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# Board Paper of Class 10 2014 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 ∆PQR, the bisector of ∠P intersects $\overline{\mathrm{QR}}$ in D. If QD : RD = 4 : 7, PR = 14, Then PQ = ……………
(A) 4
(B) 8
(C) 12
(D) 15 VIEW SOLUTION
• Question 2
If cosec $\mathrm{A}=\frac{4}{3}$ and A + B = 90, then sec B = …………..
(A) $\frac{16}{9}$

(B) $\frac{4}{3}$

(C) $\frac{3}{4}$

(D) $\frac{7}{3}$ VIEW SOLUTION
• Question 3
From the top of a building h metre high, the angle of depression of an object on the ground has a measure θ. The distance of the object from the building is
(A) h cos θ metre
(B) h sin θ metre
(C) tan θ metre
(D) h cot θ metre VIEW SOLUTION
• Question 4
For A (1, 2) and B (3, −2), the coordinates of the midpoint of AB are is ……………………
(A) (2, 2)
(B) (0, 0)
(C) (2, 0)
(D) (0, 2) VIEW SOLUTION
• Question 5
On walking …………… metres on a slope at an angle of measure 30° with the ground, one can reach the height 'a' metres from the ground.

(A) $\frac{2a}{\sqrt{3}}$

(B) $\frac{\sqrt{3}}{2}a$

(C) 2a

(D) $\frac{a}{2}$ VIEW SOLUTION
• Question 6
$\frac{{\mathrm{sin}}^{4}\theta -{\mathrm{cos}}^{4}\theta }{{\mathrm{sin}}^{2}\theta -{\mathrm{cos}}^{2}\theta }=$

(A) 3
(B) 2
(C) 0
(D) 1 VIEW SOLUTION
• Question 7
From the natural number of single digit, the probability of getting an even number is ………………….

(A) $\frac{5}{9}$

(B) $\frac{5}{10}$

(C) $\frac{4}{9}$

(D) $\frac{1}{9}$ VIEW SOLUTION
• Question 8
In ∆ABC, correspondence ABC ↔ BAC is similarity. From the following …………………… is true.
(A) ∠C ≅ ∠A
(B) ∠B ≅ ∠C
(C) ∠A ≅ ∠B
(D) ∠A ≅ ∠B ≅ ∠C VIEW SOLUTION
• Question 9
If sin 7θ = cos 2θ for acute angles 7θ and 2θ, then θ = …………………….
(A) 10
(B) 90
(C) 20
(D) 30 VIEW SOLUTION
• Question 10
In a two digit number, the digit at the units place is x and the digit at tens place is y. If y = 5, then the number is …………………
(A) 50x + 5
(B) 5x
(C) 30x + 5
(D) x + 50 VIEW SOLUTION
• Question 11
The chord of a ⊙ (0, 5) touches ⊙ (0, 3). The length of the chord is ………………………..
(A) 8
(B) 6
(C) 7
(D) 2 VIEW SOLUTION
• Question 12
The perimeter of an equilateral triangle is 6. The length of an altitude drawn on any of its sides is …………..

(A) $2\sqrt{3}$

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

(C) 2

(D) $\sqrt{3}$ VIEW SOLUTION
• Question 13
As shown in the following figure, the area of square ABCD is 16 cm2 and the area of square CIPO is 9cm2. If $\overline{)\mathrm{BC}}\perp \overline{)\mathrm{CO}}$ then the length of  ………………………… cm.

(A) 7
(B) 25
(C) 625
(D) 5 VIEW SOLUTION
• Question 14
In a given A.P., T25 – T20 = 15. ∴ d = ………………. for the A.P.
(A) 5
(B) 3
(C) 25
(D) 120 VIEW SOLUTION
• Question 15
3 years ago, the sum of the ages of a father and his son was 40 years. After 2 years, the sum of their ages will be…………….
(A) 46 years
(B) 40 years
(C) 50 years
(D) 60 years VIEW SOLUTION
• Question 16
The ratio of the areas of two similar triangles is 25 : 16. The ratio of their perimeters is …………………….
(A) 625 : 256
(B) 5 : 6
(C) 25 : 16
(D) 5 : 4 VIEW SOLUTION
• Question 17
If $y=\frac{2}{3}$ is a root of the quadratic equation 3y2 − ky + 8 = 0, then the value of k is ……………………..
(A) 13
(B) −14
(C) 14
(D) −13 VIEW SOLUTION
• Question 18
From the equations given below, a root of one equation is 3. The equation is…………………
(A) x2 + x − 6 = 0
(B) x2 − x − 6 = 0
(C) x2 − x + 6 = 0
(D) x2 + x + 6 = 0 VIEW SOLUTION
• Question 19
If α , β, γ are the zeros of a polynomial P(x) = ax3 + bx2 + cx + d (a 0) then

$\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=\dots \dots \dots \dots \dots .$

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

(B) $-\frac{c}{d}$

(C) $\frac{c}{d}$

(D) $-\frac{c}{a}$ VIEW SOLUTION
• Question 20
With the help of match-sticks, Zalak prepared a pattern as shown below. When 97 matchsticks are used, the serial number of the figure will be ___________ .

(A) Figure 32
(B) Figure 95
(C) Figure 49
(D) Figure 48 VIEW SOLUTION
• Question 21
The volume of a cylinder is 550 cm3. If its radius is 5 cm, then its height is ………….. cm.
(A) 12
(B) 9
(C) 7
(D) 14 VIEW SOLUTION
• Question 22
In the following figure ∆ABC is an equilateral triangle and AC = x cm. $\overline{)\mathrm{AD}}$ is median on $\overline{)\mathrm{BC}}$, D$\overline{)\mathrm{BC}}$ . If AD = y cm, then y = ……………………. cm.

(A) $\sqrt{\frac{3}{2}·x}$

(B) $\frac{\sqrt{3}}{2}·x$

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

(D) $\frac{3}{2}·x$VIEW SOLUTION
• Question 23
In any A.P., S n – 2 Sn – 1 + S n – 2 = ................(n > 2).
(A) a + d
(B) 2d
(C) d
(D) a VIEW SOLUTION
• Question 24
The foot of the perpendicular drawn from P(−3, 2) to the y-axis is M. The coordinates of M are…………….

(A) (0, 2)

(B) (3, 0)

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

(D) (–3, 2) VIEW SOLUTION
• Question 25
If 7 cos2θ + 3 sin2θ = 4, then cot θ =

(A) $\frac{7}{3}$

(B) 7

(C) $\sqrt{3}$

(D) $\frac{1}{\sqrt{3}}$ VIEW SOLUTION
• Question 26
The formula to find the total surface area of a Rs. 5 coin is …………………..
(A) πr2h
(B) πr(r + h)
(C) πr3h
(D) 2πr(h + r) VIEW SOLUTION
• Question 27
If the area and the circumference of a circle are numerically equal, then the radius of the circle is ………………………

(A) $\frac{5}{2}$

(B) 2

(C) 1

(D) $\frac{2}{5}$ VIEW SOLUTION
• Question 28
If the ratio of the areas of two circles is 1 : 4, then the ratio of their circumferences is ……………………….
(A) 1 : 4
(B) 2 : 3
(C) 1 : 2
(D) 3 : 2 VIEW SOLUTION
• Question 29
The product of the zeroes of polynomial x2 − 4x + 3 is …………………….
(A) 4
(B) 1
(C) −4
(D) 3 VIEW SOLUTION
• Question 30
When the length of the shadow of a pole is equal to the height of the pole, the angle of elevation of the Sun has a measure of…………...........
(A) 30°
(B) 45°
(C) 60°
(D) 75° VIEW SOLUTION
• Question 31
The area of a minor sector of ⊙ (P, 30) is 300 cm2. The length of the corresponding arc in ……………. cm.
(A) 20
(B) 10
(C) 30
(D) 40 VIEW SOLUTION
• Question 32
The volume of a sphere with radius 3 cm is ………………… cm3.
(A) 14π
(B) 18π
(C) 2π
(D) 36π VIEW SOLUTION
• Question 33
Two consecutive even numbers can be………………..
(A) x, x + 1
(B) x, x + 2
(C) x, x – 1
(D) x, 2x VIEW SOLUTION
• Question 34
The area of a sector formed by two mutually perpendicular radii in ⊙ (0, 5 cm) is………………….. cm2.

(A) 4π

(B) 25π

(C) $\frac{4}{25}\mathrm{\pi }$

(D) $\frac{25}{4}\mathrm{\pi }$ VIEW SOLUTION
• Question 35
▭ABCD is cyclic. If m ∠B = 60°, then m∠D = ……………………….
(A) 120°
(B) 100°
(C) 30°
(D) 90° VIEW SOLUTION
• Question 36
The sum of two numbers is 10 and their positive difference is 2. The bigger number is ……………
(A) 8
(B) 4
(C) 2
(D) 6 VIEW SOLUTION
• Question 37
If ………………. then the roots of the quadratic equation are equal.
(A) D = 0
(B) D ≠ 0
(C) D < 0
(D) D > 0 VIEW SOLUTION
• Question 38
In usual notations, Z − M = ………………
(A) 3
(B) 2
(C) 4
(D) 1 VIEW SOLUTION
• Question 39
If then $\mathrm{P}\left(\overline{)\mathrm{C}}\right)=$ ………………………
(A) $\frac{5}{7}$

(B) $\frac{2}{7}$

(C) 0

(D) 1 VIEW SOLUTION
• Question 40
For 2x + 3y = 7 and 3x + 2y = 3, x − y = …………………….
(A) 4
(B) −4
(C) 10
(D) 21 VIEW SOLUTION
• Question 41
Distance between the points (2, −3) and (5, a) is 5. Hence a = …………………
(A) −1
(B) 6
(C) 1
(D) 7 VIEW SOLUTION
• Question 42
The modal class of the following frequency distribution is ………………
 Class 0–10 10–20 20–30 30–40 40–50 Frequency 7 15 13 17 10
(A) 20–30
(B) 10–20
(C) 30–40
(D) 40–50 VIEW SOLUTION
• Question 43
A show-piece, as shown in the figure, is made of a cube and a hemisphere. If the measure of the total surface area of the cube is represented by A, the curved surface area of the hemisphere is represented by B and the area of the base of the hemisphere is represented by C, then …………. is true for the total surface area of the show-piece.

(A) A + B + C
(B) A + B – C
(C) B + C – A
(D) A + C – B VIEW SOLUTION
• Question 44
The distance between A(−6, 7) and B (−1, −5) is ………………
(A) 12
(B) 13
(C) 7
(D) $\sqrt{37}$ VIEW SOLUTION
• Question 45
The discriminant (D) of the equation $5x-6+\frac{1}{x}=0$ is ……………
(A) 4
(B) $\sqrt{56}$
(C) 16
(D) 56 VIEW SOLUTION
• Question 46
In the formula of mean,
(A) fi – A
(B) A – xi
(C) A – f
(D) xi – A VIEW SOLUTION
• Question 47
For $\sqrt{4+\sqrt{83}},$  the correct option is ……………….

(A) does not exist as quadratic surd

(B) does not exist as real numbers

(C)

(D) VIEW SOLUTION
• Question 48
2m . 5n (m, n ∈ N) ends with
(A) 5
(B) 0
(C) 25
(D) 125 VIEW SOLUTION
• Question 49
From the graph given below, y = P(x) has …………… zeros.

(A) 1
(B) 5
(C) 3
(D) 4 VIEW SOLUTION
• Question 50
The zero of the polynomial

(A) $\sqrt{5}$

(B)  $-\sqrt{5}$

(C) $\frac{\sqrt{5}}{5}$

(D) −5 VIEW SOLUTION
• Question 51
Find the square root of $6+4\sqrt{2}.$ VIEW SOLUTION
• Question 52
Find the sum of the zeroes and the product of the zeroes of the quadratic polynomial p(x) = 3x2 + 7x + 4, without finding the zeroes. VIEW SOLUTION
• Question 53
Solve the following pair of equations by cross-multiplication method:
2x – 5y = 4, 3x – 8y = 5 VIEW SOLUTION
• Question 54
(−100) + (−92) + (−84) + . . . + 92

OR

In a given A.P. a = 8, Tn = 33, Sn = 123. Find d and n. VIEW SOLUTION
• Question 55
In ΔABC, m∠B = 90°, $\overline{)\mathrm{BM}}\perp \overline{)\mathrm{AC}},$ M ∈ AC. If AM – MC = 7 and AB2 – BC2 = 175, then find AC. VIEW SOLUTION
• Question 56
Find the distance between A(a + b, b – a) and B(a – b, a + b). VIEW SOLUTION
• Question 57
If A + B = 90°, then prove that

OR

Prove that: $\sqrt{\frac{1-\mathrm{sin\theta }}{1+\mathrm{sin\theta }}}=\mathrm{sec\theta }-\mathrm{tan\theta }$ VIEW SOLUTION
• Question 58
The mean of a data is $\overline{)\mathrm{x}}=35.8.$ If and c = 10, then find the assumed mean (A). VIEW SOLUTION
• Question 59
Solve the given pair of linear equations:

VIEW SOLUTION
• Question 60
The angles of elevation of the top of a tower from two points at a distance a and b from the base, and in the same straight line with it, are complementary. Prove that the height of the tower is $\sqrt{\mathrm{ab}}$. VIEW SOLUTION
• Question 61
There are 5 red, 2 yellow and 3 white roses in a flowerpot. One rose is selected from it at random. What is the probability that the selected rose is (1) red (2) yellow (3) not white colour? VIEW SOLUTION
• Question 62
Find the mean of the following frequency distribution:
 Class 0-50 50-100 100-150 150-200 200-250 250-300 300-350 Frequency 10 15 30 20 15 8 2

OR

Find the median of the following frequency distribution:
​  Class 0-100 100-200 200-300 300-400 400-500 500-600 Frequency 64 62 84 72 66 52
VIEW SOLUTION
• Question 63
Prove that a tangent to a circle is perpendicular to the radius drawn from the point of contact. VIEW SOLUTION
• Question 64
OA and OB are two mutually perpendicular radii of a circle with radius 10.5 cm.
D ∈ OB and OD = 6 cm. Find the area of the shaded region in the figure given below:

VIEW SOLUTION
• Question 65
The cost of painting the surface of a sphere is Rs. 1526 at the rate of Rs. 6 per m2. Find the radius of the sphere. (π = 3.14)

OR

A well of diameter 7 m and 30 m deep is dug and the soil obtained by digging the well is evenly spread out to form a platform of size 30 m × 10 m. Find the height of the platform. VIEW SOLUTION
• Question 66
⊙ (P, 4 cm) is given. Draw a pair of tangents through A, which is in the exterior of the ⊙ (P, 4 cm) such that the measure of the angle between the tangents is 60°. Write the construction steps. VIEW SOLUTION
• Question 67
Prove that the areas of two similar acute triangles are proportional to the squares of the corresponding sides.

OR

In ΔABC, m∠B = 90°. Prove that AC2 = AB2 + BC2. VIEW SOLUTION
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