# Board Paper of Class 10 2016 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
In ΔABC, D, E and F are midpoints of $\overline{)\mathrm{AB}},\overline{)\mathrm{BC}}\mathrm{and}\overline{)\mathrm{AC}}$ respectively. If ABC = 40 then DEF = ----- .

(A) 10

(B) $\frac{40}{3}$

(C) 20

(D) 5 VIEW SOLUTION

- Question 2
From the given figure AD = ----

(A) 2

(B) $2\sqrt{3}$

(C) $\frac{\sqrt{3}}{2}$

(D) $\sqrt{3}$ VIEW SOLUTION

- Question 3
In ΔABC, if m∠B = 90, & AC = 10 then length of median $\overline{\mathrm{BM}}$ = ---------.

(A) 6

(B) $5\sqrt{2}$

(C) 5

(D) 8 VIEW SOLUTION

- Question 4

- Question 5
From the graph OA
^{2}= --------.

(A) 0

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

(C) $\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}$

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

- Question 6
(1, 0), (0, 1), (1, 1) are the co-ordinates of vertices of a triangle. The triangle is ------ triangle.

(A) Isosceles

(B) Obtuse angled

(C) Acute angled

(D) Equilateral VIEW SOLUTION

- Question 7
In given figure, co-ordinates of foot of perpendicular P are -----------.

(A) (5, 0)

(B) (–5, 1)

(C) (–1, 0)

(D) (0, –1) VIEW SOLUTION

- Question 8
If O (0, 0) and P (–8, 0) then co-ordinates of its midpoint at ----------.

(A) (–4, 0)

(B) (4,0)

(C) (0, –4)

(D) (0, 0) VIEW SOLUTION

- Question 9

- Question 10

- Question 11
From given figure tan A$\xb7$tan C = ____________.

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

(B) 2

(C) $\sqrt{2}$

(D) 1 VIEW SOLUTION

- Question 12
If cos
^{2}45° – cos^{2}30° = x. cos 45°sin 45° then x = ________.

(A) –1/2

(B) 1/2

(C) 2

(D) 3/4 VIEW SOLUTION

- Question 13
In given figure, the minimum distance to reach from point “C” to point “A” will be _________.

(A) a^{2}

(B) $\sqrt{2}$

(C) 2

(D) 2a VIEW SOLUTION

- Question 14
To find the length of a ladder making an angle θ with wall ________ trigonometric ratio is used.

(A) tan θ

(B) cot θ

(C) cosec θ

(D) none VIEW SOLUTION

- Question 15
If the ratio of the height of tower and the length of its shadow is 1: $\sqrt{3}$, then the angle of elevation of the sun has measure __________.

(A) 60°

(B) 45°

(C) 30°

(D) 75° VIEW SOLUTION

- Question 16
___________ is true for a tangent of a circle.

P: line intersect circle in one & only one point

Q: line and circle must be in same plane.

R: line passes from the centre of a circle.

(A) Q and R

(B) Only Q

(C) Only P

(D) P and Q VIEW SOLUTION

- Question 17
In an isosceles right angled triangle ΔABC, r = _______.

(A) 2

(B) 1

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

(D) $2-\sqrt{2}$ VIEW SOLUTION

- Question 18
In the figure, from one handkerchief from four corners, sectors P, Q, R, S each of 2 cm are cut, whose sum is P + Q + R + S = A
_{1}and a circle of diameter 4 cm from the centre is cut whose area is A_{2 }, then __________ is possible.

(A) A_{1}≠ A_{2}

(B) A_{1}< A_{2}

(C) A_{1}> A_{2}

(D) A_{1}= A_{2}VIEW SOLUTION

- Question 19

- Question 20
If radius of circle is decreased by 10% then there is __________ decrease in its area.

(A) 10%

(B) 21%

(C) 19%

(D) 20% VIEW SOLUTION

- Question 21
Formula to find area of sector is ____________.

(A) $\frac{1}{2}\mathrm{rl}$

(B) $\frac{\mathrm{\pi r\theta}}{360}$

(C) $\frac{{\mathrm{\pi r}}^{2}\mathrm{\theta}}{180}$

(D) πr^{2}VIEW SOLUTION

- Question 22
If the circumference of base of a hemisphere is 2π then it volume is ______________ cm
^{3}.

(A) $\frac{2\mathrm{\pi}}{3}{\mathrm{r}}^{3}$

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

(C) $\frac{8\mathrm{\pi}}{3}$

(D) $\frac{\mathrm{\pi}}{12}$ VIEW SOLUTION

- Question 23
In the given cylinder the stick of maximum length __________ can be kept inside it.

(A) 10

(B) 12

(C) 13

(D) 17 VIEW SOLUTION

- Question 24
Total area of the given closed figure will be ____________ square units.

(A) 31

(B) 45

(C) 25

(D) 40 VIEW SOLUTION

- Question 25
The ratio of the radii of two cones having equal height is 2:3 then ratio of their volume is ____________.

(A) 3:2

(B) 9:4

(C) 8:27

(D) 4:9 VIEW SOLUTION

- Question 26
The conjugate surd of $2-\sqrt{3}$ is __________.

(A) $-2+\sqrt{3}$

(B) $2-\left(-\sqrt{3}\right)$

(C) $3+\sqrt{2}$

(D) $\frac{1}{2-\sqrt{3}}$ VIEW SOLUTION

- Question 27
To get the terminating decimal expansion of a rational number $\frac{\mathrm{p}}{\mathrm{q}}$. If
*q*= 25^{m}then m and n must belong to __________.^{n}

(A) Z

(B) N U {0}

(C) N

(D) R VIEW SOLUTION

- Question 28
For polynomial p(x) = x
^{2}– 4x + 3, α + β = _______.

(A) Positive fraction

(B) Negative integer

(C) Positive integer

(D) Zero VIEW SOLUTION

- Question 29
The graph of polynomial P(x) = ax – b, where a ≠ 0; a, b ∈ R intersect

(A) $\left(\frac{\mathrm{b}}{\mathrm{a}},0\right)$

(B) $\left(0,\frac{\mathrm{b}}{\mathrm{a}}\right)$

(C) $\left(-\frac{\mathrm{b}}{\mathrm{a}},0\right)$

(D) $\left(-\frac{\mathrm{a}}{\mathrm{b}},0\right)$ VIEW SOLUTION

- Question 30
The graph of p(x) = 5x + 3, x ∈ R is __________.

(A) ray

(B) parabola

(C) line segment

(D) line VIEW SOLUTION

- Question 31
For the zeroes α & β of polynomial P(x) = ax
^{2}+ bx + c, $\frac{1}{\mathrm{\alpha}}+\frac{1}{\mathrm{\beta}}$ = ____________.

(A) $-\frac{\mathrm{b}}{\mathrm{c}}$

(B) $-\frac{\mathrm{b}}{\mathrm{a}}$

(C) $\frac{\mathrm{c}}{\mathrm{a}}$

(D) none VIEW SOLUTION

- Question 32
The probability of complement event of impossible event is ___________.

(A) 0.5

(B) 0

(C) 1

(D) 0.46 VIEW SOLUTION

- Question 33
Find the probability of having 5 Sundays in the month of February in leap year 2004.

(A) 2/7

(B) 0

(C) 1/7

(D) 1 VIEW SOLUTION

- Question 34
Class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 Frequancy 5 15 13 17 10

For the information given above, $\left(\frac{\mathrm{n}}{2}-\mathrm{cf}\right)$ will be = ____________.

(A) 10

(B) 20

(C) 30

(D) 25 VIEW SOLUTION

- Question 35
For the measures of central tendency, __________ of the following is not true.

(A) $\mathrm{Z}=3\mathrm{M}\u20132\overline{)\mathrm{x}}$

(B) $2\overline{)\mathrm{x}}+\mathrm{Z}=3\mathrm{M}$

(C) $2\overline{)\mathrm{x}}-3\mathrm{M}=-\mathrm{Z}$

(D) $2\overline{)\mathrm{x}}=\mathrm{Z}-3\mathrm{M}$ VIEW SOLUTION

- Question 36
If $\overline{)\mathrm{x}}-\mathrm{Z}=3\mathrm{and}\overline{)\mathrm{x}}+\mathrm{Z}=45$, then M = ________.

(A) 26

(B) 22

(C) 24

(D) 23 VIEW SOLUTION

- Question 37
If before three years sum of ages of father and son was 59 years then before five years the sum of their ages would have been __________.

(A) 55

(B) 61

(C) 69

(D) 57 VIEW SOLUTION

- Question 38
To eliminate
*x*from equations*x + y*+ 3*x*= 12 → ① & 8*x*+ 3*y*= 17 → ②, equation ② is to be multiplied by ______________.

(A) 8

(B) $-\frac{1}{2}$

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

(D) 3 VIEW SOLUTION

- Question 39
Equation $\frac{\mathrm{x}}{3}-\frac{\mathrm{y}}{2}=1$ can be written in standard form as _______.

(A) 3x – 2y – 6 = 0

(B) 2x – 3y – 6 = 0

(C) 2x – 3y = 6

(D) 2x – 3y = 1 VIEW SOLUTION

- Question 40
In a two digit number, if number in unit place is 8 and number in tens place is y then that number is ____________.

(A) y + 8

(B) y + 80

(C) 10y + 8

(D) 80 y VIEW SOLUTION

- Question 41
The factors of quadratic polynomial P(x) = x
^{2}+ 4x – 5 are ____________.

(A) (x + 5) (x – 1)

(B) (x + 5) (x + 1)

(C) (x + 1) (x – 5)

(D) (x – 1) (x – 5) VIEW SOLUTION

- Question 42
______________ is true for discriminate of quadratic equation x
^{2}+ x + 1= 0.

(A) D = 0

(B) D < 0

(C) D > 0

(D) D is a perfect square VIEW SOLUTION

- Question 43
If one of the roots of the equation kx
^{2}– 7x + 3 = 0 is 3, then k = ___________.

(A) –3

(B) 3

(C) –2

(D) 2 VIEW SOLUTION

- Question 44
______________ cannot be the sum of a non zero number and its reciprocal.

(A) 0

(B) 2

(C) $\mathrm{x}+\frac{1}{\mathrm{x}}$

(D) $\frac{5}{2}$ VIEW SOLUTION

- Question 45
For a quadratic equation, if discriminate D = 0, then _____________ is not possible for its roots α + β.

(A) α = β

(B) α – β = 0

(C) α + β = 2α

(D) α + β = 0 VIEW SOLUTION

- Question 46
_____________ can be one of the term in Arithmetic progression 4, 7, 10, ------.

(A) 103

(B) 123

(C) 171

(D) 99 VIEW SOLUTION

- Question 47
(
*n*– 2)th term of an arithmetic progression will be ___________.

(A) a + (n – 1)d

(B) a + (n – 3)d

(C) a + (n – 2)d

(D) none VIEW SOLUTION

- Question 48
In the AP, 5, 7, 9, 11, 13, ------ the sixth term which is prime is __________.

(A) 15

(B) 19

(C) 17

(D) 23 VIEW SOLUTION

- Question 49
From the given BD = __________.

(A) x + y

(B) $\sqrt{\mathrm{xy}}$

(C) xy

(D) $\sqrt{\mathrm{x}+\mathrm{y}}$ VIEW SOLUTION

- Question 50
For ΔABC & ΔPQR, if m∠A = m∠R and m∠C = m∠Q, then ABC ↔ ___________ is a similarity.

(A) RQP

(B) PQR

(C) RPQ

(D) QP VIEW SOLUTION

- Question 51
If 7 is prime, then prove that $\sqrt{7}$ is irrational. VIEW SOLUTION

- Question 52

- Question 53
Find the solution of given pair of linear equation by elimination method.

3x + 4y = – 17 → ①

5x + 2y = – 19 → ② VIEW SOLUTION

- Question 54
For an AP, $\frac{1}{3},\frac{4}{3},\frac{7}{3},\frac{{\displaystyle 10}}{3}.......,\mathrm{Find}{\mathrm{T}}_{18}$

ORFor an AP, 3, 9, 15, 21......., Find S

_{10}VIEW SOLUTION

- Question 55
Write statement of Pythagoras theorem and show that 6, 8, and 10 are Pythagorean triplets. VIEW SOLUTION

- Question 56
If X(3, 1), Y(4, 5) and Z(–2, –1) are co-ordinates of vertices of ΔXYZ then find area of ΔXYZ. VIEW SOLUTION

- Question 57
If sin θ = a then find the value of cot θ + sec θ.

**OR**

For ΔABC, prove that tan $\left(\frac{\mathrm{A}+\mathrm{C}}{2}\right)=\mathrm{cot}\frac{\mathrm{B}}{2}$. VIEW SOLUTION

- Question 58
Find mode given data:

Class 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 Frequency 15 20 50 30 10

- Question 59
Write standard form of quadratic equation and find the roots of the equation $3{\mathrm{x}}^{2}+5\sqrt{2}\mathrm{x}+2=0$ using general formula. VIEW SOLUTION

- Question 60
From top of a tower the angle of depression of top and bottom of a multistoreyed building are 30° and 60° respectively. If the height of the building is 100 m. find the height of a tower. VIEW SOLUTION

- Question 61
The mean of the following frequency distribution of 100 observations is 148. Find missing frequencies F
_{1}and F_{2}.

Class 0 – 49 50 – 99 100 – 149 150 – 199 200 – 249 250 – 299 300 – 349 Frequency 10 15 F _{1}20 15 F _{2}2 **OR**

The median of 230 observations is 46. Find a and b.Class 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 Frequency 12 30 a 65 b 25

- Question 62
A box contains 100 cards marked with numbers 1 to 100. If one card is drawn randomly from the box. Find the probability that it bears.

(1) Even prime number.

(2) A number divisible by 7.

(3) The number at unit place is 9. VIEW SOLUTION

- Question 63
Prove that, tangents drawn to the end points of diameter of the circle are parallel to each other. VIEW SOLUTION

- Question 64
A cylindrical container having diameter 16 cm and height 40 cm is full of ice cream. The ice-cream is to be filled into cones of height 12 cm and diameter 4 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.
**OR**

A cylinder has hemispherical ends having radius 7 cm and total height of solid is104 cm. If its outer surface is to be polished & cost of polish is Rs 100 per sq. mtr. find the total cost of polish. VIEW SOLUTION

- Question 65
The length of a side of a square field is 20 mtr. A cow is tied at the corner by means of a 7 mtr long rope. Find the area of the field which cow can graze. If the length of rope is doubled then find the increase in the grazing area. VIEW SOLUTION

- Question 66
Divide a given line segment in ratio of 3:5 and write steps of construction. VIEW SOLUTION

- Question 67
If for acuted angled ΔABC and ΔPQR ABC ↔ PQR is similarity then prove that $\frac{\mathrm{A}\left(\u25b3\mathrm{ABC}\right)}{\mathrm{A}\left(\u25b3\mathrm{PQR}\right)}=\frac{{\mathrm{AB}}^{2}}{{\mathrm{PQ}}^{2}}=\frac{{\mathrm{BC}}^{2}}{{\mathrm{QR}}^{2}}=\frac{{\mathrm{AC}}^{2}}{{\mathrm{PR}}^{2}}$

OR

Write converse of Pythagoras theorem and prove it. VIEW SOLUTION