Other Solved Mathematics Board Papers
MATHEMATICS (ICSE – Class X Board Paper 2010)
Two and Half Hour. Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the question paper.
The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions form Section A and any four questions from Section B. All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the Answer. Omission of essential working will result in the loss of marks.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables are provided.
SECTION A [40 Marks]
(Answer all questions from this Section.)
Question 1:
(a) Solve the following inequation and. represent the, solution set on the number line.
(b) Tarun bought an article for and, spent
for transportation. He marked the article at
and, sold it to a customer. If the customer had. To pay
sales tax, find
(i) the customer’s price
(ii) Tarun’s profit percent [3]
(c) Mr. Gupta opened a recurring deposit account in a bank. He deposited per month for two years. At the time of maturity he got
. Find. :
(i) the total interest earned by Mr Gupta.
(ii) the rate of interest per annum [4]
Answers:
(b) Cost Price
Overheads
Listed Price
Sales Tax rate
Question 2:
(b) Nikita invests for two years at a certain rate of interest compounded annually, At the end of first year it amounts to
. Calculate;
(i) The rate of interest;
(ii) The amount at the end of the second year. [3]
(c) are two points on the
respectively.
is the mid-point of
. Find the
(i) Co-ordinates of
(ii) Slope of line
(iii) Equation of line [4]
Answers:
(a)
(b) Compound Interest for 1 year
Amount at the end of second year
(iii) Equation of
Question 3:
(a) Cards marked with numbers 1, 2, 3, 4 … 20 are well shuffled and a card is drawn at random. What is the probability that the number of the cards is
(i) a prime number
(ii) divisible by 3
(iii) a perfect square [3]
(b) Without using trigonometric tables evaluate :
(c) Use graph paper for this question are the vertices of
.
(i) Plot the reflection of
in the
, and write its co-ordinates.
(ii) Give the geometrical name of the figure .
(iii) Write the equation of the line of symmetry of the line [4]
Answers:
(a) Given: Cards marked with numbers 1,2, … , 20
(i) Prime Numbers
(ii) No. divided by
(iii) No. perfect square
(c)
(i) Plotted above
(ii) The coordinates of
(iii) Arrow Head
(iv) is the equation of line of symmetry
Question 4:
(a) When divided by the polynomials
leave the same remainder. Find the value of
(b) In the figure, given below,
are two parallel chords and
is the center. If the radius of the circle is
, find the distance
between the two chords of lengths
respectively. [3]
(c) The distribution given below shows the marks obtained by 25 students in an aptitude test. Find the mean, median and mode of the distribution. [4]
Answers:
(a) When
Remainder1
Remainder2
Given Remainder1 = Remainder 2
(b) We know that perpendicular drawn from the center of the circle will bisect the chord.
Hence
(c)
Since is odd,
Mode (maximum frequency)
Section B [40 Marks]
Answer any four questions in this section.
Question 5:
(a) Without solving the following quadratic equation, find the value of ‘p’ for which the roots are equal. [3]
(b) Rohit borrows Rs. 86000 from Arun for2 years at 5% per annum simple interest. He immediately lends his money to Akshay at 5% compounded interest annually for the same period. Calculate Rohit’s profit at the end of two years. [3]
(c) Mrs. Kapoor opened a Saving Bank Account in State Bank of India on 9th January 2008. Her passbook entries for the year 2008 are given below:
Answers:
Mrs. Kapoor closed the account on 31st December 2008. If the bank pays interest at 4% per annum, find the interest he receives on closing the account. Give your answer correct to the nearest rupee. [4]
(a) Comparing with
, we get
For roots to be equal, we should have
(b) Simple Interest for 2 years
Compound Interest for 2 years
Gain
(c) Qualifying principal for various months:
Month | Principal (Rs.) |
January | 10000 |
February | 10000 |
March | 25500 |
April | 20000 |
May | 20000 |
June | 20000 |
July | 20000 |
August | 19800 |
September | 19800 |
October | 19800 |
November | 19800 |
Total | 204700 |
Question 6:
(a) A manufacturer marks an article at . He sells this article to a wholesaler at a discount of
on the marked price and the wholesaler sells it to a retailer at a discount of
on its marked price. If the retailer sells the article without any discount and at each stage the sales-tax is
, calculate the amount of VAT paid by:
- The Wholesaler
- The Retailer [3]
(b) In the following figure O is the center of the circle and AB is a tangent to it at point B. . Find
(c) A doorway is decorated as shown in the figure. There are four semi-circles.
, the diameter of the larger Semi-circles.
the diameter of the larger semi-circle is of length
. Center of the three equal semi-circles lie on
.
is an isosceles triangle with
. If
, find the area of the shaded, region. [4]
Answers:
(b) is a tangent to the circle.
(given)
(angle at the center)
(c) Let
As angle in semi-circle is
In right angled , by Pythagoras theorem, we get
Diameter of semicircle
Area of shaded region = Area of semi-circles + Area of three equal circles – Area of
Question 7:
(a) Use ruler and compasses only for this question :
(i) Construct , where
(ii) Construct the locus of points inside the triangle which are equidistant from
(iii) Construct the locus of points inside the triangle which are equidistant from
(iv) Mark the point which is equidistant from
and also equidistant from
. Measure and record the length of
(b) The equation of a line is Find:
(i) Slope of the line.
(ii) The equation of a line perpendicular to the given line and passing through the intersection of the lines [3]
(c) The mean of the following distribution is 52 and the frequency of class interval 30-40 is . Find
. [4]
Class Interval | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
Frequency | 5 | 3 | 7 | 2 | 6 | 13 |
Answers:
(a)
(i) Step 1: Draw a line The with
as the vertex, draw an angle of
using a compass.
Step 2: Draw of length
Join
That gives is the
(ii) Points which are equidistant from are on the angle bisector of
. Therefore draw angle bisector of
(iii) Points which are on the perpendicular bisector of are equidistant from point
Draw a perpendicular bisector of
(iv) The point where the angle bisector and the perpendicular bisector intersect is the point that is equidistant from but also equidistant form points
For point of intersection solve
Therefore intersection
Therefore equation of line
(c)
Interval | Frequency |
|||
10-20 | 5 | 15 | -30 | -150 |
20-30 | 3 | 25 | -20 | -60 |
30-40 | F | 35 | -10 | -10f |
40-50 | 7 | 45 (A) | 0 | 0 |
50-60 | 2 | 55 | 10 | 20 |
60-70 | 6 | 65 | 20 | 120 |
70-80 | 13 | 75 | 30 | 390 |
Question 8:
(a) Use the remainder theorem to factorize the following expression:
(b) If are in continued proportion, prove that:
(c) From the top of a light house high the angles of depression of two ships on opposite sides of it are
respectively. Find the distance between the two ships to the nearest meter. [4]
Answers:
(a) Let
Remainder
Hence is a factor of
Hence
(b) are in continued proportion, then
Applying componendo and dividendo
Squaring both sides
Substituting
(c)
Question 9:
(a) Evaluate
(b) In the given figure,
is a triangle with
Prove that
If
and area of
Calculate the:
(i) length of
(ii) area of
(c) Vivek invests in
,
shares at
He sells the shares when the price rises to
, and invests the proceeds in
shares at
Calculate; i) The sale proceeds ii) The number of
shares he buys; iii) The change in his annual income from dividend. [4]
Answers:
(a)
(b) Consider
(given)
(common)
(AAA postulate)
(i)
(ii)
(c) First Investment
Let the amount invested
Nominal Value of the share
Market Value of the share
Dividend earned
Sale Proceed
Second Investment
Therefore the amount invested
Nominal Value of the share
Market Value of the share
Dividend earned
Hence the change in income
Question 10:
(a) A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part. Taking x as the smaller part of the two parts, find the number. [4]
(b) The monthly income of a group of 320 employees in a company is given below:
Monthly Income | No. of Employees |
6000-7000 | 20 |
7000-8000 | 45 |
8000-9000 | 65 |
9000-10000 | 95 |
10000-11000 | 60 |
11000-12000 | 30 |
12000-13000 | 5 |
Draw an ogive of the given. distribution on a graph sheet taking on one axis and
on the other axis. From the graph determine:
(i) the median wage
(ii) the number of employees whose income is below
(iii) If the salary of a senior employee is above find the number of senior employees in the company.
(iv) The upper quartile [6]
Answers:
(a) Let the two parts be
Also
Substituting it back
(ignore this as the number is positive)
Therefore the larger part is
Hence the number is
(b)
Monthly Income | No. of Employees | Cumulative Frequency |
6000-7000 | 20 | 20 |
7000-8000 | 45 | 65 |
8000-9000 | 65 | 130 |
9000-10000 | 95 | 225 |
10000-11000 | 60 | 285 |
11000-12000 | 30 | 315 |
12000-13000 | 5 | 320 |
Here n (no. of employees) (even)
Required median (from graph)
(ii) Number of employees whose income is below approx
(iii) Number of senior employees in the company
Upper Quartile
Question 11:
(a) Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon. [3]
(b) A hemispherical bowl of diameter 7.2 cm is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius 4.8 cm. Find the height of the cone. [3]
(c) . Use componendo and dividendo to prove that:
Answers:
(a) Given side of the hexagon is
. Construct the hexagon as follows:
- First draw a line using a ruler of length 5 cm. Mark it AB.
- The using a compass, make an arc of
- Using compass, draw 120 degree angle and cut the line into 5 cm lengths using the compass. Continue this until the hexagon is completed
- One the hexagon is completed, draw perpendicular bisectors of each of the arms of the hexagon. You will get mid point for each of the arms of hexagon as marked
- Now join the mid points of the opposite sides to get lines
- Now draw the diagonals passing through the center to get lines
The six lines of symmetry
(b) Given: Diameter of hemispherical bowl
Radius of hemispherical bowl
Volume of cone $latex \displaystyle = Volume of hemi-sperical bowl
(c)
Applying componendo and dividendo
Simplifying
Square both sides
Applying componendo and dividendo
Simplifying