The roots of quadratic equation x² – 9 = 0 are

The first and last terms of an AP are 1 and 11. If the sum of all its terms is 36, then the number of terms will be

in figure , if ㄥAOB = 125^o, then ㄥCOD is equal to

In the given figure, AC: CB is

An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from his eyes is 45°. The height of the tower is

The quadratic equation \(x^{2} - 10 x + 2 = 0\) has ........... roots.

If the nth term of an A.P. is 2n + 1, then the sum of first n-terms of the A.P. is .............

A line segment drawn through the end of a radius and perpendicular to it, is a ___________to the circle.

To divide a line segment AB in the ratio 2 : 5, a ray. AX is drawn such that \(\angle BAX\) is acute. Then points are marked at equal intervals at AX. What is the minimum number of these points?

If the angle of elevation of the top of a tower from two points distance and t from its foot are complementary, the height of the tower is ........

A quadratic equation \(ax^{2} + bx + c = 0\) has real roots if \(b^{2} \ge 4ac.\)

If nth term of an A.P. is 6n + 2, then common difference is 6.

The length of the tangent is the length of the segment from an external point to the point of contact.

In a \(\Delta ABC\) , we draw \(\Delta AB'C'\sim \Delta ABC\) with scale  factor \(\frac { 13 }{ 15 } \) . Then, perimeter of \(\Delta ABC\)  > perimeter of \(\Delta AB'C'\)

The length of shadow of a tree 7 m high, when Sun's elevation is \({ 45 }^{ ° }\)is 7 m.

A girl of height 100cm stands in front of lamppost and cast a shadow of length \(100\sqrt{3}\)cm on the ground.The angle of elevation of the top of the lamppost is 

Write the nature of roots of the quadratic equation 9x²-6x-2=0

Find the 10th term of the AP : 2, 7, 12, . . .

At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY and at a distance 8 cm from A.

To divide a line segment AB in the ratio p : q, draw a ray AX so that \(\angle BAX\)  is an acute angle. How many points mark on ray AX?

A straight highway leads to the foot of a 100 m tall tower. From the top of the tower, angle of depression of a car on the highway is 30^o . Find the distance of the car from foot of the tower.

Check whether the following are quadratic equations: (x+2)³=2x(x²-1)

What will be the value of p for which the equation \(x^{ 2 }-4x+p=0\) has equal roots?

solve for x : \(9x^{ 2 }-6ax+(a^{ 2 }+b^{ 2 })=0\)

Which of the following are A.P.s? If they from an A.P., find the common difference d and write three more terms: a = ...., d = -5, n = 20, a_n = -15

Find the common difference of an AP, whose first term is \(\frac {1}{2}\) and 8th term is \(\frac {17}{6}.\)

Two APs have the same common difference. The first term of an AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms? Why?

Prove that the angle between the two tangents to a circle drawn from an external point, is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length.

In the given figure, from a point P, two tangents PT and PS are drawn to a circle with centre O such that \(\angle SPT\) = 120°, Prove that OP = 2 PS

Construct tangents to a circle of radius 3 cm from a point on concentric circle of radius 5 cm and measure its length.

Draw a circle of radius 3 cm.  From a point P, 6 cm away from its centre, construct a pair of tangents to the circle.  Measure the lengths of the tangents.

To locate a point Q on PR such that \(QR=\frac { 2 }{ 3 } PQ\),at what ratio the line segment PR should be divided?

If a pole of height 12 m throws shadow of \(4\sqrt { 3 } \) m. then find the angle of elevation of sun.

From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot of the tower, are \(\alpha\) and \(\beta\) \((\beta > \alpha)\) .Find the distance between the two objects.

The angel of elevation of an aeroplane from a point on the ground is 45^o. After a flight for 15 seconds, the elevation changes to 30^o. If the aeroplane is flying at a height of 3000m, find the speed of the aeroplane.

The product of two successive integral multiples of 5 is 300. Determine the multiples.

A journey of 192 krn from a town A to town B takes 2 hours more by an ordinary passenger train than a super fast train. If the speed of the faster train is 16 krn!h more, find the speed of the faster and the passenger train.

The first term of an A.P. is -5 and the last term 45. If the sum of the terms of the A.P. is 120, then find the number of terms and the common difference.

The interior angles of a polygon are in AP. The smallest angle is \(120^0\) and the common difference is \({ 5 } ^ { 0 }\) . Find the number of sides of the polygon.

Two tangents are drawn to a circle from an exterior point A, touching the circle at B and C. From another point R, on circle a third tangent is drawn to the circle intersecting AB in P and AC in Q and touching the circle at R. If AB = 20 units, find the perimeter of \(\triangle APQ\).

Point A(-1, y) and B(5, 7) lie on a circle with centre 0(2, - 3y). Find the values of y. Hence find the radius of the circle.

Draw an equilateral \(\Delta ABC\) of each side 4 cm.Construct a triangle similar to it and of scale factor \(\frac{3}{5}\).Is the new triangle also an equilateral?

Draw a circle of diameter AB = 6 cm with centre O and then draw a tangent to the circle at point A or B.

The top of a broken tree has its top touching the ground (shown in the following figure) at a distance of 10 m from the bottom. If the angle made by the broken part with ground is 30°, then find the length of the broken part.

Determine the height of a mountain, if the elevation of its top at an unknown distance from the base is 30\(°\)  and at a distance 10 km further off from the mountain, along the same line, the angle of elevation is 15\(°\) (Take tan 15\(°\) = 0.27) 

Solve the following quadratic equation for x ;
\((a + b^{2})x^{2} - 4abx - (a-b)^{2} = 0\)

If the equations 2x²-7x+3=0 and 4x² +ax-3=0 have common roots then what is the value of a?

Deepak repays his total loan of Rs.1,18,000 by paying every month starting with the first instalment of Rs.1000. If he increase the instalment by Rs.100 every month, what amount will be paid as the last instalment of loan? What amount of loan he still have to pay after the 30th instalment?

In the given figure, AB is a diameter of the circle. The length of AB = 5 cm. If O is the centre of the circle and the length of tangent segment AT = 12cm, determine CT.

In figure, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.

Construct a \(\Delta ABC\) , in which BC = 5cm, \(\angle CAB=120°\)and \(\angle ABC=30°\) . Then, construct another triangle whose sides are \(\frac{4}{5}\)times of the corresponding sides of \(\Delta ABC\) . Justify your construction.

The shadow of a tower, when the angle of elevation of the sun is 45⁰, is found to be 40 40m longer than when it is 60⁰.Find the height of the tower.

Anand is watching a circus artist climbing a 15m long rope, which is tightly stretched and tired from the top of a vertical pole to the ground.
(i) Find the height of the pole, if the angle made by rope with ground level is \({ 45 }^{ ° }\) .
(ii) Which mathematical concept is used to solve this question?
(iii) What value is experienced by Anand?