Mix Test - 5

Let the third proportional be ‘x’.

6 : 12 ∷ 12 : x

$\Rightarrow \frac{6}{12}=\frac{12}{\mathrm{x}}$

⇒ x = 24

∴ The third proportional to 6 and 12 = 24

Hence, the correct option is (C).

Given that: 

\(\mathrm{A}: \mathrm{B}: \mathrm{C}=2: 3: 4\)

Let \(\mathrm{A}=2 x, \mathrm{B}=3 x, \mathrm{C}=4 x\)

\(\therefore \frac{\mathrm{A}}{\mathrm{B}}: \frac{\mathrm{B}}{\mathrm{C}}: \frac{\mathrm{C}}{\mathrm{A}}=\frac{2 x}{3 x}: \frac{3 x}{4 x}: \frac{4 x}{2 x}\)

\(\Rightarrow \frac{2}{3}: \frac{3}{4}: \frac{2}{1}\)

Multiply by the L.C.M of denominator to remove fraction

So,

L.C.M of \((3,4,1)=12\)

\(\therefore \frac{\mathrm{A}}{\mathrm{B}}: \frac{\mathrm{B}}{\mathrm{C}}: \frac{\mathrm{C}}{\mathrm{A}}\)

\(\frac{2}{3} \times 12: \frac{3}{4} \times 12: \frac{2}{1} \times 12\)

\(\Rightarrow 8: 9:24\)

Hence, the correct option is (C).

For no solution, \(\frac{a _{1}}{a _{2}}=\frac{b _{1}}{b _{2}} \neq \frac{c _{1}}{c _{2}}\)

\(\Rightarrow a _{1}=3, b _{1}=- k\) and \(c _{1}=-20\) for \(3 x - ky -20=0\)

\(\Rightarrow a _{2}=6, b _{1}=-10\) and \(c _{1}=40\) for \(6 x -10 y +40=0\)

\(\Rightarrow \frac{a _{1}}{a _{2}}=\frac{b _{1}}{b _{2}}\)

\(\Rightarrow \frac 36=\frac{- k}{-10}\)

\(\Rightarrow k =5\)

Checking if \(\frac{b _{1}}{b _{2}} \neq \frac{c _{1}}{c _{2}}\)

\(\Rightarrow \frac 5 {-10}(=2) \neq-\frac{20}{40}(=-2)\)

\(\therefore\) The equations have no solution for \(k =5\)

Hence, the correct option is (C).

Let denominator of fraction = x

Then, numerator = x – 4 

∴ Fraction = $\frac{(x -4)}{x}$

Now, according to the question, 

$\frac{(x-4)-2)}{(x+1)}=\frac{1}{8}$

⇒ x-6 = $\frac{(x+1)}{8}$

⇒ 8x – 48 = x + 1 

⇒ 8x – x = 48 + 1 

7x = 49 

∴ x = 7 

∴ Fraction = $\frac{(7 – 4 )}{7}=\frac{3}{7}$ 

Hence, the correct option is (A).

Given:

C.P of bicycle = 1250

Expend 20% of C.P for repairing.

S.P = 1650

Formula:

Profit/Loss % = $\frac{\mathrm{S}.\mathrm{P}-\mathrm{C}.\mathrm{P}}{\mathrm{C}.\mathrm{P}}\times 100$

Calculation:

Expent is 20% of C.P = 1250 × 0.20

= Rs. 250

Total C.P = 1250 + 250

= 1500

Profit % = $\frac{1650-1500}{1500}\times 100$

$=\frac{150}{1500}\times 100$

= 10% gain

Hence, the correct option is (B).

Given,

\(x=(633)^{24}-(277)^{38}+(266)^{54}\)

For the unit digit,

\(24 = 4\times 6+0\) (Remainder)

\(38 = 4\times 9+2\) (Remainder)

\(54 = 4\times 13+2\) (Remainder)

Now, 

(Base number unit digit)^remainder

\(\Rightarrow (3)^{0}-(7)^{2}+(6)^{2}\)

On consider unit digit,

\(\Rightarrow 1-9+6\)

\(\Rightarrow 7-9\)

or \(17-9=8\)

\(8\) is the units digit of \(x\).

Hence, the correct option is (C).

Given:

Principal, P = Rs. 3000

Rate, R = 15%

Time, T = 8 months = $\left(\frac{8}{12}\right)$ year

Formula:

Simple Interest = $\frac{\mathrm{PRT}}{100}$

Calculation:

According to the formula,

Simple Interest $=\frac{3000\times 15\times \left(\frac{8}{12}\right)}{100}$

Simple Interest $=\frac{30000}{100}$

Simple Interest = Rs. 300

∴ The Simple interest is Rs. 300.

Hence, the correct option is (A).

Given:

C.P. = Rs. 220

S.P. = Rs. 206.8

Formula:

D = M.P – S.P

D% = $\left(\frac{\mathrm{D}}{\mathrm{M}.\mathrm{P}}\right)\times 100$

Where,

D → Discount

M.P → Marked price

S.P → Selling price

D% → Discount%

Calculation:

C.P. = Rs. 220

S.P. = Rs. 206.8

Discount = 220 – 206.8 = 13.2

∴ Discount% = $\left(\frac{13.2}{220}\right)\times 100$

⇒ 0.6 × 10 = 6%

∴ A discount of 6% was offered.

Hence, the correct option is (B).

Given

On selling article for Rs 51 more, Vishnu is getting 3% more profit.

Formula

Selling price = Cost price × $\left[\frac{\left(100 + Profit\%\right)}{100}\right]$

Let CP be 100x.

SP₁ = 100x × $\left(\frac{125}{100}\right)$ = 125x

SP₂ = 100x × $\left(\frac{128}{100}\right)$ = 128x

According to question

128x - 125x = 51

⇒ 3x = 51

⇒ x = 17

∴ Cost price = 100x = 100 × 17 = 1700.

Hence, the correct option is (B).

Concept:

If N = a^p × b^q × c^r...

Then number of factors = (p + 1)(q + 1)(r + 1).....

According to the question,

First prime factorize number 360,

360 = 2³ × 3² × 5¹

On comparing, with concept,

p = 3, q = 2 and r = 1

No. of factors = (3 + 1)(2 + 1)(1 + 1)

⇒ 4 × 3 × 2 = 24

∴ The number of factors of 360 is 24.

Hence, the correct option is (B).