**Question 1 to Question 4 is Compulsory**

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

If b is the mean proportion between a and c, show that `(a^4 + a^2b^2 + b^4)/(b^4 + b^2c^2 + c^4) = a^2/c^2`

Concept: Proportions

Solve the equation 4x^{2} – 5x – 3 = 0 and give your answer correct to two decimal places

Concept: Quadratic Equations

AB and CD are two parallel chords of a circle such that AB = 24 cm and CD = 10 cm. If the

radius of the circle is 13 cm. find the distance between the two chords.

Concept: Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)

Evaluate without using trigonometric tables,

`sin^2 28^@ + sin^2 62^@ + tan^2 38^@ - cot^2 52^@ + 1/4 sec^2 30^@`

Concept: Trigonometry Problems and Solutions

if A = `[(1,3),(3,4)]` and B = `[(-2,1),(-3,2)]` and A^{2} - 5B^{2} = 5C. Find matrix C, where C is a 2 by 2

Concept: Matrices Examples

Jaya borrowed Rs. 50,000 for 2 years. The rates of interest for two successive years are 12% and 15% respectively. She repays 33,000 at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt.

Concept: Compound Interest as a Repeated Simple Interest Computation with a Growing Principal

The catalogue price of a computer set is Rs. 42,000. The shopkeeper gives a discount of

10% on the listed price. He further gives an off-season discount of 5% on the discounted

price. However, sales tax at 8% is charged on the remaining price after the two successive

discounts. Find

1)the amount of sales tax a customer has to pay

2)the total price to be paid by the customer for the computer set.

Concept: Introduction to Sales Tax and Value Added Tax

P(1, -2) is a point on the line segment A(3, -6) and B(x, y) such that AP : PB is equal to 2 : 3. Find the coordinates of B.

Concept: Co-ordinates Expressed as (x,y)

The marks of 10 students of a class in an examination arranged in ascending order is as follows:

13, 35, 43, x, x + 4, 55, 61, 71, 80

If the median marks is 48, find the value of x. Hence find the mode of the given data.

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

What must be subtracted from 16x^{3} – 8x^{2} + 4x + 7 so that the resulting expression has

2x + 1 as a factor?

Concept: Solutions of Quadratic Equations by Factorization

In the given figure ABCD is a rectangle. It consists of a circle and two semi-circles each of

which are of radius 5 cm. Find the area of the shaded region. Give your answer correct to

three significant figures

Concept: Simple Applications of All Co-ordinate Geometry.

Solve the following inequation and represent the solution set on a number line.

`-8 1/2 < -1/2 -4x <= 7 1/2`, x ∈ 1

Concept: Equation of a Line

Given matrix B = `[(1,1),(8,3)]`, find the matrix X if X = B^{2} - 4B. Hence solve for a and b given `X[(a),(b)] = [(5),(50)]`

Concept: Matrices Examples

How much should a man invest in Rs. 50 shares selling at Rs. 60 to obtain an income of Rs. 450, if the rate of dividend declared is 10%. Also find his yield percent, to the nearest whole number.

Concept: Shares and Dividends Examples

Sixteen cards are labelled as a, b, c, ……………. m, n, o, p. They are put in a box and shuffled.a boy is asked to draw a card from the box. What is the probability that the card drawn is:

(1) a vowel

(2) a consonant

(3) none of the letters of the word median

Concept: Simple Problems on Single Events

Using a ruler and a compass construct a triangle ABC in which AB = 7 cm, CAB = 60o and

AC = 5 cm. Construct the locus of

1) points equidistant from AB and AC

2) points equidistant from BA and BC

Hence construct a circle touching the three sides of the triangle internally.

Concept: Circumscribing and Inscribing a Circle on a Triangle

A conical tent is to accommodate 77 persons. Each person must have 16 m3 of air to breathe. Given the radius of the tent as 7 m, find the height of the tent and also its curved surface area.

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

if `(7m + 2n)/(7m - 2n) = 5/3`use properties of proportion to find

1) m:n

2)`(m^2 + n^2)/(m^2 - n^2)`

Concept: Componendo and Dividendo Properties

A page from a savings bank account passbook is given below:

Date | Particulars | Amount withdrawn (Rs.) |
Amount Deposited (Rs.) |
Balance (Rs.) |

Jan 7, 2016 | B/F | 3,000.00 | ||

Jan 10, 2016 | By Cheque | 2600.00 | 5600.00 | |

Feb 8, 2016 | To Self | 1500.00 | 4100.00 | |

Apr 6, 2016 | By Cheque | 2100.00 | 2000.00 | |

May 4, 2016 | By Cash | 6500.00 | 8500.00 | |

May 27, 2016 | By Cheque | 1500.00 | 10000.00 |

1) Calculate the interest for the 6 months from January to June 2016, at 6% per annum.

2) If the account is closed on 1st July 2016, find the amount received by the account holder.

Concept: Computation of Interest

Use a graph paper for this question (Take 2 cms = 1 unit on both x and y-axis)

1) Plot the following points:A(0, 4), B(2, 3), C(1, 1) and D(2, 0)

2) Reflect points B, C, D on the y-axis and write down their coordinates. Name the images as B’, C’, D’ respectively

3) Join the points A, B, C, D, D’, C’, B’ and A in order, so as to form a closed figure. Write

down the equation of the line of symmetry of the figure formed

Concept: Lines of Symmetry

Calculate the mean of the following distribution using step deviation method.

Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

Number of students |
10 | 9 | 25 | 0 | 16 | 10 |

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

In the given figure PQ is a tangent to the circle at A, AB and AD are bisectors of `angleCAQ` and `angle PAC`. if `angleBAQ = 30^@. prove that:

1) BD is a diameter of the circle

2) ABC is an isosceles triangle

Concept: Construction of Tangents to a Circle

The printed price of an air conditioner is Rs. 45000/-. The wholesaler allows a discount of 10%

to the shopkeeper. The shopkeeper sells the article to the customer at a discount of 5% of the

marked price. Sales tax (under VAT) is charged at the rate of 12% at every stage. Find:

1) VAT paid by the shopkeeper to the government

2) The total amount paid by the customer inclusive of tax.

Concept: Introduction to Sales Tax and Value Added Tax

In the figure given, O is the centre of the circle. `angle DAE = 70^@`, Find giving suitable reasons the measure of:

1) `angle BCD`

2) `angle BOD`

3) `angle OBD`

Concept: Cyclic Properties

A(-1, 3), B(4, 2) and C(3, -2) are the vertices of a triangle.

1) Find the coordinates of the centroid G of the triangle

2) Find the equation of the line through G and parallel to AC

Concept: Equation of a Line

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`

Concept: Trigonometric Identities

The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their

ages in years is 550. Find their ages.

Concept: Nature of Roots

The daily wages of 80 workers in a project are given below.

Wages (in Rs.) |
400-450 | 450-500 | 500-550 | 550-600 | 600-650 | 650-700 | 700-750 |

No. of Workers |
2 | 6 | 12 | 18 | 24 | 13 | 5 |

Use a graph paper to draw an ogive for the above distribution. (Use a scale of 2 cm = Rs.

50 on x-axis and 2 cm = 10 workers on y-axis). Use your ogive to estimate:

1) the median wage of the workers

2) the lower quartile wage of workers

3) the numbers of workers who earn more than Rs. 625 daily

Concept: Ogives (Cumulative Frequency Graphs)

The angles of depression of two ships A and B as observed from the top of a lighthouse 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the lighthouse, find the distance between the two ships, Give your answer correct to the nearest whole number.

Concept: Trigonometric Identities

PQR is a triangle. S is a point on the side QR of ΔPQR such that `angle PSR = angle QPR`. Given QP = 8 cm,PR = 6 cm and SR = 3 cm.

1) Prove ΔPQR ~ ΔSPR

2) Find the length of QR and PS

3) `" area of ΔPQR"/"area of ΔSPR"`

Concept: Axioms of Similarity of Triangles

Mr Richard has a recurring deposit account in a bank for 3 years at 7.5% p. a. simple interest. If he gets Rs. 8325 as interest at the time of maturity, find

1) The monthly deposit

2) The maturity value

Concept: Types of Accounts