Sample Paper 9 – 2009
Class – X
Subject - Mathematics
Marks: 80 Time: 3hrs
Section - A (10x1=10)
1. Sate the Euclid’s division lemma.
2. The graph of y= f(x) is given below. Find f(x).
Y
X’ -4 -1 2 X
Y’
3. On dividing x2 + 7x + 3 by a polynomial g(x) the quotient and remainder were
x+5 and -7 respectively. Find g(x).
4. What is the nature of roots of the quadratic equation x + 1 = 3?
x
5. In the adjoining figure OACB is a quadrant of a circle with centre O and radius
7cm. If OD = 4cm, find the area of the shaded region
O
D
DD B
C
A
A
6. In ∆ABC, AB = 6 √3, AC= 12cm and BC= 6cm. Find the angle B.
7. Write down the empirical relationship between the three measures of central
tendency.
8. One card is drawn from a well-shuffled deck of 52 cards.
Calculate the proobability that the cards will not be an ace
9. Two tangents TP and TQ are drawn to circle with centre O from an external
point T, and ∟ PTQ = 600, find ∟ OPQ.
P
T
Q
O
10. Find the probability of getting 53 Tuesdays in a leap year.
Section B (5x2=10)
11. How many two-digit numbers are divisible by 7.
12. Evaluate sin700 + tan10 tan40 tan50 tan80
cos 20 2 cos 430 cosec 470
13. In the figure, ABCD is a trapezium in which AB || DC and 2AB = 3CD. Find
the ratio of the areas of ∆AOB and ∆COD.
CCC
D
A
B
O
14. Find the ratio in which the line segment joining the points (6 , 4) and (1 , -7) is
divided by x-axis.
15. Cards numbered 3,4,5,6 ………, 17 are put in a box and mixed thoroughly.
A card is drawn at random from the box. Find the probability that the card
drawn bears
(i) A number divisible by 3 or 5
(ii) A number divisible by 3 and 5.
Section C (10x3=30)
16. Find the zeros of the quadratic polynomial x2 + 7x + 10 and verify the
relationship between the zeros and the coefficients.(OR)
Find all the zeroes of x4 – 5x3 + 3x2 + 15x -18, if two of its zeroes are √ 3 and
-√3.
17. Prove that 7√ 5 is irrational.
(Or)
Explain why 7 x11 x 13 + 13 and 7 x 6 x 5 x 4 x3 x 2 x 1 + 5 are composite
numbers.
18. For which value of k will the following pair of linear equations have no
solution?
3x + y = 1 ; (x-1) 2k – 1(x + y) = 1 – ky.
(Or)
Solve : 6x + 3y = 6xy
2x + 4y = 5xy.
19. Determine the A.P whose 5th term is 15 and the sum of its 3rd and 8th terms
is 34.
(Or)
Find the sum of all three digit numbers which leave the remainder 2 when
divided by 7.
20. Prove that cosA – sinA + 1 = cosecA + cotA
cosA + sinA – 1
21. The line joining the points (2, 1) & (5, -8) is trisected at the points P & Q. If the
point P lies on the line 2x – y +k = 0, Find the value of k.
22. If the points p(x, y) is equidistant from the points A(5, 1) and B(-1, 5),
prove that x = 2.
y 3
(OR)
Show that the points (-3, 2) (1, -2) & (9, -10) can never be the vertices of a
triangle.
23. Draw a pair of tangents to a circle of radius 5cm which are inclines to each
other at an angle of 600.
24. In an equivalateral triangle, prove that three times the square of one side is
equal to four times the square of one of its altitudes.
25. Find the area of the designed region in fig given below between the two
quadrants of radius 7cm each.
Section D (5x6=30)
26. The cost of 5 oranges and 3 apples is Rs 25 and the cost of 3 oranges and 4
Apples is Rs 26 , find the cost of an orange and an apple graphically.
27. In a triangle, if the square on one side is equal to the sum of the squares on
the other two sides, prove that the angle, opposite to the first side is a right
angle. Use the above theorem and prove the following.
In a ∆ABC, AD ┴ BC and BD = 3CD. Prove that 2AB2 = 2AC2 + BC2
(Or)
In an equilateral triangle ABC, D is a point on side BC such that BD = 1 BC.
Prove that 9AD2 = 7AB2. 3
28. A man is standing on the deck of a ship, which is 8cm above of the water
level. He observes the angle of the elevation of the top of the
hill as 60º and the angle of depression of the base of the hill as 30º.
Calculate the height of the hill from the water level.
(Or)
The angle of elevation of the top of a tower from a point A on the ground is
30º. On moving a distance of 20m towards the foot of the tower to a point B,
the angle of elevation increases to 60º. Find the height of the tower.
29. A farmer connects a pipe of internal diameter 20cm from a canal into a
cylindrical tank in her field, which is 10m in diameter and 2m deep. if water
flows through the pipe at the rate of 3km/hr, in how much time will the tank to
be filled?
30. The median of the following data is 28.5.Find the missing frequencies x and
y, if the total frequency is 60
Class interval | Frequency |
0-10 | 5 |
10-20 | X |
20-30 | 20 |
30-40 | 15 |
40-50 | Y |
50-60 | 5 |
(Or)
Verify the relation Mode = 3median – 2 mean from the following data
C.I | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | Total |
Frequency | 5 | 8 | 20 | 15 | 7 | 5 | 60 |
QUESTION NUMBER | ANSWERS |
1 | a = bq+r |
2 | X3+3x2-6x-8 |
3 | X+2 |
4 | Real and distinct |
5 | 24.5 cm2 |
6 | B=90˚ |
7 | Mode=3median-2 mean |
8 | 12/13 |
9 | 30˚ |
10 | 2/7 |
11 | 13 |
12 | 2 |
13 | 9:4 |
14 | 4:7 |
15 | 7/15, 1/15 |
16 | -5,-2 sum=-7 =-b/a product = 10=c/a |
17 |
|
18 | K=2 x=1, y= 2 |
19 | -1,3,7…. ;67404 |
20 |
|
21 | P(4,-5) k= -13 |
22 | AB+BC=CA=12√2 or Area=0 |
23 |
|
24 |
|
25 | 77-49=28cm2 |
26 | (2,5) |
27 |
|
28 | 32m , 10√3m |
29 | 100 mts |
30 | x =8 ,y =7 |
SAMPLE QUESTION PAPER-III
MATHEMATICS-CLASS-X TIME:3 HOURS MARKS:80
SECTION-A(10X1=10MARKS)
1.If HCF(24,x)=6 and LCM(24,x)=144 then find the value of x.
2.If α, β ,γ are the zeroes of P(x)= 3x3 +x2 -10x -8 , find α-1 +β-1 +γ- -1.
3.For what values of β will the system of linear equations β+3y =β -3 ; 12x +βy =β have a unique solution ? (B+C) = cos A 2 2 4.Prove that sin where where A,B and C are interior angles of ∆ABC. (( (or) If sinA =1/2 and A+B=90˚ ,then what is the value of cotB.
5. Evaluate 5cos260˚ +4sec230˚ - tan230˚ Sin230˚ + cos230˚ 6.Find the ratio of the areas of a square and triangle in the figure given below where M is the mid point of AB. A B C D M 7.Find AD, if AB=3cm, BC =6cm and CD = 7cm. A B C
D
8.Find AB, given that ∟ACB = α ,∟BPA = 90-α ,BP =4cm and PC =5cm. A
B bbB P C
0, 2, 4, 3, 3, 4, 2, 3, x, 3, 4, 0, 2, 2.
SECTION-B ( 5x2=10 MARKS)
11.In an A.P prove that t2 + t8 = 2 t5 (or) Prove that tp + t p+2q = 2 t p+q.
12.It is known that a box of 550 bulbs contains 4 ℅ defective bulbs. One bulb is taken out at random from the box. Find the probability of getting a good bulb.
13.Solve ax +by = a2 +b2 bx –ay = 0.
14.The two vertices of a triangle are (6,7) and (4,-5). If the centroid of a triangle is origin , find the co-ordinates of the third vertex. (or) Find the value of p for which the points (-5,1), (1,p) and (4,2) are collinear.
15.If p and q are real and p≠ q, then show that the roots of the equation (p-q)x2 +5(p+q)x – 2(p-q) = 0 are real and unequal.
SECTION –C (10x3 =30 MARKS)
16.Draw the graph 2x-y = 6; and 2x –y +2 =0.Shade the region bounded by these lines and x-axis . Find the area of the shaded region.
17.Find the roots of the equation 2x2 -5x + 3 =0 by the method of completing the square.
18.Show that 3 -√5 is an irrational number.
19.If -4 is a root of the quadratic equation x2 +p x – 4 =0 and the quadratic equation x2 +p x + k =0 has equal roots, find the value of k.
20.Construct a circle whose radius is equal to 4cm. Let P be a point whose distance from its centre is 6cm. Construct two tangents to it from P.
cotA +cosecA -1 1 +cosA ------------------- = ------------ cotA –cosecA +1 sinA 21.Prove that
22. In the given figure , base BC of a triangle ABC is bisected at D and ∟ADB ,∟ADC are bisected by DE and DF respectively , meeting AB in E and AC in F. Show that EF║ BC
A B D C E F (or) In the figure, PQR is a right angled triangle with PQ =12cm, QR = 5cm and ∟Q =90˚. A circle with centre O and radius x is inscribed in triangle PQR. Find the value of x P
o
x
Q R |
23.Three vertices of a parallelogram ABCD are (0,0) (a,0) and (b,c). Find the co-ordinates of the fourth vertex.
24.If PA and PB are two tangents to the circle whose centre is O, then prove that the quadrilateral AOBP is cyclic.
25. ABCD is a square whose each side is 14cm. find the area of the shaded region. B A C D SECTION-D ( 5x6 = 30 MARKS)
26.State and prove Pythagoras Theorem. In triangle ABC ; ∟B =90˚ and BD ┴ AC ; AD =4cm and DC =9cm .Find BD. A
D D
B C
(or)
State and prove Basic Proportionate Theorem and hence show that the diagonals of a trapezium divide each other proportionally.
27.From a point on the ground 40m away from the foot of a tower , the angles of elevation of the top of the tower and top of the water tank,which is fixed at the top nof the tower are respectively 30˚and 45˚. Find the height of the tower and the depth of the water tank.
28.A cylindrical bucket 32cm high and 18cm of radius of the base ,is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24cm, find the radius and slant height of the heap.
29.A two digit number is such that the product of the digits is 20. If 9 is subtracted from the number , the digits interchange their places . Find the number.
30.Find the mean, median and mode of the following data.
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