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Mathematics Set 1 2012-2013 ICSE Class 10 Question Paper Solution

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Mathematics [Set 1]
Marks: 80Date: 2012-2013 March

[10]1
[3]1.1

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
Chapter: [0.0208] Matrices
[3]1.2

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

Concept: Conept of Compound Interest - Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
Chapter: [0.011000000000000001] Compound Interest
[4]1.3

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
Chapter: [0.06] Statistics
[10]2
[3]2.1

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

Concept: Proportions
Chapter: [0.0204] Ratio and Proportion
[3]2.2

If (x – 2) is a factor of the expression 2x3 + ax2 + 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
Chapter: [0.0203] Factorization
[4]2.3

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
Chapter: [0.06] Statistics
[10]3
[3]3.1

Without using tables evaluate:
3cos 80°. cosec 10° + 2sin 59° sec 31°

Concept: Trigonometry Problems and Solutions
Chapter: [0.05] Trigonometry
[3]3.2

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
Chapter: [0.032] Circles
[4]3.3

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
Chapter: [0.034] Symmetry
[10]4
[3]4.1

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

x2 – 5x – 10 = 0.

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[3]4.2

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
Chapter: [0.035] Similarity
[4]4.3

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.
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[10]5
[3]5.1

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
Chapter: [0.020499999999999997] Linear Inequations
[3]5.2

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
Chapter: [0.013000000000000001] Banking
[4]5.3

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
Chapter: [0.012] Shares and Dividends
[10]6
[3]6.1

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

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[3]6.2

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
Chapter: [0.032] Circles
[4]6.3

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
Chapter: [0.013000000000000001] Banking
[10]7
[3]7.1

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

Concept: General Equation of a Line
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[3]7.2

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
Chapter: [0.013999999999999999] Gst (Goods and Services Tax)
[4]7.3

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
Chapter: [0.05] Trigonometry
[10]8
[3]8.1

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

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[3]8.2

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
Chapter: [0.04] Mensuration
[4]8.3

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

x2 + (p – 3)x + p = 0.

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[10]9
[3]9.1

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
Chapter: [0.032] Circles
[3]9.2

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
Chapter: [0.07] Probability
[3]9.3

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
Chapter: [0.06] Statistics
[10]10
[4]10.1

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
Chapter: [0.033] Constructions
[6]10.2

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
Chapter: [0.06] Statistics
[10]11
[3]11.1

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
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[3]11.2

Using the properties of proportion, solve for x, given

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

Concept: Componendo and Dividendo Properties
Chapter: [0.0204] Ratio and Proportion
[4]11.3

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
Chapter: [0.0202] Quadratic Equations

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