430/2/1 2020 Class 10 Maths Basic Question Paper Solution
MATHEMATICS
(BASIC)
Section – A
Question numbers 1 to 10 are multiple choice questions of 1 mark each. Select the correct option.
1) HCF of two numbers is 27 and their LCM is 162. If one of the number is 54, then the other number is
(a) 36 (b) 35 (c) 9 (d) 81
Ans: (d) 81
2) The cumulative frequency table is useful in determining
(a) Mean (b) Median (c) Mode (d) All of these
Ans: (b) Median
3) In Fig. 1, O is the centre of circle. PQ is a chord and PT is tangent at P which makes an angle of 50° with PQ. Ð POQ is
(a) 130° (b) 90° (c) 100° (d) 75°
Ans: (c) 100°
4) 2√ 3 is
(a) an integer (b) a rational number
(c) an irrational number (d) a whole number
Ans: (c) an irrational no.
5) Two coins are tossed simultaneously. The probability of getting at most one head is
(a) 1/4 (b)1/ 2 (c) 2/3 (d) 3/4
Ans: (d) 3/4
6) If one zero of the polynomial (3x2 + 8x + k) is the reciprocal of the other, then value of k is
(a) 3 (b) –3 (c) 1/3 (d) – 1/3
Ans: (a) 3
7) The decimal expansion of 23/25×52 will terminate after how many places of decimal ?
(a) 2 (b) 4 (c) 5 (d) 1
Ans: (c) 5
8) The maximum number of zeroes a cubic polynomial can have, is
(a) 1 (b) 4 (c) 2 (d) 3
Ans: (d) 3
9) The distance of the point (–12, 5) from the origin is
(a) 12 (b) 5 (c) 13 (d) 169
Ans: (c) 13
10) If the centre of a circle is (3, 5) and end points of a diameter are (4, 7) and (2, y), then the value of y is
(a) 3 (b) –3 (c) 7 (d) 4
Ans: (a) 3
Question numbers 11 to 15, fill in the blanks :
11) The area of triangle formed with the origin and the points (4, 0) and (0, 6) is ________.
Ans: 12 sq units
OR
The co-ordinate of the point dividing the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2 : 1 is ________.
Ans: (3, 5)
12) Value of the roots of the quadratic equation, x 2 – x – 6 = 0 are ________.
Ans: 3 and –2
13) If sin q =5/13 then the value of tan q is ________.
Ans: tan θ =5/12
14) The value of (tan260° + sin245°) is ________.
Ans: 7/2 or 3.5
15) The corresponding sides of two similar triangles are in the ratio 3 : 4, then the ratios of the area of triangles is _________.
Ans: 9 : 16
Question numbers 16 to 20, answer the following :
16) Find the value of (cos 48° – sin 42°).
Ans: cos 48° – cos (90 – 42°)
cos 48° – cos 48° = 0
OR
Evaluate : (tan 23°) × (tan 67°)
Ans: tan (90 – 67°) × tan 67°
cot 67° × tan 67°
=1
17)
Ans: Area of shaded region =22/7×30°/360°(72-(3.5)2)
= 9.625 cm2
18) A card is drawn at random from a well shuffled deck of 52 playing cards. What is the probability of getting a black king ?
Ans: P(Black king) =2/52 or 1/26
19) A ladder 25 m long just reaches the top of a building 24 m high from the ground. What is the distance of the foot of ladder from the base of the building ?
Ans: Distance =√(25)2-(24)2=7m
20) If 3k – 2, 4k – 6 and k + 2 are three consecutive terms of A.P., then find the value of k.
Ans: (4k – 6) – (3k – 2) = (k + 2) – (4k – 6)
=>k = 3
Section – B
Question numbers 21 to 26 carry 2 marks each.
21) In a lottery, there are 10 prizes and 25 blanks. What is the probability of getting a prize ?
Ans: Total = 10 + 25 = 35, P(getting prize) =10/35 or 2/7
22) In a family of three children, find the probability of having at least two boys.
Ans: Total outcomes = 8 {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
P(atleast 2 boys) =4/8 or 1/2
OR
Two dice are tossed simultaneously. Find the probability of getting
(i) an even number on both dice.
(ii) the sum of two numbers more than 9.
Ans: Total outcomes = 36
P(even no. on both side) =9/36 or 1/4
P(sum > 9) = 6/36 or 1/6
23) Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of larger circle which touches the smaller circle.
Ans:
In ΔOCB
BC=√52-32=4cm
AB = 2 × BC = 8 cm
24) Prove that :1/1+ sinθ+1/1-sinθ=2 sec2θ
Ans: L.H.S=1/1+sinθ+1/1-sinθ=1-sin θ+1+sinθ/(1+sinθ)(1-sinθ)
= 2/1-sin2θ=2/cos2θ
=2sec2 θ
OR
Prove that :1-tan2θ/1+ tan2θ=cos2θ-sin2θ
Ans: L.H.S=1-tan2 θ/1+tan2 θ=1-sin2 θ/cos2 θ/1+sin2 θ/cos2 θ=cos2 θ-sin2 θ/cos2 θ+sin2 θ
= cos2 θ – sin2 θ
25) The wheel of a motorcycle is of radius 35 cm. How many revolutions are required to travel a distance of 11 m ?
Ans: Distance in 1 revolution =2 ×22/7 ×35=220cm
No. of revolution = 1100/220=5
26) Divide (2x 2 – x + 3) by (2 – x) and write the quotient and the remainder.
Ans:
Quotient =-2x- 3
R=9
Section – C
Question numbers 27 to 34 carry 3 marks each.
27) If α and β are the zeroes of the polynomial f(x) = 5x 2 – 7x + 1, then find the value of(α/ β+ β/ α)
Ans: α+β=7/5 and αβ=1/5
α/ β+ β/ α= α2+ β2/ αβ=( α+β)2-2 αβ/ αβ
=(7/5)2-2 ×1/5/1/5
= 39/5 or 7.8
28) Draw a line segment of length 7 cm and divide it in the ratio 2 : 3.
Ans: Correct construction
OR
Draw a circle of radius 4 cm and construct the pair of tangents to the circle from an external point, which is at a distance of 7 cm from its centre.
Ans: Correct construction
29) The minute hand of a clock is 21 cm long. Calculate the area swept by it and the distance travelled by its tip in 20 minutes.
Ans: Angle in 20 min = 120°
Area =22/7×120°/360°×(21)2=462cm2
Distance=120°/360°×2πr=44cm
30) If x = 3 sin q + 4 cos q and y = 3 cos q – 4 sin q then prove that x 2 + y2 = 25.
Ans: x2 = 9 sin2 q + 16 cos2 q+ 24 sin q cosq
y2 = 9 cos2q + 16 sin2q – 24 sinq cos q
x2 + y2 = 25
OR
If sin q + sin2q = 1; then prove that cos2q + cos4q = 1.
Ans: sin θ = 1 – sin2 θ = cos2 θ
L.H.S = cos2 θ + (cos2 θ) 2 = cos2 θ + sin2 θ
= 1 = R.H.S
31) Prove that √3 is an irrational number.
Ans: Let √3 be a rational number
√3=p/qp, q are coprime, q ≠ 0
3q2 = p2 => 3 | p2 => 3 | p Let p = 3 m
3q2 = 9m2 => q2 = 3m2 => 3 | q2 => 3 |q
3 is common factor of p and q
Contraction to our assumption
Hence √3 is irrational No.
OR
Using Euclid’s algorithm, find the HCF of 272 and 1032.
Ans: 1032 = 272 × 3 + 216
272 = 216 × 1 + 56
216 = 56 × 3 + 48
56 = 48 × 1 + 8
48 = 8 × 6 + 0
HCF(1032, 272) = 8
32) In a rectangle ABCD, P is any interior point. Then prove that PA2 + PC2 = PB2 + PD2
Ans:
Correct figure & Construction
In rt △APX AP2 = AX2 + PX2
In rt △PCY PC2 = PY2 + YC2
In rt △PBY PB2 = PY2 + BY2
In rt △PXD PD2 = DX2 + PX2
PA2 + PC2 = AX2 + PX2 + PY2 + YC2
= BY2 + PY2 + PX2 + XD2
= PB2 + PD2
33) In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 3. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square ?” Chameli disagrees. Using distance formula, find which of them is correct.
Ans: A = (3, 4), B = (6, 7), C = (9, 4), D = (6, 1)
AB = 3 √2, BC= 3√ 2, CD= 3 √2, DA= 3√ 2
AC = 6 unit BD = 6 unit
AB = BC = CD = DA and AC = BD
ABCD is a square
∴ Champa is correct
34) Solve graphically :
2x – 3y + 13 = 0; 3x – 2y + 12 = 0
Ans: Correct graph of 2x – 3y + 13 = 0, 3x – 2y + 12 = 0
Solution x = –2, y = 3
Section – D
35) The product of two consecutive positive integers is 306. Find the integers.
Ans: Let two consecutive integers x, x + 1
x(x + 1) = 306 => x2 + x – 306 = 0
=> (x + 18) (x – 17) = 0
=> x = –18, (Rejected), 17
∴ Two consecutive integers 17, 18
36) The 17th term of an A.P. is 5 more than twice its 8th term. If 11th term of A.P. is 43; then find its nth term.
Ans: a17 = 2a8 + 5 => a + 16d = 2(a + 7d) + 5
=>2d – a = 15
a11 = 43 => a + 10d = 43
Solving (1) & (2) a = 3 d = 4
an = 4n – 1
OR
How many terms of A.P. 3, 5, 7, 9, … must be taken to get the sum 120 ?
Ans: a = 3, d = 3, Sn = 120
n/2[2 ×3+(n-1)2]=120=>n2+2n-120=0
(n + 12) (n – 10) = 0
n = –12, n = 10
Reject n = –12, n = 10
37) A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on opposite bank is 60°. When he moves 30 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and width of the river. [Take √3 = 1.732]
Ans:
Correct figure
In right ΔABC
tan 60° = h/ x
√3x = h
In rt ΔABD tan 30°=h/30+x=>30+x/√3=h
Solving (1) & (2) x = 15m, h = 15√3m=25.98m
38) Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Ans: Correct Fig., given, to prove, construction
Correct proof given, to prove, construction,
OR
Prove that the length of tangents drawn from an external point to a circle are equal.
Ans: Correct Fig., given, to prove, construction
Correct proof given, to prove, construction,
39) From a solid cylinder whose height is 15 cm and the diameter is 16 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of remaining solid. (Give your answer in terms of π)
Ans:
l = 17
r = 8 cm
Total S.A. of remaining solid= C.S.A of cylinder + C.S.A of cone + Area of base
= 2πrh + πrl + πr2 = πr(2h + l + r)
= π × 8(2 × 15 + 17 + 18) = 8π(55) = 440π cm2
OR
The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts.
Ans:
ΔOAB ~ ΔOCD
OA /OC=AB/CD=>5/10=r/R
=>R = 2r
V of cone /V of frustum =1/3πr25/1/3π(r2+R2+rR)5=r2/7r2=1/7
or
7:1
40) The mode of the following frequency distribution is 36. Find the missing frequency (f).
Class |
0 – 10 | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 | 50 – 60 | 60 – 70 |
Frequency |
8 | 10 | f | 16 | 12 | 6 | 7 |
Ans: Modal class 30 – 40
l = 30 f0 = f f1 = 16 f2 = 12 h = 10
Mode=1+f1-f0/2f1-f0-f2×h
36 = 30+16-f /32 -f -12 ×10
f = 10
CBSE Class 10 Previous Question Paper 2020 Solution
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Solution |
430/2/2 | |