2/9
= {2 - (b/a)}/{2 + (b/a)} + 2/9
= (2 - 0.25)/(2 + 0.25) + 2/9
= 1.75/2.25 + 2/9
= 7/9 + 2/9
= 9/9
= 1.

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

Solution:
The nth term of an arithmetic progression is T_n = 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.

Question: What is the slope of a line perpendicular to the line whose equation is 3x + 4y = 12?

Solution:
প্রদত্ত সরল রেখার সমীকরণ: 3x + 4y = 12
y = mx + c আকারে লিখি, যেখানে m হলো রেখার ঢাল।

4y = - 3x + 12
⇒ y = (- 3/4)x + 3

অতএব, মূল রেখার ঢাল (m) = - 3/4

আমরা জানি, কোনো রেখার উপর লম্ব রেখার ঢাল m₁ = - 1/m
= - 1/(- 3/4)
= 4/3

∴ লম্ব রেখার ঢাল = 4/3

Question: If 5 ≥ x ≥ - 1 and y ≥ - 1, which of the following cannot be a value of x - y?

Solution: 
Here, 5 ≥ x ≥ - 1 and y ≥ - 1

Now,
i) If, x = - 1 and y = - 1 then, x - y = - 1 - (- 1) = -1 + 1 = 0 
ii) If, x = 2 and y = 1 then, x - y = 2 - 1 = 1 
iii) If, x = 5 and y = 0 then, x - y = 5 - 0 = 5 
iv) If, x = 5 and y = -1 then, x - y = 5 - (- 1) = 5 + 1 = 6

∴ Any value greater than 6 cannot be a value of x - y.  

Here, 2 × 2 + 1 = 5
5 × 2 - 1 = 9
9 × 2 + 1 = 19
19 × 2 - 1 = 37
37 × 2 + 1 = 75

If he scored a three-point basket, then remaining points are = 26 - 3 = 23, which is not divisible by 2
∴ As, there is no one-point basket, by scoring two three-point basket, the remaining points are: 26 - 3×2 = 20
∴ He scored maximum number of = 20/2 = 10 two-point baskets

Question: What is the sum of the first 12 terms of an arithmetic progression if the 3^rd term is - 13 and the 6^th term is - 4?

Solution:
In an arithmetic progression. We know
n^th term = a + (n - 1)d
where a = first term,
d = common difference

Given that,
3^rd term = a + 2d = - 13  … (1)
6^th term = a + 5d = - 4   … (2)

Subtract equation (1) from equation (2) then we get,
⇒ (a + 5d) - (a + 2d) = - 4 - (- 13)
⇒ 3d = 9
⇒ d = 9/3
∴ d = 3

Equation (1) we get,
⇒ a + 2(3) = - 13
⇒ a + 6 = - 13
⇒ a = - 13 - 6
∴ a = - 19

Sum of first n terms of an arithmetic progression.
Sₙ = (n/2) × [2a + (n - 1)d]
S₁₂ = (12/2) × [2(- 19) + (12 - 1)3]
= 6 × [- 38 + 11 × 3]
= 6 × [- 38 + 33]
= 6 × (- 5)
= - 30

Question: If 1 < p < 4 and 2 < q < 6, which of the following best describes p - q?

Solution: দেয়া আছে:
1 < p < 4 -------------(1)
2 < q < 6 -------------(2)

এখন, আমরা p - q এর সীমা বের করতে চাই।

(2)⇒
2 < q < 6
⇒ - 2 > -q > - 6 (যদি -1 দ্বারা গুণ করি)
⇒ - 6 < -q < - 2 ----------(3)

(1) এবং (3) যোগ করি,
⇒ (1 + (- 6 )) < p - q < (4 + (- 2))
⇒ - 5 < p - q < 2

Let, - 2x – 8 < 9
Or, - 2x < 17
Or, x > -8.5
So, the required value is -8

Students like all the three subjects = 60 - (20 + 25 + 30 - 5 - 7 - 8 + 3) = 2

As, d days is greater than 7 days
So, charge for first week is c cents
∴ For (d - 7) days the charge is = f(d - 7) cents
So, total cost = c + f(d - 7)

Question: If p and q are the roots of the equation 2x² − 9x + 7 = 0, then what is the value of (1/p) + (1/q)?

 
Solution:
Given equation:
2x² − 9x + 7 = 0
⇒ 2x² − 7x − 2x + 7 = 0
⇒ x(2x − 7) − 1(2x − 7) = 0
⇒ (x − 1)(2x − 7) = 0

So the roots are:
x = 1 = p
x = 7/2 = q

Now,
1/p + 1/q
= 1/1 + 1/(7/2)
= 1 + 2/7
= 9/7