All questions of Achievers Section for Class 8 Exam

52

Understanding the Statements
The question examines three statements related to quadratic equations, specifically of the form ax² + bx + c = 0. Let's analyze each statement to understand why option 'A' is the correct answer.
Statement (i):
- The roots of the quadratic equation ax² + bx + c = 0 are given by the formula (-b ± √(b² - 4ac)) / 2a.
- This statement is True (T). This is the standard formula known as the quadratic formula, used to find the roots of a quadratic equation.
Statement (ii):
- The discriminant of the quadratic equation ax² + bx + c = 0 is b² - 4ac.
- This statement is also True (T). The discriminant helps determine the nature of the roots of the quadratic equation.
Statement (iii):
- If the discriminant is negative, the quadratic equation has two real and distinct roots.
- This statement is False (F). When the discriminant is negative, it indicates that the quadratic equation has no real roots; instead, it has two complex roots.
Conclusion
Based on the analysis:
- Statement (i) is True (T).
- Statement (ii) is True (T).
- Statement (iii) is False (F).
Thus, the correct answer is option 'A', which states: (i)-T, (ii)-T, (iii)-F. This indicates a clear understanding of the properties of quadratic equations and their discriminants.

Understanding the Problem
To solve the problem, we need to identify a two-digit number whose digits add up to 9, and when 9 is subtracted from it, the digits are reversed.
Step 1: Define the Two-Digit Number
Let the two-digit number be represented as:
- 10a + b, where
- a = tens digit
- b = units digit
Step 2: Set Up the Equations
From the problem, we have the following conditions:
- The sum of the digits is 9:
- a + b = 9
- When 9 is subtracted from the number, the digits are interchanged:
- 10a + b - 9 = 10b + a
Step 3: Simplify the Second Equation
Rearranging the second equation gives:
- 10a + b - 9 = 10b + a
- This simplifies to:
- 9a - 9b = 9
- or, a - b = 1
Step 4: Solve the Equations
Now we have a system of equations:
1. a + b = 9
2. a - b = 1
Adding these equations:
- (a + b) + (a - b) = 9 + 1
- 2a = 10
- a = 5
Now substitute a = 5 into a + b = 9:
- 5 + b = 9
- b = 4
Step 5: Find the Number
The two-digit number is:
- 10a + b = 10(5) + 4 = 54
Step 6: Calculate Half of the Number
Now, to find half of the number:
- Half of 54 = 27
Conclusion
Thus, the half of that number is 27, confirming that the correct answer is option 'B'.

Explanation of the Statements
Let's analyze each statement one by one:
Statement (i): The sum of the first n natural numbers is given by n(n+1)/2.
- This statement is True.
- The formula n(n+1)/2 accurately calculates the sum of the first n natural numbers.
- For example, for n=3, the sum is 1 + 2 + 3 = 6, and using the formula: 3(3+1)/2 = 6.
Statement (ii): The sum of the squares of the first n natural numbers is given by n(n+1)(2n+1)/6.
- This statement is also True.
- The formula n(n+1)(2n+1)/6 correctly gives the sum of the squares of the first n natural numbers.
- For n=3, the squares are 1^2 + 2^2 + 3^2 = 14, and using the formula: 3(3+1)(2*3+1)/6 = 14.
Statement (iii): The sum of the cubes of the first n natural numbers is given by (n(n+1)/2)².
- This statement is True.
- The sum of the cubes of the first n natural numbers can be expressed as the square of the sum of the first n natural numbers.
- For n=3, the cubes are 1^3 + 2^3 + 3^3 = 36, and using the formula: (3(3+1)/2)² = 36.
Conclusion
- Since all three statements are true, the correct answer is option a: (i)-T, (ii)-T, (iii)-T.

Understanding the Statements
Let's analyze each statement regarding matrices to determine their truth values.
Statement (i): The determinant of a 2x2 matrix [a b; c d] is ad - bc.
- This statement is True.
- The formula for the determinant of a 2x2 matrix is indeed calculated as ad - bc.
Statement (ii): The inverse of a non-singular matrix always exists.
- This statement is True.
- A non-singular matrix is one that has a non-zero determinant, which guarantees that an inverse can be calculated.
Statement (iii): If a matrix is singular, its determinant is zero.
- This statement is True.
- A singular matrix is defined as one that does not have an inverse, which occurs when its determinant equals zero.
Conclusion
Thus, evaluating each statement gives us:
- (i) - T
- (ii) - T
- (iii) - T
Therefore, the correct option is (c): (i) - T, (ii) - T, (iii) - T.
This reflects a solid understanding of matrix properties, crucial for solving mathematical problems related to linear algebra.

The formula for the number of diagonals in a polygon with 'n' sides is n(n-3)/2. A regular hexagon can be divided into 6 equilateral triangles. Thus, both statements are true.

The sum of the interior angles of a polygon is not always greater than the sum of its exterior angles, as the sum of exterior angles is always 360 degrees for any polygon. However, for any triangle, the sum of its exterior angles is indeed always 360 degrees.

If the sides of a triangle are in the ratio 3:4:5, it is a right-angled triangle by the Pythagorean theorem. The area of an equilateral triangle with side length 'a' is indeed (√3/4)a². Hence, both statements are true.

In any parallelogram, the diagonals do bisect each other, so Statement-1 is true. However, the diagonals of a rhombus are not equal in length; they are only equal in a square, so Statement-2 is false.