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A bird is flying at a height of 30 m above the ground. The angle of elevation of the bird from an observer is ${30}^{\circ }$. Find the distance(in meters), between the observer and the bird.

$ta{n}^{2}A-ta{n}^{2}B=\frac{si{n}^{2}A-si{n}^{2}B}{co{s}^{2}A.co{s}^{2}B}$

The first and the last terms of an A. P. are 34 and 700 respectively. If the common difference is 18, how many terms are there and what is their sum?

A metallic right circular cone 20 cm high and whose vertical angle is ${60}^{\circ }$ is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{16}cm$, find the length of the wire. [4 MARKS]

Question 11

Write five pairs of prime numbers less than 20 whose sum is divisible by 5.
[Hint: 3 + 7 = 10]

Show that the given expression is a composite number. 41 ×3 ×15 + 15

In the given figure, RS = QT and QS = RT. Then which of the following options is correct?

O is any point inside a triangle ABC. The bisector of $\angle$AOB, $\angle$BOC and $\angle$COA meet the sides AB. BC and CA in point D. E and F respectively. Show that AD $\times$ BE $\times$ CF = DB $\times$ EC $\times$ FA. [4 MARKS]

Find the sums given below:
(i) 7+$\frac{21}{2}$ +14+...+84
(ii) 34+32+30+...+10
(iii) -5 + (-8) + (-11) + .... + (-230)

The diagonals of a quadrilateral bisect each other at right angles. Which type of quadrilateral is formed?

A fair die is thrown until a score of less than five points is obtained. The probability of obtaining less than three points on the last throw is

O is the centre of a circle of radius 8 cm. The tangent at a point A on the circle cuts a line through O at B such that AB=15 cm. Find OB.

The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. Find its capacity and total surface area. $[Use\pi =\mathrm{3.14.}]$

Tow coins are tossed simultaneously. What is the probability of getting at most one head?
(a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) $\frac{2}{3}$ (d) $\frac{3}{4}$

The polynomial $a{x}^2+bx+c$ is a linear polynomial if

If an SUV can transport 7 people, car can transport 5, jeep can transport 8 and auto can transport 2 people, which of the following combination can be used to exactly transport 25 people?

The height of a conical tent is 14 m and its floor area is 346.5 $m^2$. How much canvas, 1.1 m wide, will be required for it ?

(a) 490 m (b) 525 m (c) 665 m (d) 860 m

Question 2
During the medical check-up of 35 students of a class, their weights were recorded as follows:
$\begin{array}{cc}Weight\left(inkg\right)& Numberofstudents\\ Lessthan38& 0\\ Lessthan40& 3\\ Lessthan42& 5\\ Lessthan44& 9\\ Lessthan46& 14\\ Lessthan48& 28\\ Lessthan50& 32\\ Lessthan52& 35\end{array}$

Draw a less than type ogive for the given data. Hence, obtain the median weight from the graph verify the result by using the formula.

Explain the relationship between zeroes and coefficients of linear, quadratic and cubic polynomial.

Prove that $\sqrt{2}+\sqrt{3}$ is irrational. [3 MARKS]

Three consecutive positive integers are taken such that sum of square of the first number and the product of other two numbers is 232. Find the integers.

${\cos }^2{30}^{\circ }+si{n}^2{45}^{\circ }-\frac{1}{3}ta{n}^2{60}^{\circ }=$

Length of the altitude on the hypotenuse of a right angled triangle with area A $c{m}^2$ and one of the sides equal to b cm, is

Express $\frac{1}{2}$ and $\frac{1}{4}$ on the number lines.

Ajeet invests ₹ 9000 in a company paying a dividend of 6% per annum when its ₹ 100 shares can be bought for ₹ 150. Find :-

i) his annual income

ii) If he sells 50% of his shares at ₹ 200 each, what is his gain in this transaction?

Construct a $3\times 4$ matrix whose elements $a_{ij}$ are given by $a_{ij}=i+2j$

Question 113 (iii)

Study the layout given in the figure and answer the question.

If the cost of carpeting is Rs 50 per $m^2$, write an expression for calculating the total cost of carpeting both the bedrooms and the living room.

If a sector of radius 8cm with central 120⁰ is folded into a cone, the slant height of cone is:

The person who purchases the shares of a company is called a ____.

Find the probability that a number selected at random from the numbers 1 to 25 is not a prime number.

O is the centre of the circle. AB is a minor arc of the circle. The angle subtended by AB at centre, $\angle$AOB = ${110}^{\circ }$, then angle subtended by the arc at any point on the circle say $\angle$APB is ____, where P is any point on the circle.

A person bought an ice cream cone shown in the above figure. His son looks at the top view of the ice cream cone. Which 2-D figure will his son see?

1. 20

The circumference of a circle is 8 cm. Find the area of the sector whose central angle is 72${}^{\circ }$.

Let y(x) be a solution of the differential equation $(1+e^x)y^{\prime }+y{e}^x=1.Ify\left(0\right)=2,$ then which of the following statement (s) is/are true ?

If $x$ and $y$ are the roots of the polynomial $4k^2+5k+6$, then match the follwing with their values.

What will the ratio $AB:AC$ be if $C$ divides the line segment $AB$ in the ratio $5:12$?

Product of two consecutive natural number is 56. Find the smaller number.

If $5x-3\le 5+3x\le 4x+2$, express it as $a\le x\le b$.

If a and b are positive real numbers such that 2 log (3a - 2b) = log a + log b, then the value of $\frac{a}{b}$=

A vertical tower stands on the horizontal ground and is surmounted by a vertical flagstaff of height 'h' metre. At a point on the ground, the angle of elevation of the bottom of the flagstaff is $\alpha$ and that of the top of the flagstaff is $\beta$. The height of the tower is ___ m.

AB is diameter and AC is a chord of a circle such that $\angle BAC={30}^{\circ }$. If the tangent at C intersects AB produced in D, prove that BC = BD [4 MARKS]

A shopkeeper sells an article at its marked price which is Rs. 8000 and charges VAT at the rate of 12% from the customer. If the shopkeeper pays a VAT of Rs. 120, then the price paid by shopkeeper inclusive of the tax will be :

If two equal chords of a circle intersect, then prove that their segments are equal. [3 MARKS]

In which of the following frequency distributions, the class intervals overlap?

In a $△ABC,\angle A={60}^{\circ },\angle B={30}^{\circ }and\angle C={90}^{\circ }.$

Then the ratio AC:BC:AB will be equal to ___.