#### Alternate Sets

If x = 3 is one root of the quadratic equation x^{2} – 2kx – 6 = 0, then find the value of k.

Concept: Polynomials Examples and Solutions

What is the HCF of the smallest prime number and the smallest composite number?

Concept: Fundamental Theorem of Arithmetic

Find the distance of a point P(*x*, *y*) from the origin.

Concept: Distance Formula

In an AP, if the common difference (d) = –4, and the seventh term (a_{7}) is 4, then find the first term.

Concept: nth Term of an AP

What is the value of (cos^{2} 67° – sin^{2}^{ }23°)?

Concept: Trigonometric Ratios of Complementary Angles

Given `triangle ABC ~ triangle PQR`, if `(AB)/(PQ) = 1/3`, then find `(ar triangle ABC)/(ar triangle PQR)`

Concept: Similarity of Triangles

Given that `sqrt2` is irrational prove that `(5 + 3sqrt2)` is an irrational number

Concept: Revisiting Irrational Numbers

In Fig. 1, ABCD is a rectangle. Find the value of *x* and *y*.

Concept: Equations Reducible to a Pair of Linear Equations in Two Variables

Find the sum of first 8 multiples of 3

Concept: Sum of First n Terms of an AP

Find the ratio in which P(4, *m*) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find *m*.

Concept: Section Formula

Two different dice are tossed together. Find the probability :

1) of getting a doublet

2) of getting a sum 10, of the numbers on the two dice.

Concept: Probability Examples and Solutions

An integer is chosen at random between 1 and 100. Find the probability that it is

1) divisible by 8.

2) not divisible by 8.

Concept: Probability Examples and Solutions

Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Concept: Fundamental Theorem of Arithmetic

Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)` and `(2 - sqrt3)`

Concept: Relationship Between Zeroes and Coefficients of a Polynomial

If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides

Concept: Concepts of Coordinate Geometry

If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD

Concept: Area of a Triangle

A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.

Concept: Polynomials Examples and Solutions

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals

Concept: Areas of Similar Triangles

If the areas of two similar triangles are equal, prove that they are congruent

Concept: Areas of Similar Triangles

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

Concept: Number of Tangents from a Point on a Circle

If 4 tan θ = 3, evaluate `((4sin theta - cos theta + 1)/(4sin theta + cos theta - 1))`

Concept: Trigonometric Ratios

If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A

Concept: Trigonometric Ratios of Complementary Angles

Find the area of the shaded region in Fig. 3, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14]

Concept: Surface Area of a Combination of Solids

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article [Use π = 22/7]

Concept: Surface Area of a Combination of Solids

A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much is canvas cloth required to just cover the heap?

Concept: Volume of a Combination of Solids

The table below shows the salaries of 280 persons :

Salary (In thousand Rs) |
No. of Persons |

5 – 10 | 49 |

10 – 15 | 133 |

15 – 20 | 63 |

20 – 25 | 15 |

25 – 30 | 6 |

30 – 35 | 7 |

35 – 40 | 4 |

40 – 45 | 2 |

45 – 50 | 1 |

Calculate the median salary of the data.

Concept: Median of Grouped Data

A motorboat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Concept: Polynomials Examples and Solutions

A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed?

Concept: Quadratic Equations

The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7: 15. Find the numbers.

Concept: Arithmetic Progression

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

Concept: Pythagoras Theorem

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

Concept: Similarity of Triangles

Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are `3/4` of the corresponding sides of the ∆ABC.

Concept: Constructions Examples and Solutions

Prove that `(sin A - 2sin^3 A)/(2cos^3 A - cos A) = tan A`

Concept: Trigonometric Ratios

The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find:

1) The area of the metal sheet used to make the bucket.

2) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14]

Concept: Surface Area of a Combination of Solids

As observed from the top of a 100 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use `sqrt3`=1.732]

Concept: Heights and Distances

The mean of the following distribution is 18. Find the frequency *f* of class 19 – 21.

Class |
11-13 | 13-15 | 15-17 | 17-19 | 19-21 | 21-23 | 23-25 |

Frequency |
3 | 6 | 9 | 13 | f | 5 | 4 |

Concept: Mean of Grouped Data

The following distribution gives the daily income of 50 workers of a factory.

Daily income (in Rs |
100 − 120 | 120 − 140 | 140 − 160 | 160 − 180 | 180 − 200 |

Number of workers |
12 | 14 | 8 | 6 | 10 |

Convert the distribution above to a less than type cumulative frequency distribution, and draw its ogive.

Concept: Graphical Representation of Cumulative Frequency Distribution