Mathematics Questions & Answers

Q.1 Medium

In GP: 2,6,18,54... 5th term?

A 108

B 162

C 216

D 180

Correct Answer: B. 162

EXPLANATION

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

Step 1: Identify the first term and common ratio.

First term: a = 2

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

Step 2: Write the formula for the nth term of a GP.

The nth term of a GP is given by:

a_n = a \cdot r^{n-1}

Step 3: Substitute values for the 5th term (n = 5).

a_5 = 2 \cdot 3^{5-1}

a_5 = 2 \cdot 3^4

Step 4: Calculate the result.

a_5 = 2 \cdot 81 = 162

Therefore, the 5th term = 162

The answer is (B) 162

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Q.2 Medium

Determinant of [[1,2],[3,4]]?

A -2

B 2

C -4

D 4

Correct Answer: A. -2

EXPLANATION

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

Step 1: Identify the matrix elements.

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

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

Step 2: Apply the determinant formula for 2×2 matrix.

\text{Det} = ad - bc

Step 3: Substitute the values.

\text{Det} = (1)(4) - (2)(3)

Step 4: Calculate.

\text{Det} = 4 - 6 = -2

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

The answer is (A) -2

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Q.3 Medium

10th term of AP: 2,5,8,11...?

A 29

B 32

C 28

D 30

Correct Answer: A. 29

EXPLANATION

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.

Step 1: Identify the first term and common difference.

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

First term a = 2

Common difference d = 5 - 2 = 3

Step 2: Write the formula for the 10th term.

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

\[a_n = a + (n-1)d\]

Step 3: Substitute n = 10, a = 2, and d = 3.

a_{10} = 2 + (10-1) \times 3

a_{10} = 2 + 9 \times 3

a_{10} = 2 + 27

a_{10} = 29

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

The answer is (A) 29.

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Q.4 Medium

Integrate ∫2x dx?

A x²+C

B 2x²+C

C x+C

D 2x+C

Correct Answer: A. x²+C

EXPLANATION

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

Step 1: Identify the integral form

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

Step 2: Apply the power rule of integration

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

Rewrite 2x as: ∫2x^1 dx

Step 3: Factor out the constant

∫2x dx = 2∫x^1 dx

Step 4: Apply the power rule

2∫x^1 dx = 2 \times \frac{x^{1+1}}{1+1} + C = 2 \times \frac{x^2}{2} + C = x^2 + C

Step 5: Verify by differentiation

\(\frac{d}{dx}\)(x² + C) = 2x ✓

Therefore, ∫2x dx = x² + C

Answer: (A) x²+C

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Q.5 Medium

45° in radians?

A π/4

B π/3

C π/2

D π/6

Correct Answer: A. π/4

EXPLANATION

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

Step 1: Recall the conversion formula from degrees to radians.

\text{Radians} = \text{Degrees} \times \frac{\pi}{180°}

Step 2: Substitute 45° into the formula.

\text{Radians} = 45° \times \frac{\pi}{180°}

Step 3: Simplify the fraction.

= \frac{45\pi}{180} = \frac{\pi}{4}

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

The answer is (A) π/4

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Q.6 Medium

7! (7 factorial) = ?

A 2520

B 5040

C 720

D 1260

Correct Answer: B. 5040

EXPLANATION

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

Step 1: Understand factorial notation

7! means 7 × 6 × 5 × 4 × 3 × 2 × 1

Step 2: Multiply step by step

7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1

Step 3: Calculate from left to right

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

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Q.7 Medium

Solve: x² - 5x + 6 = 0

A x=2,3

B x=1,6

C x=3,4

D x=2,4

Correct Answer: A. x=2,3

EXPLANATION

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

Step 1: Write the equation in standard form.

\[x^2 - 5x + 6 = 0\]

Step 2: Find two numbers that multiply to give 6 (constant term) and add to give -5 (coefficient of x).

The numbers are -2 and -3, because:

(-2) × (-3) = 6

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

Step 3: Factorize the quadratic.

\[(x - 2)(x - 3) = 0\]

Step 4: Solve for x by setting each factor equal to zero.

x - 2 = 0 → x = 2

x - 3 = 0 → x = 3

Step 5: Verify by substituting back.

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).

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Q.8 Medium

Differentiate y = 3x²

A 6x

B 3x

C 6x²

D 3x²

Correct Answer: A. 6x

EXPLANATION

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

Step 1: Identify the function and the rule to apply.

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

Step 2: Recall the Power Rule.

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

Step 3: Apply the Power Rule to our function.

Here, a = 3 and n = 2

\frac{dy}{dx} = 2 \cdot 3 \cdot x^{2-1}

Step 4: Simplify.

\frac{dy}{dx} = 6 \cdot x^1 = 6x

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

Answer: A) 6x

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Q.9 Medium

Median of: 5,8,3,9,1,7,6?

A 6

B 5

C 7

D 8

Correct Answer: A. 6

EXPLANATION

To find the median, we must first arrange all numbers in ascending order, then locate the middle value.

Step 1: Count the total numbers

We have 7 numbers: 5, 8, 3, 9, 1, 7, 6

Step 2: Arrange in ascending order

1, 3, 5, 6, 7, 8, 9

Step 3: Find the position of median

Since n = 7 (odd number), median position = \(\frac{n+1}{2} = \frac{7+1}{2} = 4\)

Step 4: Identify the 4th element

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

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Q.10 Medium Geometry

A regular hexagon is inscribed in a circle of radius 6 cm. What is the difference between the perimeter of the hexagon and the circumference of the circle?

A 36 - 12π cm

B 12π - 36 cm

C 36 - 6π cm

D 6π - 36 cm

Correct Answer: B. 12π - 36 cm

EXPLANATION

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).

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