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# CBSE 10th Board mathematics exam paper 2013 solutions

##### These are the solutions to the mathematics questions which CBSE Xth Board examinees had to solve in 2013.

These are the solutions to the questions in the mathematics examination paper which CBSE Xth Board examinees had to solve in 2013. Students are advised to try to solve the questions themselves before consulting the solutions. The solutions have been answered by the http://www.topperlearning.com/ faculty.

Paper:X Math – CBSE Past Year Paper – 2013 – Set 3
Total marks of the paper:90
Total time of the paper:3 hrs
1. All questions are compulsory.

2. The question paper consists of 34 questions divided into four sections A, B, C and D.

3. Section A contains 8 questions of 1 mark each, which are multiple choice type question, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 10 questions of 4 marks each.

4. Use of calculators is not permitted.

Questions:
1] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">The
angle of depression of a car, standing on the ground, from the top of a 75 m
high tower, is 30o. The distance of the car from the base of the
tower (in m.) is:

[Marks:1]
 A. B. C. D.
2] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">The
probability of getting an even number, when a die is thrown once, is:

[Marks:1]
 A. B. C. D.
3] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">A
box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random
from the box, the probability that it bears a prime-number less than 23, is:

[Marks:1]
 A. B. C. D.
4] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">In
fig., a circle with centre O is inscribed in a quadrilateral ABCD such that,
it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively,
If AB = 29 cm, AD = 23 cm, <span style="position:relative;top:2.0pt;
mso-text-raise:-2.0pt">
B = 90o and DS = 5 cm, then the radius of
the circle (in cm) is:

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:
0in;margin-bottom:10.0pt;margin-left:0in;text-align:center" align="center"><span style="font-family:"Verdana","sans-serif";color:black;mso-themecolor:text1;
mso-no-proof:yes">

[Marks:1]
 A. 18 B. 6 C. 15 D. 11
5] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">In
fig., PA and PB are two tangents drawn from an external point P to a circle
with centre C and radius 4 cm. If PA <span style="position:relative;
top:2.0pt;mso-text-raise:-2.0pt">
PB, then the
length of each tangent is:

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:
0in;margin-bottom:10.0pt;margin-left:0in;text-align:center" align="center"><span style="font-family:"Verdana","sans-serif";color:black;mso-themecolor:text1;
mso-no-proof:yes">

[Marks:1]
 A. 3 cm B. 5 cm C. 6 cm D. 4 cm
6]

In fig., the area of triangle ABC (in sq. units) is:

[Marks:1]
 A. 2.5 B. 10 C. 15 D. 7.5
7] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">If
the difference between the circumference and the radius of a circle is 37 cm,
then using , the circumference (in cm) of the circle is:

[Marks:1]
 A. 7 B. 14 C. 154 D. 44
8] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">The
common difference of AP … is:

[Marks:1]
 A. q B. -q C. 2 D. -2
9] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Prove
that the parallelogram circumscribing a circle is a rhombus.

[Marks:2]
10] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Two
circular pieces of equal radii and maximum area, touching each other are cut
out from a rectangular card board of dimensions 14 cm <span style="position:
relative;top:2.0pt;mso-text-raise:-2.0pt">
7 cm. Find
the area of the remaining card board.

[Marks:2]
11] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">In
fig., a circle is inscribed in triangle ABC touches its sides AB, BC and AC
at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm,
then find the length of AD, BE and CF.

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:
0in;margin-bottom:10.0pt;margin-left:0in;text-align:center" align="center"><span style="font-family:"Verdana","sans-serif";color:black;mso-themecolor:text1;
mso-no-proof:yes">

[Marks:2]
12]

How many three-digit natural numbers are divisible by 7?

[Marks:2]
13] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Solve
the following quadratic equation for x:

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in"><span style="font-family:"Verdana","sans-serif";
color:black;mso-themecolor:text1">

[Marks:2]
14] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">A
card is drawn at random from a well shuffled pack of 52 playing cards. Find
the probability that the drawn card is neither a king nor a queen.

[Marks:2]
15] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">A
vessel is in the form of hemispherical bowl surmounted by a hollow cylinder
of same diameter. The diameter of the hemispherical bowl is 14 cm and the
total height of the vessel is 13 cm. Find the total surface area of the
vessel.

[Marks:3]
16] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">A
wooden toy was made by scooping out a hemisphere of same radius from each end
of a solid cylinder. If the height of the cylinder is 10 cm, and its base is
of radius 3.5 cm, find the volume of wood in the toy.

[Marks:3]
17] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">In
a circle of radius 21 cm, an arc subtends an angle of 60o at the
centre. Find: (i) the length of the arc (ii) area of the sector formed by the
arc.

[Marks:3]
18] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">In
Fig., AB and CD are two diameters of a circle with centre O, which are
perpendicular to each other. OB is the diameter of the smaller circle. If OA
= 7 cm, find the area of the shaded region.

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:
0in;margin-bottom:10.0pt;margin-left:0in;text-align:center" align="center">

[Marks:3]
19] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Find
the ratio in which the y-axis divides the line segment joining the points
(-4, -6) and (10, 12). Also, find the coordinates of the point of division.

[Marks:3]
20] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">The
horizontal distance between two poles is 15 m. The angle of depression of the
top of first pole as seen from the top of second pole is 30o. If
the height of the second pole is 24 m, find the height of the first pole.

[Marks:3]
21] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">For
what values of k, the roots of the quadratic equation (k + 4) x2 +
(k + 1)x + 1 = 0 are equal?

[Marks:3]
22] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">The
sum of first n terms of an AP is 3n2 + 4n. Find the 25th
term of this AP.

[Marks:3]
23]

Construct a tangent of a circle of radius 4 cm from a point on the concentric circle of radius 6 cm.

[Marks:3]
24] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Show
that the points (-2, 3), (8, 3) and (6, 7) are the vertices of a right triangle.

[Marks:3]
25] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Water
is flowing through a cylindrical pipe, of internal diameter 2 cm, into a
cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the
rise in level of water in the tank in half an hour.<span style="color:black;
mso-themecolor:text1">

[Marks:4]
26] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">A
Group consists of 12 persons, of which 3 are extremely patient, other 6 are
extremely honest and rest are extremely kind. A person from the group is
selected at random. Assuming that each person is equally likely to be
selected, find the probability of selecting a person who is (i) extremely
patient (ii) extremely kind or honest. Which of the above values you prefer
more?

[Marks:4]
27] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">A
bucket open at the top, and made up of a metal sheet is in the form of a
frustum of a cone. The depth of the bucket is 24 cm and the diameters of its
upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost
of metal sheet used in it at the rate of Rs 10 per 100 cm2.

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:
0in;margin-bottom:10.0pt;margin-left:0in;text-align:center" align="center"><span style="font-family:"Verdana","sans-serif";color:black;mso-themecolor:text1;
mso-no-proof:yes">

[Marks:4]
28] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">In
fig., l and m are two parallel tangents to a circle with centre O, touching
the circle at A and B respectively. Another tangent at C intersects the line
l at D and m at E. Prove that <span style="position:relative;top:3.0pt;
mso-text-raise:-3.0pt">

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:
0in;margin-bottom:10.0pt;margin-left:0in;text-align:center" align="center">

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in"><span style="font-family:"Verdana","sans-serif";
color:black;mso-themecolor:text1">

[Marks:4]
29]

Sum of the areas of two squares is 400 cm2. If the difference of their perimeters is 16 cm, find the sides of the two squares.

[Marks:4]
30] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Solve
that following for x:

<p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in"><span style="font-family:"Verdana","sans-serif";
color:black;mso-themecolor:text1">

[Marks:4]
31]

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

[Marks:4]
32] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">Find
the number of terms of the AP -12, -9, -6,… 12. If 1 is added to each term
of this AP, then find the sum of all terms of the AP thus obtained.

[Marks:4]
33]

Two poles of equal heights are standing opposite each other on either side of the roads, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60o and 30o respectively. Find the height of the poles and the distances of the point from the poles.

[Marks:4]
34] <p class="MsoListParagraph" style="margin-top:10.0pt;margin-right:0in;
margin-bottom:10.0pt;margin-left:0in">If
the area of triangle ABC formed by A(x,y), B(1,2) and C(2,1) is 6 square
units, then prove that x + y = 15.<span style="color:black;mso-themecolor:
text1">

[Marks:4]
First Published: March 6, 2013, 6:23 PM IST
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