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Which is incorrect out of the following?
Answer: (d) Democratic government is efficient. Explanation: While democracy has many advantages, efficiency is not always its strongest trait. Democratic governments often face delays in decision-making due to: Bureaucratic procedures and multiple levels of approval. Public debates and discussions,Read more
Answer: (d) Democratic government is efficient.
Explanation: While democracy has many advantages, efficiency is not always its strongest trait. Democratic governments often face delays in decision-making due to: Bureaucratic procedures and multiple levels of approval.
Public debates and discussions, which slow down policy implementation. Political disagreements, which can lead to gridlock in governance. In contrast, dictatorships or authoritarian regimes may make decisions faster, but they often lack accountability, transparency, and public participation.
Thus, while democracy ensures accountability, dignity, and fairness, it is not always the most efficient form of governance.
This question related to Chapter 1 Social Science Class 9th NCERT. From the Chapter 1 What is Democracy? Why Democracy?. Give answer according to your understanding.
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If the sum and product of the zeros of a quadratic polynomial are 3 and -10, respectively, then the quadratic polynomial is:
Step 1: Understanding Zeros and Quadratic Form - A quadratic polynomial has the general form: x² + bx + c - Its zeros are the roots that make the polynomial equal to zero - Let the zeros be p and q Step 2: Given Conditions - Sum of zeros (p + q) = 3 - Product of zeros (p * q) = -10 Step 3: RelationsRead more
Step 1: Understanding Zeros and Quadratic Form
– A quadratic polynomial has the general form: x² + bx + c
– Its zeros are the roots that make the polynomial equal to zero
– Let the zeros be p and q
Step 2: Given Conditions
– Sum of zeros (p + q) = 3
– Product of zeros (p * q) = -10
Step 3: Relationship to Quadratic Coefficients
In the standard form x² + bx + c:
– b = -(sum of zeros)
– c = product of zeros
Step 4: Calculating Coefficients
– b = -(p + q) = -3
– c = p * q = -10
Step 5: Constructing the Quadratic Polynomial
The polynomial becomes:
x² – 3x – 10
Verification:
– Coefficient of x²: 1
– Coefficient of x: -3 (negative of zero sum)
– Constant term: -10 (product of zeros)
Mathematical Reasoning:
The coefficients directly reflect the given conditions about the zeros:
– Sum of zeros: p + q = 3
– Product of zeros: p * q = -10
Conclusion:
The quadratic polynomial is x² – 3x – 10.
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The polynomial x³ + 3x² + 3x + 1 is divisible by:
Step 1: Polynomial Structure Given polynomial: f(x) = x³ + 3x² + 3x + 1 Step 2: Divisibility Test Method - To check if a polynomial is divisible by (x + a), we can use the remainder theorem - The polynomial is divisible by (x + a) if f(-a) = 0 Step 3: Checking Possible Divisors Let's evaluate f(-1):Read more
Step 1: Polynomial Structure
Given polynomial: f(x) = x³ + 3x² + 3x + 1
Step 2: Divisibility Test Method
– To check if a polynomial is divisible by (x + a), we can use the remainder theorem
– The polynomial is divisible by (x + a) if f(-a) = 0
Step 3: Checking Possible Divisors
Let’s evaluate f(-1):
f(-1) = (-1)³ + 3(-1)² + 3(-1) + 1
= -1 + 3 – 3 + 1
= 0
Mathematical Insight:
– When f(-1) = 0, it means (x + 1) is a factor of the polynomial
– This indicates the polynomial is completely divisible by (x + 1)
Step 4: Verification
Dividing x³ + 3x² + 3x + 1 by (x + 1):
– The result is a quadratic: x² + 2x + 1
– (x + 1)(x² + 2x + 1) = x³ + 3x² + 3x + 1 ✓
Conclusion:
The polynomial is divisible by x + 1.
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If α and β are the zeros of the polynomial x² – 5x + 6, then the value of α² + β² is:
Step 1: Understanding the Polynomial Polynomial: x² - 5x + 6 - Zeros are α and β - Standard form: x² - (α + β)x + (α * β) = 0 Step 2: Vieta's Formulas We already know from the given polynomial: - Sum of zeros: α + β = 5 - Product of zeros: α * β = 6 Step 3: Goal Calculation We are trying to calculatRead more
Step 1: Understanding the Polynomial
Polynomial: x² – 5x + 6
– Zeros are α and β
– Standard form: x² – (α + β)x + (α * β) = 0
Step 2: Vieta’s Formulas
We already know from the given polynomial:
– Sum of zeros: α + β = 5
– Product of zeros: α * β = 6
Step 3: Goal Calculation
We are trying to calculate: α² + β²
Step 4: Algebraic Manipulation
(α + β)² = α² + 2αβ + β²
α² + β² = (α + β)² – 2αβ
Step 5: Substitution
– (α + β)² = 5²
= 25
– αβ = 6
– α² + β² = (α + β)² – 2(αβ)
= 25 – 12
= 11
Mathematical Insight
This approach applies Vieta’s formulas to connect the coefficients of a quadratic with the characteristics of its zeros without actually solving for the zeros themselves.
Conclusion:
The value of α² + β² is 11.
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If the system of equations 2x + 3y = 5, 4x + ky = 10 has infinitely many solutions, then k =
Step 1: Understanding Infinitely Many Solutions • In a system of linear equations, infinitely many solutions occur when the equations represent the same line. • This means the equations are scalar multiples of each other. Step 2: Mathematical Representation Given equations: 1. 2x + 3y = 5 (EquationRead more
Step 1: Understanding Infinitely Many Solutions
• In a system of linear equations, infinitely many solutions occur when the equations represent the same line.
• This means the equations are scalar multiples of each other.
Step 2: Mathematical Representation
Given equations:
1. 2x + 3y = 5 (Equation ₁)
2. 4x + ky = 10 (Equation ₂)
Step 3: Condition for Infinitely Many Solutions
For the lines to be the same, the coefficients must maintain a consistent ratio.
Coefficient Analysis:
• 1st equation: 2x + 3y = 5
– Coefficient of x: 2
– Coefficient of y: 3
• 2nd equation: 4x + ky = 10
– Coefficient of x: 4
– Coefficient of y: k
Step 4: Ratio Consistency
Coefficient of x ratio: 4 ÷ 2 = 2
Coefficient of y ratio: k ÷ 3 = 2
Therefore:
k = 3 * 2 = 6
Step 5: Verification
• When k = 6, the second equation becomes: 4x + 6y = 10
• Dividing by 2: 2x + 3y = 5 (exactly the same as the first equation)
Conclusion:
The value of k must be 6 to create infinitely many solutions.
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