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
MODEL TEST PAPER – 5 (UNSOLVED)
Maximum Marks : 90 Maximum Time : 3 hours
General Instructions : Same as in CBSE Sample Question Paper.
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. When the polynomial x3 + 3x2 + 3x + 1 is divided by x + 1,the remainder is :
(a) 1 (b) 8 (c) 0 (d) –6
2. One of the factors of (9x2 – 1) – (1 + 3x)2 is :
(a) 3 + x (b) 3 – x (c) 3x – 1 (d) 3x + 1
3. An exterior angle of a triangle is 110° and the two interior opposite angles are equal.
Each of these angles is :
(a) 70° (b) 55° (c) 35° (d) 110°
4. Two sides of a triangle are of lengths 7 cm and 3.5 cm. The length of the third side
of the triangle cannot be :
(a) 3.6 cm (b) 4.1 cm (c) 3.4 cm (d) 3.8 cm
5. A rational number between 2 and 3 is :
(a) 2.010010001... (b) 6 (c)
5
2
(d) 4 – 2
6. The coefficient of x2 in (2x2 – 5) (4 + 3x2) is :
(a) 2 (b) 3 (c) 8 (d) –7
7. The adjacent sides of a parallelogram are 3 cm and 7 cm. The ratio of their altitudes
is :
(a) 7 : 3 (b) 4 : 3 (c) 3 : 4 (d) 49 : 9
8. The percentage increase in the area of a triangle, if its each side is doubled, is :
(a) 200% (b) 300% (c) 400% (d) 500%
SECTION B
(Question numbers 9 to 14 carry 2 marks each)
9. Let OA, OB, OC and OD be the rays in the anticlockwise direction starting from OA,
such that ∠AOB = ∠COD = 100°, ∠BOC = 82° and ∠AOD = 78°. Is it true that AOC
and BOD are straight lines? Justify your answer.
OR
In ∆PQR, ∠P = 70°, ∠R = 30°. Which side of this triangle is the longest? Give reasons
for your answer.

2. 2 Mathematics IX (Term - I)
10. Is
8
15
1
3
1
5
8
75
3 3 3
⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜
⎞
⎠⎟ = ?
How will you justify your answer, without actually calculating the cubes?
11. Evaluate
−⎛
⎝⎜
⎞
⎠⎟
−
1
27
2
3
12. In an isosceles triangle, prove that the altitude from the vertex bisects the base.
13. Write down the coordinates of the points A, B, C and D as shown in figure.
14. Find the value of a, if (x + a) is a factor of the polynomial x4 – a2x2 + 3x – 6a.
SECTION C
(Question numbers 15 to 24 carry 3 marks each)
15. Simplify the following by rationalising the denominators :
2 6
2 3
6 2
6 3+
+
+
OR
If
5 3
5 3
15
+
= a b , find the values of a and b.
16. If a = 9 – 4 5 , find the value of a –
1
a
.
17. If (x – 3) and x –
1
3
are both factors of ax2 + 5x + b, show that a = b.

3. Mathematics IX (Term - I) 3
OR
Factorise : ( )3 3
– 2 2a b
18. Find the value of x3 + y3 + 15xy – 125, when x + y = 5.
19. In the given figure, QP || ML and other angles are
shown. Find the values of x.
OR
If two lines interesect, prove that vertically opposite angles
are equal.
20. In the given figure, QT ⊥ PR,
∠TQR = 40° and ∠SPR = 30°. Find
the values of x and y.
21. In the given figure, D and E are points
on the base BC of a ∆ABC such that
BD = CE and AD = AE.
Prove that ∆ABE ≅ ∆ACD.
22. Find the area of a triangle, two sides of which are 18 cm and 10 cm and the perimeter
is 42 cm.
23. In the figure lines AB and CD are parallel and P is any
point between the two lines. Prove that
∠ABP + ∠CDP = ∠DPB.
24. In the figure, if AD is the bisector of ∠A, show that
AB > BD.
SECTION D
(Question numbers 25 to 34 carry 4 marks each)
25. In the given figure, ABC is an equilateral
triangle with coordinates of B and C as
B(–3, 0) and C(3, 0).
Find the coordinates of the vertex A.

4. 4 Mathematics IX (Term - I)
26. Let p and q be the remainders, when the polynomials x3 + 2x2 – 5ax – 7 and
x3 + ax2 – 12x + 6 are divided by (x + 1) and (x – 2) respectively. If 2p + q = 6, find
the value of a.
OR
Without actual division, prove that x4 – 5x3 + 8x2 – 10x + 12 is divisible by
x2 – 5x + 6.
27. Prove that :
(x + y)3 + (y + z)3 + (z + x)3 – 3(x + y) (y + z) (z + x) = 2 (x3 + y3 + z3 – 3xyz).
28. Factorise : x12 – y12.
29. In the given figure, PS is bisector of ∠QPR;
PT ⊥ RQ and ∠Q > ∠R. Show that
∠TPS =
1
2
(∠Q – ∠R).
OR
In ∆ABC, right angled at A, see figure given,
AL is drawn perpendicular to BC.
Prove that ∠BAL = ∠ACB.
30. In the given figure, AB = AD, AC = AE and
∠BAD = ∠CAE. Prove that BC = DE.
31. In the given figure, if ∠x = ∠y and
AB = BC, prove that AE = CD.
32. In the given figure, three coplanar lines l,
m, n are concurrent. They form angles a,
b, c, d, e and f. If a = 30° and c = 50°,
find the values of b , d, e and f.
33. Prove that n is not a rational number, if n is not a perfect square.
34. Represent 3.5 on the number line.