ICSE Class 9 Maths Sample Question Paper 3 with Answers

Section – A
(Attempt all questions from this Section)

Question 1.
(a) Factorize : 8(a – 2b)2 – 2a + 4b – 1
Answer:
8 (a – 2b)2 -2a + 4b – 1 = 8 (a- 2b)2 -2(a – 2b) – 1
a – 2b = x
Then, the expression becomes
= 8x2 – 2x – 1
= 8x2 – 4x + 2x – 1
= 4x (2x – 1) + 1 (2x – 1)
= (2x – 1) (4x + 1)
= {2 (a- 2b) – 1} {4 (a – 2b) + 1} (Putting x-a-2b)
= (2a – 4b – 1) (4a- 8b + 1)

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(b) The mean of 6 observations is 17.5. If five of them are 14, 9, 23, 25 and 10, find the sixth observation.
Answer:
Mean of 6 observations is 17.5
5 observations are 14, 9, 23, 25, 10
Let 6th observation be x
ICSE Class 9 Maths Sample Question Paper 3 with Answers 5

(c) If θ is an acute angle and sin θ = cos θ, find the value of 2 tan2 θ + sin2 θ-1.
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 6

ICSE Class 9 Maths Sample Question Paper 3 with Answers

Question 2.
(a) A man borrowed ₹15,000 for 2 years. The rate of interest for the two successive years are 8% and 10% respectively. If he repays ₹6,200 at the end of first year, find the outstanding amount at the end of the second year.
Answer:
(a) For 1st year : P = ₹ 15,000, R = 8% p.a.
\(\mathrm{I}=\frac{15000 \times 8 \times 1}{100}\) = ₹ 16,200
A = P + I = 15,000 + 1,200 =₹16,200
Amount of money repaid = ₹ 6,200
For 2nd year : P = 16,200 – 6,200 = ₹10,000, R = 10% p.a.
\(\mathrm{I}=\frac{10,000 \times 10 \times 1}{100}\) = = ₹1,000
∴ A =P+I=10,000+1,000=11,000
∴ The amount outstanding at the end of 2nd year = 11, 000.

(b) Solve for x if log2 (x2 – 4) = 5.
Answer:
Given: log2(x2 – 4) = 5
x2 – 4 =25
x2 =32 + 4
X = ±√36= ±6.

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(c) In the following figure, AB is a diameter of a circle with centre O. If chord AC = chord AD,
prove that (i) arc BC = arc DB (ii) AB is bisector of ∠CAD.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 1

Answer:
(i) Given: chord AC = chord AD
⇒ arc AC = arc AD …(i)
Also, arc ACB = arc ADB (AB is a diameter) …(ii)
Subtracting (i) from (ii),
arc ACB – arc AC = arc ADB – arc AD
⇒ arc BC = arc DB. Hence Proved.

(ii) ∵ arc BC = arc BD (Proved above)
∴ ∠BAC = ∠DAB
⇒ AB is bisector of ∠CAD.
Hence Proved.

ICSE Class 9 Maths Sample Question Paper 3 with Answers

Question 3.
(a) Prove that: \(\frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}=\frac{3}{2}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 7

(b) Given that 16 cot A = 12, find the value of \(\frac{\sin A+\cos A}{\sin A-\cos A}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 8

(c) If the altitudes from two vertices of a triangle to the opposite sides are equal, prove that the triangle is isosceles.
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 9

ICSE Class 9 Maths Sample Question Paper 3 with Answers

Question 4.
(a) If a + b + 2c = 0, prove that a3 + b3 + 8c3 – 6abc = 0.
Answer:
Given : a + b + 2c = 0
⇒ a + b = – 2c
Cubing both sides, we get
(a + b)3 = (- 2c)3
⇒ a3 + b3 + 3ab (a + b) = – 8c3
⇒ a3 + b3 + 3ab (- 2c) = – 8c3
⇒ a3 + b3 + 8c3 – 6abc = 0.
Hence Solved.

(b) Express \(0.1 \overline{34}\) in the form \(\frac{p}{q}\) ,p, q ∈ Z and q ≠ 0.
Answer:
Let x = \(0.1 \overline{34}\) = 0.1343434…
Multiplying both sides of (i) by 10, we get
10x = 1.343434 ………(i)
Multiplying both sides of (ii) by 100, we get
1000x = 134.3434 ………….(ii)
Subtracting (ii) from (iii), we get
1000x – 10x= 134.3434 … – 1.3434 ……………
ICSE Class 9 Maths Sample Question Paper 3 with Answers 10

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(c) If the sides are in the ratio 5 : 3 : 4, prove that it is a right angled triangle.
Answer:
Ratio of sides = 5:3:4
Let the length of sides be 5x, 3x, 4x.
Here,(3x)2 + (4x)2 = 9x2 + 16x2 = 25x2 = (5x)2
∴ By Pythagoras theorem, the triangle is right angled
Hence Proved.

Section – B
(Attempt any four questions from this Section)

Question 5.
(a) On what sum of money will the difference between compound interest and simple inter­est for 2 years be equal to ₹25 if the rate of interest charged for both is 5% p.a. ?
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 12

(b) Show by distance formula that the points A (-1, -1), B (2, 3) and C (8,11) are collinear.
Answer:
Given points are A (-1, -1), B (2, 3), C (8, 11).
Now
ICSE Class 9 Maths Sample Question Paper 3 with Answers 11
AB + BC = 5 + 10 = 15
⇒ AB + BC = AC
The points are collinear.
Hence Proved.

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(c) Factorize : a6 – 26a3 – 27.
Answer:
a6 – 26a3 – 27 = a6 – (27 – 1) a3-27 = a6 – 27a3 + a3 – 27
= a3 (a3 – 27) + 1 (a3 – 27) = {a3 – 27) (a3 + 1)
= (a3 – 33) (a3 + 13)
= (a – 3) (a2 + 3a + 9) (a + 1) (a2 – a + 1)

Question 6.
(a) Simplify: \(\frac{\left(x^{a+b}\right)^{2} \cdot\left(x^{b+c}\right)^{2} \cdot\left(x^{c+a}\right)^{2}}{\left(x^{a} \cdot x^{b} \cdot x^{c}\right)^{4}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 13

(b) In the given figure AABC, D is the mid-point of AB, E is the mid-point of AC. Calculate :
(i) DE, if BC = 8 cm.
(ii) ∠ADE, if ∠DBC = 125°.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 2
Answer:
Given : D is mid-point of AB, E is the mid-point of AC, BC = 8 cm, ∠DBC = 125°.
The line joining the mid-points of any two sides of a triangle is parallel to the third and is equal the half of it.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 14

(c) If a and b are rational numbers, find the values of a and b :
\(\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a+b \sqrt{3}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 16
ICSE Class 9 Maths Sample Question Paper 3 with Answers 17

ICSE Class 9 Maths Sample Question Paper 3 with Answers

Question 7.
(a) Draw a histogram from the following data :

Weight (in kg)40-4445-4950-5455-5960-6465-69
No. of students28121064

Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 18
ICSE Class 9 Maths Sample Question Paper 3 with Answers 19

(b) Solve:
83x – 67y = 383
67x – 83y = 367.
Answer:
83x – 67y = 383 ………….(i)
67a – 83y = 367 ………….(ii)
Adding (i) and (ii), we get
150x – 150y = 750
x – y = 5 ………. (iii)
Subtracting (ii) from (i), we get
16x + 16y = 16
x + y = 1 ……… (iv)
Adding (iii) and (iv), we get
2x = 6
X = 3
Putting x = 3 in (iv), we get
3 + y = 1
⇒ y = 1 – 3 = -2.
x = 3, y = – 2

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(c) In the following figure, area of parallelogram AFEC is 140 cm2. State, giving reason, the area of (i) parallelogram BFED (ii) ABFD.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 3
Answer:
Given : Area of parallelogram AFEC = 140 cm2.
(i)  Area of parallelogram BFED = Area of parallelogram AFEC
( ∵ They are on same base and between same parallels)
= 140 cm2

(ii) Area of Δ BFD = \(\frac{1}{2}\) x Area of parallelogram BFED
(∵ They are on same base and between same parallels)
\(\frac{1}{2}\) x 140 cm2 = 70 cm2.

Question 8.
(a) If log10 a = m and log10 b = n, express \(\frac{a^{3}}{b^{2}}\) in terms of m and n
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 20

(b) Draw the graph of 2x + y = 6 and 2x – y + 2 = 0. Hence, find the area of the region bounded by these lines and X-axis.
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 21
ICSE Class 9 Maths Sample Question Paper 3 with Answers 22
ICSE Class 9 Maths Sample Question Paper 3 with Answers 23

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(c) Factorize : \(8 x^{3}-\frac{1}{27 y^{3}}\)
Answre:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 24

Question 9.
(a) If \(\frac{x^{2}+1}{x}=2 \frac{1}{2}\) find the values of \((i) x-\frac{1}{x}
(ii) x^{3}-\frac{1}{x^{3}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 25

(b) In the following figure, OAB is a quadrant of a circle. The radius OA = 3.5 cm and OD = 2 cm. Calculate the area of the shaded portion.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 4
Answer:
Given : OA = 3.5 cm, OD = 2 cm
Area of shaded region = Area of quadrant AOB – Area of ΔAOD.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 26
= 9.625 – 3.5 = 6.125 cm2.

(c) If a + b + c = 9 and ab + be + ca = 40, find the value of a2 + b2 + c2.
Answer:
Given: a + b + c = 9 ab + bc + ca = 40
We know, (a+b+c)2 =a2+ b2 + c2 +2(ab+bc+ca)
(9)2 =a2+ b2 + c2+ 2 x 40
a2+ b2 + c2 = 81 – 80 = 1

ICSE Class 9 Maths Sample Question Paper 3 with Answers

Question 10.
(a) Solve: \((\sqrt{2})^{2 x+4}=8^{x-6}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 27

(b) Construct a rhombus whose diagonals are 5 cm and 6.8 cm.
Answer:
Given, diagonals are 5 cm and 6.8 cm.
Steps of construction :
(1) Draw AC = 5 cm.
(2) Draw perpendicular bisector PQ of AC which intersect AC at O.
(3) From POQ, cut-off OB = OD = \(\frac{6.8}{2}\) = 3.4 cm.
(4) Join the points A, B, C, D.
Then, ABCD is the required rhombus.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 28

(c) In a quadrilateral ABCD, AO and BO are the bisectors of ∠A and ∠B respectively.
Prove that ∠AOB = \(\frac{1}{2}\) (∠C + ∠D).
Answer:
Given, AO and BO are bisectors of ∠A and ∠B respectively.
ICSE Class 9 Maths Sample Question Paper 3 with Answers 29
ICSE Class 9 Maths Sample Question Paper 3 with Answers 30

ICSE Class 9 Maths Sample Question Paper 3 with Answers

Question 11.
(a) Find the value of log5√5 (125).
Answer:
ICSE Class 9 Maths Sample Question Paper 3 with Answers 31

(b) The sum of a two-digit number and the number obtained by reversing the order of its digits is 165. If the digits differ by 3, find the number.
Answer:
Let the digits in tens and units place be x and y respectively.
The number = 10x + y
The number obtained by reversing digits = 10y + x By 1st condition,
(10 + y) + (10y + x) = 165
⇒ 11x + 11y = 165
⇒ x + y =15 …(ii)
By 2nd condition, x-y =3 …(iii)
or y-x =3 …(iv)
Adding (ii) and (iii), we get 2x = 18
⇒ x =9
Putting x = 9 in (ii), we get y = 15 – 9 = 6
Again, adding (ii) and (iv), we get
2 y =18
⇒ y =9
Putting y = 9 in (ii), we get x = 15 – 9 = 6.
Substituting these values in (i), we get
The number = 10 x 9 + 6 or 10 x 6 + 9 = 96 or 69

ICSE Class 9 Maths Sample Question Paper 3 with Answers

(c) If the area of an equilateral triangle is 81√3 cm2, find its perimeter.
Answer:
Given : Area of equilateral triangle = 81√3 cm2
Let the length side of each of equilateral triangle be a cm.ICSE Class 9 Maths Sample Question Paper 3 with Answers 32

ICSE Class 9 Maths Question Papers with Answers

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