Question 01 BODMAS Operations Order

Simplify: \(100 + 8 \div 2 + \{(3 \times 2) - 6 \div 2\}\)

(A) \(123\)

(B) \(107\) ✓

(C) \(113\)

(D) \(103\)

💡 Explanation:

Step 1 (Inner Brackets): \((3 \times 2) = 6\) and \(6 \div 2 = 3\) \[\{(3 \times 2) - 6 \div 2\} = \{6 - 3\} = 3\]
Step 2 (Division): \(8 \div 2 = 4\)
Step 3 (Addition): \[100 + 4 + 3 = \mathbf{107}\]

Correct Option: (B) 107

Question 02 Mixed Fraction Summation

Simplify: \(999\frac{1}{7} + 999\frac{2}{7} + 999\frac{3}{7} + 999\frac{4}{7} + 999\frac{5}{7} + 999\frac{6}{7}\)

(A) \(2997\)

(B) \(5979\)

(C) \(5994\)

(D) \(5997\) ✓

💡 Explanation:

Separate the whole numbers and fractional parts: \[= (999 + 999 + 999 + 999 + 999 + 999) + \left(\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7}\right)\]
Whole number part: \(999 \times 6 = 5994\)
Fractional part: \(\frac{1 + 2 + 3 + 4 + 5 + 6}{7} = \frac{21}{7} = 3\)
Total Sum: \(5994 + 3 = \mathbf{5997}\)

Correct Option: (D) 5997

Question 03 Nested Square Root Radical

Simplify: \(\sqrt{49} + \sqrt{220 + \sqrt{16 + \sqrt{81}}}\)

(A) \(15\)

(B) \(12\)

(C) \(22\) ✓

(D) \(7\)

💡 Explanation:

Evaluate step-by-step from the innermost radical: \[\sqrt{81} = 9\] \[\sqrt{16 + 9} = \sqrt{25} = 5\] \[\sqrt{220 + 5} = \sqrt{225} = 15\]
Outer term: \(\sqrt{49} = 7\)
Total value: \(7 + 15 = \mathbf{22}\)

Correct Option: (C) 22

Question 04 Infinite Multiplicative Nested Radical

Find the value of \(\sqrt{3\sqrt{3\sqrt{3\cdots}}}\):

(A) \(\sqrt{3}\)

(B) \(3\) ✓

(C) \(\sqrt{3\sqrt{3}}\)

(D) \(27\)

💡 Explanation:

Let \(x = \sqrt{3\sqrt{3\sqrt{3\cdots}}}\).
Squaring both sides: \[x^2 = 3\sqrt{3\sqrt{3\cdots}}\] \[x^2 = 3x \implies x^2 - 3x = 0 \implies x(x - 3) = 0\]
Since \(x \neq 0\), we have \(x = \mathbf{3}\).

Correct Option: (B) 3

Question 05 Natural Number to Cubic Series Transformation

If \(1 + 2 + 3 + \dots + k = 55\), then find the value of \(1^3 + 2^3 + 3^3 + \dots + k^3\):

(A) \(3025\) ✓

(B) \(3125\)

(C) \(166375\)

(D) \(171875\)

💡 Explanation:

Identity: The sum of cubes of first \(k\) natural numbers equals the square of their linear sum: \[1^3 + 2^3 + 3^3 + \dots + k^3 = (1 + 2 + 3 + \dots + k)^2\]
Given \(\sum k = 55\): \[\sum k^3 = (55)^2 = \mathbf{3025}\]

Correct Option: (A) 3025

Question 06 Law of Indices (Base Inversion)

If \(\left(\frac{p}{q}\right)^{1-3x} = \left(\frac{q}{p}\right)^{\frac{1}{2}}\), then find the value of \(x\):

(A) \(4^{-1}\)

(B) \(3^{-1}\)

(C) \(2^{-1}\) ✓

(D) \(1^{-1}\)

💡 Explanation:

Invert the right-hand base using \(\left(\frac{q}{p}\right)^a = \left(\frac{p}{q}\right)^{-a}\): \[\left(\frac{p}{q}\right)^{1-3x} = \left(\frac{p}{q}\right)^{-\frac{1}{2}}\]
Equating the exponents: \[1 - 3x = -\frac{1}{2} \implies 3x = 1 + \frac{1}{2} = \frac{3}{2}\] \[x = \frac{3}{2 \times 3} = \frac{1}{2} = \mathbf{2^{-1}}\]

Correct Option: (C) 2⁻¹

Question 07 Cyclic Exponent Indices Product

Simplify: \(\left(\frac{m^a}{m^b}\right)^{a+b} \times \left(\frac{m^b}{m^c}\right)^{b+c} \times \left(\frac{m^c}{m^a}\right)^{c+a}\)

(A) \(\frac{1}{m}\)

(B) \(m\)

(C) \(1\) ✓

(D) \(m^{(a+b+c)}\)

💡 Explanation:

Simplify each cyclic factor using \(\frac{m^a}{m^b} = m^{a-b}\) and \((m^{a-b})^{a+b} = m^{(a-b)(a+b)} = m^{a^2 - b^2}\): \[= m^{a^2 - b^2} \times m^{b^2 - c^2} \times m^{c^2 - a^2}\]
Adding the exponents: \[= m^{(a^2 - b^2) + (b^2 - c^2) + (c^2 - a^2)} = m^0 = \mathbf{1}\]

Correct Option: (C) 1

Question 08 Sum of First \(n\) Odd Numbers

Find the sum of \(1 + 3 + 5 + \dots\) to \(40\text{ terms}\):

(A) \(800\)

(B) \(4000\)

(C) \(1600\) ✓

(D) \(8000\)

💡 Explanation:

Formula: The sum of the first \(n\) consecutive odd natural numbers is: \[S_n = n^2\]
For \(n = 40\text{ terms}\): \[S_{40} = 40^2 = \mathbf{1600}\]

Correct Option: (C) 1600

Question 09 Solving for \(n\) in Triangular Sum

If \(1 + 2 + 3 + \dots + n = 666\), then find the value of \(n\):

(A) \(37\)

(B) \(36\) ✓

(C) \(38\)

(D) \(35\)

💡 Explanation:

Formula for sum of first \(n\) natural numbers: \[\frac{n(n+1)}{2} = 666 \implies n(n+1) = 1332\]
Express \(1332\) as product of two consecutive integers: \[1332 = 36 \times 37 \implies n(n+1) = 36 \times 37 \implies n = \mathbf{36}\]

Correct Option: (B) 36

Question 10 Rational Symmetric Sign Simplification

Simplify: \(\frac{x^3}{x - y} + \frac{y^3}{y - x}\)

(A) \(x^2 + xy + y^2\) ✓

(B) \(x^2 - xy + y^2\)

(C) \(x^2 - xy - y^2\)

(D) \(x^2 + xy - y^2\)

💡 Explanation:

Note that \(y - x = -(x - y)\): \[\frac{x^3}{x - y} - \frac{y^3}{x - y} = \frac{x^3 - y^3}{x - y}\]
Apply the algebraic cubic expansion \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\): \[= \frac{(x - y)(x^2 + xy + y^2)}{x - y} = \mathbf{x^2 + xy + y^2}\]

Correct Option: (A) x² + xy + y²

Question 11 Rational Algebraic Polynomial Division

If \(\frac{x^2 - 25}{x + 3}\) is divided by \(\frac{x + 5}{x^2 - 9}\), then the quotient is:

(A) \((x - 5)(x + 3)\)

(B) \((x - 5)(x - 3)\) ✓

(C) \((x + 5)(x - 3)\)

(D) \((x + 5)(x + 3)\)

💡 Explanation:

Invert the divisor and factorize differences of squares: \[\frac{x^2 - 25}{x + 3} \div \frac{x + 5}{x^2 - 9} = \frac{(x - 5)(x + 5)}{x + 3} \times \frac{(x - 3)(x + 3)}{x + 5}\]
Cancel the common factors \((x + 5)\) and \((x + 3)\): \[= \mathbf{(x - 5)(x - 3)}\]

Correct Option: (B) (x - 5)(x - 3)

Question 12 Rational Quadratic Fractions Addition

Simplify: \(\frac{x+2}{x^2 + 3x + 2} + \frac{x-3}{x^2 - 2x - 3}\)

(A) \(\frac{1}{x+1}\)

(B) \(\frac{-1}{x+1}\)

(C) \(\frac{2}{x+1}\) ✓

(D) \(\frac{-2}{x+1}\)

💡 Explanation:

Factorize the denominators: \[x^2 + 3x + 2 = (x + 1)(x + 2)\] \[x^2 - 2x - 3 = (x - 3)(x + 1)\]
Substitute and cancel common terms: \[\frac{x+2}{(x+1)(x+2)} + \frac{x-3}{(x-3)(x+1)} = \frac{1}{x+1} + \frac{1}{x+1} = \mathbf{\frac{2}{x+1}}\]

Correct Option: (C) \(\frac{2}{x+1}\)

Question 13 Cubic Identity Decimal Reduction

Find the value of \(\frac{8.94 \times 8.94 \times 8.94 - 3.56 \times 3.56 \times 3.56}{8.94 \times 8.94 + 8.94 \times 3.56 + 3.56 \times 3.56}\):

(A) \(0.538\)

(B) \(5.38\) ✓

(C) \(0.0538\)

(D) \(53.8\)

💡 Explanation:

Let \(a = 8.94\) and \(b = 3.56\). The expression is in the form: \[\frac{a^3 - b^3}{a^2 + ab + b^2}\]
Using algebraic identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\): \[= \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} = a - b\]
Calculate: \[8.94 - 3.56 = \mathbf{5.38}\]

Correct Option: (B) 5.38

Question 14 Nested Multi-Bracket Fraction Simplification

Simplify: \(\left(-\frac{1}{3}\right) - \left(\frac{1}{3} \div \left(\frac{2}{3} \times \frac{5}{7}\right) + 8 - \left[5 - \frac{1}{2} - \frac{1}{4}\right]\right)\)

(A) \(-5\frac{41}{60}\)

(B) \(-5\frac{1}{60}\)

(C) \(5\frac{41}{60}\)

(D) \(-3\frac{41}{60}\) ✓

💡 Explanation:

Step 1: Evaluate square bracket: \[5 - \frac{1}{2} - \frac{1}{4} = 5 - \frac{3}{4} = \frac{17}{4}\]
Step 2: Evaluate division part: \[\frac{2}{3} \times \frac{5}{7} = \frac{10}{21} \implies \frac{1}{3} \div \frac{10}{21} = \frac{1}{3} \times \frac{21}{10} = \frac{7}{10}\]
Step 3: Combine inside main bracket: \[\frac{7}{10} + 8 - \frac{17}{4} = \frac{14 + 160 - 85}{20} = \frac{89}{20}\]
Step 4: Outer subtraction: \[-\frac{1}{3} - \frac{89}{20} = \frac{-20 - 267}{60} = \frac{-287}{60} = \mathbf{-3\frac{41}{60}}\]

Correct Option: (D) \(-3\frac{41}{60}\)

Question 15 Geometric Series Term Count Determination

How many terms of the geometric series \(1 + 4 + 16 + \dots\) make the sum \(1365\)?

(A) \(4\)

(B) \(6\) ✓

(C) \(8\)

(D) \(16\)

💡 Explanation:

Here, first term \(a = 1\), common ratio \(r = \frac{4}{1} = 4 > 1\).
Formula for sum of GP: \[S_n = \frac{a(r^n - 1)}{r - 1} \implies 1365 = \frac{1(4^n - 1)}{4 - 1}\] \[1365 \times 3 = 4^n - 1\] \[4095 = 4^n - 1 \implies 4^n = 4096\]
Express \(4096\) as power of \(4\): \[4^6 = 4096 \implies n = \mathbf{6}\]

Correct Option: (B) 6

Simplification

Simplification (சுருக்குதல்) • 15 Model Questions

Level 1: Basic BODMAS, Mixed Fraction Series & Standard Surds

Level 2: Indices Laws, Series Summations & Rational Reductions

Level 3: Factored Rational Fractions, Cubic Decimal Identities & Series Limits

Practice quiz for this topic