Download goa board of sec

GOA BOARD OF SEC. & HIGHER SEC. EDUCATION
ALTO – PORVORIM, GOA.
STD: X
MARKS: 100
Total No. of Questions: 10
SUB: MATHEMATICS
SAMPLE PAPER
Time: 3 hours
Instructions:
(1) Answer each main question on a fresh page.
(2) All questions are compulsory.
(3) The question paper consists of TEN questions, each of 10 marks
(4) There is no overall choice. However, internal choice has been
provided in TWO questions of THREE marks each.
(5) In questions on constructions, the drawing should be near and
exactly as per the given measurements. The construction lines
and arcs should also be maintained.
(6) Graph paper will be supplied on request.
(7) Use of calculators and Mathematical tables is not permitted.
Q. 1 (A) Select and write the most appropriate alternative from those
provided in the bracket.
(1)
If x + y = 4 and x – y = 8, then its solution is __________.
[ x = 8, y = -4; x = 6, y = -2; x = -2, y = 6; x = -4, y = 8 ]
(B) Following is a given pair of linear equations
3x + 2y – (k – 4) = 0
and 6x + 4y –
8
=0
Answer the following questions, with reference to the given
equations.
i)
Write down the conditions for infinitely many solutions.
ii)
Find the value of ‘k’
(2)
(C) Attempt ANY ONE of the following
(3)
i) 3x + 10y = -14 and 8x – 3y = 22 ( By elimination method)
ii) 3x + 8y = 4 and 2x – 3y = 11 ( By cross multiplication method)
(D) A fraction obtained by dividing a two digit number by the number
obtained by reversing the digits is 8. The sum of the digits of the
3
original number is 9. Find the original number.
(4)
Q. 2 (A) Select and write the most appropriate alternative from those
provided in the bracket.
(1)
The roots of the quadratic equation x2 – 5x = 0 are ___________.
[ 0 and 5, 0 and -5, 5 and -5, -5 and -5 ]
(B) Attempt each of the following:
i)
ii)
(2)
If 3 is one of the roots of a quadratic equation x2 - 2x + K = 0
find the value of ‘K’.
Find the discriminant of the equation 2x2 – 5x + 3 = 0
(C) Find the roots of ANY ONE of the following quadratic
equations:
(3)
i) 18x2 – 19x + 5 = 0 ( By factorization method)
ii) 3x2 – 7x + 2 = 0
( By using the ‘quadratic formula’ )
(D) The present age of Mr. Ashok is equal to the square of his son’s
present age. One year ago, Mr. Ashok was eight times as old as his
son. Find their present ages.
(4)
Q. 3 (A) Select and write the most appropriate alternative from those
provided in the bracket.
(1)
th
The 10 term of the arithmetic progression 4,7,10…. is ______
[12, 13, 21, 31]
(B) Attempt the following:
(3)
i)
The sum and product of the zeros of a quadratic polynomial in
‘x’ are -5 and 3 respectively. Find the quadratic polynomial.
ii)
Determine whether -1 is a zero of the polynomial x2 – 3x – 4
iii)
Find the sum of the zeros of the quadratic polynomial
2x2 - x-6
(C) Divide the polynomial x3 – 3x2 + 5x – 3 by x2 – 2 and find the
quotient and the remainder. Hence write the result in the form
Dividend = Divisor X Quotient + Remainder
(3)
(D) A sum of Rs. 400 is to be used to give four cash prizes to students
of a school for their overall performance. If each prize is Rs. 20
less than its preceding prize, find the value of each prize.
(3)
Q. 4 (A) Select and write the most appropriate alternative from those
provided in the bracket.
(1)
The product of two numbers is 108. If their H. C. F. is 3,
then their L. C. M. is __________________.
(10,12,14,36)
(B) One card is drawn from a well shuffled deck of 52 cards.
Find the probability of getting
i)
a king of red colour
ii)
a black card
(2)
(C) Without performing the long division method, determine whether the
rational number 17 has a terminating or non - terminating decimal
625
representation. Also write its decimal representation.
(3)
(D) Prove that √5 is irrational.
(4)
Q. 5 (A) Select and write the most appropriate alternative from those
provided in the bracket.
(1)
Two coins are tossed simultaneously. Therefore, the probability of
getting at least one head is ______
( 1, 1, 3, 1 )
4
2
4
(B) Two dice are thrown simultaneously. Find the probability of
i) the sum of the numbers appearing on the top of the two dice
being 6.
ii) the numbers appearing on the top of both dice being same.
(2)
(C) The following table shows the marks scored by a class in Mathematics
Marks scored
(Classinterval)
No.of
students
fi
Classmark
xi
Deviation
di = xi - a
fi . di
0 - 10
2
_______
______
_______
10 - 20
6
_______
______
_______
20 - 30
12
________
______
_______
30 - 40
10
_______
______
_______
40 - 50
7
________
______
_______
50 - 60
3
_______
______
_______
Taking the class-mark (denoted by ‘a’) of the interval 20 – 30 as the
‘assumed mean’, rewrite and complete the table, and also find the
mean of the marks obtained.
(4)
(D) The following table shows the area of land possessed by some families
Area of land
( in sq. mts)
No. of
families
0-50 50-100
1
100-150
3
5
150-200
200-250
8
250-300
6
4
Convert the distribution to a ‘less than’ type cumulative frequency distribution.
Then taking a suitable scale draw its ogive on a graph paper provided.
(3)
Q. 6 (A) Select and write the most appropriate alternative from those
provided in the bracket.
(1)
∆ ABC ~ ∆ XYZ. If ar(∆ ABC) = 25 sq.cms. and ar(∆ XYZ) = 64 sq. cms.
then BC =____ .
YZ
[ 25 ,
5 , √5 ,
8
]
64
8
√8
5
(B) Two vertical poles of heights 6m and 11m stand on a plane ground.
The distance between their feet is 12m. A rope is tied tightly from the
top of one pole to the top of the other. Answer the following
(2)
i) Draw an appropriate figure.
ii) Find the length of the rope.
P
(C) Given: In ∆ PQR,
Q = 900
QM _|_ PR
M
M
Prove that: PQ2 + QR2 = PR2
Q
R
(D) In the given ∆ABC, D is the mid-point of BC,
MN // BC and AD intersects MN at pt. E.
Prove that: ME = EN
M
(4)
A
E
(3)
N
B
D
C
Q. 7 (A) Select and write the most appropriate alternative from those
provided in the bracket
(1)
If cosec 18º = sec A, then the measure of
( 18º, 36º, 72º, 162º )
A is ___________
(B) In ∆ PQR,
Q = 90º. Draw an appropriate figure and prove that
2
2
sin P+cos P=1
(3)
B = 900.
6, find
7
i) the length of AB
ii) the value of cosA.
ii) The value of cotA
(C) In ∆ABC,
If sinA =
C
(3)
A
B
(D) From the top of a 9 metres high building AB,
the angle of elevation of the top of a tower CD is 30º
and the angle of depression of the foot of the
tower is 60º. Find the height of the tower
300
(Do not substitute the value of √3)
A
600
C
E (3)
9m
B
D
Q. 8 (A) Select and write the most appropriate alternate from those
provided in the bracket
(1)
The radius of a circle is 21 cms .
Therefore its circumference is _________cms.
( 66, 132, 264, 1386)
(B) In the adjoining figure, pt. O is the centre of two concentric circles
whose radii are 4 cms. and 6 cms. respecticvely. Points A, Y and B
are on the outer circle while points R, X and S are on the inner circle.
If AOB = 90 º, find (Do not substitute value of Л)
(2)
i) Area of sector ORXS
ii) Length of arc AYB
O
R 900 S
A
X
B
Y
(C) Construct a ∆ABC with side BC = 4 cms., AB = 5 cms. and B = 60º.
Then, using a pair of compasses and ruler only, construct a triangle
A’BC’ whose sides are 5 of the corresponding sides of ∆ABC. Measure
3
and state of the length of the longest side of the triangle
(3)
(D) On a rectangular table cloth, 12 circular designs of radius 15 cms.
each are made, as shown in the
P
Q
figure. Find the area of the
remaining portion of the table cloth.
(4)
(Take Л = 3.14 )
R
S
9 (A) Select and write the most appropriate alternative from those
provided in the bracket
(1)
The volume of water filled in a cubical container of side 3 cms.
is_____cms.3
( 9, 12, 15, 27 )
(B) A solid metallic object is in the form of a cone whose base is joined to a
hemisphere as shown in the figure. If the radius of the hemisphere, the
radius of the base of the cone and the height of the cone are 10 cms.
each, find each of the following, without substituting the value of Л.
i) The volume of the metal used for the cone.
(3)
ii) The volume of the metal used for the hemisphere
10
iii) The total volume of the metal used for the whole
10
object
(C) Draw a circle with centre O and radius 3 cms. Then, from a point P,
8 cms. away from the centre, construct a pair of tangents to the circle.
Measure and state the lengths of the tangents segments.
(3)
(D) A vessel is in the form of a hollow cylinder mounted on a hollow
hemisphere. The diameter of the hemisphere is 14 cms, and the total
height vessel is 13 cms. Find the outer surface area of the vessel.
(3)
Q. 10 (A) Select and write the most appropriate alternative from those
provided in the bracket
(1)
Point A is at a distance of 5 cms. from centre O of a circle having
radius 3 cms. Therefore the length of the tangent segment from pt. A
to the circle is ______ cms.
(2,4,6,8)
(B) A point A(x,y) divides the segment joining point B(5,-1)
and C(-10,4) in the ratio 3:2. Find
i) the value of x
ii) the value of y
(2)
(C) A circle touches all the four sides of a quadrilateral PQRS.
If PQ = 6 cms., QR = 7 cms., and RS = 4 cms. draw an appropriate
figure and find the length of PS
(3)
(D) Find the area of quadrilateral PQRS whose vertices are P(1,1), Q(7,-3),
R(12,2) and S(7,21).
(4)