APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.

1. Mathematics IX (Term - I) 1
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).
1. If mn
= 32, where m and n are natural numbers, then nm
equals :
(a) 12 (b) 15 (c) 25 (d) 64
2. If p(x) = x2
– 2 2 x + 1, then p(2 2 ) is equal to :
(a) 0 (b) 1 (c) 4 2 (d) 8 2 +1
3. The sides of a triangular flower bed are 5 m, 8 m and 11 m. The area of the flower
bed is :
(a) 2
4 21 m (b) 2
21 4 m (c) 2
330 m (d) 2
300 m
4. Area of the given traingle is :
(a) 60 cm2
(b) 30 cm2
(c) 78 cm2
(d) 32.5 cm2
5. On factorising 4x2
+ y2
+ 1 + 4xy + 2y + 4x, we get :
(a) (2x + y)2
(b) (2x + 2y + 1)2
(c) (x + y + 1)2
(d) (2x + y + 1)2
6. If x51
+ 51 is divided by (x + 1), the remainder is :
(a) 0 (b) 1 (c) 50 (d) 49
7. In the figure, l || m and t is a transversal. If ∠1 = (110° – x) and
∠5 = 4x, then the measures of ∠1 and ∠5 respectively are :
(a) 55°, 55° (b) 22°, 88°
(c) 55°, 88° (d) 88°, 88°
8. In the figure, ABC is a triangle, having sides BC and
CA produced to D and E respectively. Which of the
following is correct?
(a) AB > BC (b) AB > AC
(c) AC > BC (d) BC > AC
MODEL TEST PAPER – 3 (UNSOLVED)
Maximum Marks : 90 Maximum Time : 3 hours
General Instructions : Same as in CBSE Sample Question Paper.

2. 2 Mathematics IX (Term - I)
SECTION B
(Question numbers 9 to 14 carry 2 marks each)
9. Express 1.27 in the form
p
q
, where p and q are integers and q ≠ 0.
10. Verify whether
m
l
is a zero of the polynomial p(x) = lx + m.
11. Using a suitable identity, expand (–2x + 3y + 2z)2
.
12. What is the perpendicular distance of the point A(7, – 4) from
(i) x-axis (ii) y-axis?
13. In ∆ABC and ∆DEF, ∠A = ∠D, ∠B = ∠E and AB = EF. Will the
two triangles be congurent? Justify your answer.
OR
In the figure, if OX =
1
2
XY, PX =
1
2
XZ and OX = PX, show
that XY = XZ.
14. For what value of x + y (see the figure) will ABC be a line?
Justify your answer.
SECTION C
(Question numbers 15 to 24 carry 3 marks each)
15. Simplify by rationalising the denominator :
4 4
4 5
4 5
4 5
+
+
+
16. Write the following in descending order of magnitude : 3 4 23 3 5
, ,
OR
Simplify :
3
5
8
5
32
3
4 12 6
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
17. The polynomial p(x) = x4
– 2x3
+ 3x2
– ax + 3a – 7 when divided by (x + 1), leaves
the remainder 19. Find the value of a. Also, find the remainder when p(x) is divided
by (x + 2).
18. If x + y + z = 9 and xy + yz + zx = 26, find x2
+ y2
+ z2
.
OR
Factorise : 4x2
+ 9y2
+ 16z2
+ 12xy – 24yz – 16zx
19. In the figure, find the area of the trapezium PQRS with height
PQ.
X
Y
PO
Z

3. Mathematics IX (Term - I) 3
20. In the figure, if PQ || ST, ∠PQR = 110° and
∠RST = 130°, find ∠QRS.
21. Prove that the sum of the angles of a triangle is 180°.
22. In the figure, diagonal AC of a quadrilateral ABCD bisects the
angles A and C. Prove that AB = AD and CB = CD.
23. AD and BC are equal perpendiculars to a line segment AB (see
fig). Show that CD bisects AB.
OR
ABC and DBC are two isosceles triangles on the same base
BC (see fig.) Show that ∠ABD = ∠ACD
24. In the figure, ∠Q > ∠R. If QS and RS are bisectors of ∠Q and
∠R respectively, then show that SR > SQ.
SECTION D
(Question numbers 25 to 34 carry 4 marks each)
25. Without actual division prove that (x – 2) is a factor of the polynomial
3x3
– 13x2
+ 8x + 12. Also, factorise it completely.
26. If x2
+ 2
1
x
= 7, find the value of x3
+ 3
1
x
.
27. If two parallel lines are intersected by a transversal, then prove that the bisectors of
the interior angles form a rectangle.
OR
In the given figure, AB || CD. Find the value of x

4. 4 Mathematics IX (Term - I)
28. In a ∆ABC, AD the angle bisector of ∠BAC intersects BC at D. Show that AB > BD.
29. In the figure, ABC is a triangle in which AB = AC and
BE = CD. Prove that AD = AE.
30. Prove that if the line bisecting the vertical angle of a triangle is perpendicular to the
base, the triangle is isosceles.
31. Factorise : 12(x2
+ 7x)2
– 8(x2
+ 7x) (2x – 1) – 15(2x – 1)2
.
OR
Factorise :
1
27
2 5
5
3
3
4
3
4
2
3
3
3 3
( )x y y z z x+ + +
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
32. On the coordinate axes, draw a triangle PQR whose vertices are P(1 ,– 6), Q(7, 4) and
R (– 4, 4)
33. Represent 9 3. on the number line.
34. If x =
3 2
3 2
3 2
3 2
+
=
+
and y , find the value of x2
+ y2
.
A
B C
D E