ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

সিলেবাস

Math - 11:
Algebra, Determining Algebraic Formula and Value, Quadratic and Polynomial Equations etc.

প্রশ্ন দেখুন অনুশীলন

ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স · তারিখ অনির্ধারিত · ২১ প্রশ্ন

১.

If x + 2y = 6 and xy = 4 what is (2/x) + (1/y)?

ক
3/2
খ
1/3
গ
1/2
ঘ
1

ব্যাখ্যা

Question: If x + 2y = 6 and xy = 4 what is (2/x) + (1/y)?

Solution:
Given, x + 2y = 6
xy = 4

Now,
2/x + 1/y
= (2y + x)/xy
= (x + 2y)/xy
= 6/4
= 3/2

২.

If (2a - 2b) = 2 and (3a + 2b) = 8 then find the value of (3a + 4).

ক
5
খ
6
গ
8
ঘ
10

ব্যাখ্যা

Question: If (2a - 2b) = 2 and (3a + 2b) = 8 then find the value of (3a + 4).

Solution:
2a - 2b = 2 ...........(1)
3a + 2b = 8 .............(2)

From (1) we get,
2a - 2b = 2
⇒ a - b = 1
⇒ a = b + 1 .......(3)

From (2) we get,
3(b + 1) + 2b = 8
⇒ 3b + 3 + 2b = 8
⇒ 5b = 5
∴ b = 1
From (3) we get,
a = 1 + 1
∴ a = 2

Now,
3a + 4
= (3 × 2) + 4
= 6 + 4
= 10

৩.

For what values of m are the roots of the quadratic equation mx(x - 2√5) + 10 = 0 real and equal?

ক
(1, 2)
খ
(2, 3)
গ
(0, 2)
ঘ
(1, 0)

ব্যাখ্যা

Question: For what values of m are the roots of the quadratic equation mx (x - 2√5) + 10 = 0 real and equal?

Solution:
mx (x - 2√5) + 10 = 0
⇒ mx² - 2√5 mx + 10=0

Compare given equation with the general form of quadratic equation, which ax²+ bx + c=0
a = m, b = - 2√5m, c = 10

Since roots are real and equal, discriminant, D = 0
b² - 4ac = 0
⇒ (- 2√5m)² - 4 × m × 10 = 0
⇒ 20m² - 40m = 0
⇒ 20(m² - 2m) = 0
⇒ m² - 2m = 0
⇒ m(m - 2) = 0

Either, m = 0

Or, m - 2 = 0
∴ m = 2

৪.

If (x² - x + 2)/2 = 4 then x could be equal to -

ক
3
খ
2
গ
4
ঘ
- 3

ব্যাখ্যা

Question: If (x² - x + 2)/2 = 4 then x could be equal to -

Solution:
(x² - x + 2)/2 = 4
⇒ x² - x + 2 = 8
⇒ x² - x + 2 - 8 = 0
⇒ x² - x - 6 = 0
⇒ x² - 3x + 2x - 6 = 0
⇒ x(x - 3) + 2(x - 3) = 0
⇒ (x - 3) (x + 2) = 0

Either,
x - 3 = 0
∴ x = 3

Or,
x + 2 = 0
∴ x = - 2

∴ x = (3, - 2)

৫.

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

৬.

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.

৭.

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

৮.

If (x + 3)² = 225, Which of the following can be the value of (x + 1)?

ক
11
খ
13
গ
15
ঘ
16

ব্যাখ্যা

Question: If (x + 3)² = 225, Which of the following can be the value of (x + 1)?

Solution:
Given,
(x + 3)² = 225
⇒ (x + 3)² = (15)²
∴ x + 3 = ± 15

When, x + 3 = 15
⇒ x = 15 - 3
∴ x = 12

∴ x + 1 = 12 + 1 = 13

৯.

If L + M = 2N and N + O = 2L, then-

ক
L = M
খ
L + O = N + M
গ
M + O = N + L
ঘ
L = M + O

ব্যাখ্যা

Question: If L + M = 2N and N + O = 2L, then-

Solution:
Let
L + M = 2N .............(1)
N + O = 2L .............(2)

(1) + (2)⇒
L + M = 2N
N + O = 2L
L + M + N + O = 2N + 2L
⇒ L + M + N + O - L - N = 2N + 2L - L - N
∴ M + O = N + L

১০.

If α and β are the zeros of the polynomial f(x) = x² - 5x + k such that α - β = 1, find the value of k.

ক
6
খ
5
গ
3
ঘ
8

ব্যাখ্যা

Question: If α and β are the zeros of the polynomial f(x) = x² - 5x + k such that α - β = 1, find the value of k.

Solution:
Given,
f(x) = x² - 5x + k
α - β = 1 .................... (1)
α and β are the zeros of the polynomial.
α + β = - (- 5)
∴ α + β = 5......................... (2)

(1) + (2)
α - β = 1
α + β = 5
2α = 6
∴ α = 3

From equ. (2)
β = 5 - 3
∴ β = 2

Now, αβ = k
⇒ k = 3 × 2
∴ k = 6

১১.

ক
a + b
খ
(1 + b)²
গ
1 + b
ঘ
1 - b

ব্যাখ্যা

Question:

Solution:

১২.

Find the root of the quadratic equation: 3x² - 2√6x + 2 = 0

ক
√3/2, - (√2/3)
খ
√(2/3), √(2/3)
গ
1/√3, √(2/3)
ঘ
√(2/5), √2/3

ব্যাখ্যা

Question: Find the root of the quadratic equation: 3x² - 2√6x + 2 = 0

Solution:
3x² - 2√6x + 2 = 0
⇒ 3x² - √6x - √6x + 2 = 0
⇒ √3x(√3x - √2) - √2(√3x - √2) = 0
⇒ (√3x - √2)(√3x - √2) = 0
∴ x = √2/√3, √2/√3

১৩.

If a = 1 + √2 and b = 1 - √2, find the value of a² + b².

ক
2
খ
4
গ
6
ঘ
8

ব্যাখ্যা

Question: If a = 1 + √2 and b = 1 - √2, find the value of a² + b².

Solution:
Given that,
a = 1 + √2,
b = 1 - √2

∴ a + b = 1 + √2 + 1 - √2
= 2

And,
ab = (1 + √2)(1 - √2)
= 1² - (√2)²
= 1 - 2
= - 1

Now,
a² + b² = (a + b)² - 2ab
= (2)² - 2(- 1)
= 4 + 2
= 6

১৪.

a = 2b = 3c and abc = 36, then find the value of c.

ক
√2
খ
2
গ
2√2
ঘ
4

ব্যাখ্যা

Question: a = 2b = 3c and abc = 36, then find the value of c.

Solution:
Given,
a = 2b = 3c
∴ a = 3c
and b = 3c/2

Now, abc = 36
⇒ 3c . (3c/2) . c = 36
⇒ 9c³/2 = 36
⇒ 9c³ = 72
⇒ c³ = 72/9
⇒ c³ = 8
⇒ c³ = 2³
∴ c = 2

১৫.

If a = 3 + 2√2, then the value of (√a − 1/√a) is?

ক
2
খ
√2
গ
2√2
ঘ
0

ব্যাখ্যা

Question: If a = 3 + 2√2, then the value of (√a − 1/√a) is?

Solution:
a = 3 + 2√2
⇒ a = 2 + 1 + 2√2
⇒ a = (√2)² + 2 . √2 . 1 + 1²
⇒ a = (√2 + 1)²
⇒ √a = √2 + 1
⇒ 1/√a = 1/(√2 + 1)
⇒ 1/√a = 1(√2 - 1)/(√2 + 1)(√2 - 1)
⇒ 1/√a = √2 - 1

∴√a - 1/√a =√2 + 1 - √2 + 1 = 2

১৬.

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

১৭.

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

১৮.

If a + b + c = 15 and a² + b² + c² = 75, then find ab + bc + ca.

ক
25
খ
50
গ
75
ঘ
100

ব্যাখ্যা

Question: If a + b + c = 15 and a² + b² + c² = 75, then find ab + bc + ca.

Solution:
Given,
a + b + c = 15
a² + b² + c² = 75

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - (a² + b² + c²)
⇒ 2(ab + bc + ca) = (15)² - 75
⇒ 2(ab + bc + ca) = 225 - 75
⇒ 2(ab + bc + ca) = 150
∴ ab + bc + ca = 75

১৯.

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.

২০.

If then what is a/b?

ক
25/3
খ
9/25
গ
25/16
ঘ
25/9

ব্যাখ্যা

Question: If then what is a/b?

Solution:
(3a + 5b)/(3a - 5b) = 4
⇒ 3a + 5b = 4(3a - 5b)
⇒ 3a + 5b = 12a - 20b
⇒ 5b + 20b = 12a - 3a
⇒ 25b = 9a
∴ a/b = 25/9

২১.

If a² - 5a - 1 = 0; what is the value of a² + (1/a²)?

ক
23
খ
25
গ
27
ঘ
29

ব্যাখ্যা

Question: If a² - 5a - 1 = 0; what is the value of a² + (1/a²)?

Solution:
a² - 5a - 1 = 0
⇒ a² - 1 = 5a
⇒a - 1/a = 5
⇒ (a - 1/a)² = (5)²
⇒ a² + 1/a² - 2 = 25
∴ a² + 1/a² = 27