Mix Test - 7

Given:

\(x^{2}+18 x=-p\)

\(x^{2}+18 x+p=0\)

Concept Used:

If a quadratic equation \(\left(a x^{2}+b x+c=0\right)\) has equal roots, then discriminant should be zero i.e. \(b^{2}-\) \(4 {ac}=0\)\(x^{2}+18 x+p=0\)

Since, the equation has only one solution, therefore, roots will be equal.

Therefore, \(b^{2}-4 a c=0\)

\(\Rightarrow(18)^{2}-4(1)(p)=0\)

\(\Rightarrow 4 p=324\)

\(\Rightarrow {p}=\frac{324}{4}\)

\(\Rightarrow {p}=81\)

\(\therefore\) The value of \(p\) is 81 .

Given,

The volume of the cube = 216 cm³

All sides of a cube are equal.

The volume of the cube = a³

The total surface area of the cube = 6a²

Where a is the side of the cube.

According to the question,

a³ = 216

\(\Rightarrow\) a = \((216)^{\frac{1}{3}}\)

\(\Rightarrow\) a = \((6 \times 6 \times 6)^{\frac{1}{3}}\)

\(\Rightarrow\) a = \(\left(6^{3}\right)^{\frac{1}{3}}\)

⇒ a = 6 cm

The total surface area of cube = 6 × 6²

= 6 × 36

= 216 cm²

Hence, the correct option is (A).

Given:

Mark price of the article is 800 and the consecutive discounts are 20% and 10%.

Formula:

Overall discount percent = $\left(\mathrm{x}+\mathrm{y}-\frac{\mathrm{xy}}{100}\right)\%$

Here x = First discount percent

y = Second discount percent

Calculation:

Overall discount percent = $\left(20+10-20\times \frac{10}{100}\right)\%$

= (30 - 2)%

= 28%

So, discount on mark price = 800 × 28%

$=800\times \frac{28}{100}$

= 224

So, Selling price = Mark price - Discount

= 800 - 224

= 576

∴ The Selling price of the article is 576.

Hence, the correct option is (C).

Ram scored \(=80 \%\) marks

Total marks \(=300\)

Ram scored \(=80 \%\) of \(300\)

\(=\frac{80}{100}\times300\)

\(=80\times3\)

\(=240\) marks

So, Ram scored \(240\) marks.

Hence, the correct option is (A).

Given:

$\sqrt{0.09}$

$=\sqrt{\frac{9}{100}}$

$=\frac{3}{10}$

$=0.3$

Hence, the correct option is (D).

Given:

The number = 540

Factorization = 2 × 270

= 2 × 3 × 90

= 2 × 3 × 2 × 45

= 2 × 3 × 2 × 3 × 15

=  2 × 3 × 2 × 3 × 3 × 5

= 2² × 3³ × 5¹

Number of odd factors of N = (q + 1) × (r + 1)

Here, q = 3 and r = 1

Number of odd factors of 540 = (3 + 1)(1 + 1) = 8

∴ The number of odd factors of 540 is 8.

Hence, the correct option is (B).

Given:

Discount% = 10%

Profit% = 25%

CP = 360

Formula:

Discount% = $\left(\frac{\mathrm{MP}–\mathrm{SP}}{\mathrm{MP}}\right)\times 100$     [where MP = Marked Price and SP = Selling Price] 

Profit% = $\left(\frac{\mathrm{CP}-\mathrm{SP}}{\mathrm{CP}}\right)\times 100$    [where CP = Cost Price and SP = Selling Price]

Let the Marked Price be 100x.

SP = $100\mathrm{x}\left(1-\frac{10}{100}\right)$

= 90x

According to Condition,

$90\mathrm{x}=360\times \left(1+\frac{25}{100}\right)$

$\Rightarrow 90\mathrm{x}=360\times \frac{5}{4}$

⇒ 90x = 450

⇒ x = 5

∴ Marked Price of the article = 100 × 5

= Rs. 500

Hence, the correct option is (D).

Let the amount received by P, Q, R and S be $\frac{x}{5}$, $\frac{x}{4}$, $\frac{x}{6}$ and $\frac{x}{5}$. 

So,

= $\frac{x}{5}$ + $\frac{x}{4}$ + $\frac{x}{6}$ + $\frac{x}{5}$ = 9996

⇒ x = $\left(\frac{60}{49}\right)$ × 9996

Thus the amount that R will get = $\left(\frac{60}{49}\right)$ × $\frac{1000}{6}$

= $\frac{99960}{49}$

= 204

Hence, the correct option is (C).

Calculation:

\(x^{2}-5|x|+4=0\)

If \(x \geq 0\) i.e. \(|x|=x\)

\(\therefore\) the given equation can be written as:

\(x^{2}-5 x+4=0\)

\(\Rightarrow x^{2}-4 x-x+4=0\)

\(\Rightarrow(x-4)(x-1)\)

\(\Rightarrow \mathrm{x}=4,1\)

Similarly, for \(x<0, x^{2}-5|x|+4=0\)

\(x^{2}+5 x+4=0\)

\(\Rightarrow x^{2}+4 x+x+4=0\)

\(\Rightarrow(x+4)(x+1)\)

\(\Rightarrow x=-4,-1\)

\(\therefore\) there are four solutions \(1,-1,4,-4 .\)

Given:

Principal = Rs.13000 

Rate of interest = 15%

Effective rate of interest = 15 × \(\frac{8}{12}\) = 10%

and 2 years = 24 months = 3 eight monthly

As we know,

Compound interest \(=P(1+\frac{R }{100})^{3}-P\)

Where 

P = Principal

R = rate of interest and 

N = time period

\(\therefore\) Compound interest \(=13000(1+\frac{10}{100})^{3}-13000\) 

= Rs. 4303

Hence, the correct option is (D).