Trigonometry

Q. The incircle of a scalene triangle $ABC$ touches side $AB,BC$ and $AC$ at $P,Q$ and $R$ respectively. If $O$ is the incentre of triangle $ABC,$ then which of the following statements is (are) correct?

$\text{I.}$ Area of $△OBC$ is $\frac{1}{2}\times BC\times \text{radius of incircle}$
$\text{II.}$ $AB+CQ=BQ+AC$
$\text{III.}$ Area of $△ABC$ is $\frac{1}{2}\times \text{perimeter of}△ABC\times \text{radius of incircle}$

1. $\text{II}$ and $\text{III}$
2. $\text{I}$ and $\text{III}$
3. $\text{I}$ only
4. $\text{I},\text{II}$ and $\text{III}$

Q. In $△PQR$, $A$ and $B$ are the mid point of the sides $PQ$ and $PR$ respectively, then the ratio of area of $(△GAQ+△GBR+△GQR)$ to the area of $△PQR$, where $G$ is the centriod is

Q. The circles $x^2+y^2-4x+6y+8=0$ and $x^2+y^2-10x-6y+14=0$

1. touch externally
2. intersect
3. do not meet
4. touch internally

Q. Let $A=\left\{\right(x,y):y^2\le 4x,y-2x⩾-4\}$. The area (in square units) of the region A is :

1. 8
2. 9
3. 10
4. 11

Q. In a trapezium, the vector $\overline{BC}=\lambda \overline{AD}$ and $\overline{P}=\overline{AC}+\overline{BD}=\mu \overline{AD}$, then

1. $\mu =2+\lambda$
2. $\lambda =\mu +1$
3. $\lambda +\mu =1$
4. $\mu =\lambda +1$

Q. In an equilateral triangle ABC, AD is drawn perpendicular to BC meeting BC in A. If $(AD{)}^2=$x$(BD{)}^2$. Find $x$.

Q. The incircle of a scalene triangle $ABC$ touches side $AB,BC$ and $AC$ at $P,Q$ and $R$ respectively. If $O$ is the incentre of triangle $ABC,$ then which of the following statements is (are) correct?

$\text{I.}$ Area of $△OBC$ is $\frac{1}{2}\times BC\times \text{radius of incircle}$
$\text{II.}$ $AB+CQ=BQ+AC$
$\text{III.}$ Area of $△ABC$ is $\frac{1}{2}\times \text{perimeter of}△ABC\times \text{radius of incircle}$

1. $\text{I}$ and $\text{III}$
2. $\text{I}$ only
3. $\text{II}$ and $\text{III}$
4. $\text{I},\text{II}$ and $\text{III}$

Q. The incircle of a scalene triangle $ABC$ touches side $AB,BC$ and $AC$ at $P,Q$ and $R$ respectively. If $O$ is the incentre of triangle $ABC,$ then which of the following statements is (are) correct?

$\text{I.}$ Area of $△OBC$ is $\frac{1}{2}\times BC\times \text{radius of incircle}$
$\text{II.}$ $AB+CQ=BQ+AC$
$\text{III.}$ Area of $△ABC$ is $\frac{1}{2}\times \text{perimeter of}△ABC\times \text{radius of incircle}$

1. $\text{I}$ and $\text{III}$
2. $\text{I}$ only
3. $\text{II}$ and $\text{III}$
4. $\text{I},\text{II}$ and $\text{III}$

Q. Radius of the circle $x^2+y^2-4x+2y-45=0$

1. $5\sqrt{2}$ units
2. $4\sqrt{2}$ units
3. $3\sqrt{5}$ units
4. $4\sqrt{5}$ units

Q. PQ is a tower standing on a horizontal plane, Q being its foot. A and B are two points on the plane such that the angle QAB is 90${}^o$ and AB is 40m. It is found that $cotPAQ=\frac{3}{10}$ and $cotPBQ=\frac{1}{2}$. Show that the height of the tower is 100 m.