Board Paper of Class 10 2014 Maths (SET 3) - Solutions

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.
(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.
(iv) Use of calculators is not permitted.

Question 1
Two different coins are tossed simultaneously. The probability of getting at least one head is
(A) $\frac{1}{4}$

(B) $\frac{1}{8}$

(C) $\frac{3}{4}$

(D) $\frac{7}{8}$ VIEW SOLUTION

Question 2
A bag contains cards numbered from 1 to 25. A card is drawn at random from the bag. The probability that the number on this card is divisible by both 2 and 3 is

(A) $\frac{1}{5}$

(B) $\frac{3}{25}$

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

(D) $\frac{2}{25}$ VIEW SOLUTION

Question 3
If the height of a vertical pole is $\sqrt{3}$ times the length of its shadow on the ground, then the angle of elevation of the Sun at that time is
(A) 30°
(B) 60°
(C) 45°
(D) 75° VIEW SOLUTION

Question 4
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is
(A) $7 + \sqrt{5}$
(B) 5
(C) 10
(D) 12 VIEW SOLUTION

Question 5
A rectangular sheet of paper 40 cm × 22 cm, is rolled to form a hollow cylinder of height 40 cm. The radius of the cylinder (in cm) is
(A) 3·5

(B) 7

(C) $\frac{80}{7}$

(D) 5 VIEW SOLUTION

Question 6
In Figure 1, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm and AS = 5 cm, find x.

(A) 10
(B) 9
(C) 8
(D) 7 VIEW SOLUTION

Question 7
The next term of the A.P. $\sqrt{7}, \sqrt{28}, \sqrt{63},$ ... is
(A) $\sqrt{70}$
(B) $\sqrt{84}$
(C) $\sqrt{97}$
(D) $\sqrt{112}$ VIEW SOLUTION

Question 8
Two concentric circles are of radii 5 cm and 3 cm. Length of the chord of the larger circle, (in cm), which touches the smaller circle is
(A) 4
(B) 5
(C) 8
(D) 10 VIEW SOLUTION

Question 9
In Figure 2, XP and XQ are two tangents to the circle with centre O, drawn from an external point X. ARB is another tangent, touching the circle at R. Prove that XA + AR = XB + BR.
VIEW SOLUTION

Question 10
In Figure 3, OABC is a quadrant of a circle of radius 7 cm. If OD = 4 cm, find the area of the shaded region. $[Useπ = \frac{22}{7}]$
VIEW SOLUTION

Question 11
Solve for x:
$\sqrt{3} x^{2} - 2 \sqrt{2} x - 2 \sqrt{3} = 0$ VIEW SOLUTION

Question 12
Two different dice are rolled simultaneously. Find the probability that the sum of numbers appearing on the two dice is 10. VIEW SOLUTION

Question 13
Prove that the tangents drawn at the ends of any diameter of a circle are parallel. VIEW SOLUTION

Question 14
The sum of the first n terms of an A.P. is 4n² + 2n. Find the n^th term of this A.P. VIEW SOLUTION

Question 15
In Figure 4, from a rectangular region ABCD with AB = 20 cm, a right triangle AED with AE = 9 cm and DE = 12 cm, is cut off. On the other end, taking BC as diameter, a semicircle is added on outside the region. Find the area of the shaded region. $[Useπ = 3.14]$
VIEW SOLUTION

Question 16
In Figure 5, ABCD is a quadrant of a circle of radius 28 cm and a semi circle BEC is drawn with BC as diameter. Find the area of the shaded region. $[Useπ = \frac{22}{7}]$
VIEW SOLUTION

Question 17
Points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R. VIEW SOLUTION

Question 18
The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28^th term of this A.P. VIEW SOLUTION

Question 19
A 5 m wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of Rs 25 per metre.
$[Use π = \frac{22}{7}]$ VIEW SOLUTION

Question 20
A girl empties a cylindrical bucket, full of sand, of base radius 18 cm and height 32 cm, on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct upto one place of decimal. VIEW SOLUTION

Question 21
Two ships are approaching a light-house from opposite directions. The angles of depression of the two ships from the top of the light-house are 30° and 45°. If the distance between the two ships is 100 m, find the height of the light-house. $[Use \sqrt{3} = 1.732]$ VIEW SOLUTION

Question 22
If 1 is a root of the quadratic equation 3x² + ax – 2 = 0 and the quadratic equation a(x² + 6x) – b = 0 has equal roots, find the value of b. VIEW SOLUTION

Question 23
Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°. VIEW SOLUTION

Question 24
Find the value(s) of p for which the points (3p + 1, p), (p + 2, p – 5) and (p + 1, –p) are collinear. VIEW SOLUTION

Question 25
The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 m, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be 100 m. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question? VIEW SOLUTION

Question 26
A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal. VIEW SOLUTION

Question 27
The mid-point P of the line segment joining the points A(−10, 4) and B(−2, 0) lies on the line segment joining the points C(−9, −4) and D(−4, y). Find the ratio in which P divides CD. Also find the value of y. VIEW SOLUTION

Question 28
A hemispherical depression is cut out from one face of a cubical block of side 7 cm, such that the diameter of the hemisphere is equal to the edge of the cube. Find the surface area of the remaining solid. [Use π = $\frac{22}{7}$ ] VIEW SOLUTION

Question 29
If S_n denotes the sum of the first n terms of an A.P., prove that S₃₀ = 3 (S₂₀ − S₁₀). VIEW SOLUTION

Question 30
A metallic bucket, open at the top, of height 24 cm is in the form of the frustum of a cone, the radii of whose lower and upper circular ends are 7 cm and 14 cm respectively. Find :
(i) the volume of water which can completely fill the bucket.
(ii) the area of the metal sheet used to make the bucket.
[Use π = $\frac{22}{7}$ ] VIEW SOLUTION

Question 31
The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples. VIEW SOLUTION

Question 32
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION

Question 33
A piggy bank contains hundred 50 p coins, fifty Rs 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability that the coin which fell
(i) will be a 50 p coin.
(ii) will be of value more than Rs 1.
(iii) will be of value less than Rs 5.
(iv) will be a Rs 1 or Rs 2 coin. VIEW SOLUTION

Question 34
Solve for x:
$3 (\frac{7 x + 1}{5 x - 3}) - 4 (\frac{5 x - 3}{7 x + 1}) = 11; x \neq \frac{3}{5}, - \frac{1}{7}$ VIEW SOLUTION