Summative assessment- I class-10th guess papers

Sample Test Paper
(for Summative Assessment-1 Class - X)
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SUMMATIVE ASSESSMENT-I 2014-15
Class-X MATHEMATICS
TIME : 3 Hrs Maximum Marks : 90
Instructions:
General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A,B,C and D.
Section A comprises of 10 questions of 1 mark each. Section B comprises of 8 questions of 2 marks
each. Section C comprises of 10 questions of 3 marks each and section D comprises of 6 questions of
4 marks each.
(iii) Use of calculator is not permitted.
SECTION - A
Question numbers 1 to 10 carry 1 mark each. For each of the questions 1-10,
fouralternative choices have been provided of which only one is correct. You
have to select the correct choice.
1. If x2 + 2kx + 4 =0 has a root x=2, then the value of k is ?
(A) -1 (B) - 2 (C) 2 (D) -4
2. Which term of the A.P. 92, 88, 84, 80, ..... is 0 ?
(A) 23 (B) 32 (C) 22 (D) 24
3. If the height and length of the shadow of a man are the same, then the angle of elevation of
the sun is
(A) 30° (B) 60° (C) 45° (D) 15°
4. If sun's elevation is 60 then a pole of height 6 m will cast a shadow of length.
(A) m (B) m (C) m (D) m
5. The diameter of a circle whose area is equal to the sum of the areas of the two circles of
radii 40 cm and 9 cm is :
A) 41 cm (B) 49 cm (C) 82 cm (D) 62 cm
6. The curved surface area of a right circular cone of height 15 cm and base diameter16 cm
Class – X (MATHEMATICS) Sample Paper Page 1 of 17 Summative Assessment-I (2014-15)

is.
(A) 60 TT cm2 (B) 68 TT cm2 (C) 120 TT cm2 (D) 136 TT cm2
7. The length of the tangent drawn from a point 8 cm away from the centre of a circle, of radius
6 cm, is :
(A) cm (B) cm (C) 10 cm (D) 5 cm
8. In Fig. 1, if from an external point T, TP and TQ are two tangents to a circle with centre 0
so that POQ = 110°, then, PTQ is :
(A) 60° (B) 70° (C) 80° (D) 90°
9. In Fig, P divides AB internally in the ratio :
(A) 3 : 4 (B) 4 : 3 (C) 3 : 7 (D) 4 : 7
10. To draw a pair of tangents to a circle which are inclined to each other at an angle of 45° it
is required to draw tangents at the end points of those two radii of the circle, the angle
between which is :
(A) 70° (B) 145° (C) 135° (D) 105°
SECTION - B
Question numbers 11 to 18 carry marks each.
11. For what value of k the equation 4 x2-2 (k+1) x+(k+1)=0 has real and equal roots ?

12 . Find the common difference of an A.P. whose first term is and the 8th term is . Also
write its 4th term.
13. A tower stands vertically on the ground. From a point on the ground which is 60 m away
from the foot of the tower, the angle of elevation of the top of the tower is found to be 60° .
Find the height of the tower.
14. In Fig, arcs are drawn by taking vertices A, B and C of an equilateral triangle of side10 cm,
to intersect the sides BC, CA and AB at their respective mid-points D, E and F. Find the area
of the shaded region (Use TT=3.14)
15. Prove that the line segment joining the points of contact of two parallel tangents to a circle
is a diameter of the circle.OR In Fig. 4, 0 is the centre of a circle and BCD is tangent to it at C.
Prove that BAC + ACD = 90° .
16. Show that the point P (-4, 2) lies on the line segment joining the points A (-4, 6) andB (-4, -
6).
17. Find the point on x-axis which is equidistant from the points (-2, 5) and (2, -3).
18. The radius and slant height of a right circular cone are in the ratio of 7 : 13 and its curved
surface area is 286 cm2. Find its radius. (Use TT= 22/7)
SECTION - C
Question numbers 19 to 8 carry marks each.
19. Find the roots of the quadratic equation : a2 b2 x2 + b2 x - a2 x - 1=0

OR
Solve for x : 6 x2 + 7x - 10=0
20. How many terms of the A.P. 9, 17, 25, ..... , must be taken to get a sum of 450 ?
21. Draw a right triangle with sides of length 5 cm and 4 cm making an right angle.
Construct another triangle whose sides are times the corresponding sides of the
first triangle.
22. A cylindrical pipe has inner diameter of 4 cm and water flows through it at the rate of20 m
per minute. How long would it take to fill a conical tank, with diameter of base as 80 cm and
depth 72 cm ?
OR
A hemispherical bowl of internal diameter 36 cm. is full of liquid. This liquid is to be filled in
cylindrical bottles of radius 3 cm and height 6 cm. How many such bottles are required to
empty the bowl ?
23. Find the probability that a non leap year chosen at random has
(i) 52 Sundays (ii) 53 Sundays
24. All the face cards of spades are removed from a pack of 52 playing cards and then the
pack is shuffled well. A card is then drawn at random from the remaining pack of cards. Find
the probability of getting (i) a black face card, (ii) a queen.
OR
An urn contains 8 red, 6 white, 4 black balls. A ball is drawn at random from the urn. Find the
probability that the drawn ball is.
(i) red or white (ii) neither black nor white
25. In Fig.5, in ΔABC, D and E are the mid-points of the sides BC and AC respectively.1Find
the length of DE. Prove that DE= AB.
26. Show that the points (-4, 0), (4, 0) and (0, 3) are vertices of an isosceles triangle.
27. In Fig. 6, XY and X' Y' are two parallel tangents to a circle with centre 0 and another
tangent AB, with point of contact C intersects XY at A and X' Y' at B. Prove that ? AOB =
90° .

28. Find the area of the shaded region in Fig. 7 if BC=BD=8 cm, AC=AD=15 cm and 0 is the
centre of the circle. (Take TT=3.14)
SECTION - D
Question numbers 9 to carry marks each.
29. The product of Tanay's age (in years) five years ago and his age ten year later is
16.Determine Tanay's present age.
30. In November 2009, the number of visitors to a zoo increased daily by 20. If a total of12300
people visited the zoo in that month, find the number of visitors on 1st Nov.2009.
31. Two tangents TP and TQ are drawn to a circle with centre 0 from an external point
T. Prove that ? PTQ = 2OPQ .
32. A tent is of the shape of a right circular cylinder upto a height of 3 metres and conical
above it. The total height of the tent is 13.5 metres above the ground. Calculate the cost of
painting the inner side of the tent at the rate of Rs. 2 per square metre, if the radius of the
base is 14 metres.
OR

A solid is in the form of a right circular cylinder with hemispherical ends. The total height of
the solid is 19 cm and the diameter of the cylinder and the hemispheres is7 cm. Find the
volume and total surface area of the solid.
33. In Fig. 8, two circular flower beds have been shown on two sides of a square lawnABCD
of side 56 m. If the centre of each circular flower bed is the point of intersection0 of the
diagonals of the square lawn, find the sum of the areas of the lawn and flower beds
34. A tree is broken by the wind. The top struck the ground at an angle of 30 and at a
distance of 30 metres from its root. Find the whole height of the tree. (Use √3 =1.732)
OR
Find the height of a mountain if the elevation of its top at an unknown distance from the base
is 60° and at a distance 10 km further off from the mountain, along the same line, the angle of
elevation is 30° .

Class-X MATHEMATICS
Instructions:
(ii) The question paper consists of 34 questions divided into four sections A,B,C and D.
Section A comprises of 10 questions of 1 mark each. Section B comprises of 8 questions of 2 marks
each. Section C comprises of 10 questions of 3 marks each and section D comprises of 6 questions of
4 marks each.
SECTION - A
Question numbers 1 to 10 carry 1 mark each. For each question four choices are
provided of which only one is correct. You have to select the correct choice.
1. For what value of k will be a root of 3x2 -13x-k=0.
(A) 14 (B) (C) (D) -14
2. Which of the following is not an A.P.
(A) 13, 8, 3, -2, -7, -12 (B) 10.8, 11.2, 11.6, 12, 12.4
(C) (D)
3. If a pole of height 6 m casts a shadow m long on the ground, then the sun's elevation
is.
(A) 30° (B) 60° (C) 45° (D) 0°
4. From a point P which is at a distance of 13 cm from the centre 0 of a circle of radius 5 cm,
the pair of tangents PQ and PR to the circle are drawn. What are the lengths (in cm) of
tangents PQ and PR ?
(A) 13, 12 (B) 13, 13 (C) 12, 12 (D) 12, 18
5. To draw a pair of tangents to a circle which are inclined to each other at an angle of 75°. It
is required to draw tangents at the end points of those two radii of the circle, the angle between

them should be.
(A) 105° (B) 65° (C) 95° (D) 75°
6. The angle of elevation of the top of a tower from a point on the ground is 45° . If the
observer is 42 m away from the foot of the tower, the height of the tower is.
(A) 63 m (B) 21 m (C) 84 m (D) 42 m
7. In the figure, if the semiperimeter of .:ΔABC= 23 cm then AF+BD+CE is :
(A) 46 cm (B) 11.5 cm (C) 23 cm (D) 34.5 cm
8. In the figure, PT is a tangent to the circle with centre O. If PT=30 cm and diameter of circle
is 32 cm, then length of line segment 0P will be.
(A) 68 cm (B) 34 cm (C) 17 cm (D) 34.8 cm
9. The circumference of a circle is 100 cm. The side of a square inscribed in the circle is :
(A) cm (B) cm (C) (D)
10. The volume of the largest right circular cone that can be cut out from a cube of edge 4.2
cm is :
(A) 9.7 cm3 (B) 77.6 cm3 (C) 58.2 3 (D) 19 .4 cm3
SECTION - B
Question numbers 11 to 18 carry 2 marks each.
11. Using quadratic formula, determine the roots of following equation :

12. Find the sum of the first 23 terms of the A.P : 7, 10 2 , 14, ..........
13 . A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of
60 with the wall, find the height of the wall.
14. The minute hand of a clock is 21 cm long. Find the area swept by the minute hand on the
face of the clock from 7.00 am to 7.05 am.
15. Prove that the tangents drawn at the ends - points of a diameter of a circle are parallel.
OR
In the figure, a quadrilateral ABCD is drawn to circumscribe a circle. Prove
thatAB+CD=AD+BC.
16. Find the value of x, if the distance between the points (x, -1) and (3, -2) is x+5.
17. If the point C (-1, 2) divides internally the line segment joining A (2, 5) and B (x, y) in the
ratio 3 : 4, then find the coordinates of B.
18. The radii of the circular bases of a right circular cylinder and a cone are in the ratio of 3 : 4
and their heights are in the ratio of 2 : 3. What is the ratio of their volumes ?
SECTION - C
19. Solve the following quadratic equation :x2-3x-10=0
OR
Solve the following equation using method of factorisation:

20. Determine 'a' so that 2a+1, a2+a+1 and 3a2-3a+3 are consecutive terms of an A.P.
21. Draw a pair of tangents to a circle of radius 3.5 cm which are perpendicular to each other.
22. Show that the points (7, 3), (3, 0), (0, -4) and (4, -1) are the vertices of a rhombus.
23. Find the area of .:lABC whose vertices are A (4, 4), B (0, 0) and C (6, 2).
24. A child has a die whose six faces show the letters as given below :
A B C D E A
The die is thrown at random once. What is the probability of getting (i) A (ii) E.
25. Three coins are tossed together. Find the probability of getting at least two heads.
26. A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller
cones, each of diameter 7 cm and height 3 cm. Find the number of cones so formed.
27. A piece of wire that has been bent in the form of a semicircle including the bounding
diameter is straightened and then bent in the form of a square. The diameter of the semicircle
is 14 cm. Which has a larger area, the semi-circle or the square. Also find the difference
between them.
OR
In the figure, ABC is a triangle right angled at A. Semicircles are drawn on AB, ACand BC as
diameters. Find the area of the shaded region.
28. Prove that the parallelogram circumscribing a circle is rhombus.
OR

In the figure PO ? QO . The tangents to the circle with centre 0 at P and Q intersect at a point
T. Prove that PQ and 0T are right bisectors of each other.
SECTION - D
29. Two pillars of equal heights are on either side of a road, which is 100 m wide. The angles
of elevation of the top of the pillars are 60 and 30 at a point on the road between the pillars.
Find the position of the point between the pillars on the road and the height of the pillars.
OR
From the top of a building 60 m high the angles of depression of the top and the bottom of a
tower are observed to be 30° and 60° respectively. Find the height of the tower.
30. For what value of n, the nth terms of the A.P. 63, 65, 67, .......... and 3, 10, 17, .......... are
equal ? Also find that term.
OR
A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the
bottom to 25 cm at the top (see figure). If the top and bottom rungs are 2.5 m apart, what is
the length of the wood required for the rungs ?
31. A plane left 30 minutes later than the schedule time and in order to reach its
destination1500 km away in time, it has to increase its speed by 250 km/h from its usual
speed. Find its usual speed.

32. A building is in the form of a right circular cylinder surmounted by a hemispherical2 dome
both having the same base radii. The base diameter of the dome is equal to of the total
height of the building. Find the height of the building, if it contains m3 of air.
OR
A well whose diameter is 7 m has been dug 22.5 m deep and the earth dugout is used to form
an embankment around it. If the height of the embankment is 1.5 m. Find the width of the
embankment.
33. A shuttle cock used for playing Badminton has the shape of a frustum of a cone mounted
on a hemisphere (see figure). The diameters of the ends of the frustrum are 5 cm and2 cm,
the height of the entire shuttle cock is 7 cm. Find the external surface area.
34. Prove that the tangents to a circle from an external point are equal.

Class-X MATHEMATICS
Instructions:
(ii) The question paper consists of 34 questions divided into four sections A,B,C and D. Section A
comprises of 10 questions of 1 mark each. Section B comprises of 8 questions of 2 marks each.
Section C comprises of 10 questions of 3 marks each and section D comprises of 6 questions of 4
marks each.
SECTION - A
Questions number 1 to 10 carry 1 mark each. For each of the questions 1 to 10 four
alternative choices are given, of which only one is correct. Select the correct choice.
1. One of the roots of the quadratic equation 6x2 - x - 2=0 is :
(A) (B) (C) (D) -1
2. Which term of the A.P. 100, 90, 80, . . . . . . . is zero ?
(A) 5th (B) 6th (C) 10th (D) 11th
3. In Fig. 1, AQ, AR and BC are tangents to circle with centre 0, if AB=7 cm, BC=5 cm, AC=5
cm, then length of the tangent AQ is :

(A) 5 cm (B) 7 cm (C) 8.5 cm (D) 17 cm
4. In Fig. 2, AP and AQ are tangents to the circle with centre 0, from an external point A. If
∠PAQ = 70° then ∠APQ is equal to :
(A) 35° (B) 55° (C) 110° (D) 125°
5. The angle between two tangents drawn from an external point to a circle is 110°. The angle
subtended at the centre by the segments joining the points of contact to the centre of circle is :
(A) 70° (B) 110° (C) 90° (D) 55°
6. The angle of depression from the top of a tower 12 m high, at a point on the ground is30°.
The distance of the point from the top of the tower is :
(A) 12 m (B) 6 m (C) m (D) 24 m
7. Which one of the following cannot be the probability of an event ?
(A) 1.1 (B) 0.1 (C) 0.9 (D) 5 %
8. The outer and inner diameters of a circular ring are 34 cm and 32 cm respectively. The area
of the ring is :
(A) 66π (B) 60π (C) 33π (D) 29π
9. The volume of a sphere (in cu. cm) is equal to its surface area (in sq. cm). The diameter of
the sphere (in cm) is :
(A) 3 (B) 6 (C) 2 (D) 4

10. Given a triangle with side AB=8 cm. To get a line segment AB'= of AB, it is requiredto
divide the line segment AB in the ratio :
(A) 3 : 4 (B) 4 : 3 (C) 1 : 3 (D) 3 : 1
SECTION - B
Questions number 11 to 18 carry 2 marks each.
11. Find the values of k for which the following quadratic equation has two equal roots.2x2 + kx
+ 3=0
12. Find the sum of first twelve multiples of 7.
13. Prove that the tangents drawn at the end points of a chord of a circle make equal angles
with the chord.
14. Two coins are tossed together. Find the probability of getting at least one tail.
15. The points A (3, 2) and B (2, -3) are equidistant from a point P(x, y). Find the relation
between x and y.
16. Find the value of k for which the points A (7, -2), B (5, 1) and C (3, k) are collinear.
OR
In what ratio is the line segment joining the points P (-2, -3) and Q (3, 7) divided by the y-axis
?
17. In Fig. 3 the chord AB of a circle of radius 10 cm subtends an angle of 90° at the centre O.
Find the area of the segment ACBA. (Take π=3.14)
18. Three cubes of volume 64 cm3 each are joined end to end to form a solid. Find the surface
area of the cuboid so formed.
SECTION - C
Question number 19 to 8 carry 3 marks each.

19. Solve for x : a(a2 + b2)x2 + b2x - a=0
OR
A two digit number is such that the product of the digits is 18. When 63 is subtracted from the
number, the digits interchange their places. Find the numbers.
20. Solve for x :
21. Construct an isosceles triangle ABC with Base BC=6cm, AB=AC and4 ∠A =90°. Draw
another similar triangle whose sides are of the sides of ΔABC.
22. Prove that the parallelogram circumscribing a circle is a rhombus.
23. An electric pole is 10 m high. If its shadow is m in length. Find the angle of elevation
of the sun at that time.
OR
A circus artist is climbing a rope 12 m long which is tightly stretched and tied from the top of a
vertical pole to the ground. Find the height of the pole if the angle made by the rope with the
ground is 30°.
24. Cards numbered from 1 to 64 are placed in a box. A card is drawn at random from the box.
Find the probability that the card number on the card drawn is a perfect cube.
25. Find the area of the rhombus ABCD whose vertices are A(3, 0), B (4, 5), C (-1, 4) andD (-
2, -1).
26. Find the area of the triangle formed by the mid points of the sides of the triangle ABC
whose vertices are A (0, -1), B (2, 1) and C (0, 3).
27. In Fig. 4, a circle of radius 7 cm is inscribed in a square. Find the area of the shaded
region.

28. A toy is in the form of a cone of radius 3.5 cm surmounted on a hemisphere of same
radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
OR
How many solid spheres of diameter 6 cm are required to be melted to form a solid metal
cylinder of height 45 cm and diameter 4 cm ?
SECTION - D
Questions 9 to carry 4 marks each.
29. Solve the following equation for x.
9x2 - 9(a+b)x + (2a2 + 5ab + 2b2)=0
OR
The sum of the squares of two consecutive odd natural numbers is 130. Find the numbers.
30. Find the sum of the first 31 terms of an A.P. whose nth term is given by 3 + n
OR
The sum of the third and seventh term of an A.P. is 6 and their product is 8. Find the sum of
the first sixteen terms of the A.P.
31. Prove that tangent at any point of a circle is perpendicular to the radius through the point of
contact.
32. A solid sphere of radius 3 cm is melted and drawn into a long wire of uniform circular
cross-section. If the length of the wire is 36 m, find its radius.
33.An aeroplane at an altitude of 200 m observes the angles of depression of two opposite
points on two banks of the river to be 45° and 60°. Find, in metres, the width of the river. (use
=1.732)
34. A bucket is 18 cm in diameter at the top and 6 cm in diameter at the bottom. If it is 8 cm
high, find its capacity. Also find the area of sheet used in making the bucket.

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