Tag: maths

Questions Related to maths

If the hyperbolas, $ x^2+3xy+2y^2+2x+3y+2=0 $ and $ x^2+3xy+2y^2+2x+3y+c=0 $ are conjugate of each other, the value of $c$ is equal to

conjugate hyperbola hyperbola conic section maths

  1. $-2$

  2. $4$

  3. $0$

  4. $1$

Show answer
Correct Option: C
Explanation:

The given hyperbola is $x^2+3xy+2y^2+2x+3y+2=0$         ...(1)

We already know that the equation of the asymptote of a hyperbola differs from the hyperbola by a constant.
$\therefore$ Let $x^2+3xy+2y^2+2x+3y+k=0$            ...(2)
be the equation of the asymptotes of the given hyperbola.
Hence, equation (2) must represent a pair of straight lines, the condition for which is
$abc+2fgh-af^2-bg^2-ch^2 = 0$
$\Rightarrow (1)(2)(k)+2(\cfrac 32)(1)(\cfrac 32)-1(\cfrac 32)^2-2(1)^2-k(\cfrac 32)^2=0$
$\Rightarrow 2k+\cfrac 92-\cfrac 94-2-\cfrac 94 k=0$
$\Rightarrow k=1$
Therefore, the asymptotes are given by $x^2+3xy+2y^2+2x+3y+1=0$ .

The equation of conjugate hyperbola is $2A-H=0$
where, $A$ is the equation of asymptotes
            $H$ is the equation of given hyperbola
$2(x^2+3xy+2y^2+2x+3y+1) -$$(x^2+3xy+2y^2+2x+3y+2)=0$
$\therefore x^2+3xy+2y^2+2x+3y=0$

Hence, $c=0$.

If the line $lx+my+n=0$ meets the hyperbola $\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ at the extermities of a pair of conjugate diameters, then

conjugate hyperbola hyperbola conic section maths

  1. $a^{2}l^{2}-b^{2}m^{2}=0$

  2. $a^{2}l^{2}-b^{2}m^{2}=1$

  3. $a^{2}l^{2}-b^{2}m^{2}=2$

  4. $a^{2}l^{2}-b^{2}m^{2}=3$

Show answer
Correct Option: A
Explanation:

Let $ \theta$ and $\phi$ be the essentric angles of the conjugate diameters, 
Then,
$\theta +\phi =\dfrac{\pi}{2}$
$(a\sec \theta,b\tan \theta)   , (a\sec \phi, b \tan\phi)$
$\Rightarrow (a \sec \theta,b \tan \theta)   , (a\ \text{cosec }\theta ,b \cot \theta)$

$(y-b\tan \theta)=\dfrac{b\cot \theta-b\tan\theta}{a\ \text{cosec }\theta-a\sec\theta}(x-a\sec\theta)$

$\Rightarrow ya-ab\tan \theta=xb(\sin\theta+\cos \theta)-ab(\tan\theta+1)$
$ya=xb(\sin\theta+\cos\theta) - ab$
$ ya - xb (\sin\theta+\cot\theta)+ab=0$
$lx+my+n=0$

Comparing both, we get

$\dfrac{a}{m}=\dfrac{-b(\sin\theta+\cos\theta)}{l}=\dfrac{ab}{n}$

On elliminating $\theta $, we can get
$a^{2}l^{2}-b^{2}m^{2}=0$

Find the equation to the hyperbola,conjugate to the hyperbola $ 2x^2+3xy-2y^2-5x+5y+2=0 $.

conjugate hyperbola hyperbola conic section maths

  1. $ 2x^2+3xy-2y^2-5x+5y-8=0 $

  2. $ x^2+3xy-y^2-5x+5y-8=0 $

  3. $ x^2+3xy-y^2-5x+5y+8=0 $

  4. None of these

Show answer
Correct Option: A
Explanation:

Let Asymptotes :
$ 2x^2+3xy-2y^2-5x+5y+\lambda=0 $.
$ \therefore abc+2fgh-af^2-bg^2-ch^2=0 $
$ \lambda=-5 $
Equation Hyperbola + Conjugate Hyperbola$=$2 (Asympototes)
$ \therefore $ Conjugate Hyperbola$=$2 (Asymptotes) $- $Hyperbola
Therefore, equation of conjugate hyperbola is $  2x^2+3xy-2y^2-5x+5y-8=0 $

The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y=0$ is?

conjugate hyperbola hyperbola conic section maths

  1. $x^2+3xy+2y^2+2x+3y+1=0$

  2. $x^2+3xy+2y^2+2x+3y+2=0$

  3. $x^2+3xy+2y^2+2x+3y+3=0$

  4. $x^2+3xy+2y^2+2x+3y+4=0$

Show answer
Correct Option: B
Explanation:
$H:x^2+3xy +2y^2+2x+3y=0$
Let the pair of asympt at is be
$A:x^2 +3xy+2y^2+2x+3y+k=0$
$\therefore \ $ It represent a apir of straight lines, satisfying the condition :
$abc+2fgh-af^2 -bg^2-ch^2=0$
$\Rightarrow \ 1\times 2\times k+2\left (\dfrac {3}{2}\right) \times 1\times \left (\dfrac {3}{2}\right) -1 \left (\dfrac 32\right)^2 -2(1)^2 -k\left (\dfrac 32 \right)^2 =0$
$\Rightarrow \ 2k+\dfrac {9}{4}-2-\dfrac {9}{4}k=0\ \Rightarrow \ k=1$
$\Rightarrow \ A^2 .x^2 +3xy+2y^2 +2x+3y+1=0$
As $H+C=2A\ \Rightarrow \ C$ (conjugate $=2A-H$ hypergate) 
$\Rightarrow \ C:x^2 +3xy+2y^2+2x+3y+2=0$

1 cm = ______ kilometre.

maths tenths and hundredths more money decimal fractions in the units of currency, length, weight, capacity understanding tenth and hundredth part of a number

  1. $100$

  2. $10^{5}$

  3. $10^{-5}$

  4. $10^{-2}$

Show answer
Correct Option: C
Explanation:

$1\ cm=10^{-2}m\1\ m=10^{-3}km$

$\Rightarrow 1\ cm=10^{-2}m=10^{-2}*10^{-3}km=10^{-5}km$

${1m}^{2}=..........{cm}^{2}$

maths tenths and hundredths more money decimal fractions in the units of currency, length, weight, capacity understanding tenth and hundredth part of a number

  1. $100$

  2. $10000$

  3. $10$

  4. $1$

Show answer
Correct Option: B
Explanation:

${1m}^{2} = 1m \times 1m = 100 cm \times 100 cm = 10000cm^2$


Option B is correct.

The number of 25 paise coins in Rs 100 is ______.

maths tenths and hundredths more money decimal fractions in the units of currency, length, weight, capacity understanding tenth and hundredth part of a number

  1. 4

  2. 40

  3. 100

  4. 400

Show answer
Correct Option: D
Explanation:

$Rs.100=100\times 100\ \mathrm{paisa}=10000\ \mathrm{paisa}$


$\dfrac{10000}{25}=400$

Number of $25$ paisa coins in $Rs.100$ is $400$

Which of the symbols are never repeated ?

maths tenths and hundredths more money decimal fractions in the units of currency, length, weight, capacity understanding tenth and hundredth part of a number

  1. V, X and C

  2. V, X and D

  3. V, L and D

  4. L, K and C

Show answer
Correct Option: C
Explanation:
$"D", "L"$, and $"V"$ can never be repeated. 

Only one small-value symbol may be subtracted from any large-value symbol.

Rajiv is setting off from London for a business trip to France. He converts 190 to euros. How many euros will he receive?

maths tenths and hundredths more money decimal fractions in the units of currency, length, weight, capacity understanding tenth and hundredth part of a number

  1. $204.56$ euro

  2. $210.53$ euro

  3. $220.73$ euro

  4. $212.8$ euro

Show answer
Correct Option: D
Explanation:

$190$ Pound$=190\times1.12=212.8$ euros

How much will you receive, if you are changing $250$ US dollars into Indian rupees?

maths tenths and hundredths more money decimal fractions in the units of currency, length, weight, capacity understanding tenth and hundredth part of a number

  1. $10047.52$ Rs.

  2. $12047.52$ Rs.

  3. $14047.52$ Rs.

  4. $16047.52$ Rs.

Show answer
Correct Option: D
Explanation:

$\$1=64.2$ Rs.

$\$250=250\times64.2=16047.52$ Rs.
So option $D$ is correct.