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

This question tests your knowledge of standard trigonometric values for common angles.

The trigonometric ratios for 0°, 30°, 45°, 60°, and 90° are fixed values that must be memorized for competitive exams.

For 60°, the cosine value is:

At 60°, if you place a point on the unit circle, the x-coordinate (which represents cosine) equals \(\frac{1}{2}\).

- \(\sin 60° = \frac{\sqrt{3}}{2}\) (option B is sine, not cosine)

- \(\cos 0° = 1\) (option C)

- \(\cos 90° = 0\) (option D)

Therefore, \(\cos 60° = \frac{1}{2}\)

The correct answer is A) 1/2

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

HCF (Highest Common Factor) is the largest number that divides both given numbers without leaving a remainder.

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

Common factors = 1, 2, 3, 4, 6, 12

The largest common factor = 12

Alternative Method - Using Prime Factorization:

24 = 2³ × 3

36 = 2² × 3²

HCF = 2² × 3 = 4 × 3 = 12

Therefore, HCF of 24 and 36 = 12

The answer is (C) 12

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

The slope of a line is the coefficient of x in the standard linear equation form y = mx + c.

The equation of a line is written as: y = mx + c, where m is the slope and c is the y-intercept.

Given equation: y = 3x + 5

Standard form: y = mx + c

Here, m = 3 (coefficient of x) and c = 5 (constant term)

Therefore, the slope is 3.

The answer is (B) 3

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).