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Mathematics 2011-2012 ICSE Class 10 Question Paper Solution

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Mathematics
2011-2012 March
Marks: 80

Attempt all Question From Question 1 to Question 4

Attempt Any Four From Question 5 to Question 11


[10]1
[3]1.1

if A = `[(3,1),(-1,2)]` and `I = [(1,0),(0,1)]`, find `A^2 - 5A + 7I`

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[3]1.2

The monthly pocket money of Ravi and Sanjeev are in the ratio 5:7. Their expenditures are in the ratio 3:5. If each saves Rs. 80 every month, find their monthly pocket money.

Concept: Ratio and Proportion Example
Chapter: [0.0204] Ratio and Proportion
[4]1.3

Using the Remainder Theorem factorise completely the following polynomial.

3x3 + 2x2 – 19x + 6

Concept: Remainder Theorem
Chapter: [0.0203] Factorization
[10]2
[3]2.1

On what sum of money will the difference between the compound interest and simple interest for 2 years be equal to Rs. 25 if the rate of interest charged for both is 5% p.a.?

Concept: Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
Chapter: [0.011000000000000001] Compound Interest
[3]2.2

ABC is an isosceles right-angled triangle with ∠ABC = 90°. A semi-circle is drawn with AC as the diameter. If AB = BC = 7 cm, find the area of the shaded region. [Take Π = 22/7]

Concept: Perimeter and Area of a Circle
Chapter: [0.04] Mensuration
[4]2.3

Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find

1) The ratio in which AB is divided by y-axis.

2) Find the coordinates of the point of intersection.

3) The length of AB.

Concept: Distance Formula
Chapter: [0.020099999999999996] Co-ordinate Geometry Distance and Section Formula
[10]3
[3]3.1

In the given figure O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm. Calculate the radius of a circle.

Concept: Tangent to a Circle
Chapter: [0.032] Circles
[3]3.2

Evaluate without using trigonometric tables:

`cos^2 26^@ + cos 64^@ sin 26^@ + (tan 36^@)/(cot 54^@)`

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[4]3.3

Marks obtained by 40 students in a short assessment is given below, where a and b are two missing data.

Marks 5 6 7 8 9
Number of Students 6 a 16 13 b

If the mean of the distribution is 7.2, find a and b.

Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
Chapter: [0.06] Statistics
[10]4
[3]4.1

Kiran deposited Rs. 200 per month for 36 months in a bank’s recurring deposit account. If the bank pays interest at the rate of 11% per annum, find the amount she gets on maturity.

Concept: Computation of Interest
Chapter: [0.013000000000000001] Banking
[3]4.2

Two coins are tossed once. Find the probability of getting

1) 2 heads

2) at least 1 tail.

Concept: Simple Problems on Single Events
Chapter: [0.07] Probability
[4]4.3

Using graph paper and taking 1 cm = 1 unit along both x-axis and y-axis.

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

2) Reflect A and B in the origin to get the images A’ and B’ respectively.

3) Write down the coordinates of A’ and B’.

4) Give the geometrical name for the figure ABA’B’.

5) Draw and name its lines of symmetry.

Concept: Lines of Symmetry
Chapter: [0.034] Symmetry
[10]5
[3]5.1

In the given figure, AB is the diameter of a circle with centre O. ∠BCD = 130o. Find:

1) ∠DAB

2) ∠DBA

Concept: Cyclic Properties
Chapter: [0.032] Circles
[3]5.2

Given `[(2, 1),(-3,4)] X = [(7),(6)]. Write

1) The order of the matrix X

2) The matrix X.

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[4]5.3

A page from the Savings Bank Account of Mr Prateek is given below:

Date Particulars Withdrawal
(In Rs.)
Deposit
(In Rs.)
Balances
(In Rs.)
January 1st, 2006 B/F - - 1270
January 7th, 2006 By Cheque - 2310 3580
March 9th, 2006 To Self 2000 - 1580
June 26th, 2006 By Cash - 6200 7780
June 10th 2006 To Cheque 4500 - 3280
July 15th, 2006 By Clearing - 2630 5910
October 18th, 2006 To Cheque 530 - 5380
October 27th, 2006 To Self 2690 - 2690
November 3rd, 2006 By Cash - 1500 4190
December 6th, 2006 To Cheque 950 - 3240
December 23rd, 2006 By Transfer - 2920 6260

If he receives Rs. 198 as interest on 1st January 2007, find the rate of interest paid by the bank.

Concept: Types of Accounts
Chapter: [0.013000000000000001] Banking
[10]6
[3]6.1

The printed price of an article is Rs. 60,000. The wholesaler allows a discount
of 20% to the shopkeeper. The shopkeeper sells the article to the customer at the
printed price. Sales tax (under VAT) is charged at the rate of 6% at every stage. Find:
(1) The cost to the shopkeeper inclusive of tax.
(2) VAT paid by the shopkeeper to the Government.
(3) The cost to the customer inclusive of tax.

Concept: Computation of Tax
Chapter: [0.013999999999999999] Sales Tax and Value Added Tax
[3]6.2

Solve the following inequation and represent the solution set on the number line:

4x - 19 < (3x)/5 - 2 <= (-2)/5 + x, x ∈ R

Concept: Representation of Solution on the Number Line
Chapter: [0.020499999999999997] Linear Inequations
[4]6.3

Without solving the following quadratic equation, find the value of m for which the given equation has equation has real and equal roots.

`x^2 + 2(m - 1)x + (m + 5) = 0`

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[10]7
[3]7.1

A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 2 cm and height 8 cm. Find the number of cones.

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
Chapter: [0.05] Trigonometry
[3]7.2

Solve the following equation and give your answer correct to 3 significant figure:

`5x^2 - 3x - 4 = 0`

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[4]7.3

As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[10]8
[3]8.1

A man invests Rs. 9600 on Rs. 100 shares at Rs. 80. If the company pays him 18%
dividend find:
(1) The number of shares he buys.
(2) His total dividend.
(3) His percentage return on the shares

Concept: Shares and Dividends
Chapter: [0.012] Shares and Dividends
[3]8.2

In the given figure ΔABC and ΔAMP are right angled at B and M respectively. Given AC = 10 cm, AP = 15 cm and PM = 12 cm.

1) Prove ΔABC ~ ΔAMP

2) Find AB and BC.

Concept: Similarity of Triangles
Chapter: [0.035] Similarity
[4]8.3

if x = `(sqrt(a + 1) + sqrt(a-1))/(sqrt(a + 1) - sqrt(a - 1))` using properties of proportion show that `x^2 - 2ax + 1 = 0`

Concept: Componendo and Dividendo Properties
Chapter: [0.0204] Ratio and Proportion
[10]9
[3]9.1

The line through A(–2, 3) and B(4, b) is perpendicular to the line 2x – 4y = 5. Find the value of b.

Concept: Slope of a Line
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[3]9.2

Prove that `(tan^2 theta)/(sec theta - 1)^2 = (1 + cos theta)/(1 - cos theta)`

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[4]9.3

A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[10]10
[4]10.1

Construct a triangle ABC in which base BC = 6 cm, AB = 5.5 cm and ∠ABC = 120°.

Construct a circle circumscribing the triangle ABC.

Draw a cyclic quadrilateral ABCD so that D is equidistant from B and C.

Concept: Circumscribing and Inscribing a Circle on a Triangle
Chapter: [0.033] Constructions
[6]10.2

The following distribution represents the height of 160 students of a school.

Height (in cm) No. of Students
140 – 145 12
145 – 150 20
150 – 155 30
155 – 160 38
160 – 165 24
165 – 170 16
170 – 175 12
175 – 180 8

Draw an ogive for the given distribution taking 2 cm = 5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine:

(1) The median height.
(2) The interquartile range.
(3) The number of students whose height is above 172 cm.

Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
Chapter: [0.06] Statistics
[10]11
[3]11.1

In triangle PQR, PQ = 24 cm, QR = –7 cm and ∠PQR = 90°. Find the radius of the inscribed circle.

Concept: Circumscribing and Inscribing a Circle on a Triangle
Chapter: [0.033] Constructions
[3]11.2

Find the mode and median of the following frequency distribution

x 10 11 12 13 14 15
f 1 4 7 5 9 3
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
Chapter: [0.06] Statistics
[4]11.3

The line through P(5, 3) intersects y-axis at Q.

(1) Write the slope of the line.
(2) Write the equation of the line.
(3) Find the coordinates of Q.

Concept: Slope of a Line
Chapter: [0.021] Co-ordinate Geometry Equation of a Line

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