# Summative Assessment Paper-4

APEX INSTITUTE Managing director um APEX INSTITUTE

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.

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### Summative Assessment Paper-4

• 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.
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