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Summative Assessment-I Topper Sample Paper - 2 MATHEMATICS CLASS IX Time: 3 to 3 1 hours 2 Maximum Marks: 80 GENERAL INSTRUCTIONS: 1. All questions are compulsory. 2. The question paper is divided into four sections Section A: 8 questions (1 mark each) Section B: 6 questions (2 marks each) Section C: 10 questions (3 marks each) Section D: 10 questions (4 marks each) 3. There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3 questions of three marks and 2 questions of four marks each. 4. Use of calculators is not allowed. SECTION – A 1. If x is an irrational number then x is: (A) rational 2. (D)positive real One of the factors of (x – 1) – (x2 – 1) is: (A) x2 – 1 3. (B) irrational (C) 0 (B) x + 1 (c) x – 1 (D) x + 4 If p(x) = 2x2 − 3x + 1 does not have (x - a) as a factor, then p(a) is: (A) Equal to zero (B) is a non-zero number (C) 4a-1 (D) 4a + 1 4. Two lines PR and QS intersect each other at O. If ∠POQ: ∠QOR = 2:3. Find ∠POS. (A) 144o (B) 72o (C) 108o (D) 216o 5. Two sides of a triangle are 12cm and 13cm. The length of the third side cannot be: (A) 0.8 cm 6. (B) 5 cm (C) 4 cm (D) 6 cm What should be added to x2 + 2x + 0.5 to make it a perfect square? (A) 0.5 (B) 0.6 (C) 0.4 (D) 0.1 7. A measure of the number of square units needed to cover the outside of a figure is called ……… (A) Volume 8. (B) Area (C) Surface area (D) Curved surface area A square sheet whose perimeter is 32 cm is painted at the rate of Rs. 5 per cm2. The cost of painting is: (A) Rs.500 (B) Rs.320 (C) Rs.640 (D) Rs.550 SECTION – B 9. 10. Show that the irrational numbers 5 and 6 lie between 2 and 3. In ∆ABC, ∠ A = 60°, ∠ B = 40°. Which side of this triangle is the smallest? Give reasons for your answer. 0R If OA, OB, OC and OD are the rays such that ∠AOB= ∠COD= 150°, ∠BOC=30° and ∠AOD= 30°. Is it true that AOC and BOD are straight lines? Justify your answer. 11. Show that, if two circles are equal then their radii are equal 12. Without actually calculations the cubes, find the value of 303 + 203 – 503. 13. In figure 4, write the co-ordinates of the points P, Q, R and S. 14. Find the remainder when x11 +1 is divided by x+1. SECTION – C 3 15. Simplify: 3 (25)2 × (243)5 5 4 (16 ) 4 × (8) 3 16. If x – 1 is a factor of kx2- 3x + k then find the value of k. also find the other factor for this value of k. 17. Simplify: x2 + y2 ( 2 ) ( )( ) ( − 2 x2 + y2 x2 − y2 + x2 − y2 2 ) OR If x + y + z=9 then find the value of (3- x )3 + (3 – y)3 + (3 – z)3 – 3 (3 – x) (3 – y) (3 – z). 18. In a parallelogram ABCD, the diagonals bisect each other at 90 degrees. Prove that it is a rhombus. 19. In figure 5, BD⊥ AC, ∠DCB=30° and ∠EAC=50°. Find the value of x and y. 20. In the figure given below, AP ⊥ l and PR > PQ. Show that AR > AQ. 21. In figure 8, AB||CD and CD ||EF. Also,EA ⊥ AB. If ∠BEF= 55°.Find the values of x, y and z. 22. If a and b are rational numbers, find the value of a and b. 3 −1 3 +1 =a+b 3 OR Simplify the following by rationalising the denominators. 3 5− 3 + 2 5+ 3 23. A jigsaw puzzle is made of triangular pieces. Each piece is an isosceles triangle with base 8 cm and perimeter 18 cm. Find the number of pieces that can be fitted on 16x 9 cm rectangular board 24. In the given figure PQ is a straight line. OP bisects ∠AOB. Find the relation between ∠BOQ and ∠AOQ. . OR In figure, AB||CD and CD||EF. Also EA ⊥AB. If ∠BEF = 55°, find the value of x, y and z? SECTION – D 25. Factorize: a12x4-a4x12. 26. In figure 9, ABC is a triangle with ∠BAC=90° and AL ⊥ BC. Prove that ∠CAL= ∠BAC. 27. Using suitable identity, find the value of the following: 3 3 3 (i) (20) + ( −8) + ( −12) 28. (ii) 105×95 Find the value of: 1 3− 8 1 − 8− 7 + 1 7− 6 − 1 6− 5 + 1 5 −2 OR Simplify: 4+ 5 4− 5 + 4− 5 4+ 5 29. A car starts from the center of city and in each consecutive hour it covers a distance of 20km (along north), 16 km (along east), 24 km (along south) and 20 km (along west) respectively. Assuming the centre of city to be the origin, north-south direction is along y axis and west-east direction is along x axis; show the various position of the car on the Cartesian plane. Also, find how far is the car from x and y axis respectively at its final position. 30. Find the values of a and b so that the polynomial x3 – ax2 – 13x + b has x-1 and x + 3 as factors. OR Without actual division, show that the polynomial 2x4-5x3+2x2-x+2 is exactly divisible by x2-3x+2 31. In figure 10, the sides AB and AC of a ∆ABC are produced to P and Q respectively. If the bisectors of ∠PBC and ∠QCB meet at O, then prove that 1 ∠BOC= 90° - ∠A. 2 32. In figure 11, PQRS is a square and SRT is an equilateral triangle. Prove that : a. ∠PST= ∠QRT b. PT = QT c. ∠QTR = 15° 3+ 2 , find the value of x2 + y2 If x = 34. Prove that medians of an equilateral triangle are equal. 3− 2 and y = 3− 2 33. 3+ 2