Board Paper of Class 10 2016 Maths - Solutions

• 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
  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) $\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$
  
  (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
  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) $\frac{1}{\sqrt{2}}$
  
  (B) 2
  
  (C) $\sqrt{2}$
  
  (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) $\sqrt{2}$
  
  (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 = 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) $\frac{1}{2}\mathrm{rl}$
  
  (B) $\frac{\mathrm{\pi r\theta }}{360}$
  
  (C) $\frac{{\mathrm{\pi r}}^2\mathrm{\theta }}{180}$
  
  (D) πr² VIEW SOLUTION

• Question 22
  If the circumference of base of a hemisphere is 2π then it volume is ______________ cm³.
  
  (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 = 2^m 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) $\left(\frac{\mathrm{b}}{\mathrm{a}}, 0\right)$
  
  (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) = ax² + 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) $\mathrm{Z}=3\mathrm{M} – 2\overline{\mathrm{x}}$
  
  (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 + 3x = 12 → ① & 8x + 3y = 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) = 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) $\mathrm{x}+\frac{1}{\mathrm{x}}$
  
  (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
  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, $\frac{1}{3}, \frac{4}{3}, \frac{7}{3}, \frac{{\displaystyle 10}}{3} ......., \mathrm{Find} {\mathrm{T}}_{18}$
   

  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.   

   

  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 $3x^2+5\sqrt{2}x+2=0$ 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 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

  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.     VIEW SOLUTION

• Question 65
  The length of a side of a square field is 20 m. A cow is tied at the corner by means of a 7 m 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 $\frac{\mathrm{A}\left(△\mathrm{ABC}\right)}{\mathrm{A}\left(△\mathrm{PQR}\right)}=\frac{{\mathrm{AB}}^2}{{\mathrm{PQ}}^2}=\frac{{\mathrm{BC}}^2}{{\mathrm{QR}}^2}=\frac{{\mathrm{AC}}^2}{{\mathrm{PR}}^2}$ 

  
  OR

  
  Write converse of Pythagoras theorem and prove it. VIEW SOLUTION