In Radius

Q. If in a triangle $ABC$, $r_1=2,r_2=3$ and $r_3=6$, then $a=$

Q. If r is the radius and R is the circumradius. Then $2R\le r$

1. True
2. False

Q. Which of the following holds true in a Right angled triangle?

1. r + 2R = s
2. r + R = s
3. 2r + R = s
4. r + R = 2s

Q. The perpendicular distance of incentre from the sides of a triangle is equal for all the three sides

1. False
2. True

Q.

Two sides of a triangle are 2 and $\sqrt{3}$ and the included angle is ${30}^{\prime }$ then the in-radius r of the triangle is equal

1. $\frac{1}{4}(\sqrt{3}-1)$

2. $\frac{1}{2}(\sqrt{3}+1)$

3. $\frac{1}{2}(\sqrt{3}-1)$

4. $\frac{1}{4}(\sqrt{3}+1)$

Q. $\mathrm{If}{r}_1,r_2\mathrm{and}{r}_3\mathrm{are}\mathrm{the}\mathrm{corresponding}\mathrm{exradii}\mathrm{of}a\Delta \mathrm{ABC}\mathrm{and}r\mathrm{is}\mathrm{the}\mathrm{inradii}\mathrm{of}\mathrm{the}\mathrm{same}\mathrm{triangle},\mathrm{then}-\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}$

1. True
2. False

Q. In a triangle ABC, a:b:c=4:5:6, then the ratio of circumradius to inradius will be .

1. 15:7
2. 15:9
3. 16:7
4. 16:9

Q. In $△ABC$, the incircle touches the sides $BC,CA$ and $AB$, respectively, $D,E,$ and $F$. If the radius of the incircle is $4$ units and $BD,CE$ and $AF$ are consecutive integers, then the value of $\frac{s}{3}$, where $s$ is a semi-perimeter of triangle, is

Q. $\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If A and B are two sets with elements 4 and 3},& \text{(P)}& 1\\ & \text{respectively. Then the number of onto function}& & \\ & \text{from A to B is}& & \\ \text{(II)}& \text{The number of points in the interval}[-\sqrt{13},\sqrt{13}]& \left(Q\right)& 3\\ & \text{at which}f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)\text{attains its}& & \\ & \text{maximum value, is}& & \\ \text{(III)}& \text{In a}\Delta XYZ,y^2\sin \left(2Z\right)+z^2\sin \left(2Y\right)=2yz,& \text{(R)}& 4\\ & \text{where}y=15,z=8\text{. Then the length of inradius is}& & \\ \text{(IV)}& {\log }_{x+1}(4x^3+9x^2+6x+1)+{\log }_{4x+1}(x^2+2x+1)& \text{(S)}& 8\\ & =5,\text{then the number of solution is}& & \\ & & \text{(T)}& 24\\ & & \text{(U)}& 36\end{array}$

Which of the following has INCORRECT pair of combination?

1. $\left(I\right)\to \left(U\right)$
2. $\left(II\right)\to \left(S\right)$
3. $\left(IV\right)\to \left(P\right)$
4. $\left(III\right)\to \left(Q\right)$

Q. $\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If A and B are two sets with elements 4 and 3},& \text{(P)}& 1\\ & \text{respectively. Then the number of onto function}& & \\ & \text{from A to B is}& & \\ \text{(II)}& \text{The number of points in the interval}[-\sqrt{13},\sqrt{13}]& \left(Q\right)& 3\\ & \text{at which}f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)\text{attains its}& & \\ & \text{maximum value, is}& & \\ \text{(III)}& \text{In a}\Delta XYZ,y^2\sin \left(2Z\right)+z^2\sin \left(2Y\right)=2yz,& \text{(R)}& 4\\ & \text{where}y=15,z=8\text{. Then the length of inradius is}& & \\ \text{(IV)}& {\log }_{x+1}(4x^3+9x^2+6x+1)+{\log }_{4x+1}(x^2+2x+1)& \text{(S)}& 8\\ & =5,\text{then the number of solution is}& & \\ & & \text{(T)}& 24\\ & & \text{(U)}& 36\end{array}$

Which of the following has CORRECT pair of combination?

1. $\left(I\right)\to \left(T\right)$
2. $\left(II\right)\to \left(S\right)$
3. $\left(III\right)\to \left(R\right)$
4. $\left(IV\right)\to \left(P\right)$