In Radius

Q.

Find $\tan \left(22.5^{\circ }\right)$ using the half-angle formula.

Q.

Evaluate $\tan A+\cot (180°+A)+\cot \left(90°+A\right)+\cot \left(360°-A\right)$

1. $0$

2. $2\tan A$

3. $2\cot A$

4. $2\left(\tan A-\cot A\right)$

Q.

Let $\tan \alpha$ and $\tan \beta$ be two real roots of the equation $(k+1){\tan }^{2}x-\sqrt{2}\lambda \tan x=(1-k)$, where $(k\ne 1)$and $\lambda$ are real numbers. If ${\tan }^2(\alpha +\beta )=50$, then a value of $\lambda$ is:

1. $5\sqrt{2}$

2. $10\sqrt{2}$

3. $10$

4. $5$

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. Let $f\left(x\right)=\{\begin{array}{cc}\frac{1-si{n}^{3}x}{3co{s}^{2}x}:& x<\frac{\pi }{2}\\ p:& x=\frac{\pi }{2}\\ \frac{q(1-sinx)}{(x-2x{)}^2}:& x>\frac{\pi }{2}\end{array}$ If f(x) is continuous at $x=\frac{\pi }{2},(p,q)=$

1. $(\frac{1}{2},2)$
2. $(1,4)$
3. $(\frac{1}{2},4)$
4. $(\frac{1}{2},7)$

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.

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. Let there exist a unique point $P$ inside a $△ABC$ such that $\angle PAB=\angle PBC=\angle PCA=\alpha .$ If $PA=x,PB=y,PC=z,\Delta =$ area of $△ABC$ and $a,b,c,$ are the sides opposite to the angle $A,B,C$ respectively, then $\tan \alpha$ is equal to

1. $\frac{a^2+b^2+c^2}{4\Delta }$
2. $\frac{a^2+b^2+c^2}{2\Delta }$
3. $\frac{4\Delta }{a^2+b^2+c^2}$
4. $\frac{2\Delta }{a^2+b^2+c^2}$

Q. $ABC$ is a triangle. $3$ Circles with radii $1,4$ and $9$ as shown are drawn inside the triangle each touching two sides and the incircle. Then the radius of the incircle of the $△ABC$ is

Q. Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\{x:f(x)=0\},Y=\{x:f^{\prime }(x)=0\},$
$Z=\{x:g(x)=0\},W=\{x:g^{\prime }(x)=0\},$.
$List-I$ contains the sets $X,Y,Z$ and $W$. $List-II$ contains some information regarding these sets.

$\begin{array}{cccc}& ListI& & ListII\\ \left(I\right)& X& \left(P\right)& \supseteq \{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \}\\ \left(II\right)& Y& \left(Q\right)& \text{an arithmetic progression}\\ \left(III\right)& Z& \left(R\right)& \text{NOT an arithmetic progression}\\ \left(IV\right)& Z& \left(S\right)& \supseteq \{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\}\\ & & \left(T\right)& \supseteq \{\frac{\pi }{3},\frac{2\pi }{3},\pi \}\\ & & \left(U\right)& \supseteq \{\frac{\pi }{6},\frac{3\pi }{4}\}\end{array}$
Q.15 which of the following is the only CORRECT combination?

1. $\left(I\right),\left(P\right),\left(R\right)$
2. $\left(II\right),\left(Q\right),\left(T\right)$
3. $\left(I\right),\left(Q\right),\left(U\right)$
4. $\left(II\right),\left(R\right),\left(S\right)$

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. If a, b, c denote lengths of triangle, then the possible integral value of a /b+c +b/a+c +c/a+b is(2)

Q. If length of the side $BC$ of a $\Delta ABC$ is $4$ cm and $\angle BAC={120}^{\circ },$ then the distance between the incenter and excentre of the circle touching the side $BC$ is:

1. $8$
2. $9$
3. $7$
4. $6$

Q. The median to the base of an isosceles triangle is:

1. Perpendicular to base
2. Inclined ${60}^{\circ }$ to the base
3. Inclined ${45}^{\circ }$ to the base
4. Inclined ${30}^{\circ }$ to the base

Q. The median $\overrightarrow{AD}$ of $△ABC$ is bisected at $E$ and $\overrightarrow{BE}$ is producted to meet the side $\overrightarrow{AC}$ in $F.$ Then the ratio $\overrightarrow{AF}:\overrightarrow{FC}$ is

1. $1:3$
2. $1:4$
3. $1:2$
4. $3:1$

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

Q. IF DELTA=|A1 B1 C1 THEN THE VALUE OF |2A1+3B1+4C1 B1 C1 IS EQUA; A2 B2 C2 2A+3B2+4C2 B2 C2 A3 B3 C3| 2A3 +3B3 +4C3 B3 C3|

Q. The incircle of $△ABC$ having integral side lengths and least perimeter, touches $AB,BC,CA$ at $R,P,Q$ respectively. If $\frac{2}{AR}+\frac{5}{BP}+\frac{5}{CQ}=\frac{6}{r}$, where $r$ is the radius of the incircle, then

1. $\Delta ABC\text{is isosceles}$
2. $\Delta ABC\text{is equilateral}$
3. $\text{Area of}△ABC\text{is 27 sq. units}$
4. $\text{Inradius of}△ABC\text{is 4 units}$

Q. Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\{x:f(x)=0\},Y=\{x:f^{\prime }(x)=0\},$
$Z=\{x:g(x)=0\},W=\{x:g^{\prime }(x)=0\},$.
$List-I$ contains the sets $X,Y,Z$ and $W$. $List-II$ contains some information regarding these sets.

$\begin{array}{cccc}& ListI& & ListII\\ \left(i\right)& X& \left(P\right)& \supseteq \{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \}\\ \left(ii\right)& Y& \left(Q\right)& \text{an arithmetic progression}\\ \left(iii\right)& Z& \left(R\right)& \text{NOT an arithmetic progression}\\ \left(iv\right)& Z& \left(S\right)& \supseteq \{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\}\\ & & \left(T\right)& \supseteq \{\frac{\pi }{3},\frac{2\pi }{3},\pi \}\\ & & \left(U\right)& \supseteq \{\frac{\pi }{6},\frac{3\pi }{4}\}\end{array}$
Q.16 which of the following is the only CORRECT combination?

1. $\left(III\right),\left(R\right),\left(U\right)$
2. $\left(IV\right),\left(P\right),\left(R\right),\left(S\right)$
3. $\left(III\right),\left(P\right),\left(Q\right),\left(U\right)$
4. $\left(IV\right),\left(Q\right),\left(T\right)$

Q. Let $f\left(x\right)=[n+p\sin x],x\in (0,\pi ),n\in Z$ and $p$ is prime number, where $[.]$ denotes the greatest integer function. Then, the number of points where $f\left(x\right)$ is not differentiable, are

1. $0$
2. $2(p-1)$
3. None of these
4. $2p-1$

Q. Prove that :

$\left|\begin{array}{ccc}1& a^2+bc& a^3\\ 1& b^2+ca& b^3\\ 1& c^2+ab& c^3\end{array}\right|=-\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a^2+b^2+c^2\right)$

Q. If a, b, c are real numbers such that $\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|=0$, then show that either $a+b+c=0\mathrm{or},a=b=c$.

Q.

In a $\Delta ABC,tan\frac{A}{2}=\frac{5}{6}andtan\frac{C}{2}=\frac{2}{5},$ then a, b, c are such that

1. b²=ac

2. 2ac=b (a+c)

3. 2b=a+c

4. a+b=c

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.

Find the roots of $(4+4i{)}^{\frac{p}{q}}$ . Where p and q are integers and q $\ne$ 0

1. Where n = 0, 1, 2, 3, ..........q-1

2. Where n = 0, 1, 2, 3, ..........q-1

3. Where n = 0, 1, 2, 3, ..........q-1

4. Where n = 0, 1, 2, 3, ..........q-1

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

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

Q.

let A, B , C be the sets such thatA∪B=A∪C and A∩B=A∩C .show that B=C

Q. In a non-right-angled triangle $△PQR,$ let $p,q,r$ denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If $p=\sqrt{3},q=1,$ and the radius of the circumcircle of the $△PQR$ equals $1$, which of the following options is/are correct?

1. Length of RS = $\frac{\sqrt{7}}{2}$
2. Area of $△SOE=\frac{\sqrt{3}}{12}$
3. Length of OE = $\frac{1}{6}$
4. Radius of incircle of $△PQR=\frac{\sqrt{3}}{2}(2-\sqrt{3})$