Quants 4 — BSSC Quantitative Aptitude

• Key Fact
A number divisible by 88 must be divisible by both 8 and 11. Divisibility by 8 requires the last three digits $2y8$ to be divisible by 8. Testing natural numbers for $y$: $y=1 \implies 218$ (not div), $y=2 \implies 228$ (not div), $y=3 \implies 238$ (not div), $y=4 \implies 248$ ($248 \div 8 = 31$, divisible). Thus, minimum natural number $y = 4$. For divisibility by 11 in $85x346248$: (Sum of odd places) - (Sum of even places) = $(8 + 2 + 4 + x + 8) - (4 + 6 + 3 + 5) = (22 + x) - 18 = 4 + x$. For $4 + x$ to be divisible by 11, $x = 7$. Thus, $x^2 + y^2 = 7^2 + 4^2 = 49 + 16 = 65$.

• Related Formula(s)
$ ext{Divisibility by } 88 = ext{Divisible by } 8 ext{ and } 11$.

• Shortcut / Trick
For divisibility of $2y8$ by 8, since hundreds digit is even (2), the last two digits $y8$ must be divisible by 8 directly. The smallest natural number giving a multiple of 8 ending in 8 is $48$, which immediately gives $y = 4$ without testing 1, 2, or 3.

• Why wrong options are wrong
50: Obtained if $y=1$ is incorrectly assumed.
80: Obtained by taking $x=8, y=4$ ($64+16=80$) from a parity miscount.
85: Arithmetic calculation slip.

• Time-Saving Tip
When the hundreds digit is even, the entire 3-digit number is divisible by 8 if and only if the last two digits are divisible by 8.

• Additional Info
If $y$ was allowed to be zero, $208$ would also be divisible by 8, giving $y=0$, but the question strictly specified a natural number ($y > 0$).

• Key Fact
$5 imes 19 = 95$. Subtracting 95 from 547236: $547236 - 95 = 547141$. Checking options: (1) Last digit is 1, so not divisible by 2. (2) Last digit is not 0 or 5, so not divisible by 5. (3) Sum of digits = $5 + 4 + 7 + 1 + 4 + 1 = 22$, which is not divisible by 3. By elimination, it must be divisible by 7. Verification: $547141 \div 7 = 78163$ (exact).

• Related Formula(s)
$ ext{Divisibility by } 2, 3, 5, 7$.

• Shortcut / Trick
Never divide by 7 first. Eliminate 2 and 5 in half a second by looking at the unit digit (1). Eliminate 3 in two seconds via digital sum ($22$ is not divisible by 3). The only remaining option is 7.

• Why wrong options are wrong
2: Fails because the resulting unit digit is 1 (odd).
3: Fails because the sum of digits is 22, not a multiple of 3.
5: Fails because the unit digit is neither 0 nor 5.

• Time-Saving Tip
In govt exam divisibility tests with 7 as an option, always test and eliminate 2, 5, and 3 first. Dividing by 7 takes the most time.

• Additional Info
Divisibility rule for 7: Double the last digit and subtract from the remaining truncated number ($54714 - 2 = 54712$, repeat as needed).

• Key Fact
For telescoping series \sum rac{1}{n(n+d)} = rac{1}{d}\left(rac{1}{ ext{first}} - rac{1}{ ext{last}} ight). First bracket: $d = 1$, sum = 1 - rac{1}{14} = rac{13}{14}. Term 1 = 7 imes rac{13}{14} = rac{13}{2}. Second bracket: $d = 2$, sum = $rac{1}{2}\left(1 - rac{1}{21} ight) = rac{1}{2} imes rac{20}{21} = rac{10}{21}$. Term 2 = 10 imes rac{10}{21} = rac{100}{21}. Total sum = $rac{13}{2} + rac{100}{21} = rac{273 + 200}{42} = rac{473}{42}$.

• Related Formula(s)
$ ext{Sum} = rac{1}{ ext{common difference}} imes \left(rac{1}{ ext{first factor}} - rac{1}{ ext{last factor}} ight)$.

• Shortcut / Trick
Notice the denominators in the final sum are 2 and 21, so the LCM of the denominator must be 42. Mentally calculate $7 imes (13/14) = 6.5$ and $10 imes (1/2) imes (20/21) = 100/21 \approx 4.76$. Total $\approx 11.26$. $rac{473}{42} = 11.26$. Matches instantly.

• Why wrong options are wrong
$rac{455}{42}$: Forgot to multiply the first bracket by 7.
$rac{463}{42}$: Miscalculated the numerator sum as $263 + 200$.
$rac{485}{42}$: Missed dividing the second series by the common difference of 2.

• Time-Saving Tip
Always remember to divide by the common difference between factors in each fraction denominator ($d=2$ for the second series).

• Additional Info
SSC often presents two mixed telescoping series by factoring out constants (like 7 and 10 here) to disguise standard patterns.

• Key Fact
Let the numbers be $a$ and $b$. Sum of reciprocals = $rac{1}{a} + rac{1}{b} = rac{a + b}{ab}$. We know that product of two numbers $ab = ext{HCF} imes ext{LCM} = 4 imes 168$. Thus, $rac{1}{a} + rac{1}{b} = rac{52}{4 imes 168} = rac{13}{168}$.

• Related Formula(s)
$ ext{Product of two numbers} = ext{HCF} imes ext{LCM}$.
$rac{1}{a} + rac{1}{b} = rac{ ext{Sum}}{ ext{HCF} imes ext{LCM}}$.

• Shortcut / Trick
Direct shortcut formula: Divide the given sum directly by the HCF, then place the result over the LCM. $rac{52 / 4}{168} = rac{13}{168}$. Takes 3 seconds without finding individual numbers.

• Why wrong options are wrong
$rac{11}{168}$: Miscalculation in dividing 52 by 4.
$rac{13}{84}$: Divided by 2 instead of taking full LCM.
$rac{17}{168}$: Arithmetic error.

• Time-Saving Tip
Never waste time finding the values of $a$ and $b$ using quadratic factorization when only symmetric expressions like $(a+b)$ or $(1/a + 1/b)$ are required.

• Additional Info
The individual numbers here are 12 and 40 (sum = 52, HCF = 4, LCM = 168).

• Key Fact
Order of operations follows VBODMAS (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction). Step 1 (Vinculum/Bar): $\overline{9 - 4} = 5$. Step 2 (Small bracket): $14 - 5 = 9$. Step 3 (Curly bracket): $24 - 9 = 15$. Step 4 (Square bracket): $3 + 15 = 18$. Step 5 (Division): $90 \div 18 = 5$.

• Related Formula(s)
VBODMAS Rule priority: $\overline{ ext{Bar}} o () o \{\} o [] o ext{Of} o \div o imes o + o -$.

• Shortcut / Trick
Work inside-out mentally: $9 - 4 = 5 \implies 14 - 5 = 9 \implies 24 - 9 = 15 \implies 3 + 15 = 18 \implies 90 / 18 = 5$. Can be completed within 5 seconds without pen.

• Why wrong options are wrong
3: Arithmetic slip in dividing $90 / 18$.
6: Evaluated $14 - 9 - 4 = 1$ by ignoring the vinculum scope.
9: Missed adding the 3 inside the square bracket.

• Time-Saving Tip
A vinculum (bar) acts exactly like a pair of parentheses around the numbers beneath it. Solve it before anything else.

• Additional Info
If the expression was $-(9-4)$ without a bar, signs would distribute differently if not careful.

• Key Fact
Number of bricks = $rac{ ext{Volume of wall}}{ ext{Volume of 1 brick}}$. Convert all dimensions to cm: Wall = $1600 ext{ cm} imes 75 ext{ cm} imes 600 ext{ cm}$. Brick = $20 ext{ cm} imes 10 ext{ cm} imes 7.5 ext{ cm}$. Number of bricks = $rac{1600 imes 75 imes 600}{20 imes 10 imes 7.5} = rac{1600}{200} imes rac{75}{7.5} imes 600 = 8 imes 10 imes 600 = 48000$.

• Related Formula(s)
$ ext{Number of units} = rac{ ext{Total Volume}}{ ext{Volume of one unit}}$.

• Shortcut / Trick
Pair and cancel terms directly: $rac{75 ext{ cm}}{7.5 ext{ cm}} = 10$. $rac{1600}{20} = 80$. $rac{600}{10} = 60$. $80 imes 10 imes 60 = 48000$.

• Why wrong options are wrong
40,000: Conversion error with meters to centimeters.
44,000: Calculation error.
52,000: Incorrect division of the height dimension.

• Time-Saving Tip
Do not compute the full cubic product before dividing. Keep numbers as separated factors for instant pairwise cancellation.

• Additional Info
If mortar occupies a percentage (e.g., 10%) of the wall, multiply the wall volume by 0.9 before dividing.

• Key Fact
Use identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Here, $\cot^3 A - 1 = (\cot A - 1)(\cot^2 A + \cot A + 1)$. The expression becomes: $rac{(\cot A - 1)(\cot^2 A + \cot A + 1)}{\cot A - 1} - \cot A = (\cot^2 A + \cot A + 1) - \cot A = \cot^2 A + 1$. By fundamental identity, $1 + \cot^2 A = \csc^2 A$.

• Related Formula(s)
$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$.
$1 + \cot^2 heta = \csc^2 heta$.

• Shortcut / Trick
Value putting: Put $A = 45^\circ$ causes $1 - 1 = 0$ in denominator (invalid). Put $A = 30^\circ$: $\cot 30^\circ = \sqrt{3}$. Expression = $rac{3\sqrt{3}-1}{\sqrt{3}-1} - \sqrt{3} = rac{3\sqrt{3}-1-3+\sqrt{3}}{\sqrt{3}-1} = rac{4\sqrt{3}-4}{\sqrt{3}-1} = 4$. Check options at $30^\circ$: $\csc^2 30^\circ = 2^2 = 4$. Option 2 matches uniquely!

• Why wrong options are wrong
$\sec^2 A$: Confusing $ an$ and $\cot$ identities.
$ an^2 A$: Algebraic sign mistake.
$\sin^2 A$: Inverting the identity result.

• Time-Saving Tip
Recognizing algebraic expansions like $(x^3-1)/(x-1) = x^2+x+1$ saves using pen for value substitution altogether.

• Additional Info
Similarly, $(1 - an^3 A)/(1 - an A) - an A = \sec^2 A$.

• Key Fact
Rationalize the denominator by multiplying numerator and denominator by $(\sqrt{5} + \sqrt{2})$: $rac{(\sqrt{5} + \sqrt{2})^2}{(\sqrt{5})^2 - (\sqrt{2})^2} = rac{5 + 2 + 2\sqrt{10}}{5 - 2} = rac{7 + 2\sqrt{10}}{3}$. Substitute $\sqrt{10} = 3.162$: $rac{7 + 2(3.162)}{3} = rac{7 + 6.324}{3} = rac{13.324}{3} = 4.44133... \approx 4.441$.

• Related Formula(s)
$rac{1}{\sqrt{a} - \sqrt{b}} = rac{\sqrt{a} + \sqrt{b}}{a - b}$.
$(a + b)^2 = a^2 + b^2 + 2ab$.

• Shortcut / Trick
Numerator is $a+b+2\sqrt{ab} = 7 + 2(3.16) = 13.32$. Denominator is $a-b = 3$. $13.32 / 3 = 4.44$. Direct 5-second mental math.

• Why wrong options are wrong
4.124: Subtracted $2\sqrt{10}$ instead of adding.
4.782: Divided by 2 instead of $(5 - 2 = 3)$.
5.122: Calculation slip in doubling $3.162$.

• Time-Saving Tip
For $rac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}$, directly write $rac{(a+b) + 2\sqrt{ab}}{a-b}$ in one line without intermediate rationalization steps.

• Additional Info
If the signs were inverted (minus on top), the formula is $rac{(a+b) - 2\sqrt{ab}}{a-b}$.

• Key Fact
20% more means Length : Breadth = $6 : 5$. Let Length = $6x$ and Breadth = $5x$. Area = Length $ imes$ Breadth = $6x imes 5x = 30x^2 = 750 \implies x^2 = 25 \implies x = 5$. Length = $6(5) = 30 ext{ m}$, Breadth = $5(5) = 25 ext{ m}$. Perimeter = $2(L + B) = 2(30 + 25) = 2(55) = 110 ext{ m}$.

• Related Formula(s)
$ ext{Area of Rectangle} = L imes B$.
$ ext{Perimeter of Rectangle} = 2(L + B)$.

• Shortcut / Trick
Perimeter is $2(6x + 5x) = 22x$. The answer must be a multiple of 11! Among the options (100, 110, 120, 130), only 110 is divisible by 11. Solved in 2 seconds without calculating $x$.

• Why wrong options are wrong
100 m: Calculation slip in perimeter.
120 m: Assumed length was 40 and breadth 20.
130 m: Random distractor.

• Time-Saving Tip
Check divisibility of the final ratio expression ($2(6+5) = 22$, multiple of 11) directly against the options.

• Additional Info
If asked for the diagonal, $D = \sqrt{30^2 + 25^2} = \sqrt{900 + 625} = \sqrt{1525} = 5\sqrt{61} ext{ m}$.

• Key Fact
Sum of interior angles = $(n - 2) imes 180^\circ = 1440^\circ \implies n - 2 = 8 \implies n = 10$. Each exterior angle = $rac{360^\circ}{n} = rac{360^\circ}{10} = 36^\circ$. Each interior angle = $180^\circ - 36^\circ = 144^\circ$. Difference = Interior - Exterior = $144^\circ - 36^\circ = 108^\circ$.

• Related Formula(s)
$ ext{Sum of interior angles} = (n - 2) imes 180^\circ$.
$ ext{Exterior angle} = rac{360^\circ}{n}$.
$ ext{Interior} + ext{Exterior} = 180^\circ$.

• Shortcut / Trick
Difference between interior and exterior angle = $(180^\circ - E) - E = 180^\circ - 2E$. For $n=10$, $E = 36^\circ$. Difference = $180^\circ - 2(36^\circ) = 180^\circ - 72^\circ = 108^\circ$.

• Why wrong options are wrong
$96^\circ$: Miscalculated $n=12$.
$100^\circ$: Arithmetic error from another polygon type.
$120^\circ$: Difference calculated for an octagon ($n=8$).

• Time-Saving Tip
Never calculate the individual interior angle from the total sum. First find the exterior angle ($360^\circ/n$), then use $ ext{Diff} = 180^\circ - 2E$.

• Additional Info
Number of diagonals in this 10-sided polygon (decagon) = $rac{n(n - 3)}{2} = rac{10 imes 7}{2} = 35$.