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The area of an equilateral triangle is $49\sqrt{3}c{m}^2$. Taking each angular point as centre, circles are drawn with radius equal to half the length of the side of the triangle. Find the area of the triangle not included in the circles. [Take \sqrt{3} = 1.73.]

For which one of the following subsets, $co{s}^{2}A$ + $si{n}^{2}A$ = 1 is valid?

If one zero of the polynomial $x^2-4x+1$ is $(2+\sqrt{3})$ , write the other zero.

If a, b, c are in continued proportion and a(b-c)=2b, prove that:

a-c = 2(a + b) / a

Construct a triangle ABC with BC = 6.5 cm,AB = 5.5 cm.AC = 5 cm. Construct the incircle of the triangle. Measure the radius of the incircle.(approximate)

The following data gives the information on the observed lifetime (in hours) of 225 electrical components:
$\begin{array}{ccccccc}\text{Life time (in hours)}& 0-20& 20-40& 40-60& 60-80& 80-100& 100-120\\ \text{Number of components}& 10& 35& 52& 61& 38& 29\end{array}$
Determine the modal lifetimes of the components.

Mass of two spheres is in the ratio 8:1. Assuming that both of them have same density, find the ratio of their radii.

Find the mode for the following data.

$\begin{array}{cc}\text{Class interval}& \text{Frequency}\\ 0-10& 16\\ 10-20& 20\\ 20-30& 32\\ 30-40& 48\\ 40-50& 30\\ 50-60& 14\end{array}$
[2 MARKS]

The faces of a red cube and a yellow cube are numbered from 1 to 6. Both cubes are rolled. What is the probability that the top face of each cube will have the same number?

Prove that Cot A -1 / 2- sec^{​2 A = cot A/ 1+tan A}

A flagstaff 6 metres high throws a shadow 2$\sqrt{3}$ m long on the ground. The angle of elevation of the top of the flagstaff from the point where the shadow ends?

A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in figure. Find the total length of wire required and the area of each sector. [3 MARKS]

The given figure shows the cross-section of a water channel consisting of a rectangle and a semi-circle. Assuming that the channel is always full, find the volume of water discharged through it in one minute if water is flowing at the rate of 20 cm per second. Give your answer in cubic metres correct to one place of decimal.

In figure, $△FEC\cong △GBD$ and $\angle 1=\angle 2$. Prove that $△ADE\sim △ABC$.

The diameters of the front and rear wheels of a tractor are 80 cm and 2 m respectively. Find the number of revolutions that a rear wheel makes to cover the distance which the front wheel covers in 800 revolutions.

Find the probability of getting the sum of two numbers, less than 3 or more than 11 when a pair of distinct dice is thrown together.

If A = ${45}^{\circ },$ verify that:

(i) $sin2A=2sinAcosA$ (ii) $cos2A=2co{s}^{2}A-1=1-2si{n}^{2}A$

The monthly income of 100 families are given as below:
$\begin{array}{cc}& \\ \text{Income in (in Rs)}& \text{Number of families}\\ 0-5000& 8\\ 5000-10000& 26\\ 10000-15000& 41\\ 15000-20000& 16\\ 20000-25000& 3\\ 25000-30000& 3\\ 30000-35000& 2\\ 35000-40000& 1\end{array}$
Calculate the modal income.

In the figure, tangent at a point C of a circle and a diameter AB when extended intersect at P. If $\angle PCA={110}^{\circ }$, find $\angle CBA$

Surds are the_____ roots of any equation with ___ coefficients.

Write a pair of linear equations which has the unique solution $x=-1,y=3$.How many such pairs can you write?

Prove that
$\frac{sin\theta }{1-cot\theta }+\frac{cos\theta }{1-tan\theta }=cos\theta +sin\theta$ [4 MARKS]

show that the reciprocal of 3+2 root 2 is an irrational number

All kings, jack & diamonds have been removed from a pack of 52 playing cards & the remaining cards are well shuffled. A card is drawn from remaining pack.

Find the probability that the card drawn is

a) a red queen

b) a face card

c) a heart

Vikram travels 4 m east, takes a right turn and again travels for 3 m. How far is he from the starting point?

How can we find mode if two classes have the highest and same frequency? How can we identify the median class if two classes are equally near to $\frac{n}{2}$?

using quadratic formula solve for x : a²b²x² - (4b⁴ - 3a⁴) x - 12a²b² = 0

A cylinder is having a volume of $231$ sq. cm. If the height of cylinder is $6cm,$ a ball will fit inside the cylinder if its radius is

If two digit number is four times the sum of its digits and twice the product of digits .find the number

If θ lies in Quadrant 1, $\sqrt{se{c}^2\theta +cose{c}^2\theta }$ =

The condition for equation ax² + bx + c = 0 to be quadratic is

Find the value of p+q if the equation $x^2+(p+q)x+$ $\frac{pq}{b}$=0 has equal roots.

Find the $n^{th}$ term of the series 3.4.7 + 4.5.9 + 5.6.11 + -------------

Find the sum of all three digit natural numbers, which are multiples of 7.

find the zeroes of 6x²-3

Area of the given figure will be sq. units.
(Area of each square = 1 sq. unit)

One integer is chosen out of 1, 2, 3, ..…., 100. What is the probability that it is neither divisible by 4 nor by 6?

If m times the m th term of an A.P is equal to n times it's n th term then find the(m+n)th term

Find the area of the colored region in the figure if the radius of the circle is 7 cm.

M is the mid-point of the line segment joining the points A(0, 4) and B(6, 0). M also divides the line segment OP in the ratio 1 : 3.Find:
(i) co-ordinates of M
(ii) co-ordinates of P
(iii) length of BP