10th Class CBSE Mathematics 2018-19

10th Class CBSE Mathematics 2018-19

Time: 3 hours
M.M.: 80
Class: 10th
Subject: Mathematics
Date: 07/03/2019

General Instructions:

• All questions are compulsory.
• This question paper consists of 30 questions divided into four sections – A, B, C and D.
• Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.
• There is no overall choice. However, an internal choice has been provided in two questions of 1 marks two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.
• Use of calculator is not permitted.

Section A

Question numbers 1 to 6 carry 1 mark each.

Question: 1. Two positive integers a and b can be written as a = x²y² and b = xy³. x, y are prime numbers. Find LCM (a, b).

Question: 2. How many two digits numbers are divisible by 3?

Question: 3. In fig. 1, DE||BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar (Δ ABC) to the ar (Δ ADE)?

Question: 4. Find the coordinates of a point A, where AB is diameter of a circle whose center is (2, -3) and B is the point (1, 4).

Question: 5. For what values of k, the roots of the equation x² + 4x + k = 0 are real?
Or
Find the value of k for which the roots of the equation 3x² – 10x + k = 0 are reciprocal of each other.

Question: 6. Find A if tan 2A = cot (A – 24°)
Or
Find the value of (sin² 33° + sin² 57°)

Section B

Question numbers 7 to 12 carry 2 marks each.

Question: 7. Find, how many two digit natural numbers are divisible by 7.
Or
If the sum of first n terms of an AP is n², then find its 10th terms.

Question: 8. A game consists of tossing a coin 3 times and noting the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game.

Question: 9. Find the ratio in which the segment joining the points (1, -3) and (4, 5) is divided by x-axis? Also find the coordinates of this point on x-axis?

Question: 10. A die is thrown once. Find the probability of getting a number which (i) is a prime number (ii) lies between 2 and 6.

Question: 11. Find c if the system of equations cx + 3y + (3 – c) = 0; 12x + cy – c = 0 has infinitely many solutions?

Question: 12. Find the HCF of 1260 and 7344 using Euclid’s algorithm.
Or
Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.

Section C

Question number 13 to 22 carry 3 marks each.

Question: 13. Find all zeros of the polynomial 13. 3x³ + 10x² – 9x – 4 if one its zero is 1.

Question: 14. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at p and Q intersect at a point T (see Fig. 2). Find the length TP.

Question: 15. Prove that 2+√3/5 is an irrational number, given that √3 is an irrational number.

Question: 16. Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan² θ + cot² θ.
Or
Prove that (1 + cot A – cosec A) (1 + tan A + sec A) = 2

Question: 17. A father’s age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
Or
A fraction becomes 1/3 when 2 is subtracted from the numerator and it becomes 1/2 when 1 is subtracted from the denominator. Find the fraction.

Question: 18. Find the point on y-axis which is equidistant from the points (5, – 2) and (-3. 2).
Or
The line segment joining the points A(2, 1) and B(5, -8)  trisected at the points P and Q such that Pis nearer to A. If P also lies on the line given by 2x – y + k = 0, find the value of k.

Question: 19. Find the made of the following frequency distribution.

Question: 20. Water in a canal, 6 m wide and 1.5 m deep, is flowing with speed of 10 km/hour. How much area will it irrigate in 30 minutes; If 8 cm standing water is needed?

Question: 21. In fig. 3, ∠ ACB = 90° and CD⊥ AB, prove that CD² = BD × AD.

Or
If P and Q are the points on side CA and CB respectively of Δ ABCD, right angled at C. Prove that (AQ² + BP²) = (AB² + PQ²)

Question: 22. Find the area of the shaded region in Fig. 4, if ABCD is a rectangle with sides 8 cm and 6 cm and O is the centre of circle. (Take π = 3.14)

Question: 23.

Question: 24. Prove that the ration of the two similar triangle is equal to the square of the ratio of their corresponding sides.

Question: 25. The following distribution gives the daily income of 50 workers of a factory.

Convert the distribution above to a ‘less than type’ cumulative frequency distribution and draw its ogive.
Or
The table shows the daily expenditure on food of 25 households in a locality. Find the mean daily expenditure on food.

Question: 26. Construct a Δ ABC in which CA = 6 cm, AB = 5 cm and ∠ BAC = 45°. Then construct a triangle whose sides are 3/5 of the corresponding sides of Δ ABC.

Question: 27. A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm³. The radii of the top and bottom of circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of the metal sheet used in making it. [Use π = 3.14]

Question: A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angel of elevation of the top of the light house from 60° to 30°. Find the speed of the boat in metres per minute. [Use √3 = 1.732] Or
Two pole of equal heights are standing opposite on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distances of the point from the poles.

Question: 29. Two water taps together can fill a tank in 15/8 hours. The tap with longer diameter takes 2 hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.
Or
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Question: If the sum of first four terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms.