Ex Radius

Q. The approximate value of $(0.007{)}^{\frac{1}{3}}$ is

1. $\frac{23}{120}$
2. $\frac{21}{120}$
3. $\frac{29}{120}$
4. $\frac{31}{120}$

Q. In a triangle $ABC,\frac{r_1-r}{a}+\frac{r_2-r}{b}+\frac{r_3-r}{c}=$

1. $\frac{r_1+r_2+r_3}{3}$
2. $r_1+r_2+r_3$
3. $\frac{r_1+r_2+r_3}{2}$
4. $\frac{r_1+r_2+r_3}{s}$

Q. The radii $r_1,r_2,r_3$ of the escribed circles of the triangle $ABC$ are in H.P. If the area of the triangle is $24{\text{cm}}^2$ and its perimeter is $24\text{cm}$, then the length of its largest side is

1. $10$
2. $9$
3. $8$
4. $7$

Q. If the entries in a $3\times 3$ determinant are either $0$ or $1,$ then the greatest value of their determinant is

1. $1$
2. $2$
3. $3$
4. $9$

Q.

In any $\Delta$ABC, $4(\frac{s}{a}-1)(\frac{s}{b}-1)(\frac{s}{c}-1)$ is equal to

1. r/R

2. 2r/R

3. 3r/R

4. 4r/R

Q.

If the sides fo a triangle are in A.P. as well as in G.P. Then the value of $\frac{r_1}{r_2}-\frac{r_2}{r_3}$

1. 1

2. 0

3. 2r

4. None

Q. The radii $r_1,r_2,r_3$ of escribed circle of a triangle ABC are in H.P. If its area is $24c{m}^2$ and its perimeter is $24cm$, find the length of its sides.

1. True
2. False

Q. Relation between Exradius , semiperimeter and circumradius can be given by
($r_1$ is the radius of the circle opposite the angle A)

1. $r_1=\frac{2\Delta }{s-a},r_1=4Rcos\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
2. $r_1=\frac{\Delta }{s-a},r_1=4Rsin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$
3. $r_1=\frac{2\Delta }{(s-a)(s-b)},r_1=4Rcos\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
4. $r_1=\frac{\Delta (s-a)}{(s-b)(s-c)},r_1=4Rcos\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$

Q. In $△ABC,R,r,r_1,r_2,r_3$ denote the circumradius, inradius, the exradii opposite to the vertices $A,B,C$ respectively. Given that $r_1:r_2:r_3=1:2:3$.
The sides of the triangle are in the ratio

1. $1:2:3$
2. $1:5:9$
3. $3:5:7$
4. $5:8:9$

Q. If $r_1,r_2,r_3,s,△,a,b,c$ are the standard notations then which of the following is correct?

1. All of the above
2. $r_1=\frac{△}{s-a}$
3. $r_2=\frac{△}{s-b}$
4. $r_3=\frac{△}{s-c}$

Q. If $r_1,r_2,r_3$ represent the exradii and r represents the inradius then, $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}-\frac{1}{r}$ = 0

1. True
2. False

Q. Relation between Exradius , semiperimeter and circumradius can be given by
($r_1$ is the radius of the circle opposite the angle A)

1. $r_1=\frac{2\Delta }{s-a},r_1=4Rcos\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
2. $r_1=\frac{\Delta }{s-a},r_1=4Rsin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$
3. $r_1=\frac{2\Delta }{(s-a)(s-b)},r_1=4Rcos\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
4. $r_1=\frac{\Delta (s-a)}{(s-b)(s-c)},r_1=4Rcos\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$

Q. If $r_1,r_2,r_3$ represent the exradius and r represents the in radius then, $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}-\frac{1}{r}$ = 0

1. True
2. False

Q. In $△ABC,R,r,r_1,r_2,r_3$ denote the circumradius, inradius, the exradii opposite to the vertices $A,B,C$ respectively. Given that $r_1:r_2:r_3=1:2:3$.
The sides of the triangle are in the ratio

1. $1:2:3$
2. $3:5:7$
3. $1:5:9$
4. $5:8:9$

Q. If $r_1,r_2,r_3,s,△,a,b,c$ are the standard notations then which of the following is correct?

1. $r_2=\frac{△}{s-b}$
2. $r_3=\frac{△}{s-c}$
3. $r_1=\frac{△}{s-a}$
4. All of the above

Q. Differentiate with respect to $x$ : $x^2\cos x$

1. $2x\cos x$
2. $2x\cos x-x^2\sin x$
3. $2x\cos x-x^2$
4. None of these

Q. In $\Delta ABC,\frac{b-c}{r_1}+\frac{c-a}{r_2}+\frac{a-b}{r_3}=$

1. $0$
2. $1$
3. $2$
4. $3$

Q.

In any $\Delta ABC,\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}$ is equal to

1. 3/r

2. 2/r

3. 1/r

4. 1

Q. If $r_1,r_2,r_3$ represent the exradii and r represents the inradius then, $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}-\frac{1}{r}$ = 0

1. True
2. False

Q. In a $\Delta ABC,$ the value of $a(r{r}_1+r_2{r}_3)=$

1. $\frac{ca-r_1{r}_3}{r_2}$
2. $\frac{r_{3}r+r_1{r}_2}{ab}$
3. $\frac{r_3^2}{4R-r_1-r_2}$
4. $abc$

Q.

If the sides fo a triangle are in A.P. as well as in G.P. Then the value of $\frac{r_1}{r_2}-\frac{r_2}{r_3}$

1. 1

2. 0

3. 2r

4. None

Q. The radii $r_1,r_2,r_3$ of the escribed circles of the triangle $ABC$ are in H.P. If the area of the triangle is $24{\text{cm}}^2$ and its perimeter is $24\text{cm}$, then the length of its largest side is

1. $10$
2. $9$
3. $8$
4. $7$

Q. The radii $r_1,r_2,r_3$ of escribed circle of a triangle ABC are in H.P. If its area is $24c{m}^2$ and its perimeter is $24cm$, find the length of its sides.

1. True
2. False

Q. If ${\Delta }_1=\left|\begin{array}{ccc}1& 0& 2\\ 0& 2& -1\\ 0& -1& 3\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}2& -1& 0\\ 3& 1& 0\\ 0& 0& 2\end{array}\right|,$ then the value of ${\Delta }_1{\Delta }_2$ is

1. $100$
2. $-100$
3. $50$
4. $-50$

Q. Differentiate and find the approximate value of ${\left(26\right)}^{\frac{1}{3}}$

Q. A wire of length $36cm$ is cut into two pieces, one of the pieces is turned in form of a square and the other in the form of a equilateral triangle. Find the length of each piece such that the sum of the areas of the two are minimum.

Q. In $△ABC,R,r,r_1,r_2,r_3$ denote the circumradius, inradius, the exradii opposite to the vertices $A,B,C$ respectively. Given that $r_1:r_2:r_3=1:2:3$.
The sides of the triangle are in the ratio

1. $1:2:3$
2. $3:5:7$
3. $1:5:9$
4. $5:8:9$

Q. Which pair of functions is identical?

1. ${\sin }^{-1}\left(\sin x\right)$ and $\sin \left({\sin }^{-1}x\right)$
2. ${\log }_e{e}^x,{\log }^{lo{g}_{e}x}$
3. none of these
4. ${\log }_e{x}^2,2{\log }_{e}x$

Q. In a triangle $ABC$ with usual notation, if $r=1,R=3$ and $s=5,$ then which of the following is (are) CORRECT?

1. Area of triangle $ABC$ is $5$ units
2. Product of the sides of triangle$ABC$ is $60$
3. $a^2+b^2+c^2=24$
4. Sum of the ex-radii of triangle $ABC$ is $13$

Q. The radii $r_1,r_2,r_3$ of escribed circle of a triangle ABC are in H.P. If its area is $24c{m}^2$ and its perimeter is $24cm$, find the length of its sides.

1. True
2. False