Duration: 3h

A solid metallic cuboid of dimensions 9 m × 8 m × 2 m is melted and recast into solid cubes of edge 2 m. Find the number of cubes so formed.

Chapter: [7.02] Surface Areas and Volumes

PQ is a tangent drawn from an external point P to a circle with centre O, QOR is the diameter of the circle. If ∠POR = 120°, what is the measure of ∠OPQ?

Chapter: [3.01] Circles

A ladder 15 m long makes an angle of 60° with the wall. Find the height of the point where the ladder touches the wall.

Chapter: [4.01] Heights and Distances

If one root of the quadratic equation 6*x*^{2} – *x* – *k* = 0 is

*k*.

Chapter: [2.03] Quadratic Equations

If two adjacent vertices of a parallelogram are (3, 2) and (−1, 0) and the diagonals intersect at (2, −5), then find the coordinates of the other two vertices.

Chapter: [6.01] Lines (In Two-dimensions)

In Fig., if AB = AC, prove that BE = EC

Chapter: [3.01] Circles

Find the probability that in a leap year there will be 53 Tuesdays ?

Chapter: [5.01] Probability

If seven times the 7^{th} term of an A.P. is equal to eleven times the 11^{th} term, then what will be its 18^{th} term?

Chapter: [2.02] Arithmetic Progressions

Two different dice are thrown together. Find the probability that the product of the numbers appeared is less than 18 ?

Chapter: [5.01] Probability

Solve for *x*:

3x2-2x-83=0

Chapter: [2.01] Pair of Linear Equations in Two Variables

Show that ∆ ABC with vertices A (–2, 0), B (0, 2) and C (2, 0) is similar to ∆ DEF with vertices D (–4, 0), F (4, 0) and E (0, 4) ?

Chapter: [6.01] Lines (In Two-dimensions)

In the given figure, ∆ABC is an equilateral triangle of side 3 units. Find the coordinates of the other two vertices ?

Chapter: [3.02] Triangles

In the given figure, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and AC is 14 cm. If arcs of equal radii 7 cm taking A, B, C and D as centres, have been drawn, then find the area of the shaded region ?

Chapter: [3.02] Triangles

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Chapter: [3.01] Circles

The shadow of a tower at a time is three times as long as its shadow when the angle of elevation of the sun is 60°. Find the angle of elevation of the sun at the time of the longer shadow ?

Chapter: [4.01] Heights and Distances

In the given figure, PA and PB are tangents to a circle from an external point P such that PA = 4 cm and ∠BAC = 135°. Find the length of chord AB ?

Chapter: [7.01] Areas Related to Circles

Find the coordinates of the points of trisection of the line segment joining the points (3, –2) and (–3, –4) ?

Chapter: [6.01] Lines (In Two-dimensions)

If the quadratic equation (c^{2} – ab) x^{2} – 2 (a^{2} – bc) x + b^{2} – ac = 0 in x, has equal roots, then show that either a = 0 or a^{3} + b^{3} + c^{3} = 3abc ?

Chapter: [2.03] Quadratic Equations

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

Chapter: [2.02] Arithmetic Progressions

A solid sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?

Chapter: [7.02] Surface Areas and Volumes

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Chapter: [3.01] Circles

A child puts one five-rupee coin of her saving in the piggy bank on the first day. She increases her saving by one five-rupee coin daily. If the piggy bank can hold 190 coins of five rupees in all, find the number of days she can continue to put the five-rupee coins into it and find the total money she saved.

Write your views on the habit of saving.

Chapter: [2.02] Arithmetic Progressions

A park is of the shape of a circle of diameter 7 m. It is surrounded by a path of width of 0·7 m. Find the expenditure of cementing the path, if its cost is Rs 110 per sq. m ?

Chapter: [3.03] Constructions

In a rectangular park of dimensions 50 m × 40 m, a rectangular pond is constructed so that the area of grass strip of uniform width surrounding the pond would be 1184 m^{2}. Find the length and breadth of the pond ?

Chapter: [7.02] Surface Areas and Volumes

Two circles touch internally. The sum of their areas is 116 π cm^{2} and the distance between their centres is 6 cm. Find the radii of the circles ?

Chapter: [3.01] Circles

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number, (ii) a number divisible by 5 ?

Chapter: [5.01] Probability

A well of diameter 3 m is dug 14 m deep. The soil taken out of it is spread evenly all around it to a width of 5 m to form an embankment. Find the height of the embankment ?

Chapter: [7.02] Surface Areas and Volumes

Solve for x:

4x^{2} + 4bx − (a^{2} − b^{2}) = 0

Chapter: [2.03] Quadratic Equations

Draw a circle of radius of 3 cm. Take two points P and Q on one of its diameters extended on both sides, each at a distance of 7 cm on opposite sides of its centre. Draw tangents to the circle from these two points P and Q ?

Chapter: [3.03] Constructions

From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be 45° and 30° respectivly. Find the height of the hill ?

Chapter: [4.01] Heights and Distances

A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base. Show that the volumes of the three portions starting from the top are in the ratio 1 : 7 : 19 ?

Chapter: [3.01] Circles

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