**Question 1 to Question 4 is compulsory**

**Attempt any four questions fromQuestion 5 to Question 11**

Find the value of and 'y' if:

`2[(x,y),(9 , (y - 5))] + [(6,4),(-7,5)] = [(10,7),(22,15)]`

Concept: Matrices Examples

Sonia had a recurring deposit account in a bank and deposited Rs. 600 per month for 2 1/2 years. If the rate of interest was 10% p.a., find the maturity value of this account.

Concept: Types of Accounts

Cards bearing numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card which is:

1) a prime number

2) a number divisible by 4.

3) a number that is a multiple of 6.

4) an odd number.

Concept: Simple Problems on Single Events

The circumference of the base of the cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? (1000 cm^{3} = 1*l*)

`["Assume "pi=22/7]`

Concept: Volume of a Cylinder

If (k – 3), (2k + l) and (4k + 3) are three consecutive terms of an A.P., find the value of k.

Concept: Nature of Roots

PQRS is a cyclic quadrilateral. Given ∠QPS = 73°, ∠PQS = 55° and ∠PSR = 82°, calculate:

1) ∠QRS

2) ∠RQS

3) ∠PRQ

Concept: Cyclic Properties

If (x + 2) and (x + 3) are factors of x^{3} + ax + b, find the values of 'a' and `b'.

Concept: Factor Theorem

Prove that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`

Concept: Trigonometric Identities

Using a graph paper draw a histogram of the given distribution showing the number of runs scored by 50 batsmen. Estimate the mode of the data:

Runs scored |
3000- 4000 |
4000- 5000 |
5000- 6000 |
6000- 7000 |
7000- 8000 |
8000- 9000 |
9000- 10000 |

No. of batsmen |
4 | 18 | 9 | 6 | 7 | 2 | 4 |

Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data

Solve the following inequation, write down the solution set and represent it on the real number line:

–2 + 10x ≤ 13x + 10 < 24 + 10x, x 𝜖 Z

Concept: Representation of Solution on the Number Line

If the straight lines 3x – 5y = 7 and 4x + ay + 9 = 0 are perpendicular to one another, find the value of a.

Concept: Equation of a Line

Solve x^{2} + 7x = 7 and give your answer correct to two decimal places

Concept: Quadratic Equations

The 4th term of a G.P. is 16 and the 7th term is 128. Find the first term and common ratio of the series

Concept: Ratio and Proportion Example

A man invests Rs. 22,500 in Rs. 50 shares available at 10% discount. If the dividend paid by the company is 12%, calculate:

1) The number of shares purchased

2) The annual dividend received.

3) The rate of return he gets on his investment. Give your answer correct to the nearest whole number.

Concept: Shares and Dividends Examples

Use graph paper for this question (Take 2 cm = 1 unit along both x and y-axis). ABCD is a quadrilateral whose vertices are A(2, 2), B(2, –2), C(0, –1) and D(0, 1).

1) Reflect quadrilateral ABCD on the y-axis and name it as A'B'CD

2) Write down the coordinates of A' and B'.

3) Name two points which are invariant under the above reflection

4) Name the polygon A'B'CD

Concept: Simple Applications of All Co-ordinate Geometry.

Using properties of proportion, solve for x. Given that x is positive:

`(2x + sqrt(4x^2 -1))/(2x - sqrt(4x^2 - 1)) = 4`

Concept: Proportions

if A = `[(2,3),(5,7)]`, B = `[(0,4),(-1,7)]` and c = `[(1,0),(-1, 4)]`, find AC + B^{2} – 10C.

Concept: Matrices Examples

Prove that (1 + cot θ – cosec θ)(1+ tan θ + sec θ) = 2

Concept: Trigonometric Identities

Find the value of k for which the following equation has equal roots.

x^{2} + 4kx + (k^{2} – k + 2) = 0

Concept: Nature of Roots

On a map drawn to a scale of 1: 50,000, a rectangular plot of land ABCD has the following dimensions. AB = 6 cm; BC = 8 cm and all angles are right angles. Find:

1) the actual length of the diagonal distance AC of the plot in km.

2) the actual area of the plot in sq. km.

Concept: Area and Volume of Solids - Sphere

A(2, 5), B(–1, 2) and C(5, 8) are the vertices of a triangle ABC, `M' is a point on AB such that AM: MB = 1: 2. Find the coordinates of 'M'. Hence find the equation of the line passing through the points C and M

Concept: Equation of a Line

Rs. 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received Rs. 100 more. Find the original number of children.

Concept: Quadratic Equations

If the mean of the following distribution is 24, find the value of 'a '.

Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |

Number of students |
7 | a | 8 | 10 | 5 |

Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data

Using ruler and compass only, construct a ΔABC such that BC = 5 cm and AB = 6.5

cm and ∠ABC = 120°

1) Construct a circum-circle of ΔABC

2) Construct a cyclic quadrilateral ABCD, such that D is equidistant from AB and BC.

Concept: Circumscribing and Inscribing a Circle on a Triangle

Priyanka has a recurring deposit account of Rs. 1000 per month at 10% per annum. If she gets Rs. 5550 as interest at the time of maturity, find the total time for which the account was held.

Concept: Computation of Interest

In ΔPQR, MN is parallel to QR and `(PM)/(MQ) = 2/3`

1) Find `(MN)/(QR)`

2) Prove that ΔOMN and ΔORQ are similar.

3) Find, Area of ΔOMN : Area of ΔORQ

Concept: Similarity of Triangles

The following figure represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. This common radius is 7 cm. The height of the cylinder and cone are each of 4 cm. Find the volume of the solid.

Concept: Area and Volume of Solids - Cone

Use Remainder theorem to factorize the following polynomial:

`2x^3 + 3x^2 - 9x - 10`

Concept: Remainder Theorem

In the figure given below 'O' is the centre of the circle. If QR = OP and ∠ORP = 20°. Find the value of 'x' giving reasons

Concept: Circumscribing and Inscribing a Circle on a Triangle

The angle of elevation from a point P of the top of a tower QR, 50 m high is 60o and that of the tower PT from a point Q is 30°. Find the height of the tower PT, correct to the nearest metre

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables

The 4th term of an A.P. is 22 and the 15th term is 66. Find the first terns and the common

difference. Hence find the sum of the series to 8 terms.

Concept: Nature of Roots

Use Graph paper for this question.

A survey regarding height (in cm) of 60 boys belonging to Class 10 of a school was conducted. The following data was recorded:

Height in cm |
135-140 | 140-145 | 145-150 | 150-155 | 155-160 | 160-165 | 165-170 |

No. of boys |
4 | 8 | 20 | 14 | 7 | 6 | 1 |

Taking 2 cm = height of 10 cm along one axis and 2 cm = 10 boys along the other axis draw an ogive of the above distribution. Use the graph to estimate the following:

1) the median

2) lower Quartile

3) if above 158 cm is considered as the tall boys of the class. Find the number of boys in the class who are tall.

Concept: Finding the Mode from the Lower Quartile