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 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

23

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

5 315

+ = 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.

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 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.