The Orthocentre

Q.

The value of $k$ such that the lines $2x-3y+k=0,3x-4y-13=0$ and $8x-11y-33=0$ are concurrent is

1. $20$

2. $-7$

3. $7$

4. $-20$

Q. Question 7
Three circles each of radius 3.5 cm are drawn in such a way that each of them touches the other two . Find the area enclosed between these circles.

Q.

Let $\mathrm{A}(-1,1),\mathrm{B}(3,4)$ and $\mathrm{C}(2,0)$ be given three points. A line $\mathrm{y}=\mathrm{m}\mathrm{x},\mathrm{m}>0$, intersects lines $\mathrm{AC}$ and $\mathrm{BC}$ at point$\mathrm{P}$ and $\mathrm{Q}$ respectively. Let ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ be the areas of $∆\mathrm{ABC}$ and $∆\mathrm{PQC}$ respectively, such that ${\mathrm{A}}_1=3{\mathrm{A}}_2$, then the value of $\mathrm{m}$ is equal to:

1. $\frac{4}{15}$

2. $1$

3. $2$

4. $3$

Q.

If the equation of the plane passing through the line of intersection of the planes $2x–7y+4z–3=0,3x–5y+4z+11=0$and the point$(-2,1,3)$ is $ax+by+cz–7=0$, then the value of $2a+b+c–7$ is __________

Q.

In the given, ABCD is a quadrilateral with diagonals AC and BD intersecting at point O. Hence, area of $\Delta$ AOD $\times$ area of $\Delta$ BOC = area of $\Delta$ AOB $\times$ area of $\Delta$ COD.

1. True

2. False

Q.

ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle through A, B, C and D. If $\angle ADC={130}^{\circ }$, $\angle BAC$ ___.

Q.

The sum of the three sides of a triangle is greater than the sum of its three ___.

1. interior angles

2. exterior angles

3. sides

4. medians

Q.

If $r(1-m^2)+m(p-q)=0$, then a bisector of the angle between the lines represented by the equation $p{x}^2-2rxy+q{y}^2=0$, is

1. $y=x$

2. $y=-x$

3. $y=mx$

4. $my=x$

Q.

Circumcenter is the point of intersection of ___ of the triangle.

1. angle bisectors

2. altitudes
3. perpendicular bisectors
4. medians

Q.

If $R$ and $r$ are the radii of the circumcircle and incircle of a regular polygon of $n$ sides, each side being of length $a$, then $a$ is equal to

1. $2(R+r)\sin \left(\frac{{\displaystyle \pi }}{{\displaystyle 2n}}\right)$

2. $2(R+r)\tan \left(\frac{{\displaystyle \pi }}{{\displaystyle 2n}}\right)$

3. $2(R+r)$

4. None of these

Q.

The sides of a triangle have lengths$(\mathrm{in}\mathrm{cm})10,6.5\mathrm{and}\mathrm{a},$where $\mathrm{a}$ is a whole number. The minimum value that a can take is-

1. $6$

2. $5$

3. $3$

4. $4$

Q. In figure, if DEIIBC, find the ratio of ar $\left(\Delta ADE\right)$ and are $\left(\Delta DECB\right)$.

Q.

In triangle ABC, D is a point on side BC such that 2BD = 3DC. Which of the following option is correct?

1. Area of triangle ABD = Area of triangle ABC

2. Area of triangle ABD = Area of triangle ABC

3. Area of triangle ABD = Area of triangle ABC

4. Area of triangle ABD = Area of triangle ABC

Q.

In $\Delta XYZ,\overline{XY}>\overline{YZ}>\overline{ZX}$ Which of the following is the smallest angle?

1. X

2. Z

3. Y

4. X = Y = Z

Q.

Let $R$ be the real line. Consider the following subsets of the plane $R\times R,S=\left\{(x,y):y=x+1\&0<x<2\right\},T=\left\{(x,y):x–yisaninteger\right\}$ Which one of the following is true?

1. Both $S\&T$ are equivalence relation on $R$

2. $S$is an equivalence relation on $R$ but $T$ is not

3. $T$ is an equivalence relation on $R$ but $S$ is not

4. Neither $S$ nor $T$ is an equivalence relation on $R$

Q.

Construct an equilateral triangle $XYZ$ whose side is $4.8cm.$

Q. The minimum number of altitudes that has to intersect in order to get the position of orthocentre of any triangle are.

1. 2
2. 3
3. 4

Q.

An altitude has one end point at a vertex of the triangle and the other on the line containing the opposite side, then how many altitudes does a triangle have?

1. 1

2. 6

3. 3

4. 9

Q.

In an equilateral $△ABC$, if $△DFE$ is formed by joining the midpoints of the sides, the area of $△DFE$ is ___ $\times$ the area of $△$ ABC.

1. 2

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

3. 4

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

Q.

The vertices of a triangle are $\mathrm{L}(0,1),\mathrm{M}(1,-2)$ and $\mathrm{N}(-2,1)$. Draw the figure and its image after the translation $5\mathrm{unit}$ right.

Q.

Construct a $△DEF$ such that $DE=6cm,EF=5.3cm,\angle E=60°$.

Q.

Observe the dots arranged in triangles. How many dots will be there in the fifth triangle?

1. $12$

2. $15$

3. $18$

4. $21$

Q. If $H$ is the orthocentre of a triangle $ABC$, then the radii of the circle circumscribing the triangles $BHC,CHA$ and $AHB$ respectively equal to:

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

Q.

How do you solve a hyperbola conic section $?$

Q. (a) In the given figure, POQ is a line, then the value of 'x'is [4 MARKS]

Figure


(b) In the figure below, x = ${30}^{\circ }$. The value of (6y – 3x) is ___.

Q. Find the orthocenter of the triangle the equation of whose sides are $x+y=1,2x+3y=0$ and $4x-y+4=0$.

Q. The orthocentre of obtuse angled triangle lies _________ the triangle

1. Inside
2. On vertex
3. Outside
4. None

Q.

In the given figure, AE and BD are two medians of $△$ABC meeting at F. The ratio of the area of $△$ABF and the quadrilateral FDCE is:

1. 1:1

2. 2:1

3. 1:2

4. 2:3

Q.

In which of the following triangle corresponding altitudes and medians coincides?

1. Scalene triangle

2. Isosceles triangle

3. Equilateral triangle

4. Obtuse angle triangle

Q.

Prove that supplement of an angle is an obtuse angle.