CUET UG 2023 — Geometry

64 small solid iron spheres of radius 'r' are melted to form a big sphere of radius R. If S and S' are surface areas of the small and the big sphere respectively, then find the ratio S' : S.

64 solid iron spheres of radius r are melted to form a sphere of radius R. Find R : r.

A, B, C, are three points on the circumference of a circle and if $AB = AC = 5\sqrt{2}$, angle $BAC = 90$ degrees then radius of the circle is __________.

A cone of height 24 cm and base radius 6 cm is made of modelling clay. A child reshapes it in the form of a sphere. The radius of the sphere is :

A cuboidal slab of copper having dimensions 22 cm $\times$ 10 cm $\times$ 5 cm is melted and recasted in the form of a wire of 1 mm diameter. The wire is rubber coated at the rate of Rs. 1.50 per meter. Find the cost of rubber coating (in Rs.) Use $\pi = \frac{22}{7}$.

A ladder 25 feet long is leaning against a wall such that it touches 24 feet high window. How far is the foot of the ladder from the wall ?

A man standing on the bank of a river observes that the angle subtended by a tree standing on the opposite bank is 60 degrees on his side of bank. When he moved away 24m from the bank, he finds the angle to be 30 degrees. Find the breadth of the river:

A point on the y-axis which is equidistant from the points A(6, 5) and B(-4, 3) is:

A quadrilateral has vertices in the order $(0, -1)$, $(-2, 3)$, $(6, 7)$ and $(8, 3)$. The quadrilateral is a:

A rectangle has its longer side 2 cm greater than its shorter side. Its area is 80 $cm^2$. Find the perimeter of the rectangle (in cm).

A rectangular room can be partitioned into two equal square rooms by a 7 meter long partition. What is the floor area of the rectangular room in $m^2$ ?

A sector as shown in the figure is assembled to form a cone. What is the base radius (in cm) of the cone so formed ? $\left(\pi = \frac{22}{7}\right)$

A solid metallic sphere of radius 8 cm is melted and recasted as a cone of height 8 cm. Find the base radius of the cone (in cm).

A toy is made in the shape of a hemisphere of diameter 7 cm surmounted by a cone. If this 15.5 cm high toy is polished at 20 paise per cm$^2$, then find the cost of polishing. Take $\pi = \frac{22}{7}$

A well has to be dug out 20 m deep and its radius is 3.5 m. Find the cost of plastering the inner curved surface at Rs 5 per square meter.

AB is a diameter of a circle with centre (3, 4). If coordinates of A is the point (4, 9), then find the coordinates of B.

An angle is $\frac{3}{7}$ times its supplementary angles. Find the angle.

An interior angle of a regular polygon is 135$^\circ$. The polygon is a/an:

Find how many triangle are there in the given figure? 

Find the angle of elevation of the Sun, when the length of the shadow of a tree is $\frac{1}{\sqrt{3}}$ times the height of the tree.

Find the area of an equilateral triangle each of whose sides measures 4 cm:

Find the coordinates of centroid of $\triangle ABC$ if the mid point of BC is D(2, 4) and vertex A is (2, -3).

Find the length of the arc of a circle with radius 5 cm, if its central angle is $70^\circ$.

Find the perimeter of a rhombus whose one diagonal is 16 cm long and area is 240 cm$^2$.

From a point P on the ground the angle of elevation of the top of 10 m high building is $30°$. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from the point P is $45°$. Length of the flagstaff is: (Take $\sqrt{3} = 1.732$)

From a solid cylinder whose height is 2.4 cm and diameter is 1.4 cm, a conical cavity of same height and same diameter is carved out. The total surface area of the remaining solid is: Use $\pi = \frac{22}{7}$

From the figure, find $x$.

From the figure, find the values of $x$ and $y$. 

From the figure, what is the value of x? 

If A (1, 2), B(4, y), C(x, 6) and D(3, 5) are the vertices of a parallelogram ABCD, then find the values of x and y.

If CE is parallel to DB, what is the value of $x$: 

If in $\triangle ABC$ $\angle A + \angle B = 90^\circ$ and $\sin B = \frac{4}{5}$, then find the value of $\cos A$.

If perimeter of a rhombus is 104 cm and length of one of its diagonals is 48 cm, then area of the rhombus (in cm$^2$) is :

If point B(0, 1) is equidistant from points A(5, -3) and C(x, 6), then find the values of x.

If $\sin 2\theta = \cos 40^\circ$, then the smallest positive value of $\theta$ is :

If $3\sin\theta - 4\cos\theta = 0$, then value of $\tan\theta\cdot\csc\theta$ is :

If $\sin\theta + \csc\theta = 2$, then what is the value of $\sin^2\theta + \csc^2\theta$?

In a $\Delta ABC$ right angled at A, if $\angle ABC = 60^\circ$ and AC = 4 units, then length of BC (in units) is :

In a triangle ABC right angled at B, AB=8 unit and AC=10 unit. What is the value of $\sin^2\theta - \cos^2\theta$ where theta is angle ACB ?

In the figure, ABCDE and AEPQ are a regular pentagon and a square respectively. What is the measure of angle AQB ?

In the given figure DE || BC, the value of x is: 

In $\triangle ABC$ with AB = 5 cm, BC = 12 cm, AC = 13 cm and $\angle B = 90^\circ$, which of the following is/are not correct? (a) $\tan C = \frac{12}{13}$ (b) $\text{cosec} A = \frac{13}{12}$ (c) $\sin B = \frac{5}{13}$ (d) $\tan A = \frac{12}{15}$ (e) $\cos C = \frac{12}{13}$ Choose the correct answer from the options given below:

Match List - I with List - II. | List - I (Shape) | List - II (Area) | |---|---| | (A) Square | (I) $\pi a^2$ | | (B) Triangle | (II) $b \times h$ | | (C) Circle | (III) $a \times a$ | | (D) Parallelogram | (IV) $\frac{1}{2} b \times h$ | Choose the correct answer from the options given below :

Radius of the base and height of a cone are 5 cm and 12 cm respectively. Find its slant height.

The angle of elevation of the sun, when the length of the shadow of a tree is equal to the height of the tree is :

The area of a circle is 154 cm$^2$. Find the circumference of the circle. (Take $\pi = \frac{22}{7}$)

The area of a circle is numerically equal to its circumference. Find the diameter of the circle.

The area of a right-angled triangle with base 3 m and hypotenuse 5 m is :

The area of four walls of a rectangular hall having length 18 m and height 8 m is 448 m$^2$. What is the breadth of the hall (in m)?

The circumference of a circular field is 396 m and that of the other circular field is 132 m. Find the area (in m$^2$) of the third circular field whose radius is the sum of the radii of the first two fields. Take $\pi = \frac{22}{7}$

The curved surface area of a right circular cylinder of height 14 cm is 88 cm$^2$. The diameter of the base of the cylinder is $\left(\text{Use } \pi = \frac{22}{7}\right)$ :

The distance between two parallel sides of a trapezium is 15 m and its area is 480 m$^2$. If one of the parallel sides is 20 m long, then length of the other side is:

The heights of two right circular cones are in the ratio 1 : 2 and the circumferences of their bases are in the ratio 3 : 4. Find the ratio of their volumes.

The length of the side of an equilateral triangle is $4\sqrt{3}$ cm. Find its height.

The point (-2, 3) lies in which quadrant?

The sides of a triangle are in the ratio $\frac{1}{3} : \frac{1}{4} : \frac{1}{5}$. If the semi-perimeter of the triangle is 47 cm, then what is the length of the longest side?

The surface area (in $m^2$) of a sphere of radius 7 cm is: Use $\pi = \frac{22}{7}$

The volume of a sphere of radius r is obtained by multiplying its surface area with:

The volumes of 3 solid cubes made of metal are 125 cm$^3$, 64 cm$^3$ and 27 cm$^3$ respectively. After melting all the three cubes a solid cube is made. Find the edge of the new cube.

Which of the following statements are incorrect? (a) Volume of a cone = $\frac{1}{3}\pi r^3 h$ (b) Volume of a cone = $\frac{1}{3}\pi r^2 h$ (c) Volume of a hemisphere = $\frac{2}{3}\pi r^2$ (d) Volume of a hemisphere = $\frac{2}{3}\pi r^3$ (e) Volume of a cylinder = $\frac{1}{3}\pi r^3 h$ Choose the correct answer from the options given below: