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# Board Paper of Class 10 2015 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
D, E and F are respectively mid-points of sides $\overline{)\mathrm{AB}}$$\overline{)\mathrm{BC}}$ and $\overline{)\mathrm{CA}}$ of ∆ABC. Which of the following correspondence between ∆ABC and ∆DEF is similarity.

(A) ABC ↔ EFD

(B) ABC ↔ FED

(C) ABC ↔ DEF

(D) ABC ↔ EDF VIEW SOLUTION
• Question 2
The points A(5, 2), B(3, 4) and C(x, y) are collinear points and AB = BC then find the co-ordinates of C.
(A) (1, 6)
(B) (2, –2)
(C) (4, 3)
(D) (8, 6) VIEW SOLUTION
• Question 3
The co-ordinates of the foot perpendicular from P(–3, 1) to X axis are _______

(A) (0, 1)

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

(C) (–3, 0)
(D) (1, 0) VIEW SOLUTION
• Question 4
A(2, –2), B(4, 3) and C(–3, 11) are the vertices of ∆ABC. If G is centroid of triangle, then the co-ordinate of G is

(A) (1, 4)
(B) (1, –4)
(C) (–1, 4)
(D) (–1, –4) VIEW SOLUTION
• Question 5
A(1, 2) and B(3, –2) are the end points of $\overline{)\mathrm{AB}}$ . Then the midpoints of $\overline{)\mathrm{AB}}$.

(A) (–1, 0)
(B) (2, 0)
(C) (2, 1)
(D) (0, 0) VIEW SOLUTION
• Question 6
If 15tan2θ + 4sec2θ = 23 then tan2θ = _______.
(A) 27/15
(B) 45
(C) 19/11
(D) 1 VIEW SOLUTION
• Question 7
cot2θ – cosec2θ = _______ (0 < θ < 90)
(A) –1
(B) 1
(C) 0
(D) Not defined VIEW SOLUTION
• Question 8
1 – 2sin230
(A) cosec 60
(B) tan 60
(C) sin 60
(D) cos 60 VIEW SOLUTION
• Question 9
If sin2 45 – sin2 60 = xcos2 45 then x = _______
(A) 3/2
(B) 3/4
(C) –1/2
(D) 2 VIEW SOLUTION
• Question 10
An observer from the top of the light house, the angle of depression of two ships P and Q anchored in the sea to the same side are found to have measure 35 and 50 respectively. Then from the light house _______
(A) The distance of P is more than Q.
(B) The distance of Q is more than P.
(C) P and Q are at equal distance.
(D) The relation about the distance of P and Q cannot be determined. VIEW SOLUTION
• Question 11
In right angle triangle, the side opposite to the angle having measure 30 is _______ to the hypotenuse
(A) three times
(B) half
(C) double
(D) fourth part VIEW SOLUTION
• Question 12
An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from his/her eyes has measure 45. What is the height of the tower?
(A) 28.5 m
(B) 30 m
(C) 27 m
(D) 1.5 m VIEW SOLUTION
• Question 13
∆ABC, AB = 5, BC = 12 and AC = 13, then the radius of the circle touching all the three sides is _______
(A) 2
(B) 3
(C) 1
(D) 4 VIEW SOLUTION
• Question 14
A chord of ʘ(0, 10) touches ʘ(0, 6). Therefore the length of the chord is _______
(A) 10
(B) 8
(C) 16
(D) 6 VIEW SOLUTION
• Question 15
In a circle having radius r, an arc of a circle subtends an angle θ at the centre, then the length of arc l = _______

(A) $\frac{\pi {r}^{2}\theta }{360}$

(B) πr2

(C) 2πr

(D) $\frac{\pi r\theta }{180}$ VIEW SOLUTION
• Question 16
The length of the minor arc ACB of a circle ʘ(O, r) is 1/6 part of its circumference. Then the measure of angle of minor arc ACB subtending at the centre is _______
(A) 60
(B) 90
(C) 30
(D) 120 VIEW SOLUTION
• Question 17
If the radius of a circle increased by 20% then the corresponding increase in the area of circle is _______
(A) 40%
(B) 44%
(C) 20%
(D) 21% VIEW SOLUTION
• Question 18
In figure the shaded region indicates

(A) Major segment
(B) Minor sector
(C) Minor segment
(D) Major Sector VIEW SOLUTION
• Question 19
The volume of sphere is $\frac{4}{3}$π cm3. Then the radius is _______m.
(A) 1
(B) 0.02
(C) 0.01
(D) 2 VIEW SOLUTION
• Question 20
The volume of a cone having radius 2 cm and height 6 cm is _______ cm3
(A) 14π
(B) 12π
(C) 8π
(D) 16π VIEW SOLUTION
• Question 21
The curved surface area of a sphere is 1256 cm2. Find the radius of the sphere.
(π = 3.14)
(A) 100 cm
(B) 10 cm
(C) 1 cm
(D) 314 cm VIEW SOLUTION
• Question 22
Answer the following question from the given figure.
Here the solid is composed of cube, cylinder and cone.
Here,
The curved surface area of cone is P.
The curved surface area of cylinder is Q.
The area of the base of cone and cylinder is R and the total surface area of cube is S.
Then, find the total surface area of solid.

(A) P + Q – R + S
(B) P + Q – 2R + S
(C) P + Q + R + S
(D) P + Q + 2R + S VIEW SOLUTION
• Question 23
Here is a graph of “Less than type” and “more than type” ogive is drawn.
It shows the information about the yearly salary (In lakhs Rs.) of officers in a factory.

Boundary points (Annual Salary)
Both the curves intersect (20.5, 15). Find the median of the information.
(A) 35.5 lakhs
(B) 20.5 lakhs
(C) 15 lakhs
(D) 17.75 lakhs VIEW SOLUTION
• Question 24
For some data $\mathrm{M}+\overline{)\mathrm{X}}=22$ and $\mathrm{M}-\overline{)\mathrm{X}}=2$ then Z = ________
(A) 14
(B) 15
(C) 16
(D) 12 VIEW SOLUTION
• Question 25
The wickets taken by a bowler in a one day cricket match are 4, 5, 6, 3, 4, 0, 3, 2, 3, 5.
The mode of the data is___________.
(A) 3
(B) 4
(C) 5
(D) 2 VIEW SOLUTION
• Question 26
Following is the information about the number of students studying in a high school. If one student of the school is to be elected as a representative, find out the probability of the elected student to be a boy of Std. IX.
 Std. Boys Girls Total IX 220 110 330 X 110 110 220 Total 330 220 550

(A) $\frac{1}{5}$

(B) $\frac{3}{5}$

(C) $\frac{2}{5}$

(D) $\frac{4}{5}$ VIEW SOLUTION
• Question 27
The probability that one will get 75 marks in the question paper of 100 marks is

(A) $\frac{1}{101}$

(B) $\frac{75}{100}$

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

(D) $\frac{75}{101}$ VIEW SOLUTION
• Question 28
If n is positive even integer, then n(n + 1)(n + 2) is _______
(A) a prime number
(B) divisible by 20
(C) divisible by 24
(D) divisible by 16 VIEW SOLUTION
• Question 29
The decimal expansion of $\frac{1}{32}$ is
(A) 0.03125
(B) 0.15625
(C) 0.3125
(D) 0.00625 VIEW SOLUTION
• Question 30
If α and β are the zeroes of polynomial P(x) = x2 – 3x + 2k, and α + β = α.β, then the value of k is ________.
(A) 3
(B) –3
(C) 1
(D)$\frac{3}{2}$ VIEW SOLUTION
• Question 31
The degree of polynomial (x + 1)(x2xx4 + 1) is _______.
(A) 5
(B) 4
(C) 1
(D) 3 VIEW SOLUTION
• Question 32
The number of real zeroes of polynomial P(x) = x3x is _______
(A) 2
(B) 1
(C) 0
(D) 3 VIEW SOLUTION
• Question 33
α, β, & γ are the zeroes of cubic polynomial P(x) = ax3 + bx2 + cx + d, (a ≠ 0) then product of their zeroes (α.β.γ) = _______.

(A) $\frac{-b}{a}$

(B) $\frac{c}{a}$

(C) $\frac{a}{b}$

(D) $\frac{-d}{a}$ VIEW SOLUTION
• Question 34
Equation $\frac{x}{5}-\frac{y}{3}=\frac{4}{5}$ can be expressed in the standard form as _______.
(A) 3x – 5y – 4 = 0
(B) 3x – 5y – 12 = 0
(C) 5x – 3y – 4 = 0
(D) 5x – 3y = 12 VIEW SOLUTION
• Question 35
In a two digit number, the digit at tens place is 7 and the sum of the digits is 8 times the digit at unit place. Then the number is _______.
(A) 17
(B) 71
(C) 70
(D) 78 VIEW SOLUTION
• Question 36
Two lines x + 2y + 7 = 0 and 2x + ky + 18 = 0 do not intersect each other. Find the value of k.
(A) 3
(B) 2
(C) 1
(D) 4 VIEW SOLUTION
• Question 37
Two lines y = 3x and x = 3y intersect each other in _______.
(A) (0, 0)
(B) (0, 3)
(C) (3, 3)
(D) (3, 0) VIEW SOLUTION
• Question 38
The solution set of the quadratic equation x2 – 30x + 221 = 0 is _______.
(A) {13, 17}
(B) {–13, –17}
(C) {–13, 17}
(D) {13, –17} VIEW SOLUTION
• Question 39
The discriminant of x2 – 3x + k = 0 is 1 then the value of k = _______.
(A) –2
(B) 4
(C) –4
(D) 2 VIEW SOLUTION
• Question 40
For the quadratic equation ax2 + bx + c = 0, a, b, c Q, If D = 0 then _______.
Choose the correct option in respect to the statements below.
(P) The roots of the equation are equal.
(Q) The roots of the equation are not equal.
(R) The roots of the equation are rational numbers
(S) The roots of the equation has no roots
Options:
(A) Statements P and R are correct
(B) Statements Q and R are correct
(C) Only statement S is correct
(D) Only statement P is correct VIEW SOLUTION
• Question 41
_______ is a solution of quadratic equation 2x2x – 3 = 0.
(A) 2/3
(B) –1
(C) 0
(D) 1 VIEW SOLUTION
• Question 42
The roots of quadratic equation $\frac{\mathrm{x}}{\mathrm{k}}=\frac{\mathrm{k}}{\mathrm{x}}$ are _______.
(A) k, –k
(B) –k, –k
(C) k, k
(D) k2, –k2 VIEW SOLUTION
• Question 43
A student named Akash is observing birds flying in the sky at the evening time. Birds are flying in the pattern like 1 in the first row, 3 in the second, 5 in the third and so on…. Making total 20 rows as per the figure given below, find out the total number of birds.

(A) 39
(B) 400
(C) 40
(D) 200 VIEW SOLUTION
• Question 44
A series of steps lead to a temple. The number of steps is 30. The height of each step is 20 cm. Then find the height of the temple from the base.
(A) 5.80 m
(B) 6 m
(C) 30 m
(D) 6.20 m VIEW SOLUTION
• Question 45
If k + 2, k, 3k – 2 are three consecutive terms of A.P., then k = _______.
(A) –2
(B) 6
(C) 5
(D) 8 VIEW SOLUTION
• Question 46
In the figure, $\overline{)\mathrm{XY}}\parallel \mathrm{BC}$
AX = 1 cm
XB = 3 cm
BC = 6 cm
then XY _____

(A) 2 cm
(B) 1.5 cm
(C) 1 cm
(D) 3 cm VIEW SOLUTION
• Question 47
In figure which of the following given correspondence between the vertices of two triangles are similarity.
(P) correspondence ABC ↔ DEF
(Q) correspondence ABC ↔ FDE
(R) correspondence ABC ↔ EFD

Options :
(A) P and R are correct
(B) Only P is correct
(C) Q and R are correct
(D) P, Q and R are correct VIEW SOLUTION
• Question 48
In figure, m∠C = 90 in isosceles triangle ∆ABC, then AB2 = _______.

(A) $\sqrt{2}{\mathrm{BC}}^{2}$
(B) 2BC2
(C) BC2
(D) 4BC2 VIEW SOLUTION
• Question 49
Which of the following statements are correct for the correspondence between the vertices of two triangles?
(P) The area of two similar triangles is equal.
(Q) The corresponding angles of both similar triangles are having same proportion.
(R) The corresponding sides of both similar triangles are having same proportion.
(S) The corresponding sides of both similar triangles are congruent.
(A) Only statement R is correct.
(B) Statement Q and S are correct.
(C) Statement P, Q and R are correct.
(D) Statement Q and R are correct. VIEW SOLUTION
• Question 50
Four pairs of showing measurements of sides  and $\overline{)\mathrm{CA}}$ of ∆ABC are given below.
Show which of the following pair/s is/are shows right angle triangle.
 Pair P: AB = 25 BC = 7 AC = 24 Pair Q: AB = 8 BC = 6 AC = 10 Pair R: AB = 3 BC = 4 AC = 6 Pair S: AB = 8 BC = 6 AC = 5

(A) Pairs Q and R show right angle triangle
(B) Pairs P and Q show right angle triangle
(C) Pairs P and S show right angle triangle
(D) Pairs P, Q and S show right angle triangle VIEW SOLUTION
• Question 51
The length of diagonal of square is $\sqrt{2}\left(6+2\sqrt{5}\right)$ cm. Then find its length of side of square. VIEW SOLUTION
• Question 52
Prove that (3x + 7) is a factor of (6x3 + 29x2 + 44x + 21). VIEW SOLUTION
• Question 53
$\overline{)\mathrm{AB}}$ is given. Point P lies on the perpendicular bisector of $\overline{)\mathrm{AB}}$ such that the measure of $\overline{)\mathrm{AP}}$ is double than $\overline{)\mathrm{AB}}.$ If the perimeter of ∆ABP is
35 cm. Find the measures of sides of ∆ABP. VIEW SOLUTION
• Question 54
In a game of cricket, the score of team with respect to wickets follows arithematic progression.
First wicket falls on score 35, second falls on 62, third on 89___________
What would be the final score of team. VIEW SOLUTION
• Question 55
In trapezium ABCD, M∈ $\overline{)\mathrm{AD}}$, N∈ $\overline{)\mathrm{BC}}$ are the points such that $\frac{\mathrm{AM}}{\mathrm{MD}}=\frac{\mathrm{BN}}{\mathrm{NC}}=\frac{2}{3}$

The diagonal $\overline{)\mathrm{AC}}$ intersects $\overline{)\mathrm{MN}}$ in O. Then find the value of $\frac{\mathrm{AO}}{\mathrm{AC}}.$ VIEW SOLUTION
• Question 56
A(–5, 2), B(x, –3) and C(–2, y) are the vertices of ΔABC. G is a centroid of ΔABC. The coordinates of G are G(–2, 1). Then find the value of x and y. VIEW SOLUTION
• Question 57
Prove with the help of trigonometric Identities.
2sin2θ + 4sec2θ + 5cot2θ + 2cos2θ – 4tan2θ – 5cosec2θ = 1

OR

In ∆ABC, m∠C = 90 and  , find sinA and sinB. VIEW SOLUTION
• Question 58
Find the mode for the following frequency distribution.
 Class 4 – 8 8 – 12 12 – 16 16 – 20 20 – 24 24 – 28 Frequency 9 6 12 7 15 3
VIEW SOLUTION
• Question 59
In a fifth match of India and Srilanka played in November 2014.
The sum of one third runs scored by Indian team and one seventh runs scored by Srilankan team is 137.
If India won this match by minimum required run, then find the runs scored by Indian team. VIEW SOLUTION
• Question 60
A bridge across a valley is 800 metres long. There is a temple in the valley directly below the bridge. The angle of depression of the top of the temple from the two ends of the bridge have measure 30° and 60°.
Find the height of the bridge above the top of the temple. VIEW SOLUTION
• Question 61
The mean of the following frequency distribution is 350, find the missing frequency:
 Class 100 – 200 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700 Frequency 5 3 3 – 2 1

OR

The cumulative frequency “more than type” is given in the following table.
Find the median of the following data.
 Wages ≥ 0 ≥ 50 ≥ 100 ≥ 150 ≥ 200 ≥ 250 Cumulative Frequency 90 80 60 30 15 0
VIEW SOLUTION
• Question 62
Two balance dice are thrown once. Write down all possible outcomes of this experiment.
What is the probability that
(1) The product of numbers obtained on upper face of the dice is even number.
(2) The sum of numbers on both the dice is a prime number. VIEW SOLUTION
• Question 63
Prove that, In a plane of circle, A tangent to a circle is perpendicular to the radius drawn from the point of contact. VIEW SOLUTION
• Question 64
The radius of a field in the form of a sector is 21m. The cost of constructing a wall around the field is Rs. 1500 at the rate of Rs. 20 per meter.
What will be the cost of tilling the whole field at the costs of Rs. 15 per square meter.   VIEW SOLUTION
• Question 65
In an experiment of Maths, frustum of cone is made up of seven rings.
The radius of uppermost ring is 4 cm and the radius of each ring is increased by 1 cm, so that the radius of the last ring is 10 cm. If the width of each ring is 3 cm. Then find the volume of entire frustum. (avoid the spaces between rings).
OR

A solid composed of a cylinder with hemispherical ends on both the sides.
The radius and height of the cylinder are 20 cm and 30 cm respectively. Find the total surface area of solid. (Take п = 3.14) VIEW SOLUTION
• Question 66
Draw a circle of radius 5 cm. From a point 8 cm away from the centre, construct two tangents to the circle. Measure them.
(Write the steps of construction)                     VIEW SOLUTION
• Question 67
If a line parallel to one of the sides of a triangle intersects the other two sides in distinct points, then the segment of the other two sides in one halfplane are proportional to the segments in the other halfplane.

OR

The diagonals of a convex ⃞ PQRS intersect at right angles then prove that
PQ2 + RS2 = PS2 + QR2 VIEW SOLUTION
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