09th 940114_A1 Mathematics QP

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Class - IX

MATHEMATICS

Time : 3 to 3½ hours Maximum Marks : 80

 â×Ø : 3  âð 3½ �æ�Åð ¥çÏ·¤Ì× ¥´·¤ : 80

Total No. of Pages : 13

 ·é¤Ü Âë cÆæð ´ ·¤è â´�Øæ : 13

General Instructions :

1. All questions are compulsory.

2. The question paper consists of 34 questions divided into four sections A, B, C and D.Section - A comprises of 10 questions of 1 mark each, Section - B comprises of 8 questions of2 marks each, Section - C comprises of 10 questions of 3 marks each and Section - D comprisesof 6 questions of 4 marks each.

3 . Question numbers 1 to 10 in Section - A are multiple choice questions where you are to selectone correct option out of the given four.

4. There is no overall choice. However, internal choice has been provided in 1 question of twomarks, 3 questions of three marks each and 2 questions of four marks each. You have toattempt only one of the alternatives in all such questions.

5. Use of calculator is not permitted.

6. An additional 15 minutes time has been allotted to read this question paper only.

 âæ×æ�Ø çÙÎð üàæ Ñ

1.  âÖè ÂýàÙ ¥çÙßæØü ãñ´Ð

2.  §â ÂýàÙ-Â�æ ×ð ´ 34 ÂýàÙ ãñ´, Áæð ¿æÚU ¹�Çæð´ ×ð´ ¥, Õ, â ß  Î ×ð ́ çßÖæçÁÌ ãñÐ  ¹�ÇU - ¥ ×ð ´ 10 ÂýàÙ ãñ ´ ¥æñÚ Âý�Øð·¤ ÂýàÙ 1 ¥´  ·¤ ·¤æ ãñ, ¹�ÇU - Õ   ×ð´  8 ÂýàÙ ãñ´ ¥æñÚ Âý�Øð·¤ ÂýàÙ 2 ¥´·¤æð´ ·ð¤ ãñ´, ¹�ÇU - â   ×ð´ 10 Âý àÙ ãñ  ´ ¥æñÚ Âý�Øð·¤ ÂýàÙ 3 ¥´·¤æð´ ·¤æ ãñ, ¹�ÇU - Π  ×ð´  6 ÂýàÙ ãñ´ ¥æñÚ Âý �Øð·¤ ÂýàÙ 4 ¥´  ·¤æð´ ·¤æ ãñÐ

3.  ÂýàÙ â´�Øæ 1 âð 10 Õãé  çß·¤ËÂèØ ÂýàÙ ãñ´Ð çΰ »° ¿æÚ çß·¤ËÂæð ´ ×ð ´ âð  °·¤ âãè  çß·¤Ë ¿éÙð´Ð4.  §â×ð´ ·¤æð§ü Öè âßæðüÂçÚ çß·¤Ë Ùãè´ ãñ, Üð ç·¤Ù ¥æ´ÌçÚ·¤ çß·¤Ë 1 Âý àÙ 2 ¥´·¤æð´ ×ð´, 3 Âý àÙ 3 ¥´  ·¤æð´ ×ð ´ ¥æñÚ 2 ÂýàÙ 

4 ¥´·¤æð´ ×ð  ´ çΰ »° ãñ´Ð ¥æ çΰ »° çß·¤ËÂæð´ ×ð´ âð °·¤ çß·¤Ë ·¤æ ¿ØÙ ·¤Úð´Ð

5.  ·ñ¤Ü·é¤ÜðÅÚ ·ð¤ ÂýØæð»  ßçÁü Ì ãñÐ

6.  §â Âý àÙ-Â�æ ·¤æð ÂɸÙð ·ð¤ çÜ° 15  ç×ÙÅ ·¤æ â×Ø çÎØæ »Øæ ãñÐ §â ¥ßçÏ ·ð¤ ÎæñÚæÙ Àæ�æ ·ð¤ßÜ ÂýàÙ-Â�æ ·¤æð Âɸ𠴻ð¥æñÚ ßð ©�æÚ-ÂéçSÌ·¤æ ÂÚ ·¤æð§ü ©�æÚ Ùãè´ çܹð´»ðÐ

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1 P.T.O.

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SECTION - A

Question numbers 1 to 10 carry 1 mark each.

1. Which of the following is cubic polynomial

(A) x313x224x13 (B) x214x27

(C) 3x214 (D) 3(x21x11)

2. The degree of the polynomial p(x)53 is :

(A) 3 (B) 1 (C) 0 (D) 2

3. For the given triangle PQR, which of the following is true ? Fig. 1.

(A) PQ5QR (B) PQ > QR (C) PQ < QR (D) �P5�Q

4. In DPQR PQ5QR and �R5508 then �Q is equal to :

(A) 908 (B) 808 (C) 658 (D) 1008

5. In fig. 2 AB??CD and t is transversal, the value of x is equal to :

(A) 508 (B) 708 (C) 358 (D) 208

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6. How many number of lines does pass through two distinct points.

(A) 1 (B) 2 (C) 3 (D) 4

7. In fig. 3 if AB5AC find x.

(A) 558 (B) 1108 (C) 508 (D) 708

8. Which of the following is an example of a geometrical line.

(A) Black Board (B) Sheet of paper

(C) Meeting place of two walls (D) Tip of the sharp pencil

9.p

qform of the number 0 3. is :

(A)3

10(B)

3100

(C)13

(D)12

10. The product of Quotient of a non-zero rational number with an irrational number is :

(A) Irrational number (B) Rational number

(C) Whole number (D) Natural number

SECTION - B

Question numbers 11 to 18 carry 2 marks each.

11. Find 4 rational numbers between1 4

and3 5

.

12. Find the value of x if 243255(23)x.

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13. Factorise :2 1

4 8

x

x .

14. What value of x would make XOY a line if �XOZ57x1208 and �YOZ53x in fig. 4.

15. In fig. 5, O is the mid point of AB and CD prove that AC 5BD.

16. In DABC, if �A5508 and �B5608 determine the shortest and the longest side of the triangle.

17. In which quadrants do the given points lie ?(a) (2, 21) (b) (21, 7) (c) (22, 23) (d) (4, 5)

18. ABC is an isosceles triangle with AB5AC. Draw APABC. Show that �B5�C.OR

In fig. 6, find x.

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SECTION - C

Question numbers 19 to 28 carry 3 marks each.

19. Show that :

(xa2b)a1b . (xb2c)b1c . (x c2a)c1a 5 1

20. If5 3

a b 35 3

find rational numbers a and b.

OR

Simplify : 12 18 6 20 3 50 8 45

21. If x 5 3 2 2 find the value of31

 x

x

§ ·¨ ¸© ¹

OR

If x 521 5 find the value of 22

1 x

x

§ ·¨ ¸© ¹

22. Evaluate using suitable identity : (999)3

OR

Factorise : 27a318b3154a2b136ab2

23. The polynomials kx313x228 and 3x325x1k are divided by x12. If the remainder in each

case is the same, find the value of k.

24. In fig. 7, triangle AOB with co-ordinates of A and O as (4, 0) and (0, 0). AB55. Find theco-ordinates of B.

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25. In fig. 8 TQ, TR are the bisectors of �Q and �R respectively. �QPR5808  �PRT5308

determine �TQR and �QTR.

26. If DABC is an isosceles triangle with AB5AC. Prove that the perpendiculars BD and CEfrom the vertices B and C to their opposite sides are equal.

27. In fig. 9, l ?? m � 1 and � 2 are in the ratio 5 : 4. Find � 3 and � 4.

28. A garden is in the shape of a quadrilateral. The sides of the garden are 9m, 40m, 28 m and15 m respectively in consecutive order and the angle between first two sides is a right angle.Find the area of the garden.

SECTION - D

Question numbers 29 to 34 carry 4 marks each.

29. Factorise : x323x2210x124

30. Simplify : (5a13b)3 2 (5a23b)3

OR

Find the value of �a� if (x2a) is a factor of x52a2x312x1a13, hence factorise x222ax23.

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31. If x22y511 and xy58 find the value of x328y3.

OR

If x12y58 and xy56. find the value of x318y3.

32. If the bisectors of angles¸ABC and ¸ACB of a triangle ABC meet at a point O then

¸BOC59011

2¸A

33. Prove that the two triangles are congruent if any two angles and the included side of onetriangle is equal to any two angles and the included side of the other triangle.

34. In DABC, ¸C5908 M is the mid point of (fig. 10). AB as well as CD. Show that

(i) DAMC #DBMD

(ii) ¸DBC5908

(iii) DDBC#DACB

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 ¹�ÇU - ¥ 

 Âý àÙ â´�Øæ 1 âð 10 Ì·¤ Âý�Øð·¤ ·¤æ 1 ¥´  ·¤ ·¤æ ãñÐ

1.  çÙ�Ù ×ð´ ·¤æñÙ ÌèÙ �ææÌ ·¤æ ÕãéÂÎ ãñÐ

(A) x313x224x13 (B) x214x27

(C) 3x214 (D) 3(x21x11)

2.  Õãé  ÂΠp(x)53 ·¤è �ææÌ (â´�Øæ) ãñ?

(A) 3 (B) 1 (C) 0 (D) 2

3. ¥æ·ë¤çÌ - 1 ×ð´ °·¤ ç�æÖéÁ PQR  ãñÐ çÙ�Ù ×ð ´ ·¤æñÙ ·¤ÍÙ â�Ø ãñÐ

(A) PQ5QR (B) PQ > QR (C) PQ < QR (D)�

P5�

Q

4. DPQR ×ð´ PQ5QR ¥æñÚ  �R5508 ÌÕ �Q ·¤æ ×æÙ ãæð»æ Ñ

(A) 908 (B) 808 (C) 658 (D) 1008

5. ¥æ·ë¤çÌ - 2 ×ð´, AB??CD ¥æñÚ t °·¤ çÌÂü·¤ Úð¹æ ãñÐ x ·¤æ ×æÙ ãæð »æ Ñ

(A) 508 (B) 708 (C) 358 (D) 208

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6.  Îæð çÖóæ çÕ�Îé¥æð´ âð ç·¤ÌÙè Úð¹æØ𴠹贿è Áæ â·¤Ìè ãñ´Ð

(A) 1 (B) 2 (C) 3 (D) 4

7. ¥æ·ë¤çÌ - 3 ×ð´ ØçΠAB5AC, ÌÕ x ·¤æ ×æÙ �ææÌ ·¤ÚæðÐ

(A) 558 (B) 1108 (C) 508 (D) 708

8.  çÙ�Ù ×ð´ ·¤æñÙ �Øæç×çÌØ Úð¹æ ·¤æ ©ÎæãÚ�æ ãñÐ

(A)  àØæ×Â^  (B)  ©�æÚ ÂéçSÌ·¤æ ·¤æ Âóææ 

(C)  Îæð çÎßæÚæ𴠷𤠩×ØçÙcÅ Öæ» ·¤è ¥æ·ë¤çÌ  (D)  Âñ´  çâÜ ·¤æ Ùé·¤èÜæ çâÚæ 

9. 0 3. â´�Øæ ·¤æ p

q  M¤Â ãæð »æ Ñ

(A) 310

(B) 3100

(C) 13

(D) 12

10.  °·¤ àæê  �Øð�æÚ ÂçÚ×ð Ø â´  �Øæ ¥æñÚ °·¤ ¥ÂçÚ×ðØ â´�Øæ ·¤æ »é�æÙÈ¤Ü ÌÍæ Öæ» âÎñß ãæð»æ Ñ

(A) ¥ÂçÚ×ðØ â´  �Øæ  (B)  ÂçÚ×ðØ â´�Øæ 

(C)  Âê�æü â´�Øæ  (D)  Âýæ·ë¤Ì â´�Øæ 

 ¹�ÇU - Õ 

 ÂýàÙ â´�Øæ 11 âð 18 Ì·¤ Âý�Øð ·¤ ÂýàÙ 2 ¥´·¤æð´ ·¤æ ãñÐ

11.1

3  ÌÍæ 

4

5·ð¤ Õè¿ 4 ÂçÚ×ðØ â´�ØæØð´ �ææÌ ·¤ÚæðÐ

12.  ØçΠ243255(23)x Ìæð x ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

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13.  »é�æÙ¹´Ç ·¤èçÁ° Ñ 2 1

4 8

x

x

14. x ·¤æ ßã ×æÙ �ææÌ ·¤èçÁ° Ìæç·¤ XOY  °·¤ âÚÜÚð¹æ ÕÙ ÁæØÐ ØçΠ �XOZ57x1208 ¥æñ Ú  �YOZ53x

¥æ·ë¤çÌ 4 ×ð´ ÎàææüØæ »Øæ ãñ´Ð

15. ¥æ·ë¤çÌ - 5 ×ð ´, çÕ�Îé O Úð¹æ¹´  Ç AB ß CD ·¤æ ×�Ø çÕ�Îé ãñÐ çâh ·¤Úæð AC5BD.

16.  ç�æÖéÁ ABC ×ð ´ ØçΠ�A5508 ¥æñÚ  �B5608, ç�æÖé  Á ·¤è âÕâð Àæð Åè °ß´ âÕâð ÕǸè ÖéÁæ �ææÌ ·¤ÚæðÐ

17.  çÙ�Ù çÕ�Îé ç·¤Ù ç·¤Ù ÂæÎæð´ ×ð´ çSÍÌ ãñ ´Ð(a) (2, 21) (b) (21, 7) (c) (22, 23) (d) (4, 5)

18. ABC °·¤ â×çmÕæãé ç�æÖéÁ ãñ çÁâ×ð´ AB5AC. ABA BC  ¹è´¿æÐ çâh ·¤èçÁ° �B5�C.

¥Íßæ 

¥æ·ë¤çÌ - 6 ×ð´  x ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

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 ¹�ÇU - â 

 ÂýàÙ â´�Øæ 19 âð 28 Ì·¤ Âý�Øð ·¤ ÂýàÙ 3 ¥´·¤æð´ ·¤æ ãñÐ

19.  Îàææü§° Ñ (xa2b)a1b . (xb2c)b1c . (xc2a)c1a 5 1

20.  ØçΠ5 3

a b 35 3

  ÂçÚ×ð Ø â´�Øæ a ÌÍæ b �ææÌ ·¤èçÁØðÐ

¥Íßæ 

 âÚÜ ·¤èçÁ° 12 18 6 20 3 50 8 45 .

21.  ØçΠx 5 3 2 2 ÌÕ 3

1 x

x§ ·¨ ¸© ¹

  ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

¥Íßæ 

 ØçΠx 521 5   ÌÕ  22

1 x

x

§ ·¨ ¸© ¹

  ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

22.  ©ÂØé�Ì âßüâç×·¤æ ·¤æ ©ÂØæð» ·¤ÚÌð ãé° (999)3 ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

¥Íßæ 

 »é   �æÙ¹´Ç �ææÌ ·¤èçÁ° Ñ 27a3

18b3

154a2

b136ab2

.

23.  ØçΠx12 âð ÕãéÂΠkx313x228 ÌÍæ 3x325x1k ·¤æð çßÖæçÁÌ ·¤ÚÙð ÂÚ àæðáÈ¤Ü â×æÙ Âýæ�Ì ãæð Ìæ ãñ ÌÕ k ·¤æ  ×æÙ �ææÌ ·¤èçÁ°Ð

24. ¥æ·ë¤çÌ - 7 ×ð ́, ç�æÖéÁ AOB ÎàææüØæ »Øæ ãñ çÁâ×ð´ çÕ�Îé  A ÌÍæ O ·ð¤ çÙØÌæ´·¤ (4, 0) ÌÍæ (0, 0) ãñÐ ØçΠAB55.

 ÌÕ çÕ�Îé B ·ð¤ çÙØÌæ´·¤ �ææÌ ·¤èçÁ°Ð

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25. ¥æ·ë¤çÌ  - 8 ×ð´, 345   ·ð¤ ·¤æð �æ Q  ÌÍæ ·¤æð�æ R ·ð¤ ¥hü·¤ ·ý¤×àæÑ TQ ÌÍæ TR ãñ´ ØçΠ �QPR5808 ÌÍæ �PRT5308, ÌÕ �TQR ÌÍæ  �QTR ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

26.  ç�æÖé  Á ABC °·¤ â×çmÕæãé ç�æÖéÁ ãñ, çÁâ×ð´ AB5AC.  çÕ�Îé B ÌÍæ C âð â�×é¹ Öé  Áæ¥æð´ ÂÚ BD ÌÍæ CE Ü�Õ 

 ÇæÜð »Øð ãñ´Ð çâh ·¤Úæð BD5CE.

27. ¥æ·ë¤çÌ - 9, l ?? m ÌÍæ ·¤æð�æ 1 ¥æñÚ 2 ×ð´  5 ¥æñÚ 4 ·¤æ ¥ÙéÂæÌ ãñ, ÌÕ � 3 ÌÍæ  � 4 ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

28.  °·¤ ¿ÌéÖéüÁæ·¤æÚ Õæ» ·¤è ¿æÚæð´ Öé  Áæ¥æð´ ·¤è Ü»æÌæÚ ·¤× ×ð ́ Ü�Õæ§üØæ´ 9m, 40m, 28 m ÌÍæ 15 m ãñ  ´Ð ØçÎ ÂýÍ× ÎæðÖé  Áæ¥æð´ ·ð¤ Õè¿ ·¤æ ·¤æð�æ 908  ãæð Ìæð Õæ» ·¤æ ÿæð�æÈ¤Ü �ææÌ ·¤èçÁ°Ð

 ¹�ÇU - Π

 Âý àÙ â´�Øæ 29 âð 34 Ì·¤ Âý �Øð·¤ ÂýàÙ 4 ¥´·¤æð´ ·¤æ ãñÐ

29.  »é�æÙ¹´Ç �ææÌ ·¤èçÁ° Ñ x323x2210x124

30.  âÚÜ ·¤èçÁ° Ñ (5a13b)3 2 (5a23b)3

¥Íßæ 

 ØçΠ(x2a) ÕãéÂΠx52a2x312x1a13 ·¤æ »é  �æÙ¹´Ç ãæð Ìæð a ·¤æ ×æÙ �ææÌ ·¤èçÁ° ¥æñÚ ÌÕ x222ax23 ·ð¤ »é�æÙ¹´Ç Öè �ææÌ ·¤èçÁ°Ð

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31.  ØçΠx22y511 ¥æñÚ  xy58 ÌÕ x328y3 ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

¥Íßæ 

 ØçΠx12y5

8¥æñÚ xy5

6 ÌÕ x31

8y3 ·¤æ ×æÙ �ææÌ ·¤èçÁ°Ð

32.  ØçÎ ç�æÖé  Á ABC ×ð´ ¸ABC ÌÍæ ¸ACB ·ð¤ ¥hü·¤ çÕ�Îé O ÂÚ ç×ÜÌð ãñ ́ Ìæð çâh ·¤Úæð ¸BOC59011

2¸A.

33.  ØçÎ ç·¤âè ç�æÖéÁ ·ð¤ Îæð ·¤æð �æ ¥æñÚ âç�×çÜÌ ÖéÁæ ÎêâÚð ç�æÖéÁ ·ð¤ ·ý¤×àæÑ Îæð ·¤æð�æ °ß´ âç�×çÜÌ ÖéÁæ ·ð¤ ¥Ü» ¥Ü» ÕÚæÕÚ ãæð´ Ìæð çâh ·¤Úæð ç·¤ ÎæðÙæð´ ç�æÖéÁ âßæZ»â× ãæð´»ðÐ

34.  ç�æÖé  Á ABC ×ð´, ¸C5908  ãñÐ ÖéÁæ AB ÌÍæ CD ·¤æ M ×�Ø çÕ�Îé ãñÐ Îàææü§° Ñ

(i) DAMC# DBMD

(ii) ¸DBC5908

(iii) DDBC# DACB

- o 0 o -