Mathematics

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

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

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

When flipping a fair coin, the probability of any single outcome equals the number of favorable outcomes divided by total possible outcomes.

A coin has exactly 2 possible outcomes: Head or Tail.

Number of ways to get a Head = 1

Therefore, the probability of getting a head in a coin toss is \(\frac{1}{2}\) or 0.5 or 50%.

The answer is C) 1/2

Square root is the inverse operation of squaring, and asks: what number multiplied by itself equals 144?

We need to find a number that when multiplied by itself gives 144.

We test the options by squaring them:

- 11² = 11 × 11 = 121 (not 144)

- 12² = 12 × 12 = 144 (matches!)

- 13² = 13 × 13 = 169 (not 144)

- 14² = 14 × 14 = 196 (not 144)

Since 12 × 12 = 144, we have √144 = 12

Therefore, the answer is (B) 12

To find f(2), substitute x = 2 into the given function and calculate the result.

Therefore, f(2) = 12

The answer is B) 12

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