If the equation \(9x^{2} + 6kx+4 = 0\)  has equal roots, then find the value of k.

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

In the given figure, if TP and TQ are the two tangent to a circle with centre O, so that \(\angle POQ={ 110 }^{ \circ }\), then find \(\angle PTQ\).

A box contains cards numbered 6 to 50.A card is down at random from the box.Find the probability that the drawn card has a number which is a perfect square.

Draw a line segment of length 10 cm, and divide it in the ratio 2:2.  Measure the two parts. 

Given a rhombus PQRS in which PQ = 5.5 cm and angle PQR = 70^o . Divide it into two triangles say PQR and PSR. Construct the triangle PQ'R' similar to triangle PQR with scale factor \(\frac { 3 }{ 2 } \). Draw a line segment R'S' parallel to RS, where R' lies on extended PR and S' lies on extended PS. Is PQ'R'S' a rhombus? Give reasons.

If two vertices of a triangle are (6, 3) and (-1, 7) and centroid (1, 5), then find the third vertex.

Find the sum of first 24 terms of the AP a₁, a₂, a₃ ..., if its is known that a₁ + a₅ + a₁₀ + a₁₅ + a₂₀ + a₂₄ = 225

The angle of elevation of the top of the building of an organisation working for poor and needy children at a point on level ground is 45^o. After moving 100m towards the building along the same horizontal line, the angle of elevation of the building is 60⁰ .Find the height of the building.
(i)Which mathematical concept is being used here?
(ii)Which social act is being discussed here?

Prove that (-2,3),(8,3) and (6,7) are the vertices of a right-angled triangle.

If the radius of the circle is 6 cm and the length of an arc is 12 cm. Find the area of the sector.

A child reshapes a cone of height 24 cm and base radius 6 cm, to the form of a sphere. Find the radius of the sphere.

If α and β are zeroes of the quadratic polynomial p(x)=x²-(k+6)x+2(2k-1), then find the value of k, if \(a+\beta =\frac { a\beta }{ 2 } \) .

In the given figure, \(\triangle ABC,\) is the right angles at B. \(\triangle BSC\) is right angles at S and BC = 7.5 cm, RS = 5 cm, RB = 6 cm \(\angle BSR=x^0\) and \(\angle SAB=y^0.\)  Find

(i) tan x⁰
(ii) sin y
(ii) cos y⁰

Using section formula, show that the points A(-3, -1), B(1, 3) and C(-1, 1) are collinear.

ABCDEF is a regular hexagon. With vertices A, B, C, D, E and F as the centres of circles with same radius r are drawn. Find the area of the shaded portion shown in the given figure.

A statue 1.46 m tall stand on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60^o and from the same point, the angle of elevation of the top of the pedestal is 45^o . Find the height of the pedestal. \((\sqrt { 3 } =1.73)\) .

Three unbiased coins are tossed together. What is the probability of getting
(i) two heads?
(ii) atleast two heads?
(iii) atmost two heads?

AB is a diameter of a circle. AH and BK are perpendicular from A and B respectively to the tangent at P.Prove that AH + BK = AB.

Draw a line segment AB of length 7 cm. Taking A as centre, draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2 cm. Construct tangents to each circle from the centre of the other circle

Find the mean of the following frequency distribution:

Divide 56 in four parts in AP, such that the ratio of the product of their extremes (1st and 4th) to the product of means (2nd and 4rd) is 5 : 6.

In an isosceles right angled triangle, if the hypotenuse is \(5\sqrt{2}\) cm, then find the length of the sides of the triangle.

Solve each of the following equation by factorisation.
t²+3t-10=0

From the top of a hill the angles of depression of two consecutive kilometre stones east are found to be 30^o and 60^o . Find the height of the hill.

Three horses are tethered at 3 corner of a triangular plot having sides 20 m, 30 m and 40 m with ropes of 7 m length each. Find the area of this plot which can be grazed by the horses.

If \(15\tan ^{ 2 }{ \theta } +4\sec ^{ 2 }{ \theta } =23\), then find the value of \(\left( \sec { \theta } +cosec\theta \right) ^{ 2 }-\sin ^{ 2 }{ \theta } \)

If median of the following frequency distribution is 35, then find the value of x.

If α and β are the zeroes of the quadratic polynomial f(x)=x²-px+q, then prove that \(\frac { a^{ 2 } }{ \beta ^{ 2 } } +\frac { \beta ^{ 2 } }{ a^{ 2 } } =\frac { p^{ 4 } }{ q^{ 2 } } -\frac { 4p^{ 2 } }{ q } +2.\)

Solve the following pair of linear equations graphically:
2x + 3y = 12, 2y - 1 = x.
Determine the co-ordinates of the vertices of the triangle formed by the lines represented by these equations with the x-axis.

The external length, breadth and height of a closed rectangular wooden box are 18 cm, 10 cm and 6 cm respectively and thickness of wood is 1/2 cm. When the box is empty it weighs 15 kg and when filled with sand it weighs 100 kg. Find the weight of the cubic cm of wood and cubic am of sand.

Two tankers contain 504 L and 735 L of milk, respectively. Find the maximum capacity of a container which can measure the milk of either tank is exact number of times.