DESIGN OF THE QUESTION PAPER MATHEMATICS – CLASS IX Time : 3 Hours Maximum Marks : 80 The weightage or the distribution of marks over different dimensions of the question paper shall be as follows: 1. Weightage to Content/ Subject Units S.No. Units Marks 1. 2. 3. 4. 5. 6. Number Systems Algebra Coordinate Geometry Geometry Mensuration Statistics and Probability 06 20 06 22 14 12 2 Weightage to Forms of Questions S.No. Forms of Questions Marks for each Question Number of Questions Total Marks 1. 2. 3. 4. MCQ SAR SA LA 01 02 03 06 10 05 10 05 10 10 30 30 Total 30 80 3. Scheme of Options All questions are compulsory, i.e., there is no overall choice. However, internal choices are provided in two questions of 3 marks each and 1 question of 6 marks. 4. Weightage to Difficulty Level of Questions S.No. Estimated Difficulty Level of Questions Percentage of Marks 1. 2. 3. Easy Average Difficult 20 60 20 Note A question may vary in difficulty level from individual to individual. As such, the assessment in respect of each question will be made by the paper setter/ teacher on the basis of general anticipation from the groups as whole taking the examination. This provision is only to make the paper balanced in its weight, rather to determine the pattern of marking at any stage. BLUE PRINT MATHEMATICS – CLASS IX Forms of Questions → Content Units ↓ MCQ SAR SA LA Total NUMBER SYSTEMS 1 (1) 2 (1) 3 (1) – 6 (3) ALGEBRA Polynomials, Linear Equations in Two Variables 1 (1) 4 (2) 9 (3) 6 (1) 20 (7) COORDINATE GEOMETRY 1 (1) 2 (1) 3 (1) – 6 (3) GEOMETRY Introduction to Euclid’s Geometry, Lines and Angles, Triangles, Quadrilaterals, Areas, Circles, Constructions 4 (4) – 6 (2) 12 (2) 22 (8) MENSURATION Areas, Surface areas and Volumes 2 (2) – 6 (2) 6 (1) 14 (5) STATISTICS AND PROBABILITY Statistics, Probability 1 (1) 2 (1) 3 (1) 6 (1) 12 (4) Total 10 (10) 10 (05) 30 (10) 30 (05) 80 (30) SUMMARY Multiple Choice Questions (MCQ) Number of Questions: 10 Marks: 10 Short Answer with Reasoning (SAR) Number of Questions: 05 Marks: 10 Short Answer (SA) Number of Questions: 10 Marks: 30 Long Answer (LA) Number of Questions: 05 Marks: 30 Total 30 80 MATHEMATICS CLASS IX Time: 3 hours Maximum Marks: 80 General Instructions 1. All questions are compulsory. 2. The question paper consists of four sections A, B, C and D. Section A has 10 questions of 1 mark each, section B has 5 questions of 2 marks each, section C has 10 questions of 3 marks each and section D is of 5 questions of 6 marks each. 3. There is no overall choice. However internal choices are provided in 2 questions of 3 marks each and 1 question of 6 marks. 4. Construction should be drawn neatly and exactly as per the given measurements. 5. Use of calculators is not allowed. SECTION A In Questions 1 to 10, four options of answer are given in each, out of which only one is correct. Write the correct option. 1. Which of the following represent a line parallel to x-axis? (A) x + y = 3 (B) 2x + 3 = 7 (C) 2 y − 3 = y +1 (D) x + 3 = 0 2. Zero of the polynomial p ( x)= 3x+5 is : 5 −5 (A) 0 (B) – 5 (C) (D)33 3. The abscissa of a point P, in cartesian plane, is the perpendicular distance of P from: (A) y-axis (B) x-axis (C) origin (D) line y = x 4. The reflex angle is an angle: (A) less than 90° (B) greater than 90° (C) less than 180° (D) greater than 180° 5. If the lines l, m, and n are such that l 1 m and m1 n , then (A) l 1 n (B) l ⊥ n (C) l and n are intersecting (D) l = n 6. In Fig.1, ∠ B < ∠ A and ∠D > ∠C, then: (A) AD > BC (B) AD = BC (C) AD < BC (D) AD = 2 BC Fig. 1 7. In Fig. 2, the measure of ∠ BCD is: (A) 100° (B) 70° (C) 80° (D) 30° Fig. 2 8. The height of a cone of diameter 10 cm and slant height 13cm is: (A) 69 cm (B) 12 cm (C) 13 cm (D) 194 cm 9. The surface area of a solid hemisphere with radius r is 2 3 (A) 4πr2 (B) 2πr2 (C) 3πr2 (D) πr 3 10. If the mode of the following data 10, 11, 12, 10, 15, 14, 15, 13, 12, x, 9, 7 is 15, then the value of x is: 21 (A) 10 (B) 15 (C) 12 (D) 2 SECTION B 12 11. Find an irrational number between two numbers and and justify your answer. 771 It is given that = 0.142857 7 43 212. Without actually dividing, find the remainder when x + x − 2 x + x +1 is divided by x −1, and justify your answer. 13. Give the equations of two lines passing through (2, 10). How many more such lines are there, and why? 14. Two points with coordinates (2, 3) and (2, –1) lie on a line, parallel to which axis? Justify your answer. 15. A die was rolled 100 times and the number of times, 6 came up was noted. 2 If the experimental probability calculated from this information is , then how 5 many times 6 came up? Justify your answer. SECTION C 23 16. Find three rational numbers between and . 55 17. Factorise: 54 a 3 − 250 b3 18. Check whether the polynomial p ( y)= 2 y 3 + y 2 + 4 y −15 is a multiple of (2y – 3). 19. If the point (3, 4) lies on the graph of the equation 2 y = ax + 6 , find whether (6, 5) also lies on the same graph. 20. Plot (–3, 0), (5, 0) and (0, 4) on cartesian plane. Name the figure formed by joining these points and find its area. 21. Diagonals AC and BD of a trapezium ABCD with AB 1 DC , intersect each other at O. Prove that ar(AOD) = ar(BOC). OR ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that ABCD is a square. 22. Construct a triangle PQR in which ∠Q = 60° and ∠R = 45° and PQ + QR + PR = 11 cm. 23. Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm. 24. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs 12.50 per m2. OR The height of a solid cone is 16 cm and its base radius is 12 cm. Find the total ⎛ 22 ⎞ surface area of cone. ⎜ Use π= ⎟⎝ 7 ⎠ 25. A die is thrown 400 times, the frequency of the outcomes of the events are given as under. Outcome 1 2 3 4 5 6 Frequency 70 65 60 75 63 67 Find the probability of occurence of an odd number. SECTION D 26. A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field. 27. Draw a histogram and frequency polygon for the following distribution: 28. Prove that two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side Marks Obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 No. of Students 7 10 6 8 12 3 2 2 of the other triangle. Using above, prove that CD bisects AB, in Figure 3, where AD and BC are equal perpendiculars to line segment AB. Fig. 3 29. Prove that equal chords AB and CD of a circle subtend equal angles at the centre. Use the above to find ∠ ABO in Figure 4, where O is the centre of the circle Fig. 4 30. Factorise the expression 332 28x + 27 y + 36xy + 54xy OR 9 The Linear equation that converts Fahrenheit to Celsius is F = C + 32 5 Draw the graph of the equation using Celsius for x-axis and Fahrenheit for y-axis. From the graph find the temperature in Fahrenheit for a temprature of 30°C. Marking Scheme MATHEMATICS – CLASS IX SECTION A 1. (C) 2. (D) 3. (A) 4. (D) 5. (A) 6. (C) 7. (C) 8. (B) 9. (C) 10. (B) (1 × 10 = 10) SECTION B 1 1 11. Since = 0.142857 142857 ... and ( )72 21 = 0.285714 285714 ... ( )72 12 1 Therefore, an irrational number between and ( )77 2 1 can be 0.1501500 15000 ... ( )2 12. Let p(x) = x4 + x3 – 2x2 + x + 1, then by Remainder theorem, 1 on dividing with x – 1, remainder is f (1) (1)2 1 Therefore, remainder = 1 + 1 – 2 + 1 + 1 = 2 ( )2 1 13. 3x – y + 4 = 0, x – y + 8 = 0 ()2 Through one point, infinitely many lines can pass. 1 Therefore, infinitely many such lines will be there. (1 )2 1 14. Parallel to y-axis. ( )2 Since x-coordinate of both points is 2. 1 So, both points lie on the line x = 2 which is parallel to y-axis. (1 )2 1 15. Answer is 40 ( )2 frequency of the event occurring Probability of an event = the total number of trials 2 x 1 Therefore, = ,i.e., x = 40 (1)5 100 2 SECTION C 2 8 312 16. = and = (1)520 520 9 10 11 Therefore, three rational numbers can be , , (2)20 20 20 17. 54a 3 – 250b3 = 2[27a 3 – 125b3] (1) 1 = 2[(3a)3 – (5b)3] ()2 1 = 2(3a – 5b) (9a 2 + 15ab + 25b2) (1)2 18. p(y) is a multiple of (2y – 3) if (2y – 3) is a factor of p(y). (1) ⎛ 3 ⎞ Therefore, p⎜⎟ must be zero ⎝ 2 ⎠⎛ 3 ⎞⎛ 3 ⎞3 ⎛ 3 ⎞2 ⎛ 3 ⎞ p ⎜⎟ = 2⎜⎟ + ⎜⎟ + 4⎜⎟ −15 (1)⎝ 2 ⎠⎝ 2 ⎠⎝ 2 ⎠⎝ 2 ⎠ = 27 9 6 15 9 6 – 15 0 4 4 + + − = + = Hence, p(y) is a multiple of (2y – 3) (1) 19. Since, (3, 4) lies on 2y = ax + 6. Therefore, 8 = 3a, i.e., a = 2 3 (1) Now, we have 2y = 2 6 3 x + ( 1 2 ) Putting x = 6, y = 5, we get 10 = 2 .6 6 4 6 10 3 + = + = (1) Hence (6, 5) lies on the same graph ( 1 2 ) 20. Correct plotting figure formed is a triangle (1 1 2 ) Figure formed is a triangle 1 Area = × 8 × 4 =16 sq. unit2 21. ar (ABD) = ar (ABC) (1) [Δs between same parallels and on the same base] Therefore, ar (ABD) – ar (AOB) = ar (ABC) – ar (AOB) i.e., ar (AOD) = ar (BOC) OR Given ABCD is a rectangle with ∠1 = ∠2 and ∠3 = ∠4 But ∠1 = ∠4 (alternate angles) 1 Therefore, we have ∠2 = ∠4, which means AB = BC, similarly AD = CD ( )2 Hence, ABCD is a square. (1) 22. For neat and accurate construction (3) 1 23. a = 18 cm, b = 10 cm. Therefore, c = 42 – 28 = 14 cm and s = 21 ( )2 1 Δ = s (s − a)( s − b)( s − c) ()2 = (21) (3) (11) (7) (1) = 21 11 or 69.69 cm2 (Approx) (1) 1 24. r = 25 cm, h = 3.5 m ( )2 C.S.A. = 2πrh22 25 35 11 2 1 = 2 ×× ×= m (1 )7 100 10 2 2 11 Therefore, cost = Rs ×12.50 = Rs 68.75 (1)2 OR h = 16 cm and r = 12 cm, therefore, l = h2 + r 2 = 20 cm (1) Total surface area = πrl + πr2 = πr (l + r) (1) 22 6 = ×12 × 32 =1206 cm2 (1)77 1 25. Sum of frequencies = 400 ( )2 Odd numbers are 1, 3, 5 Therefore, frequency of all odd numbers = 70 + 60 + 63 = 193 (1) Frequency of occurring of event 1 P(event) = ( )The total number of trials 2 193 Therefore, probability of occurence of odd number = (1)400 SECTION D 1 26. Let AL = x, therefore, BM = 15 – x ()2 Now 132 – x2 = (14)2 – (15 – x)2 1 1 Solving to get x = 6.6 m ( )2 1 Therefore, height DL = (13) 2 − (6.6) 2 ( )2 = 11.2 m (1) 1 Therefore, area of trapezium = (sum of parallel sides) × height (1)2 1 = (10 + 25) (11.2) m2 (1)2 1 = 196 m2 ( )2 27. For correctly making the histogram (4) For correctly making the frequency polygon (2) 28. For correct given, to prove, construction and figure ( ×4 = 2)1 2 For correct proof (2) 1 ∠A = ∠B = 90º ( )2 ∠1 = ∠2 (vert. opp. angles) 1 AD = BC (Given) ( )2 1 Therefore, Δ AOD ≅Δ BOC [AAS] ( )2 1 Therefore, AO = OB, i.e., CD bisects AB ( )2 1 29. For correct given, to prove, construction and figure ( ×4=2)2 For correct proof (2) ∠ AOB = ∠ DOC = 70° (1) Therefore, ∠ ABO = 180° – [70° + 40°] = 70° (1) 332 230. 8x + 27y + 36xy + 54xy = (2x)3 + (3y)3 + 18xy (2x + 3y) (2) = (2x)3 + (3y)3 + 3(2x) (3y) (2x + 3y) (2) = (2x + 3y)3 = (2x + 3y) (2x + 3y) (2x + 3y) (2) OR For correct graph taking Celsius on x-axis and Fahrenheit on y-axis (4) From graph getting F = 86 for C = 30 (2) Other Exemplar Problems by NCERT • Exemplar Problems in Science for Class IX • Exemplar Problems in Physics for Class XI • Exemplar Problems in Chemistry for Class XI • Exemplar Problems in Mathematics for Class XI • Exemplar Problems in Biology for Class XI