Mathematics

This question asks you to find the 5th term of a geometric progression using the formula for the nth term.

First term: a = 2

Common ratio: r = 6/2 = 3 (or 18/6 = 3, or 54/18 = 3)

The nth term of a GP is given by:

Therefore, the 5th term = 162

The answer is (B) 162

For a 2×2 matrix, the determinant is calculated using the cross-product formula: (top-left × bottom-right) minus (top-right × bottom-left).

Matrix = [[1, 2], [3, 4]]

where a = 1, b = 2, c = 3, d = 4

Therefore, the determinant of [[1,2],[3,4]] = -2

The answer is (A) -2

To find the nth term of an arithmetic progression, we use the formula \(a_n = a + (n-1)d\) where a is the first term and d is the common difference.

From the AP: 2, 5, 8, 11...

First term a = 2

Common difference d = 5 - 2 = 3

We need to find \(a_{10}\) using:

Therefore, the 10th term of the AP is 29.

The answer is (A) 29.

Integration is the reverse process of differentiation, and we use the power rule to find antiderivatives.

We need to find ∫2x dx, which means we're integrating the function 2x with respect to x.

The power rule states: ∫x^n dx = \(\frac{x^{n+1}}{n+1}\) + C

Rewrite 2x as: ∫2x^1 dx

∫2x dx = 2∫x^1 dx

\(\frac{d}{dx}\)(x² + C) = 2x ✓

Therefore, ∫2x dx = x² + C

Answer: (A) x²+C

To convert degrees to radians, use the conversion formula that relates the two angle measurement systems.

Therefore, 45° = \(\frac{\pi}{4}\) radians.

The answer is (A) π/4

Factorial is the product of all positive integers from 1 up to a given number.

7! means 7 × 6 × 5 × 4 × 3 × 2 × 1

7 × 6 = 42

42 × 5 = 210

210 × 4 = 840

840 × 3 = 2520

2520 × 2 = 5040

5040 × 1 = 5040

Therefore, 7! = 5040

The answer is (B) 5040

This question asks you to find the roots of a quadratic equation using factorization.

The numbers are -2 and -3, because:

(-2) × (-3) = 6

(-2) + (-3) = -5

x - 2 = 0 → x = 2

x - 3 = 0 → x = 3

For x = 2: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓

For x = 3: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓

Therefore, x = 2, 3

The answer is (A).

Differentiation is finding the rate of change of a function with respect to a variable.

We have y = 3x² and we need to use the Power Rule of differentiation.

The Power Rule states: If y = ax^n, then \(\frac{dy}{dx} = n \cdot a \cdot x^{n-1}\)

Here, a = 3 and n = 2

Therefore, the derivative of y = 3x² is 6x

Answer: A) 6x

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

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).