CBSE Previous Year Question Paper Class 10 Maths 2018

Time Allowed: 3 hours Maximum Marks: 80

General Instructions:

  • All questions are compulsory.

  • This question paper consists of 30 questions divided into four sections- A, B, C and D.

  • Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 8 questions of 4 marks each.

  • There is no overall choice. However, an internal choice has been provided in two questions of 1 mark each, two questions of 2 marks each, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.

  • Use of calculators is not permitted.

Section – A

Question 1.
If x = 3 is one root of the quadratic equation x2 – 2kx – 6 = 0, then find the value of k. [1]
Solution:
Given quadratic equation is, x2 – 2kx – 6 = 0
x = 3 is a root of above equation, then
(3)2 – 2k (3) – 6 = 0
⇒ 9 – 6k – 6 = 0
⇒ 3 – 6k = 0
⇒ 3 = 6k
⇒ k = =
⇒ k =

Question 2.
What is the HCF of the smallest prime number and the smallest composite number? [1]
Solution:
Smallest prime number = 2
Smallest composite number = 4
Prime factorisation of 2 is 1 × 2
Prime factorisation of 4 is 1 × 22
HCF (2, 4) = 2

Question 3.
Find the distance of a point P(x, y) from the origin. [1]
Solution:
The given point is P (x, y).
The origin is O (0, 0)
The distance of point P from the origin,

Question 4.
In an AP if the common difference (d) = -4 and the seventh term (a7) is 4, then find the first term. [1]
Solution:
Given,
d = -4, a7 = 4
a + 6d = 4
⇒ a + 6(-4) = 4
⇒ a – 24 = 4
⇒ a = 4 + 24
⇒ a = 28

Question 5.
What is the value of (cos2 67° – sin2 23°) ? [1]
Solution:
We have, cos2 67° – sin2 23°
= cos2 67° – cos2 (90° – 23°) [∵ sin (90° – θ) = cos θ]
= cos2 67° – cos2 67°
= 0

Question 6.
Given ΔABC ~ ΔPQR,
Solution:

Section – B

Question 7.
Given that √2 is irrational, prove that (5 + 3√2) is an irrational number. [2]
Solution:
Given, √2 is an irrational number.
Let √2 = m
Suppose, 5 + 3√2 is a rational number.

But is rational number, so m is rational number which contradicts the fact that m = √2 is an irrational number.
So, our supposition is wrong.
Hence, 5 + 3√2 is also irrational.
Hence Proved.

Question 8.
In fig. 1, ABCD is a rectangle. Find the values of x and y. [2]

Solution:
Given, ABCD is a rectangle.
AB = CD
⇒ 30 = x + y
or x + y = 30 …(i)
Similarly, AD = BC
⇒ 14 = x – y
or x – y = 14 …(ii)
On adding eq. (i) and (ii), we get
2x = 44
⇒ x = 22
Putting the value of x in eq. (i), we get
22 + y = 30
⇒ y = 30 – 22
⇒ y = 8
So, x = 22, y = 8.

Question 9.
Find the sum of the first 8 multiples of 3. [2]
Solution:
First 8 multiples of 3 are 3, 6, 9,….. up to 8 terms
We can observe that the above series is an AP with
a = 3, d = 6 – 3 = 3, n = 8
Sum of n terms of an A.P is given by,

Question 10.
Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, -3). Hence find m. [2]

Solution:
Let P divides line segment AB in the ratio k : 1
Coordinates of P

Question 11.
Two different dice are tossed together. Find the probability:
(i) of getting a doublet.
(ii) of getting a sum 10, of the numbers on the two dice. [2]
Solution:
Total outcomes on tossing two different dice = 36
(i) A: getting a doublet
A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
Number of favourable outcomes of A = 6

(ii) B: getting a sum 10.
B = {(4, 6), (5, 5), (6, 4)}
Number of favourable outcomes of B = 3

Question 12.
An integer is chosen at random between 1 and 100. Find the probability that it is:
(i) divisible by 8.
(ii) not divisible by 8. [2]
Solution:
The total numbers are 2, 3, 4, …….. 99
(i) Let E be the event of getting a number divisible by 8.
E = {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96} = 12

(ii) Let E’ be the event of getting a number not divisible by 8.
Then, P(E’) = 1 – P(E) = 1 – 0.1224 = 0.8756

Section – C

Question 13.
Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers. [3]
Solution:

Prime factorization of 404 = 2 × 2 × 101
Prime factorization of 96 = 2 × 2 × 2 × 2 × 2 × 3
HCF = 2 × 2 = 4
And LCM = 2 × 2 × 2 × 2 × 2 × 3 × 101 = 9696
HCF = 4, LCM = 9696
Verification:
HCF × LCM = Product of the two given numbers
4 × 9696 = 404 × 96
38784 = 38784
Hence Verified.

Question 14.
Find all zeros of the polynomial (2x4 – 9x3 + 5x2 + 3x – 1) if two of its zeroes are (2 + √3) and (2 – √3). [3]
Solution:
Here, p(x) = 2x4 – 9x3 + 5x2 + 3x – 1
And two of its zeroes are (2 + √3) and (2 – √3).
Quadratic polynomial with zeros is given by,
{x – (2 + √3)}. {x – (2 – √3)}
⇒ (x – 2 – √3) (x – 2 + √3)
⇒ (x – 2)2 – (√3)2
⇒ x2 – 4x + 4 – 3
⇒ x2 – 4x + 1 = g(x) (say)
Now, g(x) will be a factor of p(x) so g(x) will be divisible by p(x)

For other zeroes,
2x2 – x – 1 = 0
2x2 – 2x + x – 1 = 0
or 2x (x – 1) + 1 (x – 1) = 0
(x – 1) (2a + 1) = 0
x – 1 = 0 and 2x + 1 = 0
x = 1, x =
Zeros of p(x) are
1, , 2 + √3 and 2 – √3.

Question 15.
If A(-2, 1) and B(a, 0), C(4, b) and D( 1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides. [3]
OR
If A(-5, 7), B(-4, -5), C(-1, -6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.
Solution:
Given ABCD is a parallelogram.




Question 16.
A plane left 30 minutes later than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed. [3]
Solution:
Let the usual speed of plane be x km/h.
Increased speed = (x + 100) km/h.
Distance to cover = 1500 km.
Time taken by plane with usual speed = hr
Time taken by plane with increased speed =
According to the question,

x2 + 100x = 300000
x2 + 100x – 300000 = 0
x2 + 600x – 500x – 300000 = 0
x(x + 600) – 500(x + 600) = 0
(x + 600) (x – 500) = 0
Either x + 600 = 0 ⇒ x = -600 (Rejected)
or x – 500 = 0 ⇒ x = 500
Usual speed of plane = 500 km/hr.

Question 17.
Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal. [3]
OR
If the area of two similar triangles is equal, prove that they are congruent.
Solution:
Let ABCD be a square with side ‘a’.


Question 18.
Prove that the lengths of tangents drawn from an external point of a circle are equal. [3]
Solution:
Given: A circle with centre O on which two tangents PM and PN are drawn from an external point P.

To Prove: PM = PN
Construction: Join OM, ON and OP
Proof: Since tangent and radius are perpendicular at point of contact,
∠OMP = ∠ONP = 90°
In ΔPOM and ΔPON,
OM = ON (Radii)
∠OMP = ∠ONP
PO = OP (Common)
ΔOMP = ΔONP (RHS cong.)
PM = PN (C.P.C.T)
Hence Proved.

Question 19.
If 4 tan θ = 3, evaluate
or
If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.
Solution:
Given, 4 tan θ = 3
⇒ tan θ = 3/4 (= P/H)


OR
Given, tan 2A = cot (A – 18°)
⇒ cot (90° – 2A) = cot (A – 18°)
[∵ tan θ = cot (90° – θ)]
⇒ 90° – 2A = A – 18°
⇒ 90° + 18° = A + 2A
⇒ 108° = 3A
⇒ A = 36°

Question 20.
Find the area of the shaded region in Fig. 2, where arcs are drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14] [3]

Solution:
Given, ABCD is a square of side 12 cm.

P, Q, R and S are the midpoints of sides AB, BC, CD and AD respectively.
Area of shaded region = Area of square – 4 × Area of quadrant
= a2 – 4 × πr2
= (12)2 – 3.14 × (6)2
= 144 – 3.14 × 36
= 144 – 113.04
= 30.96 cm2

Question 21.
A wooden article was made by scooping out a hemisphere form each end of a solid cylinder, as shown in Fig. 3. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm. Find the total surface area of the article. [3]

OR
A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap?
Solution:

Given, Radius (r) of cylinder = Radius of hemisphere = 3.5 cm.
Total SA of article = CSA of cylinder + 2 × CSA of hemisphere
Height of cylinder, h = 10 cm
TSA = 2πrh + 2 × 2πr2
= 2πrh + 4πr2
= 2πrh (h + 2r)
= 2 × × 3.5 (10 + 2 × 3.5)
= 2 × 22 × 0.5 × (10 + 7)
= 2 × 11 × 17
= 374 cm2
OR
Base diameter of cone = 24 m.
Radius r = 12 m
Height of cone, h = 3.5 m
Volume of rice in conical heap = πr2h
= × × 12 × 12 × 3.5 = 528 cm3

Question 22.
The table below shows the salaries of 280 persons: [3]

Calculate the median salary of the data.
Solution:

= = 140
The cumulative frequency just greater than 140 is 182.
Median class is 10 -15.
l = 10, h = 5, N = 280, c.f. = 49 and f = 133

Section – D

Question 23.
A motorboat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. [4]
OR
A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed?
Solution:
Given, speed of motorboat instil
water = 18 km/hr.
Let speed of stream = x km/hr.
Speed of boat downstream = (18 + x) km/hr.
And speed of boat upstream = (18 – x) km/hr.
Time of the upstream journey =
Time of the downstream journey =
According to the question,

⇒ x2 + 48x – 324 = 0
⇒ x2 + 54x – 6x – 324 = 0
⇒ x(x + 54) – 6(x + 54) = 0
⇒ (x + 54)(x – 6) = 0
Either x + 54 = 0 ⇒ x = -54
Rejected, as speed cannot be negative
or x – 6 = 0 ⇒ x = 6
Thus, the speed of the stream is 6 km/hr.
OR
Let the original average speed of train be x km/hr.
Increased speed of train = (x + 6) km/hr.
Time taken to cover 63 km with average speed = hr.
Time taken to cover 72 km with increased speed =
According to the question,

⇒ 135x + 378 = 3(x2 + 6x)
⇒ 135x + 378 = 3x2 + 18x
⇒ 3x2 + 18x – 135x – 378 = 0
⇒ 3x2 – 117x – 378 = 0
⇒ 3(x2 – 39x – 126) = 0
⇒ x2 – 39x – 126 = 0
⇒ x2 – 42x + 3x – 126 – 0
⇒ x(x – 42) + 3(x – 42) = 0
⇒ (x – 42) (x + 3) = 0
Either x – 42 = 0 ⇒ x = 42
or x + 3 = 0 ⇒ x = -3
Rejected (as speed cannot be negative)
Thus, the average speed of the train is 42 km/hr.

Question 24.
The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7:15. Find the numbers. [4]
Solution:
Let the first term of AP be a and d be a common difference.
Let your consecutive term of an AP be a – 3d, a – d, a + d and a + 3d
According to the question,
a – 3d + a – d + a + d + a + 3d = 32
⇒ 4a = 32
⇒ a = 8 …(i)
Also,
(a – 3d) (a + 3d) : (a – d) (a + d) = 7 : 15

For d = 2, four terms of AP are,
a – 3d = 8 – 3 (2) = 2
a – d = 8 – 2 = 6
a + d = 8 + 2 = 10
a + 3d = 8 + 3(2) = 14
For d = -2, four terms are
a – 3d = 8 – 3(-2) = 14
a – d = 8 – (-2) = 10
a + d = 8 + (-2) = 6
a + 3d = 8 + 3 (-2) = 2
Thus, the four terms of AP series are 2, 6, 10, 14 or 14, 10, 6, 2.

Question 25.
In an equilateral ∆ABC, D is a point on the side BC such that BD = (1/3)BC. Prove that 9(AD)2 = 7(AB)2. [4]
OR
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Solution:

Given, ABC is an equilateral triangle and D is a point on BC such that BD = BC.
To prove: 9AD2 = 7AB2
Construction : Draw AE ⊥ BC
Proof: BD = BC …(i) (Given)
AE ⊥ BC
We know that perpendicular from a vertex of an equilateral triangle to the base divides base in two equal parts.
BE = EC = BC …(ii)
In ∆AEB,
AD2 = AE2 + DE2 (Pythagoras theorem)
or AE2 = AD2 – DE2 …(iii)
Similarly, In ∆AEB,
AB2 = AE2 + BE2


OR
Given: ∆ABC is a right angle triangle, right-angled at A.

To prove : BC2 = AB2 + AC2
Construction : Draw AD ⊥ BC.
Proof: In ∆ADB and ∆BAC,
∠B = ∠B (Common)
∠ADB = ∠BAC (Each 90°)
∆ADB ~ ∆BAC (By AA similarity axiom)
= (CPCT)
AB2 = BC × BD
Similarly,
∆ADC ~ ∆CAB
=
AC2 = BC × DC …(ii)
On adding equation (i) and (ii)
AB2 + AC2 = BC × BD + BC × CD = BC (BD + CD) = BC × BC
AB2 + AC2 = BC2
BC2 = AB2 + AC2
Hence Proved.

Question 26.
Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the ∆ABC. [4]
Solution:

  1. Draw a line segment BC = 6 cm.

  2. Construct ∠XBC = 60°.

  3. With B as centre and radius equal to 5 cm, draw an arc intersecting XB at A.

  4. Join AC. Thus, ∆ABC is obtained.

  5. Draw an acute angle ∠CBY below of B.

  6. Mark 4-equal parts on BY as B1, B2, B3 and B4

  7. Join B4 to C.

  8. From By draw a line parallel to B4C intersecting BC at C’.

  9. Draw another line parallel to CA from C’, intersecting AB at A’.

  10. ∆A’BC’ is required triangle which is similar to ∆ABC such that BC’ = BC.

Question 27.

Solution:

Question 28.
The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find:
(i) The area of the metal sheet used to make the bucket.
(ii) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14] [4]
Solution:
Given, Height of frustum, h = 24 cm.
Diameter of lower end = 10 cm
Radius of lower end, r = 5 cm.
Diameter of upper end = 30 cm
Radius of upper end, R = 15 cm.

(i) Area of metal sheet used to make the bucket = CSA of frustum + Area of base
= πl(R + r) + πr2
= π[26 (15 + 5) + (5)2]
= 3.14 (26 × 20 + 25)
= 3.14 (520 + 25)
= 3.14 × 545
= 1711.3 cm2
(ii) We should avoid the bucket made by ordinary plastic because plastic is harmful to the environment and to protect the environment, its use should be avoided.

Question 29.
As observed from the top of a 100 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use √3 = 1.732] [4]
Solution:
Let AB be the lighthouse and two ships are at C and D.

Distance between two ships = y – x
= 100√3 – 100 [from equation (i) and (ii)]
= 100 (√3 – 1)
= 100(1.732 – 1)
= 100 (0.732)
= 73.2 m

Question 30.
The mean of the following distribution is 18. Find the frequency f of the class 19-21. [4]

OR
The following distribution gave the daily income of 50 workers of a factory:

Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.
Solution: