Geometry Questions

If the distance between the centres of two circles is less than the sum of radii of the two, then they:

Choose the figure which is different from the rest based on its boundary properties:

What is the area (in cm
2
) of a sector of a circle, with central angle 75
∘
, when the diameter of the circle is 9cm?

Count the total number of squares in a standard $4\times 4$ grid matrix configuration:

Atul walks 5 km toward North and then turns to the right. After walking 3 km he turns to the right and walks 5 km. How far away is he from his starting position?

A question along with a set of statements is given. Find which of the given statements is/are sufficient to answer the question.
Question: Find the perimeter of a triangle given that:
Statements:
I. Area of the triangle is $24\text{sq. units}$.
II. Height of the triangle is $4\text{units}$.

Starting from a point, Karan moves $6\text{km}$ in one direction and then $8\text{km}$ at a right angle to his initial path. What is the direct distance between his starting point and his final position?

How many quadrilaterals are there in the given figure?

In a rhombus, one diagonal is $40\text{cm}$ and the other diagonal is $22\text{cm}$ longer than half the first diagonal. Find the length of each side of the rhombus.

The diagonals $AC$ and $BD$ of a parallelogram $ABCD$ intersect each other at the point $O$. If $\angle DAC=32^{\circ }$ and $\angle AOB=70^{\circ }$, then $\angle DBC$ is equal to

Two adjacent sides of a parallelogram are $74\text{cm}$ and $40\text{cm}$. If one of its diagonals is $102\text{cm}$, then the area of the parallelogram is

If each exterior angle of a regular polygon is $18^{\circ }$, then the number of sides of the polygon is

In a pair of adjacent angles: (i) Vertex is always common, (ii) one arm is always common and (iii) uncommon arms are always opposite rays. Then

If the area of an equilateral triangle is $a$ and its height is $b$, then $\frac{b^2}{a}$ is

A horse is tethered to a stake by a rope $30\text{meters}$ long. If the horse moves along the circumference of a circle keeping the rope tight, find how far the horse has gone from the initial point if the rope has traced an angle of $105^{\circ }$ at the centre?

The perimeter of an isosceles right-angled triangle having an area of 200cm
2
is:

If $z\ne 1$ and $\frac{z^2}{z-1}$ is real, then the point represented by the complex number $z$ lies:

The square of the diagonal of a cube is 2346 cm
2
. What is the total surface area (in cm
2
) of the cube?

In △ABC and △PQR, we are given ∠A=∠P and AC=PR.
Which of the following options needs to be satisfied for △PQR and △ABC to be congruent?

In △ABC, AB=9cm and AC=13cm, then the length of BC could be:

Find the area (in m
2
) of the rectangular field, one of whose sides is 35m and a diagonal is 37m.

What is the area of a triangle with base $8\text{meters}$ and height $16\text{meters}$?

How many rectangles are there in the given figure?

Count the number of triangles in the given figure.

When a cone is sliced by a plane parallel to its base, but not passing through its apex, what is the shape of the resulting cross-section?

Determine the number of pentagons in the following figure.

If you have a regular pentagonal prism and you slice it parallel to its bases, what shape is the cross-section?

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is:

If the vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$, then the length of the median through $A$ is:

$\text{ABCD}$ is a trapezium such that $\text{AB}$ and $\text{CD}$ are parallel and $\text{BC}\perp \text{CD}$. If $\angle \text{ADB}=\theta$, $\text{BC}=p$ and $\text{CD}=q$, then $\text{AB}$ is equal to:

A sphere has a radius of $9.21\text{cm}$. What is its volume? (Use $\pi =3.14$)

In a $\Delta PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to:

Statement-1 : The point $A(1,0,7)$ is the mirror image of the point $B(1,6,3)$ in the line $\frac{x-1}{0}=\frac{y-1}{2}=\frac{z-2}{3}$.Statement-2 : The line $\frac{x-1}{0}=\frac{y-1}{2}=\frac{z-2}{3}$ bisects the line segment joining $A(1,0,7)$ and $B(1,6,3)$.

The two circles $x^2+y^2=ax$ and $x^2+y^2=c^2$ ($c>0$) touch each other if:

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{2/5}$ is:

If the angle between the line $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z-3}{\lambda }$ and the plane $x+2y+3z=4$ is ${\cos }^{-1}\sqrt{\frac{5}{14}}$, then $\lambda$ equals:

The lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersect $L_3$ at $R$.Statement-1 : The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$.Statement-2 : In any triangle, bisector of an angle divides the opposite side in the ratio of the sides containing the angle.

For a regular polygon, let $r$ and $R$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is:

In a triangle $ABC$, medians $AD$ and $BE$ are drawn. If $AD=4$, $\angle DAB=\frac{\pi }{6}$ and $\angle ABE=\frac{\pi }{3}$, then the area of the $\Delta ABC$ is:

Let $z_1$ and $z_2$ be two roots of the equation $z^2+az+b=0$, $z$ being complex. Further, assume that the origin, $z_1$ and $z_2$ form an equilateral triangle, then:

The sum of the radii of inscribed and circumscribed circles for an $n$ -sided regular polygon of side $a$, is:

Match List-I with List-II:

The geometry and magnetic behaviour of the complex $[{\text{Ni(CO)}}_4]$ are:

Which of the following pairs of compounds is isoelectronic and isostructural?

A question along with a set of statements is given below. We have to find which of the given statements is/are sufficient to answer the given question.

Question : Find the area of the triangle PQR.
Statements :
I. PQ = 5 m., PR = 5 m.
II. Length of PO is 4 m.

In a triangle ABC, P is the midpoint of BC. If AB = (2x+4) cm, AC = 6 cm and AP ⊥ BC, then the value of x is:

A hemisphere has 14cm diameter. Find its volume.

The perimeter of the rectangle is 280m and the difference between its two sides is 40m. Find the side of a square whose area is equal to the area of this rectangle.

Find the total surface area of a hemisphere with a radius of $11\text{cm}$?

If $L$ is the mid-point of the side $YZ$ of $△XYZ$, and the area of $△XYL$ is $13{\text{cm}}^2$, then the area (in ${\text{cm}}^2$) of $△XYZ$ is:

In △ABC, D is the mid-point of BC and G is the centroid. If GD = 10 cm, then the length of AD is ____________.

Which of the following best describes a straight angle?

In a given square figure $\text{ABCD}$, both main diagonals $\text{AC}$ and $\text{BD}$ are drawn, intersecting each other at point $\text{O}$. Count the total number of triangles formed inside the figure.

The chord length of a chord made on an arc of a circle is equal to the radius of the circle. Find the length of the arc (in units), if the radius of the circle is 21 units. (Take $\pi =\frac{22}{7}$)

Two circles with centers $B$ and $D$ have radii $DA=8\text{cm}$ and $BC=x\text{cm}$, respectively. $AC$ is tangent to both circles. If $DB$ and $AC$ intersect at the point $E$, $AE=12\text{cm}$ and $EC=18\text{cm}$, then find the value of $x$ (in $\text{cm}$).

The volumes of two spheres are in the ratio of $512:3375$. The ratio of their surface areas is:

Find the area of a minor sector of a circle whose circumference is $88\text{cm}$ and the length of its minor arc is $22\text{cm}$ (use $\pi =\frac{22}{7}$).

The volume of a cube is four times the volume of a cuboid. If the sides of the cuboid are $32\text{cm}$, $8\text{cm}$ and $4\text{cm}$, then find the ratio of the total surface area of the cube to that of the cuboid.

In a parallelogram ABCD, if vertices are in respective order and diagonals AC and BD intersect at O, then which of the following is NOT always correct?

From an external point $A$, tangents $AP$ and $AQ$ are drawn to a circle with centre $O$. If $\angle APQ=40^{\circ }$, then find $\angle POQ$.

In $\Delta ABC$, $AD$ is the internal bisector of $\angle A$, meeting the side $BC$ at $D$. If $BD=5\text{cm}$ and $BC=7.5\text{cm}$, then the ratio $AB:AC$ is:

The base of a parallelogram is twice its height. If the area of the parallelogram is $338{\text{cm}}^2$, then find its height (in $\text{cm}$).

What is the total surface area of a cone whose radius is $\frac{r}{4}$ and slant height is $4l$?

In an isosceles △LMN, LM=LN, and ∠MLN=37
∘
. Find ∠MNL.

The ratio of length and breadth of a rectangle is $21:20$ and its perimeter is $164\text{cm}$. What is the length of its diagonal (in cm)?

If the surface areas of two spheres are in the ratio of 9: 49, then the ratio of their volumes is:

Three solid iron cubes of edges 8 cm, 10 cm and 12 cm are melted together to make a new cube. It is observed that 496 cm³ of the melted material is lost due to improper handling. The area (in cm²) of the whole surface of the newly formed cube is:

Find the area of a right angle whose base is 15 cm and hypotenuse is 17 cm.

If the angles of a triangle are in the ratio of 9:11:16, then the difference between the greatest angle and the smallest angle is:

In △ABC and △PQR, AB=7m, BC=8m, AC=9m, PQ=7m, QR=8m and PR=9m. Which of the following is true for these triangles?

△ABC is inscribed in a circle with centre O. If AB=35 cm, BC=12 cm and AC=37 cm, then what is the circumradius of the triangle?

In the given figure $AB=AC$ and $\angle ACD=105^{\circ }$, then $\angle BAC$ is equal to

If each edge of a cube is doubled, then the percentage increase in its total surface area is

In the following figure, $PAB$ is a secant and $PT$ is a tangent to the circle. If $PT=5\text{ cm}$ and $PA=3\text{ cm}$, then find the measurement of $AB$.

The diagonals $AC$ and $BD$ of a parallelogram $ABCD$ intersect each other at the point $O$. If $\angle DAC=32^{\circ }$ and $\angle AOB=70^{\circ }$, then $\angle DBC$ is equal to

If the distance between center to chord is $12\text{ cm}$ and the length of the chord is $10\text{ cm}$, then the diameter of the circle is

The perimeter of a rhombus whose diagonals are 10cm and 24cm is

In the given figure, ABCD is a cyclic quadrilateral. Then the ordered pair (x, y) is

The perimeter of a rhombus whose diagonals are $10\text{cm}$ and $24\text{cm}$ is

In a cyclic quadrilateral ABCD, the opposite interior angles are given as linear expressions: ∠A=(4y+20)
∘
, ∠B=(3y−5)
∘
, ∠C=(4x)
∘
, and ∠D=(7x+5)
∘
. Find the values of the ordered pair (x,y).

In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PQR to the area of the trapezium SQRT?(Note: Assume the text asks for the ratio of Area of PQR to Area of SQRT based on the mathematical constraints provided)

In which of the following molecules, all atoms are coplanar?

${\text{XeF}}_2$ is isostructural with: