A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region.

In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region. [Use π = 22/7]

The slant height of a frustum of a cone is 4 cm and the perimeters of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.

The dimensions of a solid iron cuboid are 4.4 m × 2.6 m × 1.0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe.

Two taps running together can fill a tank in 3 ^{1}/_{13} hours. If one tap takes 3 hours more than the other to fill the tank, then how much time will each tap take to fill the tank?

In an equilateral triangle ABC, D is a point on the side BC such that BD = ^{1}/_{3} BC. Prove that 9AD^{2} = 7AB^{2}

The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30° and 60°, respectively. Find the height of the tower and also the horizontal distance between the building and the tower.

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms.

An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are 45° and 60° respectively. Find the width of the river. [Use √3 = 1.732]

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k - 1, 5k) are collinear, then find the value of k.

Two different dice are thrown together. Find the probability that the numbers obtained have

- even sum
- even product

In the given figure, ABCD is a rectangle of dimensions 21 cm × 14 cm. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure.

In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3.5 m. If the tank is full, find the rainfall in cm.

If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.

How many two-digit numbers are divisible by 3?

Determine the AP whose 3rd term is 5 and the 7th term is 9.

A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find its length and breadth.

The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?

The ratio of incomes of two persons is 9:7 and the ratio of their expenditures is 4:3. If each of them manages to save Rs. 2000 per month, find their monthly incomes.

Find the zeroes of the quadratic polynomial x^{2} + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

A sweet seller has 420 *kaju barfis* and 130 *badam barfis*. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

**Use Euclid’s algorithm to find the HCF of 4052 and 12576.**

**Check whether the equation 5x ^{2} – 6x – 2 = 0 has real roots and if it has, find them by the method of completing the square. Also verify that roots obtained satisfy the given equation.**

**A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/hour more. Find the original speed of the train.**

**Find the value(s) of k for which the pair of linear equations kx + y = k ^{2} and x + ky = 1 have infinitely many solutions.**

**The following frequency distribution shows the distance (in metres) thrown by 68 students in a Javelin throw competition.**

**Draw a less than type Ogive for the given data and find the median distance thrown using this curve.**

**The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the missing frequency k.**

Two dairy owners A and B sell flavoured milk filled to capacity in mugs of negligible thickness, which are cylindrical in shape with a raised hemispherical bottom. The mugs are 14 cm high and have diameter of 7 cm as shown in given figure. Both A and B sell flavoured milk at the rate of Rs. 80 per litre. The dairy owner A uses the formula πr^{2}h to find the volume of milk in the mug and charges Rs. 43.12 for it. The dairy owner B is of the view that the price of actual quantity of milk should be charged. What according to him should be the price of one mug of milk? Which value is exhibited by the dairy owner B? (Use π = 22/7)

Find the mode of the following distribution of marks obtained by the students in an examination:

Given the mean of the above distribution is 53, using empirical relationship estimate the value of its median.

**A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the remaining solid after the cone is carved out.**

Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?

In given figure ABPC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

**Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is 3, determine the number.**

**Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.**

**A box contains 12 balls of which some are red in colour. If 6 more red balls are put in the box and a ball is drawn at random, the probability of drawing a red ball doubles than what it was before. Find the number of red balls in the bag.**

**A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the probability that the number on the drawn card is **

**a square number****a multiple of 7**

**If two positive integers p and q are written as p = a ^{2}b^{3} and q = a^{3}b; a, b are prime numbers, then verify:**

**Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.**

**Show that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.**

**An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP.**

**If sin θ + cos θ = √2, then evaluate: tan θ + cot θ**

In given figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that ∠ AOB = 90°.

**In given figure ∠1 = ∠2 and ΔNSQ ≅ ΔMTR , then prove that ΔPTS ~ ΔPRQ.**

**The points A(4, –2), B(7, 2), C(0, 9) and D(–3, 5) form a parallelogram. Find the length of the altitude of the parallelogram on the base AB.**