Duration: 3h

A number is chosen at random from the number –3, –2, –1, 0, 1, 2, 3. What will be the probability that square of this number is less then or equal to 1?

Chapter: [5.01] Probability

If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?

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

The ratio of the height of a tower and the length of its shadow on the ground is `sqrt3 : 1`. What is the angle of elevation of the sun?

Chapter: [4.01] Heights and Distances

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

Chapter: [7.02] Surface Areas and Volumes

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Chapter: [3.01] Circles

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD

Chapter: [3.01] Circles

Find the roots of the following quadratic equations by factorisation

`sqrt2 x^2 +7x+ 5sqrt2 = 0`

Chapter: [2.03] Quadratic Equations

Find how many integers between 200 and 500 are divisible by 8.

Chapter: [2.02] Arithmetic Progressions

Find the value of k for which the equation x^{2} + k(2x + k − 1) + 2 = 0 has real and equal roots.

Chapter: [2.03] Quadratic Equations

Draw a line segment of length 7 cm and divide it internally in the ratio 2 : 3.

Chapter: [3.03] Constructions

The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is (`7/2`, y). Find the value of y

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

Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles

Chapter: [3.02] Triangles

Two different dice are thrown together. Find the probability that the numbers obtained have a sum less than 7

Chapter: [5.01] Probability

Two different dice are thrown together. Find the probability that the numbers obtained have a product less than 16

Chapter: [5.01] Probability

In a simultaneous throw of a pair of dice, find the probability of getting a doublet of odd numbers

Chapter: [5.01] Probability

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.

Chapter: [3.01] Circles

Find the sum of *n* terms of the series `(4 - 1/n) + (4 - 2/n) + (4 - 3/n)+ ......`

Chapter: [2.02] Arithmetic Progressions

If the equation (1 + m^{2}) x^{2} + 2mcx + c^{2} – a^{2} = 0 has equal roots then show that c^{2} = a^{2} (1 + m^{2})

Chapter: [2.03] Quadratic Equations

In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region.

Chapter: [7.01] Areas Related to Circles

If the m^{th} term of an A.P. be `1/n` and n^{th} term be `1/m`, then show that its (mn)^{th} term is 1.

Chapter: [2.02] Arithmetic Progressions

A metallic solid sphere of radius 10.5 cm is melted and recasted into smaller solid cones, each of radius 3.5 cm and height 3 cm. How many cones will be made?

Chapter: [7.02] Surface Areas and Volumes

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

Chapter: [4.01] Heights and Distances

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park. Write your views on recycling of water.

Chapter: [7.02] Surface Areas and Volumes

In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square where O and O' are centres of the circles. Find the area of shaded region.

Chapter: [7.01] Areas Related to Circles

Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25

Chapter: [5.01] Probability

A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle.

Chapter: [3.01] Circles

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

Chapter: [3.01] Circles

Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream.

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

If a≠b≠0, prove that the points (a, a^{2}), (b, b^{2}) (0, 0) will not be collinear.

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

Draw a right triangle in which the sides (other than the hypotenuse) are of lengths 4 cm and 3 cm. Now construct another triangle whose sides are `3/5` times the corresponding sides of the given triangle.

Chapter: [3.03] Constructions

If the sum of first *m* terms of an A.P. is the same as the sum of its first *n* terms, show that the sum of its first (*m* + *n*) terms is zero

Chapter: [2.02] Arithmetic Progressions

Two points A and B are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60° and 45° respectively. If the height of the tower is 15 m, then find the distance between the points.

Chapter: [4.01] Heights and Distances

The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is `1/27` of the given cone, then at what height it is cut from its base?

Chapter: [7.02] Surface Areas and Volumes

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