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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
Answer the following questions from 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  AB, BC and AC  respectively. If ABC = 40 then DEF = ----- .


    (A) 10

    (B) 403

    (C) 20

    (D) 5 VIEW SOLUTION
  • Question 2
    From the given figure AD = ----



    (A) 2

    (B) 23

    (C) 32

    (D) 3 VIEW SOLUTION
  • Question 3
    In ΔABC, if m∠B = 90, & AC = 10 then length of median BM¯ = ---------.

    (A) 6

    (B) 52

    (C) 5

    (D) 8 VIEW SOLUTION
  • Question 4
    From the figure, AB = -------.


    (A) 12

    (B) 4

    (C) 8

    (D) 16 VIEW SOLUTION
  • Question 5
    From the graph OA2 = --------.


    (A) 0

    (B) a2 + b2

    (C) a2+b2

    (D) a+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
    If sin2 (3x + 45) + cos2 (2x + 60) = 1 then x

    (A) 60

    (B) 30

    (C) 15

    (D) 0 VIEW SOLUTION
  • Question 10
    If tan5θ·tan 4θ = 1, then θ = ______.

    (A) 10

    (B) 3

    (C) 7

    (D) 9 VIEW SOLUTION
  • Question 11
    From given figure tan A·tan C = ____________.


    (A) 12

    (B) 2

    (C) 2

    (D) 1 VIEW SOLUTION
  • Question 12
    If cos2 45° – cos2 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) a2

    (B) 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: 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) 1-12

    (D) 2-2 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 = A1 and a circle of diameter 4 cm from the centre is cut whose area is A2 , then __________ is possible.


    (A) A1 ≠ A2

    (B) A1 < A2

    (C) A1 > A2

    (D) A1 = A2 VIEW SOLUTION
  • Question 19
    According to figure of clock angle θ =_______.


    (A) 150°

    (B) 20°

    (C) 120°

    (D) 100° VIEW SOLUTION
  • 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) 12rl

    (B) πrθ360

    (C) πr2θ180

    (D) πr2 VIEW SOLUTION
  • Question 22
    If the circumference of base of a hemisphere is 2π then it volume is ______________ cm3.

    (A) 2π3r3

    (B) 2π3

    (C) 8π3

    (D) π12 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 2-3 is __________.

    (A) -2+3

    (B) 2--3

    (C) 3+2 

    (D) 12-3 VIEW SOLUTION
  • Question 27
    To get the terminating decimal expansion of a rational number pq. If q = 2m 5n then m and n must belong to __________.

    (A) Z

    (B) N U {0}

    (C) N

    (D) R VIEW SOLUTION
  • Question 28
    For polynomial p(x) = x2 – 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) ba, 0

    (B) 0, ba

    (C) -ba, 0

    (D) -ab, 0 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) = ax2 + bx + c, 1α+1β = ____________.

    (A) -bc

    (B) -ba

    (C) ca

    (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, n2-cf 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) Z=3M  2x

    (B) 2x+Z=3M

    (C) 2x -3M=-Z

    (D) 2x=Z-3M VIEW SOLUTION
  • Question 36
    If x-Z=3 and x+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 + 3x = 12 → ① & 8x + 3y = 17 → ②, equation ② is to be multiplied by ______________.

    (A) 8

    (B) -12

    (C) 12

    (D) 3 VIEW SOLUTION
  • Question 39
    Equation x3-y2=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) = x2 + 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 x2 + 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 kx2 – 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) x+1x

    (D) 52 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) xy

    (C) xy

    (D) x+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 7 is irrational. VIEW SOLUTION
  • Question 52
    For polynomial P(x) = 6x3 + 29x2 + 44x + 21 find P (–2). VIEW SOLUTION
  • 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, 13, 43, 73, 103 ......., Find T18
     
    OR
     
    For an AP, 3, 9, 15, 21......., Find S10     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 A+C2=cotB2. 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
    VIEW SOLUTION
  • Question 59
    Write standard form of quadratic equation and find the roots of the equation 3x2+52x+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 F1 and F2.  
     
    Class 0 – 49 50 – 99 100 – 149 150 – 199 200 – 249 250 – 299 300 – 349
    Frequency 10 15 F1 20 15 F2 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
    VIEW SOLUTION
  • 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 AABCAPQR=AB2PQ2=BC2QR2=AC2PR2 

    OR

    Write converse of Pythagoras theorem and prove it. VIEW SOLUTION
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