**Question 1 to Question 4 is compulsory**

**Attempt Any Four From Question 5 to Question 11**

Ranbir borrows Rs. 20,000 at 12% per annum compound interest. If he repays Rs. 8400 at the end of the first year and Rs. 9680 at the end of the second year, find the amount of loan outstanding at the beginning of the third year.

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

Find the values of x, which satisfy the inequation

`-2 5/6 < 1/2 - (2x)/3 <= 2, x in W`. Graph the solution set on the number line.

Concept: Representation of Solution on the Number Line

A die has 6 faces marked by the given numbers as shown below

The die is thrown once. What is the probability of getting

1) a positive integer

2) an integer greater than -3.

3) the smallest integer.

Concept: Simple Problems on Single Events

Find x, y if `[(-2,0),(3,1)][(-1),(-2x)] + 3[(-2),(1)] = 2[(y),(3)]`

Concept: Addition and Subtraction of Matrices

Shahrukh opened a ‘Recurring Deposit’ account in a bank and deposited Rs. 800 per month for 1½ years. If he received Rs. 15,084 at the time of maturity, find the rate of interest per annum.

Concept: Types of Accounts

Calculate the ratio in which the line joining A(-4,2) and B(3,6 is divided by a point P(x, 3). Also, find (i) x (ii) Length of AP.

Concept: Ratios

Without using trigonometric tables, evaluate

`sin^2 34^@ + sin^2 56^@ + 2tan 18^@ tan 72^@ - cot^2 30^@`

Concept: Trigonometry Problems and Solutions

Using the Remainder and Factor Theorem, factorise the following polynomial:

`x^3 + 10x^2 - 37x + 26`

Concept: Remainder Theorem

In the figure given below, ABCD is the rectangle. AB = 14 cm, BC = 7 cm. From the rectangle, a quarter circle BFEC and a semicircle DGE are removed. Calculate the area of the remaining piece of the rectangle. (Take `pi = 22/7`).

Concept: Circle - Direct Application Problems Including Inner and Outer Area

The numbers 6, 8, 10, 12, 13 and x are arranged in an ascending order. If the mean of the observations is equal to the median, find the value of x

Concept: Median of Grouped Data

In the figure, m∠DBC = 58°. BD is the diameter of the circle. Calculate:

1) m∠BDC

2) m∠BEC

3) m∠BAC

Concept: Arc and Chord Properties - Angle in a Semi-circle is a Right Angle

Use graph paper to answer the following questions. (Take 2 cm = 1 unit on both axes)

1) Plot the points A( -4, 2) and B(2, 4)

2) A' is the image of A when reflected at the y-axis. Plot it on the graph paper and write the coordinates of A'.

3) B' is the image of B when reflected on the line AA'. Write the coordinates of B'.

4) Write the geometric name of the figure ABA'B'.

5) Name a line of symmetry of the figure formed

Concept: Lines of Symmetry

A shopkeeper bought a washing machine at a discount of 20% from a wholesaler, the printed price of the washing machine being Rs. 18,000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the rate of sales tax is 8%, find:

1) the VAT paid by the shopkeeper.

2) the total amount that the consumer pays for the washing machine

Concept: Introduction to Sales Tax and Value Added Tax

if `(x^2 + y^2)/(x^2 - y^2) = 17/8`then find the value of :

1) x : y

2) `(x^3 + y^3)/(x^3 - y^3)`

Concept: Componendo and Dividendo Properties

In Δ ABC, ∠ABC = ∠DAC. AB = 8 cm, AC = 4 cm, AD = 5 cm.

1) Prove that Δ ACD is similar to Δ BCA.

2) Find BC and CD

3) Find- area of Δ ACD : area of Δ BCA

Concept: Similarity of Triangles

The value of 'a' for which of the following points A(a, 3), B (2, 1) and C(5, a) a collinear. Hence find the equation of the line.

Concept: Distance Formula

Salman invests a sum of money in Rs. 50 shares, paying 15% dividend quoted at 20% premium. If his annual dividend is Rs. 600, calculate:

1) the number of shares he bought.

2) his total investment.

3) the rate of return on his investment.

Concept: Shares and Dividends

The surface area of a solid metallic sphere is 2464 cm^{2}. It is melted and recast into solid right circular cones of radius 3.5 cm and height 7 cm. Calculate:

1) the radius of the sphere

2) the number of cones recast. (Take π = 22/17)

Concept: Area and Volume of Solids - Sphere

Calculate the mean of the distribution given below using the shortcut method.

Marks | 11-20 | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 |

No. of students | 2 | 6 | 10 | 12 | 9 | 7 | 4 |

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

In the figure given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find:

1) AB.

2) the length of tangent PT.

Concept: Construction of Tangents to a Circle

Let `A = [(2,1),(0,-2)], B = [(4,1),(-3,-2)] and C = [(-3,2),(-1,4)]`. Find `A^2 + AC - 5B`

Concept: Addition and Subtraction of Matrices

The compound interest, calculated yearly, on a certain sum of money for the second year is Rs. 1320 and for the third year is Rs. 1452. Calculate the rate of interest and the original sum of money

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

Construct a ΔABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the incircle of the triangle. Measure and record the radius of the incircle.

Concept: Circumscribing and Inscribing a Circle on a Triangle

(Use a graph paper for this question.) The daily pocket expenses of 200 students in a school are given below:

Pocket expenses (in Rs.) | Number of students (Frequency) |

0-5 | 10 |

5-10 | 14 |

10-15 | 28 |

15-20 | 42 |

20-25 | 50 |

25-30 | 30 |

30-35 | 14 |

35-40 | 12 |

Draw a histogram representing the above distribution and estimate the mode from the graph.

Concept: Finding the Mode from the Histogram

If (x - 9) : (3x + 6) is the duplicate ratio of 4: 9, find the value of x.

Concept: Ratios

Solve for x using the quadratic formula. Write your answer corrected to two significant figures. (x - 1)^{2} - 3x + 4 = 0

Concept: Nature of Roots

A page from the ‘Savings Bank’ account of Priyanka is given below:

Date | Particulars |
Amount (Rs.) |
Amount deposited (Rs.) |
Balance (Rs.) |

03/04/2006 | B/F | 4000.00 | ||

05/04/2006 | By cash | 2000.00 | 6000.00 | |

18/04/2006 | By cheque | 6000.00 | 12000.00 | |

25/05/2006 | By cheque | 5000.00 | 7000.00 | |

30/05/2006 | By cash | 3000.00 | 10000.00 | |

20/07/2006 | By self | 4000.00 | 6000.00 | |

10/09/2006 | By cash | 2000.00 | 8000.00 | |

19/09/2006 | To cheque | 1000.00 | 7000.00 |

If the interest earned by Priyanka for the period ending September 2006 is Rs. 175, the find the rate of interest.

Concept: Types of Accounts

A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.

Concept: Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem

The marks obtained by 100 students in a Mathematics test are given below:

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

No. of students |
3 | 7 | 12 | 17 | 23 | 14 | 9 | 6 | 5 | 4 |

Draw an ogive for the given distribution on a graph sheet.

Use a scale of 2 cm = 10 units on both axes.

Use the ogive to estimate the:

1) Median.

2) Lower quartile.

3) A number of students who obtained more than 85% marks in the test.

4) A number of students who did not pass in the test if the pass percentage was 35.

Concept: Ogives (Cumulative Frequency Graphs)

In the figure given below, O is the centre of the circle. AB and CD are two chords of the circle. OM is perpendicular to AB and ON is perpendicular to CD.

AB = 24 cm, OM= 5 cm, ON= 12 cm. Find the:

1) radius of the circle

2) length of chord CD.

Concept: Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal

Prove the identity (sin θ + cos θ) (tan θ + cot θ ) = sec θ + cosec θ

Concept: Trigonometric Identities

An aeroplane at an altitude of 250 m observes the angle of depression of two boats on the opposite banks of a river to be 45° and 60° respectively. Find the width of the river. Write the answer correct to the nearest whole number.

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