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Sample Question Paper
Mathematics
First Term (SA - I)
Class IX
Time: 3 to 3 ½ hours
M.M.:90
General Instructions
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D. Section A
comprises of 8 questions of 1 mark each, section B comprises of 6 questions of 2 marks each,
section C comprises of 10 questions of 3 marks each and section D comprises of 10 questions of 4
marks each.
(iii) Question numbers 1 to 8 in section A are multiple choice questions where you have to select one
correct option out of the given four.
(iv) There is no overall choice. However, internal choice has been provided in 1 question of two
marks, 3 questions of three marks each and 2 questions of four marks each. You have to attempt
only one of the alternatives in all such questions.
SECTION – A
Question numbers 1 to 8 carry 1 mark each. For each question, four alternative choices have been
provided of which only one is correct. You have to select the correct choice
Q. 1
Solution:
Ans : (D)
Q. 2
Solution:
Ans : (B)
Q. 3
Zero of the polynomial
Solution:
Ans: (D)
Q. 4
The coefficient of y in the expansion of
Solution:
Ans: (C)
Q. 5
. If each one is the supplement of the other, then the
value of a is :
Solution:
Ans : (C)
Q. 6
Solution:
Ans : (C)
Q. 7
Two sides of a triangle are 13 cm and 14 cm and its semiperimeter is 18 cm. Then third side of the
triangle is :
(A) 12 cm
(B) 11 cm
(C) 10 cm
Solution:
Let the third side of the triangle be x cm
Ans : (D)
(D) 9 cm
Q. 8
If the sides of a triangle are doubled, then its area:
(A) remains the same
(B) is doubled
(C) becomes three times
(D) becomes four times
Solution:
Let a, b and c be the sides of the original triangle and s be its semi-perimeter
The sides of the new triangle are 2a, 2b and 2c. Let s’ be its semi-perimeter.
SECTION – B
Question numbers 9 to 14 carry 2 marks each
Q. 9
Find an irrational number between
Solution:
It is given that
Q. 10
Solution:
Q. 11
Using suitable identity evaluate :
Solution:
Q. 12
In the adjoining figure, AC = XD, C is the midpoint of AB and D is the midpoint of XY. Using an
Euclid’s axiom show that AB = XY
Solution:
AB = 2AC
XY = 2 XD
Also, AC = XD
AB = XY
[ C is the midpoint of AB]
[ D is the midpoint of XY]
[Given]
[ Things which are double of the same thing are equal to one another]
Q. 13
In the given figure, O is the midpoint of AB and CD. Prove that AC = BD
OR
. Find the shortest and longest side of the triangle.
Solution:
In Δ OAC and Δ OBD
OA = OB
AOC = DOB
OC = OD
AC = BD
OR
[O is the midpoint of AB]
[Vertically opposite angles]
[O is the midpoint of CD]
[ SAS rule]
[ CPCT]
Q. 14
Which of the following points do not lie in any quadrant? Where do those points lie?
Solution:
Points (−3, 0) and (0, 7) do not lie in any quadrant.
Point (−3, 0) lies on x-axis and point (0, 7) lies on y-axis.
Section – C
Question numbers 15 to 24 carry 3 marks each
Q. 15
Represent
on the number line.
OR
Solution:
Steps of construction :
1. Take OA = 2 units, on the number line.
2. Draw BA = 1 unit, perpendicular to OA. Join OB
3. Taking O as centre and OB as radius, draw an arc intersecting the number line at C.
4.
Hence, point C represents
OR
Q. 16
Solution:
Q. 17
OR
Solution:
OR
Q. 18
Determine the value of a for which the polynomial
Solution:
Q. 19
In the given figure, lines AB and CD intersect at O.
If
OR
In the following figure,
Find the value of x
Solution:
OR
Q. 20
In the given figure, ABC is a triangle with BC produced to D. Also bisectors of
Solution:
Q. 21
The degree measure of three angles of a triangle are x, y and z. If z =
Solution:
, then find the value of z
Q. 22
In the given figure, sides AB and AC of
. Show that AC > AB
are extended to points P and Q respectively. Also
Solution:
Q. 23
ABCD is a field in the form of a quadrilateral whose sides are indicated in the figure. If
Solution:
Q. 24
In the given figure, AC = BC,
Solution:
SECTION – D
Question numbers 25 to 34 carry four marks each.
Q. 25
OR
Evaluate after rationalising the denominator of
Solution:
. It is being given that
OR
Q. 26
Prove that :
Solution:
Q. 27.
Solution:
Q. 28
Write the coordinates of the vertices of a rectangle in III quadrant whose length and breadth are 5 and
2 units respectively, one vertex is at the origin and the shorter side is on y-axis. Also, plot the points
on the graph
Solution:
Coordinates of the vertices of the rectangle are
Q. 29
Without actual division, prove that
OR
Solution:
OR
is exactly divisible by
\
Q.30
Solution:
Q. 31
Prove that if two lines intersect each other, then the vertically opposite angles are equal
Solution:
Given : Two lines AB and CD intersect each other at the point O
To prove:
Proof : Ray OA stands on the line CD at O
Q. 32
ABC and DBC are two triangles on the same base BC. Such that A and D lie on the opposite sides of
BC, AB = AC and DB = DC. Show that AD is the perpendicular bisector of BC.
Solution:
are on the same base BC such that A and D lie on the opposite sides of BC,
AB = AC and DB = DC. Let AD intersects BC at O
To prove :
AB = AC
AD = AD
[Given]
[Common side]
Q. 33
In the given figure, the sides AB and AC of
are produced to points P and Q respectively. If
bisectors BO and CO of
respectively, meet at point O, prove that
(i)
(ii)
Solution:
(i) In the given figure,
Q. 34
In the given figure, S is any point in the interior of
. Show that SQ + SR < PQ + PR
Solution:
.
Construction: Extend QS up to point T such that T lies on PR