# CBSE Board Mathematics Class X | Sample Paper -1 |

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### All questions are compulsory.

1. Answer any six questions from the following.                         6
1. Given that HCF (306,657) =9, find LCM 9306, 657).
2. The mean and median of a grouped frequency distribution are 36 and 38. Find mode by the empirical formula.
3. The graph of a pair of linear equations is a pair of parallel lines. How many solutions will the pair of linear equations have?
4. Find the value of sin210 +sin230  +sin260  + sin280 .
5. One root of the equation x2-px +2 = 0 is 2. Find p.
6. Find K so that 2k – 1, 7, and 11 are in AP.

2. Each question carries two marks.                                                        12

1. If tan A  = cot B than prove that  (A + B ) = 90
2. How many two-digit numbers are divisible by 3?
3. The following distribution given the daily income of 50 workers of a factory.
 Daily income (in Rs) Number of workers 100 – 120 12 120 – 140 14 140 – 160 8 160 -180 6 180 – 200 10

Convert the distribution above to a less than type cumulative frequency distribution

d. Find the distance between the Points (a, b) and ( -a, -b)

e. Evaluate +2sec220 -2cot270 .

f. A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the           tosses give the same result, I, e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

1. Each question carries three marks.                                                                         30
1. 3x – 2y = 8 and    2x + 3y = 1   Solve the system of equation by the cross multiplication method.
2. Prove that = sec θ + tan θ
3. Describe the nature of the roots of the quadratic equation 2x2 + x – 4 = 0
4. Find the coordinates of the points of trisection of the line segment joining (4,- 1 ) and (- 2 , -3 )
5. The median of the following data is 137. Find the values of x and y, if the total frequency is 68.
 Class intervals 65- 85 85- 105 105-125 125-145 145-165 165-185 185-205 Frequency 4 X 13 20 14 Y 4

f. If the sum of the first n terms of an AP is 4n – n2, what is the first term (what is S1 ) ? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

g. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

h. 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.

i. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

j. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.

1. Each question carries 4 marks.                                                                             32
1. In triangle PQR right angled at Q. PR+QR = 25 cm and PQ =5 cm , Determine the values of sin P . cos P and tan P
2. Prove that ; ( )2   + ( )2  =  7 + tan2 A + cot 2 A
3. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km per hr more than that of the passenger train, find the average speed of the two trains.
4. Draw the graphs of the equations 5x – y = 5 and 3x – y =3. Determine the co-ordinate of the vertices of the triangle formed by these lines and the y axis.
5. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. It they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
6. On dividing x3 – 3x2 + x + 2  by a polynomial g(x), the quotient and remainder were x – 2 and  – 2x  + 4 respectively. Find g(x).
7. Prove that   is an irrational number.
8. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are  (0, – 1) , ( 2, 1 ) and ( 0,3 ). Find the ratio of this area to the area of the given triangle.