To find the median, we must first arrange all numbers in ascending order, then locate the middle value.
We have 7 numbers: 5, 8, 3, 9, 1, 7, 6
1, 3, 5, 6, 7, 8, 9
Since n = 7 (odd number), median position = \(\frac{n+1}{2} = \frac{7+1}{2} = 4\)
Counting from left: 1st is 1, 2nd is 3, 3rd is 5, 4th is 6
Therefore, the median is 6.
The answer is (A) 6
Logarithm is the inverse operation of exponentiation; \(\log_b(x) = y\) means \(b^y = x\).
We need to find the value of \(\log_{10}(1000)\). This asks: "10 raised to what power equals 1000?"
Since \(10^3 = 1000\), we have:
Therefore, \(\log_{10}(1000) = 3\)
The answer is (B) 3
The sum of all three interior angles of any triangle is always constant, regardless of the triangle's shape or size.
Every triangle has exactly 3 interior angles. These angles are formed where two sides of the triangle meet.
The angle sum property of a triangle states:
- Equilateral triangle: 60° + 60° + 60° = 180°
- Right triangle: 90° + 45° + 45° = 180°
- Isosceles triangle: 70° + 70° + 40° = 180°
This property holds for all triangles without exception. It's a basic axiom in Euclidean geometry.
Therefore, the sum of angles in a triangle = 180°
Answer: B) 180°
To find the area of a circle, we use the formula Area = πr², where r is the radius.
- Radius r = 7 cm
- π = 22/7
Therefore, the area of the circle is 154 sq cm.
The answer is (A) 154 sq cm.
This is a linear equation problem where we need to isolate the variable x on one side.
\[2(5) + 5 = 10 + 5 = 15\] ✓
Therefore, x = 5
The answer is B) x = 5
This question tests your knowledge of standard trigonometric ratios for common angles.
Therefore, sin 30° = 0.5
Answer: B) 0.5
By the Angle Bisector Theorem, the angle bisector divides the opposite side in the ratio of the adjacent sides.
Therefore, BD/DC = AB/AC, which gives 4/DC = 6/9.
Solving: DC = (4 × 9)/6 = 36/6 = 6 cm.
This theorem is essential for solving numerous geometry problems in competitive exams.
For a regular hexagon inscribed in a circle of radius R, the side length equals R.
Here, side = 6 cm, so perimeter = 6 × 6 = 36 cm.
The circumference of the circle = 2πR = 12π cm.
Since 12π ≈ 37.7 > 36, the difference is 12π - 36 cm (circumference is greater).
The area of the ring = π(R² - r²).
Given that this equals the area of the inner circle = πr², we have R² - r² = r², which gives R² = 2r², so R = r√2.
Therefore, the ratio R:r = √2:1.
By the geometric mean altitude theorem, when an altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean of the two segments of the hypotenuse.
Therefore, altitude = √(9 × 16) = √144 = 12 cm.
This is a fundamental property used frequently in competitive exams.