Central Board Secondary Examination(CBSE) – Maths Sample Paper
Time allowed : 3 hours Maximum Marks : 80
All questions are compulsory.
The question paper consists of 30 questions divided into four sections — A, B, C and D.
Section A comprises of ten questions of 1 mark each, Section B comprises of five questions of 2 marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6 marks each
All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
There is no overall choice. However, an internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
In question on construction, the drawings should be neat and exactly as per the given measurement.
SECTION A (1 marks)
1. Complete the missing entries in the following factor tree :
2. If (x + a) is a factor of 2x2 + 2ax + 5x + 10, find a.
3. Show that x = – 3 is a solution of x + 6x + 9 = 0.
4. The first term of an A.P. is p and its common difference is q. Find its 10th term.
5. If tan A = 5/12, find the value of (sin A + cos A) sec A.
6. The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.
- In Figure 1, PQ II BC and AP : PB = 1 : 2. Find ar(ΔAPQ)/ar(ΔABC)
8. The surface area of a sphere is 616 cm . Find its radius.
9. A die is thrown once. Find the probability of getting a number less than 3.
10. Find the class marks of classes 10 – 25 and 35 – 55.
SECTION B(2 marks)
11. Find all the zeros of the polynomial x4 + x3 – 34x2 – 4x + 120, if two of its zeros are 2 and – 2.
12. A pair of dice is thrown once. Find the probability of getting the same number on each dice.
13. If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.
In a ΔABC, right-angled at C, if tan A =1/√3, find the value of sin A cos B + cos A sin B.
14. Find the value of k if the points (k, 3), (6, – 2) and (- 3, 4) are collinear.
15. E is a point on the side AD produced of a ||gm ABCD and BE intersects CD at F. Show that ΔABE ~ ΔCFB.
SECTION C (3 Marks)
16. Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or (3m + 1) for some integer m.
17. Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis :
x + 3y = 6
2x – 3y = 12
18. For what value of n are the nth terms of two A.P.’s 63, 65, 67, … and 3, 10, 17, … equal ?
If m times the mth term of an A.P. is equal to n times its nth term, find the (m + n)th term of the A.P.
19. In an A.P., the first term is 8, nth term is 33 and sum to first n terms is 123. Find n and d, the common difference.
20. Prove that :
(1 + cot A + tan A) (sin A – cos A) = sin A tan A – cot A cos A.
Without using trigonometric tables, evaluate the following :
21. If P divides the join of A(-2, -2) and B(2, -4) such that = AP/AB = 3/7, find the coordinates of P.
22. The mid-points of the sides of a triangle are (3, 4), (4, 6) and (5, 7). Find the coordinates of the vertices of the triangle.
23. Draw a right triangle in which the sides containing the right angle are 5 cm and 4 cm. Construct a similar triangle whose sides are — times the sides of the above triangle.
24. Prove that a parallelogram circumscribing a circle is a rhombus.
In Figure 2, AD 1 BC. Prove that AB2 + CD2 = BD2 + AC2.
25. In Figure 3, ABC is a quadrant of a circle of radius 14 cm and a semi-circle is drawn with BC as diameter. Find the area of the shaded region.
SECTION D (6 marks)
26. A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounces on it. If their speeds are equal, at what distance from the hole is the snake caught ?
The difference of two numbers is 4. If the difference of their reciprocals is
27. The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of 3600 V3 m, find the speed, in km/hour, of the plane.
28. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.
Using the above, prove the following :
In Figure 4, AB || DE and BC || EF. Prove that AC || DF.
Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above, prove the following :
ABC is an isosceles triangle in which AB = AC, circumscribed about a circle, as shown in Figure 5. Prove that the base is bisected by the point of contact.
- If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, find the capacity and total surface area of the bucket. [Use π = 22/7]
- Find mean, median and mode of the following data :
|80 – 100||6|
|100 – 120||5|
|120 – 140||3|