Componendo and Dividendo

Q.

Rationalize $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

Q.

What is the formula of $x^3+y^3+z^3-3xyz$?

Q.

If $\tan \left(\theta \right)=\frac{1}{\sqrt{7}}$, then $\frac{cose{c}^2\left(\theta \right)-{\sec }^2\left(\theta \right)}{cose{c}^2\left(\theta \right)+{\sec }^2\left(\theta \right)}$ is equal to

1. $\frac{1}{2}$

2. $\frac{3}{4}$

3. $\frac{5}{4}$

4. $2$

Q.

Name the property under multiplication used in the following:$\frac{-13}{17}\times \left(\frac{-2}{7}\right)=\frac{-2}{7}\times \left(\frac{-13}{17}\right)$

Q.

Solve the following equation by factorization:

$\sqrt{2}{x}^2-3x-2\sqrt{2}=0$

Q.

The largest possible set of real numbers which can be the domain of $f\left(x\right)=\sqrt{1-\frac{1}{x}}$ is

1. $(0,1)\cup (0,\infty )$

2. $(-1,0)\cup (1,\infty )$

3. $(-\infty ,-1)\cup (0,\infty )$

4. $(-\infty ,0)\cup [1,\infty )$

Q.

^{Solve for x and y}

^{ax + by = 2ab}

bx + ay = a^{​​​​​​2} + b^{​​​​​​2}

^{​​​​​​}

Q.

If $\tan \left(\beta \right)=\cos \left(\theta \right)\tan \left(\alpha \right)$ then ${\tan }^2\left(\frac{\theta }{2}\right)=?$

1. $\frac{\sin \left(\alpha +\beta \right)}{\sin \left(\alpha -\beta \right)}$

2. $\frac{\cos \left(\alpha -\beta \right)}{\cos \left(\alpha +\beta \right)}$

3. $\frac{\sin \left(\alpha -\beta \right)}{\sin \left(\alpha +\beta \right)}$

4. $\frac{\cos \left(\alpha +\beta \right)}{\cos \left(\alpha -\beta \right)}$

Q.

Find the value of $m$ for the given equation $\frac{(2m+5)}{3}=(3m-10)$.

Q.

Rationalize $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

Q.

If 7x -15y = 4x + y , find the value of x : y. Hence, use componendo and dividendo to find the values of :

(i) $\frac{9x+5y}{9x-5y}$ (ii) $\frac{3x^2+2y^2}{3x^2-2y^2}$

Q. Prove the following identities:

$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$

Q.

Using componendo and dividendo find out the value of x.
$\frac{\sqrt{3x+4}+\sqrt{3x-5}}{\sqrt{3x+4}-\sqrt{3x-5}}=9$

1. 5

2. 3

3. 7

4. 9

Q.

Solve for x:

$\sqrt{x^2+x-6}-x+2=\sqrt{x^2-7x+10},xϵR$

Q.

If $x:y=3:5$, find the ratio $3x+4y:8x+5y$.

Q.

If $x=sec\theta -\cos \theta \mathrm{and}y=se{c}^n\theta -{\cos }^n\theta$, then

1. $(x^2+4){\left(\frac{dy}{dx}\right)}^2=n^2(y^2+4)$

2. $(x^2+4){\left(\frac{dy}{dx}\right)}^2=x^2(y^2+4)$

3. $(x^2+4){\left(\frac{dy}{dx}\right)}^2=(y^2+4)$

4. None of these

Q.

If (3x + 5y)/(3x – 5y) = 7/3, find x:y.

1. 23:6

2. 25:6

3. 19:7

4. 6:25

Q.

If $3,x,12$are in continued proportion, find the value of $x$.

Q. If $x=5-2\sqrt{6}$. Find the value of $x-\frac{1}{x}$.

Q. If 2x+3y=19 and 5x+4y=37 then find value of x+y

Q.

Show that $\frac{\mathrm{sin\theta }-\mathrm{cos\theta }+1}{\mathrm{sin\theta }+\mathrm{cos\theta }-1}=\frac{1}{\mathrm{sec\theta }-\mathrm{tan\theta }}$

Q.

If $p=-2$, find the value of: $4p+7$

Q.

Are the following in proportion ?

$15,16,20,21$

Q.

If $x^2+\frac{1}{x^2}=79$ find the value of $x^3+\frac{1}{x^3}$ .

Q.

Are the following in proportion ?

$18,24,9,12$

Q. 25(a-5b)²-4(a-3b)²

Q.

Question 177(i)

Find x.
$-{\left(\frac{1}{7}\right)}^{-5}÷-{\left(\frac{1}{7}\right)}^{-7}=(-7{)}^x$

Q.

1. r = 15

2. r = 20

3. r = 25

4. r = 10

Q.

Solve the following equation :

$2p-3=7$

Q. Solve for $x$:

1. $\frac{3}{2}$
2. $\frac{3}{4}$
3. $\frac{1}{2}$
4. $\frac{1}{4}$