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In a circle, two chords AB and CD intersect at a point P inside the circle. Prove that

(a) $\Delta PAC\sim \Delta PDB$

(b) $PA.PB=PC.PD$.

A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm,find the radius and slant height of the heap.

A circle with centre O, diameter AB and a chord AD is drawn. Another circle is drawn with AO as diameter to cut AD at C, then BD = 2 OC.

If general term of an AP is given by 2 + 4n then find its common difference.

A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A first and then at B, with radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm and so on. The total length of such a spiral made up of 13 consecutive semi circles is ___. Take $\pi =\frac{22}{7}$

Express 0.99999…in the form. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

The right circular cone of largest volume that can be enclosed by a sphere of $1m$ radius has height of

From a cubical piece of wood of side 21 cm, a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece. Find the surface area and volume of the remaining piece.

Two vertices of an equilateral triangle are (0,0) and (√3,√3). Find the third vertex.

The volume of a cone of radius 7 $cm$ and slant height 14 $cm$ is ______.
$(use\pi =\frac{22}{7})$

The product of two consecutive odd numbers is 675. If the smaller odd number is $x$, then the quadratic equation formed by the given data will be.

Find the excluded values of $\frac{2x+1}{x^2-x-6}.$

Question 14

Divide 20 pens between Sheela and Sangeeta in the ratio 3 : 2.

In the figure given below AB = 3 cm, AC = 5 cm and AD = 4 cm and D is the midpoint of BC. Then the length of BD is :

In the figure below, $a={20}^{\circ }$. The value of $(4a–b)$ is ______.

The following table shows the daily wages of workers in a factory:
$\begin{array}{cccccc}\text{Daily eages (in Rs)}& 0-100& 100-200& 200-300& 300-400& 400-500\\ \text{Number of workers}& 40& 32& 48& 22& 8\end{array}$
Find the median daily wage income of the workers.

If the mean of the following distribution is 2.6, then the value of y is

A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are only given to persons having age 18 years onwards but less than 60 years.
$\begin{array}{cc}\text{Age in years}& \text{Number of policy holders}\\ \text{Below 20}& 2\\ \text{Below 25}& 6\\ \text{Below 30}& 24\\ \text{Below 35}& 45\\ \text{Below 40}& 78\\ \text{Below 45}& 89\\ \text{Below 50}& 92\\ \text{Below 55}& 98\\ \text{Below 60}& 100\end{array}$

Prove that for a cone made by bending a semicircle, the curved surface area is double the base area.

Ramkali saved $Rs.5$ in the first week of a year and then increased her weekly saving by $Rs.\mathrm{1.75.}$ If in the $n^{th}$ week, her week, her weekly savings become $Rs.20.75,$ find $n.$

A cylinder has a radius of 7 cm and height of 25 mm. If ten such cylinders are stacked up, then the total volume will be .

Locus of the point which divides double ordinates of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ in the ratio $1:2$ internally is

A(–3, 0), B(4,–1) and C(5, 2) are vertices of ΔABC. The length of altitude through A is __________.

Wire A is bent to form an equilateral triangle and wire B is bent in the form of a square, such that the area covered by the triangle and the square is the same. What is the ratio of the length of wire A to wire B?