Incentre

Q. The x-coordinate of the incentre of the triangle that has the coordinates of mid-points of its sides as (0, 1), (1, 1) and (1, 0) is

1. $1-\sqrt{2}$
2. $2+\sqrt{2}$
3. $2-\sqrt{2}$
4. $1+\sqrt{2}$

Q. If the line $3x+4y-24=0$ intersects the x-axis at the point $A$ and y-axis at the point $B$, then the incentre of the triangle $OAB$, where $O$ is the origin is :

1. $(4,3)$
2. $(2,2)$
3. $(3,4)$
4. $(4,4)$

Q.

The coordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are $(20,25),(8,16)$ and $(8,9)$. The coordinates of a vertex of the triangle are

1. (5, 10)

2. (50, -5)

3. (15, 30)

4. (10, 15)

Q. The axis of the parabola $x^2+2xy+y^2-5x+5y-5=0$ is

1. $x+y=0$
2. $x+y-1=0$
3. $x-y+1=0$
4. $x-y=\frac{1}{\sqrt{2}}$

Q.

$\text{Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are A(-36, 7), B(20, 7)and C(0, -8).}$

Q. If the area of the triangle with vertices $(2,5),(7,k)$ and $(3,1)$ is $10$, then the value of $k$ is

1. $-5$ or $35$
2. $15$ or $-5$
3. $-5$ or $-25$
4. $5$ or $-35$

Q. The incentre of the triangle formed by the vertices $A(-36,7),B(20,7)$ and $C(0,-8)$ is

1. $(1,\frac{1}{2})$
2. $(-1,0)$
3. $(0,-1)$
4. $(\frac{1}{2},1)$

Q. If the vertices of $△ABC$ are $A=(3,-5),B=(-3,3)$ and $C=(-1,-2)$ and internal angle bisector through $A$ cuts the side $BC$ at $D,$ then the length of $AD$ is

1. $\frac{7}{3}\sqrt{2}$ units
2. $\frac{14}{3}\sqrt{2}$ units
3. $\frac{3}{14}\sqrt{2}$ units
4. $\frac{3}{7}\sqrt{2}$ units

Q. If the line $3x+4y-24=0$ intersects the x-axis at the point $A$ and y-axis at the point $B$, then the incentre of the triangle $OAB$, where $O$ is the origin is :

1. $(4,3)$
2. $(2,2)$
3. $(3,4)$
4. $(4,4)$

Q. If A(0, 1, 2), B(2, -1, 3) and C(1, -3, 1) are the vertices of a triangle, then its circumcentre and orthocenter are situated at a distance of

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

Q.

Write the coordinates of the incentre of the triangle having its vertices at(0, 0), (5, 0) and (0, 12).

Q. The $x$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0,1),(1,1)$ and $(1,0)$ is :

1. $2+\sqrt{2}$
2. $1+\sqrt{2}$
3. $2-\sqrt{2}$
4. $1-\sqrt{2}$

Q.

The incentre of the triangle formed by (0, 0), (5, 12), (16, 12) is

1. (-9, 7)

2. (-7, 9)

3. (9, 7)

4. (7, 9)

Q. The incentre of the triangle whose vertices are $(1,\sqrt{3}),(0,0)$ and $(2,0)$ is

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

Q. The coordinate of base $BC$ of an isosceles triangle $ABC$ are given by $B(1,3)$ and $C(-2,7)$. Possible coordinates of the vertex $A$ can be:($\angle B=\angle C$)

1. $(\frac{\sqrt{2}}{3},\frac{\sqrt{2}}{3})$
2. $(\frac{2}{3},\frac{2}{3})$
3. $(\frac{1}{3},\frac{1}{3})$
4. None of these

Q.

The incentre of the triangle with vertices (1, $\sqrt{3}$), (0, 0) and (2, 0) is

1. 

2. 

3. 

4. 

Q. Find the area of region bounded by the triangle whose vertices are $(-1,1),(0,5)$ and $(3,2)$, using integration.

Q. Solve the following inequality:
$\frac{\sqrt{2x^2+15x-17}}{10-x}⩾0.$

Q. If the coordinates of the vertices of A triangle are (0, 0), (0, 2) and (3, 1), `then area of the triangle is

1. 3 sq. units
2. 2 sq. units
3. -3 sq. units
4. 1 sq. unit

Q. If the vertices $A,B,C$ of a triangle $ABC$ are $(1,2,3),(-1,0,0),(0,1,2)$, respectively then find $\angle ABC$. [$\angle ABC$ is the angle between the vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}$] .

Q.

Incentre of triangle whose vertices are A(-36, 7), B(20, 7), C(0, -8), is

1. (0, 0)

2. (-1, 0)

3. (-1, -1)

4. (0, -1)

Q. Let $z$ be a complex number such that the imaginary part of $z$ is non zero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

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

Q.

The incentre of the triangle formed by (0, 0), (5, 12), (16, 12) is

1. (7, 9)

2. (9, 7)

3. (-9, 7)

4. (-7, 9)

Q. The solution of ${\left(\frac{dy}{dx}\right)}^2-\left(\frac{dy}{dx}\right)(e^x+e^{-x})+1=0$ is:

1. $y+e^{-x}=const$
2. $y=e^{-x}+const$
3. $x=\log (c+y)$
4. $y=e^x+const$

Q. The system of equations $3x-y+4z=2;x+2y-z=3;4x-y+\lambda z=-1$ has no solution. The value of $\lambda$ is

1. $1$
2. $3$
3. $5$
4. $6$

Q. If the vertices of $△ABC$ are $A=(3,-5),B=(-3,3)$ and $C=(-1,-2)$ and internal angle bisector through $A$ cuts the side $BC$ at $D,$ then the length of $AD$ is

1. $\frac{7}{3}\sqrt{2}$ units
2. $\frac{14}{3}\sqrt{2}$ units
3. $\frac{3}{7}\sqrt{2}$ units
4. $\frac{3}{14}\sqrt{2}$ units

Q. Three points $\left(0,0\right),\left(3,\sqrt{3}\right),\left(3,\lambda \right)$ form an equilateral triangle, then $\lambda$ is equal to

1. $-\sqrt{3}$
2. $-4$
3. $2$
4. $-3$

Q. Vertices of a triangle $ABC$ are the points $(0,0),(a,0),(\frac{a}{2},\frac{a\sqrt{3}}{2})$. Then its centroid is the point $(\frac{a}{2},\frac{a\sqrt{3}}{6})$

1. true
2. false

Q. Find the factors of the quadractic equation given below.
$2x^2+x-1=0$

Q. Refer Diagram

1. 8
2. 9
3. 7
4. 12
5. 10