**Attempt all questions from Question 1 to Question 4**

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

Given A = `[(2,-6),(2,0)], B = [(-3,2),(4,0)], C = [(4,0),(0,2)]`

Find the matrix X such that A + 2X = 2B + C.

Concept: Addition and Subtraction of Matrices

At what rate % p.a. will a sum of Rs. 4000 yield Rs. 1324 as compound interest in 3 years?

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

The median of the following observations

11, 12, 14, (x - 2), (x + 4), (x + 9), 32, 38, 47 arranged in ascending order is 24.

Find the value of x and hence find the mean.

Concept: Median of Grouped Data

What number must be added to each of the numbers 6, 15, 20 and 43 to make them

proportional?

Concept: Proportions

If (x – 2) is a factor of the expression 2x^{3} + ax^{2} + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

Concept: Factor Theorem

Draw a histogram from the following frequency distribution and find the mode from the graph:

Class | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | |

Frequency | 2 | 5 | 18 | 14 | 8 | 5 |

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

Without using tables evaluate:

3cos 80°. cosec 10° + 2sin 59° sec 31°

Concept: Trigonometry Problems and Solutions

In the given figure, ∠BAD = 65°, ∠ABD = 70°, ∠BDC = 45°

1) Prove that AC is a diameter of the circle.

2) Find ∠ACB

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

AB is a diameter of a circle with centre C = (–2, 5). If A = (3, –7). Find

1) the length of radius AC

2) the coordinates of B.

Concept: Lines of Symmetry

Solve the following equation and calculate the answer correct to two decimal places:

x^{2} – 5x – 10 = 0.

Concept: Quadratic Equations

In the given figure, AB and DE are perpendicular to BC.

1) Prove that ΔABC ∼ ΔDEC

2) If AB = 6 cm; DE = 4 cm and AC = 15 cm. Calculate CD.

3) Find the ratio of area of ΔABC: area of ΔDEC

Concept: Similarity of Triangles

Using a graph paper, plot the points A(6,4) and B(0,4).

(1) Reflect A and B in the origin to get the images A' and B'.

(2) Write the co-ordinates of A' and B'.

(3) State the geometrical name for the figure ABA' B'.

(4) Find its perimeter.

Concept: Simple Applications of All Co-ordinate Geometry.

Solve the following inequation, write the solution set and represent it on the number line `-x/3 <= x/2 - 1 1/3 < 1/6, x ∈ R`

Concept: Representation of Solution on the Number Line

Mr Britto deposits a certain sum of money each month in a Recurring Deposit Account of a bank. It the rate of interest is of 8% per annum and Mr Britto gets Rs. 8088 from the bank after 3 years, find the value of his monthly instalment.

Concept: Computation of Interest

Salman buys 50 shares of face value Rs. 100 available at Rs. 132.

(1) What is his investment?

(2) If the dividend is 7.5%, what will be his annual income?

(3) If he wants to increase his annual income by Rs. 150, how many extra shares

should he buy?

Concept: Shares and Dividends

Show that `sqrt((1-cos A)/(1 + cos A)) = sinA/(1 + cosA)`

Concept: Trigonometric Identities

In the given circle with centre O, ∠ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.

Concept: Tangent to a Circle

Given below are the entries in a Saving Bank A/c passbook.

Date |
Particulars |
Withdrawals |
Deposit |
Balance |

Feb8 | B/F | - | - | Rs. 8500 |

Feb 18 | To self | Rs. 4000 | - | |

April 12 | By cash | - | Rs. 2230 | |

June 15 | To self | Rs. 5000 | - | |

July 8 | By cash | - | Rs. 6000 |

Calculate the interest for six months from February to July at 6% p.a.

Concept: Computation of Interest

In ΔABC, A(3, 5), B(7, 8) and C(1, –10). Find the equation of the median through A.

Concept: Mid-point Formula

A shopkeeper sells an article at the listed price of Rs. 1500 and the rate of the VAT is 12% at each stage of the sale. If the shopkeeper pays a VAT of Rs. 36 to the Government, what was the price, inclusive to TAX, at which the shopkeeper purchased the article from the wholesaler?

Concept: Introduction to Sales Tax and Value Added Tax

In the figure given, from the top of a building AB = 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to 30o and 60o respectively. Find:

1) The horizontal distance between AB and CD.

2) The height of the lamp post.

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

Find x and y if `[(x,3x),(y, 4y)] = [(5),(12)]`

Concept: Matrices Examples

A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of cones recast.

Concept: Area and Volume of Solids - Sphere

Without solving the following quadratic equation, find the value of ‘p' for which the given equation has real and equal roots:

x^{2} + (p – 3)x + p = 0.

Concept: Quadratic Equations

In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region.

Concept: Areas of Sector and Segment of a Circle

A box contains some black balls and 30 white balls. If the probability of drawing a black ball is two-fifths of a white ball, find the number of black balls in the box.

Concept: Simple Problems on Single Events

Find the mean of the following distribution by step deviation method:

Class Interval | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

Frequency | 10 | 6 | 8 | 12 | 5 | 9 |

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

Using a ruler and compasses only:

1) Construct a triangle ABC with the following data: AB = 3.5 cm, BC = 6 cm and ABC = 120°

2) In the same diagram, draw a circle with BC as diameter. Find a point P on the circumference of the circle which is equidistant from AB and BC.

3) Measure ∠BCP.

Concept: Circumscribing and Inscribing a Circle on a Triangle

The marks obtained by 120 students in a 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 |
5 | 9 | 16 | 22 | 26 | 18 | 11 | 6 | 4 | 3 |

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

Use a suitable scale for ogive to estimate the following:

(1) The median.

(2) The number of students who obtained more than 75% marks in the test.

(3) The number of students who did not pass the test if minimum marks required to pass is 40

Concept: Median from the Ogive

In the figure given below, the line segment AB meets X-axis at A and Y-axis at B. The point P(-3, 4) on AB divides it in the ratio 2:3. Find the coordinates of A and B.

Concept: Equation of a Line

Using the properties of proportion, solve for x, given

`(x^4 + 1)/(2x^2) = 17/8`

Concept: Componendo and Dividendo Properties

A shopkeeper purchases a certain number of books for Rs. 960. If the cost per book was Rs. 8 less, the number of books that could be purchased for Rs. 960 would be 4 more. Write an equation, taking the original cost of each book to be Rs. x, and solve it to find the original cost of the books.

Concept: Quadratic Equations