**Question 1 to Question 4 is Compulsory**

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

Using remainder theorem, find the value of k if on dividing 2x^{3} + 3x^{2} - kx + 5 by x - 2, leaves a remainder 7

Concept: Remainder Theorem

Given A = `[(2,0),(-1,7)]` and I = `[(1,0),(0,1)]` and `A^2 = 9A+ ml`. Find m

Concept: Matrices Examples

The mean of following numbers is 68. Find the value of ‘x’.

45, 52, 60, x, 69, 70, 26, 81 and 94

Hence estimate the median.

Concept: Median of Grouped Data

The slope of a line joining P(6, k) and Q (1-3k, 3) is `1/2` Find

1) k

2) A midpoint of PQ, using the value of ‘k’ found in (1).

Concept: Slope of a Line

Without using trigonometrical tables, evaluate:

`cosec^2 57^@ - tan^2 33^@ + cos 44^@ cosec 46^@ - sqrt2 cos 45^@ - tan^2 60^@`

Concept: Trigonometry Problems and Solutions

A certain number of metallic cones, each of radius 2 cm and height 3 cm are melted and recast into a solid sphere of radius 6 cm. Find the number of cones.

Concept: Area and Volume of Solids - Sphere

Solve the following inequation, write the solution set and represent it on the number line.

`-3(x - 7) >= 15 - 7x > (x+1)/3`, x ∉ R

Concept: Representation of Solution on the Number Line

In the figure given below, an AD is a diameter. O is the centre of the circle

AD is parallel to BC and `angle CBD` = 32°. Find

1) `angle OBD`

2) `angle AOB`

3) `angle BED`

Concept: Circumscribing and Inscribing a Circle on a Triangle

If (3a + 2b) : (5a + 3b) = 18 : 29. Find a : b

Concept: Ratio and Proportion Example

A game of numbers has cards marked with 11, 12, 13, … 40. A card is drawn at random. Find the probability that the number on the card drawn is :

1) A perfect square

2) Divisible by 7

Concept: Simple Problems on Single Events

Use graph paper for this question. (Take 2 cm = 1 unit along both x and y-axis.) Plot the points O(0, 0), A(-4, 4), B(-3, 0) and C(0, -3)

1) Reflect points A and B on the y-axis and name them A’ and B’ respectively. Write down their coordinates.

2) Name the figure OABCB’A’.

3) State the line of symmetry of this figure

Concept: Lines of Symmetry

Mr Lalit invested Rs. 5000 at a certain rate of interest, compounded annually for two years. At the end of the first year, it amounts to Rs. 5325. Calculate

1) The rate of interest

2) The amount at the end of the second year, to the nearest rupee.

Concept: Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years

Solve the quadratic equation x^{2} - 3(x + 3) = 0; Give your answer correct two significant figures

Concept: Quadratic Equations

A page from the savings bank account of Mrs Ravi is given below.

Date |
Particulars |
Withdrawal(Rs.) |
Deposit (Rs.) |
Balance (Rs.) |

April 3rd 2006 | B/F | 6000 | ||

April 7th | By cash | 2300 | 8300 | |

April 15th | By cheque | 3500 | 11800 | |

May 20th | To self | 4200 | 7600 | |

June 10th | By cash | 5800 | 13400 | |

June 15th | To self | 3100 | 10300 | |

August 13th | By cheque | 1000 | 11300 | |

August 25th | To self | 7400 | 3900 | |

September 6th 2006 |
By cash | 2000 | 5900 |

She closed the account on 30th September 2006. Calculate the interest Mrs Ravi earned

at the end of 30th September 2006 at 4.5% per annum interest. Hence, find the

amount she receives on closing the account.

Concept: Types of Accounts

In what time will Rs. 1500 yield Rs. 496.50 as compound interest at 10% per annum compounded annually?

Concept: Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years

Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them.

Concept: Lines of Symmetry

In the given figure PQRS is a cyclic quadrilateral PQ and SR produced meet at T

1) Prove ΔTPS ~ ΔTRQ.

2) Find SP if TP = 18 cm, RQ = 4 cm and TR = 6 cm

3) Find the area of quadrilateral PQRS if the area of ΔPTS = 27 cm^{2}.

Concept: Cyclic Properties

Given matrix A = `[(4sin30^@, cos0^@),(cos0^@, 4sin30^@)]`and B = `[(4),(5)]`

if AX = B

1) Write the order of matrix X.

2) Find the matrix ‘X’.

Concept: Matrices Examples

An aeroplane at an altitude of 1500 metres, finds that two ships are sailing towards it in the same direction. The angles of depression as observed from the aeroplane are 45° and 30° respectively. Find the distance between the two ships

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

The table shows the distribution of the scores obtained by 160 shooters in a shooting competition. Use a graph sheet and draw an ogive for the distribution. (Take 2 cm = 10 scores on the X-axis and 2 cm = 20 shooters on the Y-axis).

Scores | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |

No. of shooters |
9 | 13 | 20 | 26 | 30 | 22 | 15 | 10 | 8 | 7 |

Use your graph to estimate the following:

1) The median

2) The interquartile range.

3) The number of shooters who obtained a score of more than 85%.

Concept: Median from the Ogive

If `x/a=y/b = z/c` show that `x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)`

Concept: Trigonometric Identities

Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct :

1) A circle of radius 2.5 cm, passing through A and C.

2) Construct two tangents to the circle from the external point B. Measure and record the length of the tangents.

Concept: Construction of Tangents to a Circle

A line AB meets X – axis at A and Y –axis at B. P (4, -1) divides AB in the ratio 1 : 2.

1) Find the coordinates of A and B.

2) Find the equation of the line through P and perpendicular to AB.

Concept: Equation of a Line

A dealer buys an article at a discount of 30% from the wholesaler, the marked price being Rs. 6000. The dealer sells it to a shopkeeper at a discount of 10% on the marked price. If the rate of the VAT is 6%, find

1) The price paid by the shopkeeper including the tax.

2) The VAT paid by the dealer

Concept: Introduction to Sales Tax and Value Added Tax

The given figure represents a kite with a circular and a semicircular motifs stuck on it.

The radius of a circle is 2.5 cm and the semicircle is 2 cm. If diagonals AC and BD are

of lengths 12 cm and 8 cm respectively, find the area of the:

1) Shaded part. Give your answer correct to the nearest whole number.

2) Unshaded part

Concept: Simple Applications of All Co-ordinate Geometry.

A model of a ship is made to a scale 1: 300

1) The length of the model of the ship is 2 m. Calculate the lengths of the ship.

2) The area of the deck ship is 180,000 m2. Calculate the area of the deck of the model.

3) The volume of the model in 6.5 m3. Calculate the volume of the ship.

Concept: Area and Volume of Solids - Sphere

Mohan has a recurring deposit account in a bank for 2 years at 6% p.a. simple interest. If he gets Rs. 1200 as interest at the time of maturity find:

1) the monthly instalment

2) the amount of maturity

Concept: Types of Accounts

The histogram below represents the scores obtained by 25 students in a mathematics mental test. Use the data to :

1) Frame a frequency distribution table

2) To calculate mean

3) To determine the Modal class

Concept: Graphical Representation of Histograms

A bus covers a distance of 240 km at a uniform speed. Due to heavy rain, its speed gets reduced by 10 km/h and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be ‘x’ km/h, form an equation and solve it to evaluate ‘x’.

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

Prove that `cosA/(1+sinA) + tan A = secA`

Concept: Trigonometric Identities

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown

Construct a ABC in which BC = 6.5 cm, ABC = 60°, AB = 5 cm.

Concept: Circumscribing and Inscribing a Circle on a Triangle

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown.

Construct the locus of points at a distance of 3.5 cm from A.

Concept: Loci Examples

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown

Construct the locus of points equidistant from AC and BC.

Concept: Loci Examples

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown

Mark 2 points X and Y which are a distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY.

Concept: Loci Examples

Ashok invested Rs. 26,400 on 12%, Rs. 25 shares of a company. If he receives a dividend of Rs. 2,475. Find the :

1) number of shares he bought

2) The market value of each share

Concept: Shares and Dividends