ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: Find the roots of the quadratic equation 2x² - 9x - 35 = 0

Solution:
2x² - 9x - 35 = 0
⇒ 2x² - 14x + 5x -35  = 0
⇒ 2x(x - 7) + 5(x -7) = 0
⇒ (x - 7) (2x + 5) = 0
∴ x = 7, - 5/2

Question: If Discriminant > 0, then the equation has-

Solution:
Discriminant (নিশ্চায়ক),
ax² + bx + c = 0 দ্বিঘাত সমীকরণের নিশ্চায়কের মান b² - 4ac

দ্বিঘাত সমীকরণের মূলের প্রকৃতি:
1. যদি b² - 4ac = 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় বাস্তব ও সমান হবে।
2. যদি b² - 4ac > 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় বাস্তব ও অসমান হবে।
3. যদি b² - 4ac < 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় অবাস্তব ও অসমান হবে।
4. যদি b² - 4ac পূর্ণবর্গ সংখ্যা হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় মূলদ ও অসমান হবে।

Question: If x² - 2ax + a² = 0 ,  find the value of x/a.

Solution:
a is obviously a non-zero number.
Given that,
⇒ x² - 2ax + a² = 0
⇒ (x - a)² = 0
⇒ x - a = 0
⇒ x = a
⇒ x/a = a/a  ; [divide by a]
∴ x/a = 1

∴ So the value x/a is 1.

Question: If (2a + 1)(a + 3) = 12a , then find the value of a.

Solution:
⇒ (2a + 1)(a + 3) = 12a
⇒ 2a² + 6a + a + 3 - 12a = 0
⇒ 2a² - 5a + 3 = 0
⇒ 2a² - 2a - 3a + 3 = 0
⇒ 2a(a - 1) - 3(a - 1) = 0
⇒ (a - 1)(2a - 3) = 0

Now,
a - 1 = 0
∴ a = 1
or
2a - 3 = 0
∴  a = 3/2

∴ a = 1 or a = 3/2

Question: What is the value of the greater root of the equation x² - 7x - 30 = 0?

Solution:
x² - 7x - 30 = 0
⇒ x ² - 10x + 3x - 30 = 0
⇒ x(x - 10) + 3(x - 10) = 0
⇒ (x - 10)(x + 3) = 0

so the roots of the equation are x₁ = 10 and x₂ = - 3

So the greater one is obviously 10.

Question: For the function f(x) = x² - 4x + 3 find x when f(x) = 8

Solution:
f(x) = x² - 4x + 3
f(x) = 8

Now,
⇒ x² - 4x + 3 = 8
⇒ x² - 4x + 3 - 8 = 0
⇒ x² - 4x - 5 = 0
⇒ x² - 5x + x - 5 = 0
⇒ x(x - 5) + 1(x - 5) = 0
⇒ (x - 5)(x + 1) = 0
Now,
x - 5 = 0
∴ x = 5
Or,
⇒ x + 1 = 0
∴ x = - 1

∴ x = - 1, 5

Question: 5p² - (4p - 3)(3p + 2) = ?

Solution:
5p2 - (4p - 3) (3p + 2)
= 5p2 - {(4p ×  3p) + (4p × 2) - (3 × 3p) - (3 × 2)}
= 5p² - (12p² + 8p - 9p - 6)
= 5p² - (12p² - p - 6)
= 5p2 - 12p² + p + 6
= - 7p² + p + 6

Question: x³ + 5x + k = 0 solution is - 3, but what is the value of k?

Solution:
∴ একটি সমাধান = - 3
x = - 3 বসিয়ে পাই,
⇒ (- 3)³ + 5(- 3) + k = 0
⇒ - 27 - 15 + k = 0
⇒ k - 42 = 0
∴ k = 42

Question: Which of the following is the polynomial equation 5x³ + 9x² - 4 = 0

Solution:
The given polynomial equation is in terms of x.
The highest power of x is 3 and hence the degree of the equation is 3.
Hence, it is a cubic equation.

The highest degree of the polynomial is 2 is a quadratic equation.
The highest degree of the polynomial is 1 is a linear equation.
The highest degree of the polynomial is 4 is a biquadratic equation.

Question: Find the polynomial equation of the lowest degree in terms of a whose roots are -3 and 8.

Solution:
Let,
The roots are α = - 3 and β = 8 .
Thus, the corresponding polynomial equation is,
⇒ a² - (α + β)a + αβ = 0
⇒ a² - (- 3 + 8)a + (- 3)8 = 0
⇒ a² - 5a - 24 = 0

Question: If α, β are the roots of the equation x² - 4x - 5 = 0 then α² + β² equals to:

Solution:
x² - 4x - 5 = 0
⇒ x² - 5x + x - 5 = 0
⇒ x(x - 5) + 1(x - 5) = 0
⇒ (x - 5)(x + 1) = 0
∴ x = 5, -1

Hence, α = 5 , β = -1 Hence, The value of  α² +  β² = 5² + (-1)² = 25 + 1 = 26