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Mathematics Set 1 2018-2019 ICSE Class 10 Question Paper Solution

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Mathematics [Set 1]
Marks: 80Academic Year: 2018-2019
Date: March 2019

SECTION - A (40 Marks) Attempt all questions from this Section
[10]1
[3]1.a

Solve the following inequation and write down the solution set: 

11x - a <15 x + 4 ≤ 12xk + 14 , x ∈ W

Represent the solution on a real number line.

Concept: Representation of Solution on the Number Line
Chapter: [0.020499999999999997] Linear Inequations
[3]1.b

A main invest Rs. 4500 in shares of a company which is paying 7.5% dividend.  If Rs. 100 shares are available at a discount of 10%.
Find :
(i) Number of shares he purchaces.
(ii) His annual income.

Concept: Shares and Dividends
Chapter: [0.012] Shares and Dividends
[4]1.c

In a case of 40 students, marks obtained by the students in a class test (out of 10) are given below:

Marks 2 3 4 5 6 7 8 9 10
Number of students 1 2 3 3 6 10 5 4 3

Calculate the following for the given distribution:
(i) Median
(ii) Mode

Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
Chapter: [0.06] Statistics
[10]2
[3]2.a

Using the factor theorem, show that (x - 2) is a factor of `x^3 + x^2 -4x -4 .`

Hence factorise the polynomial completely.

Concept: Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem
Chapter: [0.0203] Factorization
[3]2.b

Prove that: 
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[4]2.c

In a Arithmetic Progression (A.P.) the fourth and sixth terms are 8 and 14 respectively. Find that:
(i) first term
(ii) common difference
(iii) sum of the first 20 terms. 

Concept: Arithmetic Progression - Finding Sum of Their First ‘N’ Terms.
Chapter: [0.0206] Arithmetic Progression
[10]3
[3]3.a

Simplify 

sin A `[[sinA   -cosA],["cos A"  " sinA"]] + cos A[[ cos A" sin A " ],[-sin A" cos A"]]`

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[3]3.b

M and N are two points on the X axis and Y axis respectively. P (3, 2) divides the line segment MN in the ratio 2 : 3.
Find:
(i) the coordinates of M and N
(ii) slope of the line MN.

Concept: Section Formula
Chapter: [0.020099999999999996] Co-ordinate Geometry Distance and Section Formula
[4]3.c

A solid metallic sphere of radius 6 cm is melted and made into a solid cylinder of height 32 cm. Find the:
(i) radius of the cylinder 
(ii) curved surface area of the cylinder
              Take π = 3.1

Concept: Volume of a Cylinder
Chapter: [0.04] Mensuration
[10]4
[3]4.a

The following numbers, K + 3, K + 2, 3K – 7 and 2K – 3 are in proportion. Find K. 

Concept: Proportions
Chapter: [0.0204] Ratio and Proportion
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[3]4.b

Solve for x the quadratic equation x2 - 4x - 8 = 0.  
Give your answer correct to three significant figures. 

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[4]4.c

Use rule and compass only for answering this question. 
Draw a circle of radius 4 cm. Mark the centre as O. Mark a point P outside the circle at a distance of 7 cm from the centre. Construct two tangents to the circle from the external point P.
Measure and write down the length of any one tangent.

Concept: Construction of Tangents to a Circle
Chapter: [0.033] Constructions
SECTION - B (40 Marks) Attempt any four questions from this Section
[10]5
[3]5.a

There are 25 discs numbered 1 to 25. They are put in a closed box anx shaken thoroughly. A disc is drawn at random from the box. 
Find the probability that the number on the disc is :
(i) an odd number
(ii) divisible by 2 and 3 both.
(iii) a number less than 16.

Concept: Random Experiments
Chapter: [0.07] Probability
[3]5.b

Rekha opened a recurring deposit account for 20 months. The rate of interest is 9% per annum and Rekha receives Rs. 441 as interest at the time of maturity. 
Find the amount Rekha deposited each month.

Concept: Computation of Interest
Chapter: [0.013000000000000001] Banking
[4]5.c

Use a graph sheet for this question. 
Take 1 cm = 1 unit along both x and y axis.
(i) Plot the following points:
      A(0,5), B(3,0), C(1,0)  and  D(1,–5)
(ii) Reflect the points B, C and D on the y axis and name them as  B',C'andD' respectively.
(iii) Write down the coordinates of B',C 'and D'
(iv) Join the point A, B, C, D, D ', C ', B', A in order and give a name to the closed figure ABCDD'C'B

Concept: Simple Applications of All Co-ordinate Geometry.
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[10]6
[3]6.a

In the given figure,  ∠PQR = ∠PST = 90°  ,PQ = 5cm and PS =  2cm.
(i) Prove that ΔPQR ~ ΔPST.
(ii) Find Area of ΔPQR : Area of quadrilateral SRQT. 

Concept: Similarity of Triangles
Chapter: [0.035] Similarity
[3]6.b

The first and last term of a Geometrical Progression (G.P.) and 3 and 96 respectively. If the common ratio is 2, find: 
(i) ‘n’ the number of terms of the G.P.
(ii) Sum of the n terms.

Concept: Geometric Progression - Finding Sum of Their First ‘N’ Terms
Chapter: [0.0207] Geometric Progression
[4]6.c

A hemispherical and a conical hole is scooped out of a.solid wooden cylinder. Find the volume of the  remaining solid where the measurements are as follows:
The height of the solid cylinder is 7 cm, radius of each of hemisphere, cone and cylinder is 3 cm. Height of cone is 3 cm.

Give your answer correct to the nearest whole number.Taken`pi = 22/7`.

Concept: Three-dimensional Solids Sphere
Chapter: [0.04] Mensuration
[10]7
[3]7.a

In the given figure AC is a tangent to the circle with centre O.
If  ∠ADB = 55° , find x and y. Give reasons for your answers. 

Concept: Tangent to a Circle
Chapter: [0.032] Circles
[3]7.b

The model of a building is constructed with the scale factor 1 : 30. 
(i) If the height of the model is 80 cm, find the actual height of the building in meters.
(ii) If the actual volume of a tank at the top of the building is 27m3, find the volume of the tank on the top of the model. 

Concept: Three-dimensional Solids Sphere
Chapter: [0.04] Mensuration
[4]7.c

Given `[["4 " " 2" ],[" -1 "" 1 " ]]` M = 6I , where M is a matrix and I is unit matrix of order 2×2. 
(i) State the order of matrix M.
(ii) Find the matrix M.

Concept: Matrices Examples
Chapter: [0.0208] Matrices
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[10]8
[3]8.a

The sum of the first three terms of an Arithmetic Progression (A.P.) is 42 and the product of the first and third term is 52. Find the first term and the common difference.

Concept: Arithmetic Progression - Finding Sum of Their First ‘N’ Terms.
Chapter: [0.0206] Arithmetic Progression
[3]8.b

The vertices of a ΔABC are A(3, 8), B(–1, 2) and C(6, –6). Find: 
(i) Slope of BC
(ii) Equation of a line perpendicular to BC and passing through A.

Concept: Equation of a Line
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[4]8.c

Using ruler and a compass only construct a semi-circle with diameter BC = 7cm. Locate a point A on the circumference of the semicircle such that A is equidistant from B and C. Complete the cyclic quadrilateral ABCD, such that D is equidistant from AB and BC. Measure ∠ADC and write it down.

Concept: Arc and Chord Properties - Angle in a Semi-circle is a Right Angle
Chapter: [0.032] Circles
[10]9
[3]9.a

The data on the number of patients attending a hospital in a month are given below. Find the average (mean) number of patients attending the hospital in a month by using the shortcut method.  Take the assumed mean as 45. Give your answer correct to 2 decimal places.

Number of patients 10 -  20 20 - 30 30 - 40 40 - 50 50 - 60 60 -  70
Number of Days 5 2 7 9 2 5
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
Chapter: [0.06] Statistics
[3]9.b

Using properties of proportion solve for x, given

`(sqrt(5x)+sqrt(2x -6))/(sqrt(5x)- sqrt(2x -6)) = 4`

Concept: Ratio and Proportion Example
Chapter: [0.0204] Ratio and Proportion
[4]9.c

Sachin invests Rs. 8500 in 10%, Rs 100 shares at Rs. 170. He sells the shares when the price of each share rises by Rs. 30. He invests the proceeds in 12% Rs. 100 shares at Rs. 125. Find:
(i) the sale proceeds.
(ii) the number of Rs. 125 shares he buys.
(iii) the change in his annual income.

Concept: Shares and Dividends
Chapter: [0.012] Shares and Dividends
[10]10
[6]10.a

Use graph paper for this question. 
The marks obtained by 120 students in an English 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

Draw the ogive and hence, estimate:
(i) the median marks.
(ii) the number of students who did not pass the test if the pass percentage was 50.
(iii) the upper quartile marks.

Concept: Graphical Representation of Ogives
Chapter: [0.06] Statistics
[4]10.b

A man observes the angle of elevation of the top of the tower to be 45°. He walks towards it in a horizontal line through its base. On covering 20 m the angle of elevation changes to 60°. Find the height of the tower correct to 2 significant figures. 

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
Chapter: [0.05] Trigonometry
[10]11
[3]11.a

Using the Remainder Theorem find the remainders obtained when ` x^3 + (kx + 8 ) x + k ` is divided by x + 1 and x - 2 . 

Hence find k if the sum of the two remainders is 1.

Concept: Remainder Theorem
Chapter: [0.0203] Factorization
[3]11.b

The product of two consecutive natural numbers which are multiples of 3 is equal to 810. Find the two numbers. 

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
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[4]11.c

In the given figure, ABCDE is a pentagon inscribed in a circle such that AC is a diameter and side BC//AE.If
∠ BAC=50°, find giving reasons:

(i) ∠ACB

(ii) ∠EDC

(iii) ∠BEC

Hence prove that BE is also a diameter

Concept: Lines of Symmetry
Chapter: [0.034] Symmetry

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