ICSE Class 9 Maths Sample Question Paper 6 with Answers

Section – A [40-Marks]
(Attempt all questions from this Section)

Question 1.
(a) Prove that log (1 + 2 + 3) = log 1 + log 2 + log 3.
Answer:
log (1 + 2 + 3) = log 6 = log (1 x 2 x 3)
= log 1 + log 2 + log 3.

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(b) In the given figure, CD is a diameter which meets the chord AB in E such that
AE = BE = 4 cm. If CE = 3 cm, find the radius of the circle.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 1
Answer:
Given : AE = BE = 4 cm, CE = 3 cm
Let r be the radius (OB = OC)
OE = OC – CE = r – 3.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 6
⇒ OB2 = OE2 + BE2 (Pythagoras theorem)
⇒ r2 = (r – 3)2 + 42 r2
⇒ r2 – 6r + 9 + 16
⇒ 6r = 25
\(r=\frac{25}{6}=4 \frac{1}{6} \mathrm{~cm}\)

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) If ₹ 6,400 is invested at 6 \(\frac{1}{4}\) % p.a. compound interest, find (i) the amount after 2 years (ii) the interest earned in 2 years.
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 7

Question 2.
(a) Evaluate tan x and cos y from the given figure.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 2
Answer:
In ΔACD, AC2 = AD2 + CD2
132 – 52 + CD2
⇒ CD2 = 169 – 25 = 144
⇒ CD = 12.
In A BCD, BC2 = CD2 + BD2
= 144 + 162 = 144 + 256 = 400
BC =20
ICSE Class 9 Maths Sample Question Paper 6 with Answers 8

(b) ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively. Show that the altitudes are equal.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 3
Answer:
Given: AC=AB
∴ In ΔBEC and ΔCFB,
∠C=∠B (∵ AB = AC)
∠BEC = ∠CFB (Each being a right angle)
BC = BC (Common side)
∴ ΔBFC  ≅ ΔCFB (AAS axiom)
∴ BE = CF (c.p.ct.)
Hence Proved.

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) The mean of 5 observations is 15. If the mean of first three observations is 14 and that of the last three is 17, find the third observation.
Answer:
Mean of 5 observations = 15
∴ Sum of 5 observations = 15 x 5 = 75
Mean of first 3 observations = 14
∴ Sum of first 3 observations = 14 x 3 = 42
Mean of last 3 observations = 17
∴ Sum of last 3 observations = 17 x 3 = 51
∴ The third observation = (42 + 51) – 75 = 18.

Question 3.
(a) Factorize : x4 + 4
Answer:
x4 + 4 = (x4 + 4x2 + 4) – 4x2 = {(x2)2 + 2 .x. 2 + (2)2} – (2x)2
= (x2 + 2)2 – (2x)2 = (x2 + 2 + 2x) (x2 + 2 – 2x)
= (x2 + 2x+ 2) (x2 – 2x+ 2)

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(b) Evaluate : \(\frac{\sin 30^{\circ}-\sin 90^{\circ}+2 \cos 0^{\circ}}{\tan 30^{\circ} \cdot \tan 60^{\circ}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 9

(c) Simplify:\((81)^{3 / 4}-3 \times(7)^{0}-\left(\frac{1}{27}\right)^{-2 / 3}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 10

Question 4.
(a) If x \(\frac{2}{x}\) = 5, find the value of \(x^{3}-\frac{8}{x^{3}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 11

(b) If the hypotenuse of a right angled triangle is 6 m more than twice the shortest side and third side is 2 m less than hypotenuse, find the sides of the triangle.
Answer:
Let the shortest side be x m.
Then, Hypotenuse =(2x+6)cm,thirdside=2x+6-2=(2x+4)m.
∴ (2x + 6)2 = (2x + 4)2 + x2 (Using Pythagoras theorem)
= (2x)2 + 2.2x.6 + 62 = (2x)2 + 2.2xA +42 + x2
= 4x2 + 24x + 36 = 4x2 + 16x + 16 + x2
= 24x – 16x = x2 + 16 – 36
= x2 – 8x – 20 = 0
= x2 – (10 – 2) x – 20 =0
= x2 – 10x + 2x – 20 =0
= x (x – 10) + 2 (x – 10) = 0
= (x – 10) (x + 2) = 0
= x – 10 =0 or x + 2 = 0
= x = 10 or x = – 2
∴ x = 10 (∵ x cannot be negative)
∴ 2x + 6 = 2 x 10 + 6 = 26
and 2x + 4 = 2 x 10 + 4 = 24
Therefore, the sides are 10 m, 26 m and 24 m.

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) Simplify: \(\frac{3}{\sqrt{6}+\sqrt{3}}-\frac{4}{\sqrt{6}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{2}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 12
ICSE Class 9 Maths Sample Question Paper 6 with Answers 13

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

Question 5.
(a) Solve : log10 6 + log10 (4x + 5) = log10 (2x + 7) +1
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 14

(b) 3 men and 4 women can do a piece of work in 14 days while 4 men and 6 women can do it in 10 days. How long would it take 1 woman to finish the work ?
Answer:
Let 1 man take x days and 1 woman take y days to finish the work.
∴ In 1 day, 1 man does = \(\frac{1}{x} \) work and 1 woman does = \(\frac{1}{y}\) work.
So, 3 men and 4 women do=\(3 \times \frac{1}{x}+4 \times \frac{1}{y}=\frac{3}{x}+\frac{4}{y}\)
It is given that 3 men and 4 women finish the work in 14 days.
\(\frac{3}{x}+\frac{4}{y}=\frac{1}{14}\) ………….(i)
Also, 4 men and 6 women do the work in 10 days.
= \(\frac{4}{x}+\frac{6}{y}=\frac{1}{10}\) …………(ii)
Multiplying equation (i) by 4 and equation (ii) by 3, we get
ICSE Class 9 Maths Sample Question Paper 6 with Answers 16

∴ One woman finish the work in 140 days.

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) There are two regular polygons with number of sides equal to (n – 1) and (n + 2). Their exterior angles differ by 6°. Find the value of n
Answer:
For first polygon,
ICSE Class 9 Maths Sample Question Paper 6 with Answers 17
\(\frac{3}{n^{2}+2 n-n-2}=\frac{1}{60}\)
n2  + n – 2 = 180
n2 + n- 182 = 0
n2  + (14 – 13) n – 182 = 0
n2  + 14n – 13n – 182 = 0
n(n + 14) -13 (n + 14) = 0
(n + 14) (n – 13) = 0
n + 14 = 0 or n – 13 =0
n = -14 = 0  or n = 13 (∵n cannot be negative)
∴ n = 13.

ICSE Class 9 Maths Sample Question Paper 6 with Answers

Question 6.
(a) If \(a^{2}+\frac{1}{a^{2}}=7\), find the value of \(a^{2}-\frac{1}{a^{2}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 18

(b) Construct a trapezium ABCD in which AD || BC, Z B = 60°, AB = 5 cm, BC = 6.2 cm, and CD = 4.8 cm.
Answer:
Given : AD||BC, ZB = 60°, AB = 5 cm, BC = 6.2 cm, and CD = 4.8 cm.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 19
Steps of construction :
(1) Draw BC = 6.2 cm.
(2) At B, draw ∠CBX = 60° and cut off BA = 5 cm.
(3) At A, draw exterior ∠XAY = 60° such that AY||BC.
(4) From C, cut-off AY at D such that CD = 4.8 cm and join CD.
Hence, ABCD is the required trapezium.

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) The inner dimensions of a closed wooden box are 2 m, 1.2 m and 0.75 m. The thickness of the wood is 2.5 cm. Find the cost of wood required to make the box if 1m3 of wood costs ₹ 5400.
Answer:
The inner dimensions of the closed box are 2 m, 1.2 m, 0.75 m.
Inner volume = (2 x 1.2 x 0.75) m3 = 1.8 m3
Thickness of the box = 2.5 cm = 2.5/100m= 0.025 m
∴ Outer dimensions are (2 + 2 x 0.025) m, (1.2 + 2 x 0.025) m, (0.75 + 2 x 0.025) m
i.e. 2.05 m, 1.25, 0.8 m.
∴ Outer volume = (2.05 x 1.25 x 0.8) m3 = 2.05 m3
Volume of wood = (2.05 – 1.8) m3 0.25 m3
Cost of 1 m3 of wood = ₹ 5400
Cost of 0.25 m3 of wood = ₹5400 x 0.25
= ₹ 1350

Question 7.
(a) Solve : 4x2 + 15 =16x
Answer:
4x2+15 =16x
4x2 – 16x+15 =0
4x2 – (10-t-6)x+15 =0
4x2 10x – 6x+15 =0
= 2x(2x – 5)- 3(2x – 5) =0
(2x – 5)(2x – 3) =0
2x – 5 =0 or 2x – 3=0
2x =5 or 2x=3
ICSE Class 9 Maths Sample Question Paper 6 with Answers 20

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(b) Find graphically the vertices of the triangle whose sides have equations
2y – x = 8, 5y – x = 14 and y – 2x = 1.
Answer:
Given equations are,
2y – x =8 ……….(i)
5y – x =14 …(ii)
and y – 2x =1 …(iii)
From(i), x =2y – 8
ICSE Class 9 Maths Sample Question Paper 6 with Answers 21
∴ (- 6, 1), (- 4, 2), (- 2, 3)
From (ii), x=2 y-8
ICSE Class 9 Maths Sample Question Paper 6 with Answers 22
(- 4, 2), (1, 3), (6, 4)
From (iii) y=2 x+1
ICSE Class 9 Maths Sample Question Paper 6 with Answers 23
∴ (1, 3), (2, 5), (- 1, – 1)
These points are plotted on the graph.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 24
The three lines intersect at point (- 4, 2), (1, 3) and (2, 5) which are the required vertices of triangle formed by them

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) If 3tan2 θ-1=0′, find cos 2θ, given that θ is acute.
Answer:
Given:  3tan2 θ-1=0
tan2θ = 1/3
tan θ   =\(\frac{1}{\sqrt{3}}\)]
⇒ tanθ = tan 30°
θ = 30°
cos2θ = cos (2 x 30°) = cos 60° =\(\frac{1}{2}\)

Question 8.
(a) Solve for x : 3(2x + 1) – 2x+2 + 5 = 0.
Answer:
⇒ 3(2x + 1) – 2x + 2 + 5 =0
⇒ 3.2x + 3 – 2x. 22 + 5 =0
⇒ 3.2x – 4.2x + 8=0
⇒ -2x = – 8
⇒ 2x = 23
⇒ x =3

(b) Find the area of a triangle whose perimeter is 22 cm, one side is 9 cm and the difference of the other two sides is 3 cm.
Answer:
One side = 9 cm, perimeter = 22 cm.
Let other two sides be a cm and b cm and a > b.
According to the question,
a + b + 9 = 22
⇒ a + b = 13 ………..(i)
and a – b =3 (Given) ………(ii)
Adding equations (i) and (ii), we have
2 a = 16 ⇒ a = 8
Subtracting equation (ii) from equation (i), we have
2b = 10 ⇒ b = 5
The sides are a = 8 cm, b = 5 cm, c = 9 cm
ICSE Class 9 Maths Sample Question Paper 6 with Answers 25

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(c) Insert four irrational numbers between 2√3 and 3√2
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 26

Question 9.
(a) Form a cumulative frequency distribution table from the following data by exclusive method taking 4 as the magnitude of class intervals.
31, 23, 19, 29, 20, 16, 10, 13, 34, 38, 33, 28, 21, 15, 18, 36, 24, 18, 15, 12, 30, 27, 23, 20, 17, 14, 32, 26, 25, 18, 29, 24, 19, 16, 11, 22, 15, 17, 10, 25.
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 27

(b) Solve simultaneously : \(\frac{2}{x}+\frac{2}{3 y}=\frac{1}{6} ; \frac{3}{x}+\frac{4}{y}=-\frac{1}{2}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 28
ICSE Class 9 Maths Sample Question Paper 6 with Answers 33

(c) The diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that area of ΔOAD = area of ΔOBC. Prove that ΔBCD is a trapezium.
ICSE Class 9 Maths Sample Question Paper 6 with Answers 4
Answer:
Given: Area of ΔOAD = area of ΔOBC.
Draw DM ⊥ AB,CN ⊥AB.
∴ DM || CN (∵Both DM, CN are perpendicular to AB)
Now,
Area of ΔOAD = Area of ΔOBC
Area of ΔOAD + Area of ΔOAB = Area of ΔOBC + Area of ΔOAB
ICSE Class 9 Maths Sample Question Paper 6 with Answers 34

ICSE Class 9 Maths Sample Question Paper 6 with Answers

Question 10.
(a) If the interest is compounded half yearly, calculate the amount when the principal is ₹ 7400, the rate of interest is 5% p. a. and the duration is one year.
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 35

(b) Find the coefficient of x2 and x in the product of (x – 2) (x – 3) (x – 4).
Answer:
Given :  (x -2) (x – 3) {x – 4)
Here,  a = – 2, b = – 3, c = -4
Coefficient of x2 = a + b + c = (- 2) + (- 3) + (- 4) = – 9
Coefficient of x = ab + be + ca = (- 2) (- 3) + (- 3) (- 4) + (- 4) (- 2)
= 6 + 12 + 8 = 26

(c) If the figure given, ABCD is a trapezium in which AB || DC. P is the mid-point of AD and PR || AB. Prove that PR = \(\frac{1}{2} (AB + CD)\).
ICSE Class 9 Maths Sample Question Paper 6 with Answers 5
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 36

Question 11.
(a) Factorize : a3 + 3a2b + 3ab2 + 2b3.
Answer:
a3 + 3a2b + 3 ab2 + 2b3 = (a3 + 3 a2b + 3ab2 + b3) + b3
= (a + b)3 + (b)3 = (a + b + b) {(a + b)2 – (a + b)b + b2}
= (a + 2b) (a2 + 2ab + b2 – ab – b2 + b2)
= (a + 2b) {a2 + ab + b2)

ICSE Class 9 Maths Sample Question Paper 6 with Answers

(b) In the point A (2, – 4) is equidistant from the points P (3, 8) and Q (- 10, y), find the values of y.
Answer:
Given points are A (2, – 4), P (3, 8), Q (- 10, y)
AQ = AP
⇒ AQ2 = AP2
(- 10 – 2)2 + (y + 4)2
= (3 – 2)2 + (8 + 4)2 144 + (y + 4)2
= 1 + 144 (y + 4)2 = 1 y + 4 = ±1
y + 4= 1 or y + 4 = -1 y = – 3 or y = – 5
y = – 3 or – 5

(c) Simplify: \(\sqrt[a b]{\frac{x^{a}}{x^{b}}} \cdot b \sqrt[x]{\frac{x^{b}}{x^{c}}} \cdot \sqrt[c a]{\frac{x^{c}}{x^{a}}}\)
Answer:
ICSE Class 9 Maths Sample Question Paper 6 with Answers 37

ICSE Class 9 Maths Question Papers with Answers

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