Annual Examination – Set 3 വാർഷിക പരീക്ഷ
A full annual-examination paper covering all ten chapters (Part 1 and Part 2). Print it on A4 paper and give it to the student with a timer; the answer key is printed separately.
ANNUAL EXAMINATION · MODEL PAPER · SET 3
Standard IX · MATHEMATICS ഗണിതം
Name:Roll No.:Division:
- The first 15 minutes is cool-off time. Use it to read the questions and plan your answers; do not write during this time.
- Answer all the questions. The score for each question is shown in brackets on the right.
- Read each question carefully before you answer. Answer in keeping with the score and the time.
- Show the steps of your working and give explanations wherever necessary. Calculators are not allowed.
- Draw neat figures wherever needed. For constructions use a ruler, a set square and compasses, and keep the construction lines.
Section AAnswer all questions · 10 × 1 = 10 marks
1.
The sum of two numbers is 36 and their difference is 10. The smaller number is
[1](a) 13(b) 23(c) 18(d) 26
2.
Using √2 ≈ 1.414 and √3 ≈ 1.732, the value of √2 + √3 correct to two decimal places is
[1](a) 2.24(b) 3.14(c) 3.15(d) 3.46
3.
The orthocentre (the point where the altitudes meet) of a right triangle is
[1](a) the midpoint of the hypotenuse(b) the vertex of the right angle(c) the centroid(d) a point outside the triangle
4.
Fill in the blanks: 99 × 98 = (100 − 1)(100 − 2) = 10000 − 200 − 100 + =
[1]5.
√6 × √15 =
[1](a) √21(b) 3√10(c) 9√10(d) 90
6.
The area of an equilateral triangle of side 4 cm is
[1](a) 4√3 cm²(b) 8√3 cm²(c) 16√3 cm²(d) 2√3 cm²
7.
A point moves along a number line towards the left at 6 m/s and passes the point O (zero) at time 0. Using s = vt, its position 4 seconds before it reached O was
[1](a) −24(b) 24(c) −10(d) 10
8.
True or false? Give the reason: (−1)20 + (−1)21 = 0
[1]9.
How many integers x satisfy |x − 2| < 4?
[1](a) 4(b) 7(c) 8(d) 9
10.
If p(x) = x³ − 2x + 5, then p(−1) =
[1]Section BAnswer all questions · 8 × 2 = 16 marks
11.
If 1 is added to the numerator of a fraction, it becomes ½. If 3 is added to its denominator, it becomes ⅓. Find the fraction.
[2]12.
State whether each statement is true or false, with the reason. (a) √8 + √2 is bigger than √10. (b) (√5 + 2) − (√5 − 1) is not a fraction (rational number).
[2]13.
In triangle ABC, AD is the median to BC and G is the centroid (figure). Prove that the area of triangle BGD is ⅙ of the area of triangle ABC.
[2]14.
Numbers that are 1 less than a multiple of 5 (4, 9, 14, 19, …) can be written as 5n − 1. Prove that the product of two such numbers is always 1 more than a multiple of 5. Check with 9 × 14.
[2]15.
In the roof truss ABC (figure), the rafters meet at a right angle at C and the post CD is perpendicular to the beam AB. AD = 2 m and DB = 8 m. (a) Why do triangles ADC and CDB have the same angles? (b) Find the length of the post CD and the area of triangle ABC.
[2]16.
The surface of a circular temple pond has an area of 616 m². Taking π = 22/7, find (a) the radius of the pond and (b) the length of its boundary (circumference).
[2]17.
Solve: (a) |x − 7| = |x + 2| (b) |x + 2| = 5
[2]18.
A rectangle has breadth x cm and length 2x cm. Write as expressions in x: (a) the length of its diagonal, (b) its perimeter divided by its area. Is each a polynomial? Give the reason, and the degree where it is one.
[2]Section CAnswer all questions · 6 × 3 = 18 marks
19.
Two positive numbers have sum s and difference d. (a) Show that the larger number is ½(s + d) and the smaller is ½(s − d). (b) A 7.5 m rope is cut into two pieces whose lengths differ by 2.5 m. Use (a) to find the pieces. (c) Someone says: “A rectangle has perimeter 18 cm and its length is 10 cm more than its breadth.” Show that this is impossible.
[3]20.
Two right-triangular flower beds are made in the corners of a courtyard. Bed P has perpendicular sides 1 m and 3 m; bed Q has perpendicular sides 2 m and 2 m. (a) Find the length of the third side (hypotenuse) of each bed. (b) Without adding up either perimeter, show that the perimeter of P is longer than that of Q by √10 − √8 m. (c) Using √10 ≈ 3.162 and √8 ≈ 2.828, find this difference in centimetres.
[3]21.
Draw triangle ABC with BC = 7 cm, CA = 5 cm and AB = 6 cm. Find the midpoints D of BC and E of CA by drawing their perpendicular bisectors. Draw the medians AD and BE and mark their meeting point G. Measure AD and AG, and check that AG = 2 GD.
[3]22.
A rectangular plot has length x m and breadth y m (x > 3). It is reshaped: the length is decreased by 3 m and the breadth is increased by 3 m (figure). (a) Use an identity to show that the new area is xy + 3(x − y) − 9. (b) Show that the new area is more than the old area when x − y > 3, the same when x − y = 3 and less when x − y < 3. (c) For a plot 20 m long and 12 m broad, find the new area using (a), and check by direct multiplication.
[3]23.
A straight road is taken as a number line; the bus stand is at 0 and distances are in km, east positive (figure). Bus stop A is at −7 and bus stop B is at 5. (a) Find the distance between A and B. (b) A new stop M is to be equally far from A and B. Write this condition with absolute values and find M. (c) A school bus must always stay less than 4 km from M. Write this with absolute value, then as an inequality, and show it on the number line.
[3]24.
A square photo of side x cm is put in a frame 2 cm wide all around it. (a) Write the area f(x) of the frame alone as a polynomial. (b) What is its degree? Find f(10). (c) Write the length of the diagonal of the outer square as an expression in x. Is it a polynomial? Give the reason.
[3]Section DAnswer all questions · 4 × 4 = 16 marks
25.
Meera makes a paper kite ABDC for Onam (figure). Triangle ABC is equilateral with side 12 cm. Triangle BCD on the other side of BC has BD = 9 cm and CD = 15 cm. (a) Find the height of triangle ABC from A. (b) Find the area of triangle ABC. (c) Find the area of triangle BCD using Heron’s formula. (d) Find the total area of the kite correct to two decimal places, taking √3 ≈ 1.732.
[4]26.
Children are decorating the temple ground for the festival. (a) At 4 p.m. a vertical stick 1.2 m tall casts a shadow 0.8 m long, and at the same time the flagstaff (കൊടിമരം) casts a shadow 6 m long. Draw a rough figure, explain why the two triangles have the same angles, and find the height of the flagstaff. (b) For the pandal, two bamboo poles start from a point O (figure): OA = 2 m, AC = 1 m, OB = 2.4 m, BD = 1.2 m, and the cross piece AB = 1.8 m. (i) Show that the sides of triangle OCD around ∠O are the same multiple of those of triangle OAB. (ii) Find the length of the cross piece CD. (iii) Is CD parallel to AB? Why?
[4]27.
In a Vallam Kali race the straight canal is taken as a number line in metres: the finishing post is O (zero) and east is positive. Two boats, Karichal and Chundan, pass O at the same moment; this moment is time 0, times after it are positive and times before it are negative. Position s = vt. (a) Karichal moves west at 8 m/s. Write its v and find its position 3 s after and 3 s before passing O. (b) With the distributive law (x + y)z = xz + yz, taking x = 10, y = −8, z = −3, explain why (−8) × (−3) must be 24. (c) 4 s before passing O, Chundan was at 36. Find its v. Which boat is faster? (d) Find both positions 5 s after passing O. Which boat is ahead, and by how much?
[4]28.
A kolam is painted on a square tile of side 20 cm (figure). At each corner there is a quarter circle of radius 5 cm (shaded grey), painted red. At the centre there is a ring between circles of radii 7 cm and 5 cm (shaded blue), painted yellow; the inner circle and the rest of the tile are left unpainted. (a) Find the total red area, as a multiple of π. (b) Find the yellow area, as a multiple of π. (c) Taking π ≈ 3.14, find the unpainted area. (d) A circle is to be painted with the same area as the yellow ring. Find its radius in simplest form.
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