# Board Paper of Class 10 2013 Maths - Solutions

**(PART – A)**

**General Instructions:**

1. There are

**50**objective type questions in this part and all are compulsory.

2. The questions are serially numbered from

**1**to

**50**and each carries

**1**mark.

3. You are supplied with separate OMR sheet with the alternative (A) ◯, (B) ◯, (C) ◯, (D) ◯ against each question number. For each question, select the correct alternative and darken the circle ◯ as ● completely with the pen against the alphabet corresponding to that alternative in the given OMR sheet

From the following

**1**to

**50**questions, select the correct alternative from those given and darken the circle with pen against the alphabet, against number in OMR sheet.

• Each question carries 1 mark.

**(PART – B)**

**General Instructions:**

1. There are four sections in this part of the question paper and total

**1**to

**17**question are there.

2. All the questions are compulsory. Internal options are given.

3. Draw figures wherever required. Retain all the lines of construction.

4. The numbers at right side represent the marks of the question..

**SECTION-A**

**51**to

**58**with calculations.(Each question carries

**2**marks).

**SECTION-B**

Answer the following questions from

**59**to

**62**with calculations. Each question is of

**3**marks.

**SECTION-C**

Answer the following questions from No.

**63**to

**65**, as directed with the calculations. Each question is of

**4**marks.

**SECTION-D**

Answer the following questions from No.

**66**to

**67**. Each question carries

**5**marks.

- Question 1
If the H.C.F. of (x, y) = 1, then the H.C.F. of (x – y; x + y) =

(A) 1 or 2

(B) x or y

(C) x + y or x – y

(D) 4 VIEW SOLUTION

- Question 2
In Euclid’s division Lemma for positive integers a and b, the unique integers q and r are obtained such that a = bq + r is

(A) 0 < r < b

(B) 0 ≤ r ≤ b

(C) 0 < r ≤ b

(D) 0 ≤ r < b VIEW SOLUTION

- Question 3

- Question 4

- Question 5
The product of the zeroes of the cubic polynomial p(x) is _____

(A) $\frac{\mathrm{Coefficient}\mathrm{of}-{x}^{2}}{\mathrm{Coefficient}\mathrm{of}\mathit{}{x}^{3}}\phantom{\rule{0ex}{0ex}}$

(B) $\frac{\mathrm{Coefficient}\mathrm{of}x}{\mathrm{Coefficient}\mathrm{of}{x}^{3}}$

(C) $\frac{-\mathrm{The}\mathrm{constant}\mathrm{Term}}{\mathrm{Coefficient}\mathrm{of}{x}^{2}}$

(D) None of these. VIEW SOLUTION

- Question 6
The diagram below shows two sticks – one BLACK and the other WHITE. Based on the measurements shown, what is the length of the white stick?

(A) 5 cm

(B) 8.5 cm

(C) 13.5 cm

(D) 17 cm VIEW SOLUTION

- Question 7
*x*= ___________ is identified as a GOLDEN NUMBER

(A) $\frac{1+\sqrt{5}}{2}$

(B) 0

(C) $\frac{{\displaystyle 1+\sqrt{2}}}{{\displaystyle 2}}$

(D) 1 VIEW SOLUTION

- Question 8
The Discriminant value of equation 5x
^{2}– 6x + 1= 0 is………

(A) 16

(B) $\sqrt{56}$

(C) 4

(D) 56 VIEW SOLUTION

- Question 9
If ___________, then the quadratic equation does not have real solution.

(A) D = 0

(B) D > 0

(C) D < 0

(D) D ≥ 0 VIEW SOLUTION

- Question 10
Given below is a graph showing two lines.

Which of the following statements is true about the solution(s) of the pair of equations represented by these lines?

(A) They have a unique solution.

(B) They do not have any solution.

(C) They have infinite solutions.

(D) We cannot predict the number of solutions without knowing the algebraic form of these equations. VIEW SOLUTION

- Question 11
2 years ago, the addition of ages of father – mother and their two daughters was 40 years. After 3 years the addition of their ages will be………

(A) 40

(B) 46

(C) 50

(D) 60 VIEW SOLUTION

- Question 12
In a two digit number, the digit at the ten’s place is 4 and the product of the two two digits is 4 times greater than the value of the digit in the ten’s place. What is the two digit number?

(A) 42

(B) 48

(C) 44

(D) 84 VIEW SOLUTION

- Question 13
As per the given figure, the graph of y = p(x) has _________ real zeros.

(A) 0

(B) 1

(C) 2

(D) 3 VIEW SOLUTION

- Question 14
If 2k + 1, 13, and 5k – 3 are three consecutive terms in an A.P., then k =

(A) 17

(B) 13

(C) 4

(D) 9 VIEW SOLUTION

- Question 15
The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,………., is said to be in _______________.

(A) Arithmetic Progression.

(B) Finite Sequence.

(C) Fibonacci Sequence.

(D) None of the above. VIEW SOLUTION

- Question 16

- Question 17
In ΔABC, the bisector of ∠A intersects $\overline{)\mathrm{BC}}$ at a point D. Then

(A) BD × AC = BC × AB

(B) BD × AB = DC × AC

(C) AC × AB = DC × BC

(D) BD × AC = DC × AB VIEW SOLUTION

- Question 18
In ΔABC, the measures of $\overline{)\mathrm{BC}},\overline{)\mathrm{CA}}\mathrm{and}\overline{)\mathrm{AB}}$ are in the ratio 3 : 4 : 5. Correspondence ABC ↔ PQR is congruence. If PR = 12, then the perimeter of ΔPQR is ____________

(A) 12

(B) 24

(C) 27

(D) 36 VIEW SOLUTION

- Question 19
Out of following triplets _________ is not a Pythagorean triplet.

(A) 7, 24, 25

(B) 20, 21, 29

(C) 11, 60, 61

(D) 13, 35, 37 VIEW SOLUTION

- Question 20
In ΔABC, $\overline{)\mathrm{AD}}$ is a median, hence according to the Apollonius theorem, _____________ is true.

(A) AB^{2 }+ AC^{2 }= 2 (AD^{2}+ BC^{2})

(B) AB^{2}+ AC^{2}= 2 (BD^{2}+ DC^{2})

(C) AB^{2}+ AC^{2 }= 2 (AD^{2}+ DC^{2})

(D) AB^{2}+ AC^{2}= 2 (BD^{2 }+ BC^{2}) VIEW SOLUTION

- Question 21
In Mathematics Exam, the probability of Aayushi scoring 100 out of 100 is

(A) 1

(B) 0

(C) $\frac{1}{100}$

(D) $\frac{1}{101}$ VIEW SOLUTION

- Question 22
The probability in an event k is

(A) 0 ≥ P (k) ≥ 1

(B) 0 ≤ P (k) ≤ 1

(C) 0 > P (k) > 1

(D) 0 < P (k) < 1 VIEW SOLUTION

- Question 23
If a dice is tossed once, then the probability of getting a prime number is

(A) $\frac{1}{3}$

(B) $\frac{1}{6}$

(C) $\frac{1}{2}$

(D) 1 VIEW SOLUTION

- Question 24

- Question 25
Rachna had an average score of 45 from 6 tests. Her teacher dropped her lowest score, which is 30 and calculated the average of the remaining scores to decide her grade. Which of the following gives her new average score?

(A) $\frac{\left(45\times 5-30\right)}{5}$

(B) $\frac{\left(45\times 5-30\right)}{6}$

(C) $\frac{\left(45\times 6-30\right)}{5}$

(D) $\frac{\left(45\times 6-30\right)}{6}$ VIEW SOLUTION

- Question 26
In a Math test taken by 35 students, the average score of 15 girls is 10 and that of 20 boys is also 10. Which of the following can be calculated based on the above data?

(A) The highest score in the class.

(B) The lowest score among the boys in the class.

(C) The sum of the scores of the 35 students of the whole class.

(D) All of the above. VIEW SOLUTION

- Question 27

- Question 28
Which of the following pair is a correct trigonometric inter-relationship?

(1) cos θ (a) $\frac{\mathrm{cos}\mathrm{\theta}}{\mathrm{sin}\mathrm{\theta}}$ (2) tan θ (b) $\frac{1}{\mathrm{cosec}\mathrm{\theta}}$ (3) cot θ (c) $\frac{1}{\mathrm{sec}\mathrm{\theta}}$ (4) sin θ (d) $\frac{1}{\mathrm{cot}\mathrm{\theta}}$ (e) sin θ . cos θ

(A) 1-d, 2-e, 3-b, 4-a

(B) 1-b, 2-a, 3-e, 4-d

(C) 1-c, 2-d, 3-a, 4-b

(D) 1-e, 2-b, 3-c, 4-d VIEW SOLUTION

- Question 29

- Question 30
When observed from top of a tower, the angle of depression of two houses A and B in the eastern and western directions is 30° and 60° respectively, then

(A) House A is nearer to the tower than house B.

(B) House B is nearer to the tower than house A.

(C) House A and B are equidistant from the tower.

(D) None of the above. VIEW SOLUTION

- Question 31
On walking x meters, making an angle of 30° with the ground, to find a ball fallen in a valley, one can reach a depth of ‘y’ meters below the ground, then

(A) x = y

(B) x = 2y

(C) $2\mathrm{x}=\sqrt{3}\mathrm{y}$

(D) 2x = y VIEW SOLUTION

- Question 32
If the length of a minor arc $\stackrel{\u23dc}{\mathrm{AB}}$ of a circle is $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ of its circumference, then the angle subtended by the minor arc $\stackrel{\u23dc}{\mathrm{AB}}$ at the center will be

(A) 30^{∘}

(B) 45^{∘}

(C) 90^{∘}

(D) 60^{∘}VIEW SOLUTION

- Question 33
The length of the minute hand of a clock is 14 cm. If minute hand moves from 1 to 10 on the dial, then an area of ______ cm
^{2}will be covered.

(A) 462

(B) 154

(C) 308

(D) 616 VIEW SOLUTION

- Question 34
If the radius of a circle is increased by 10%, then the corresponding area of the new circle will be _________. (π = 3.14)

(A) 121 πr^{2}

(B) 12.1 πr^{2}

(C) 1.21 π^{2}

(D) None of the given three. VIEW SOLUTION

- Question 35
The maximum area of a triangle inscribed in a semi-circle having radius 10 cm is _________ cm
^{2}.

(A) 10

(B) 50

(C) 100

(D) 200 VIEW SOLUTION

- Question 36
If the area of a circle is 38.5 m
^{2}, then its circumference will be

(A) 22

(B) 2.2

(C) 38.5

(D) 3.85 VIEW SOLUTION

- Question 37
$\square $ABCD is a Rhombus. If it is inscribed in $\odot $(O, r), then $\square $ABCD is a ___________.

(A) Square

(B) Rectangle

(C) Trapezium

(D) None of the above VIEW SOLUTION

- Question 38
In ΔABC, m∠B = 90°, AB = 4 and BC = 3, then the radius of circle touching all three sides of triangle will be

(A) 2.5

(B) 2

(C) 3.5

(D) 3 VIEW SOLUTION

- Question 39
One circle touches all sides of ⧠ABCD. If AB = 5, BC = 8, CD = 6; then AD =

(A) 3

(B) 7

(C) 4

(D) 9 VIEW SOLUTION

- Question 40
Point P is on the outer side of $\odot $(O, 15). The tangent drawn from point P touches the circle at T. If PT = 8, then OP =

(A) 7

(B) 13

(C) 17

(D) 23 VIEW SOLUTION

- Question 41
The cub (cubical) volume of a hemisphere, having diameter 1 cm, will be ______ cm
^{3}.

(A) $\frac{\mathrm{\pi}}{6}$

(B) $\frac{\mathrm{\pi}}{12}$

(C) $\frac{2\mathrm{\pi}}{3}$

(D) $\frac{4\mathrm{\pi}}{3}$ VIEW SOLUTION

- Question 42
If the frustum of a cone has a height of 6 cm and radius of 5 cm and 9 cm respectively; then its cubical volume will be ___________ cm
^{3}.

(A) 320π

(B) 151π

(C) 302π

(D) 98π VIEW SOLUTION

- Question 43
The formula for finding the total surface area of a Cylinder having cone shaped lid at both the ends, will be

(A) πr (l + 2r)

(B) πr (2h + r)

(C) 2πr (h + l)

(D) 2πr (h + 2r) VIEW SOLUTION

- Question 44

- Question 45
The two triangles in the figure are congruent by the congruence theorem. Here, it is given OQ = OR. Which of the following condition, along with the given condition, is sufficient to prove that the two triangles are congruent to each other?

(A) ∠P = ∠S

(B) ∠Q = ∠R

(C) OP = OS

(D) PQ = SR VIEW SOLUTION

- Question 46
In ΔPQR, $\frac{\mathrm{PQ}}{1}=\frac{\mathrm{PR}}{2}=\frac{\mathrm{QR}}{\sqrt{3}}$, then m∠R =

(A) 90°

(B) 60°

(C) 45°

(D) 30° VIEW SOLUTION

- Question 47
From P(−3, 2) the feet of a perpendicular drawn on the Y-axis is M. The co-ordinates of M are ______.

(A) (3, 0)

(B) (0, 2)

(C) $\left(\frac{3}{2},-1\right)$

(D) (−3, 2) VIEW SOLUTION

- Question 48
The distance of P(a, b) from the origin is

(A) a^{2}+ b^{2}

(B) |a – b|

(C) |a + b|

(D) $\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}$ VIEW SOLUTION

- Question 49
Which of the following group is true for ABCD?

(A) ABCD is Rhombus (a) $\overline{)\mathrm{AC}}\mathrm{and}\overline{)\mathrm{BD}}$ bisect each other. (B) ABCD is Parallelogram (b) $\overline{\mathrm{AC}}\mathrm{and}\overline{)\mathrm{BD}}$ bisect at each other at an right angle. (C) ABCD is Rectangle (c) $\overline{\mathrm{AC}}\mathrm{and}\overline{)\mathrm{BD}}$ are congruent and bisect each other at an right angle. (D) ABCD is Square (d) $\overline{\mathrm{AC}}\mathrm{and}\overline{)\mathrm{BD}}$ are congruent and bisect each other.

(A) 1-d, 2-a, 3-d, 4-c

(B) 1-c, 2-d, 3-a, 4-b

(C) 1-b, 2-a, 3-d, 4-c

(D) 1-b, 2-c, 3-d, 4-a VIEW SOLUTION

- Question 50
When point A(
*x*_{1},*y*_{1}) and point B(*x*_{2},*y*_{2}) are joined to form $\overline{)\mathrm{AB}},\mathrm{and}\overline{)\mathrm{AB}}$ is divided in the proportion of λ : 1; the co-ordinates of the point will be

(A) $\frac{\lambda {x}_{2}+{x}_{1}}{\lambda +1},\frac{\lambda {y}_{2}+{y}_{1}}{\lambda +1}$

(B) $\frac{{\displaystyle \lambda {x}_{2}+{x}_{1}}}{{\displaystyle \lambda -1}},\frac{{\displaystyle \lambda {v}_{2}+{v}_{1}}}{{\displaystyle \lambda -1}}$

(C) $\frac{{\displaystyle \lambda {x}_{1}+{x}_{2}}}{{\displaystyle \lambda +1}},\frac{{\displaystyle \lambda {v}_{2}+{v}_{2}}}{{\displaystyle \lambda +1}}$

(D) $\frac{{\displaystyle \lambda {x}_{2}+{x}_{1}}}{{\displaystyle \lambda -1}},\frac{{\displaystyle \lambda {y}_{2}+{y}_{1}}}{{\displaystyle \lambda -1}}$ VIEW SOLUTION

- Question 51

- Question 52
Find the quadratic equation, whose addition of zeroes is $-\frac{7}{3}$ and multiplication is $\frac{4}{3}$. VIEW SOLUTION

- Question 53
In an Arithmetic Progression T
_{7}= 18 and T_{18}= 7, obtain T_{25}.ORIn an arithmetic progression 2, 7, 12, 17……, summation of how many terms will be 990? VIEW SOLUTION

- Question 54
In ΔPQR, m∠Q = 90° and $\overline{)\mathrm{QM}}$ is an altitude; M∈ $\overline{)\mathrm{PR}}$. If QM = 12, PR = 26; then find PM and RM; If PM < RM; then find PQ and QR. VIEW SOLUTION

- Question 55
Two concentric circles having radii 73 and 55 are given. The chord of circle having a greater radius touches the smaller circle. Find the length of this chord. VIEW SOLUTION

- Question 56
Find the area of triangle ΔABC having vertices A(4, 2), B(3, 9) and C(10, 10).ORFind the co-ordinates of the points which divide the line segment joining A(−7, 5) and B(5, −1) into three congruent segments (Such points are called the points of trisection of segment). VIEW SOLUTION

- Question 57
In Hostel, one day reading hours of 20 students was observed, whose result is mentioned in the table below. Form the table, find the
**Mode**.No. of reading hours 1 – 3 3 – 5 5 – 7 7 – 9 9 – 11 Student’s strength in the hostel 7 2 8 2 1

- Question 58
A card is selected at random from a well-shuffled pack of 52 cards. Find the probability that the selected card is

(1) a black coloured queen.

(2) not a king VIEW SOLUTION

- Question 59
Prove (sinθ + cosecθ)
^{2}+ (cosθ + secθ)^{2}= 7 + tan^{2}θ + cot^{2}θ.ORFind the value of $\frac{\mathrm{cosec}38}{\mathrm{sec}52}+\frac{2}{\sqrt{3}}\mathrm{tan}38.\mathrm{tan}60.\mathrm{tan}52-3\left({\mathrm{sin}}^{2}32+{\mathrm{sin}}^{2}58\right)$ VIEW SOLUTION

- Question 60
The chord of a circle 84 cm in diameter subtends an angle of 60° at the centre of the circle. Find the area of the minor segment corresponding to the chord. (Take $\sqrt{3}$ = 1.73) VIEW SOLUTION

- Question 61
Find the solution for the following equations:

$\frac{5}{2x}+\frac{2}{3y}=7;\frac{3}{x}+\frac{2}{y}=12\left(x\ne 0;y\ne 0\right)$ VIEW SOLUTION

- Question 62
Find the median of the following frequency distribution
Class 4 – 8 8 – 12 12 – 16 16 – 20 20 – 24 24 – 28 Frequency 9 16 12 7 15 1

- Question 63
On a Hemisphere, frustum of a cone shaped shuttle-cock is used for playing Badminton. The outer radius of the frustum of cone is 5 cm and its inner radius is 2 cm. The height of the entire shuttle-cock is 7 cm. Find the outer surface area of the shuttle-cock. VIEW SOLUTION

- Question 64
A jet plane is at a vertical height of h. The angles of depression of two tanks on the horizontal ground are found to have measures α and β, α > β. Prove that the distance between the tanks is $\frac{\mathrm{h}\left(\mathrm{tan}\mathrm{\alpha}-\mathrm{tan}\mathrm{\beta}\right)}{\mathrm{tan}\mathrm{\alpha}.\mathrm{tan}\mathrm{\beta}}$ assuming both the tanks are on the same side of the jet plane. VIEW SOLUTION

- Question 65
The petrol rate is increased by Rs. 5/- per litre. Now for Rs.1320/-, 2 litres less petrol can be bought as compared to the previous rate. Find the increased price of petrol per litre.ORKailash’s age at present is 2 years less than 6 times the age of his daughter Prerna. The product of their ages 5 years later will be 330. What was the age of Kailash when his daughter Prerna was born? VIEW SOLUTION

- Question 66
Draw $\overline{)\mathrm{PQ}}$, where PQ = 10 cm. Draw circle $\odot $ (P, 4) and $\odot $(Q, 3). Draw tangents to each circle from the centre of the other circle. Write points of construction.ORDraw ΔABC, where m∠ABC = 90°; BC = 4 cm and AC = 5 cm and then construct ΔBXY with $\frac{4}{3}$ scale factor. Write points of construction. VIEW SOLUTION

- Question 67
Write the converse of the Pythagoras Theorem and prove it. VIEW SOLUTION