Maths Prev 1

8/12/2019 Maths Prev 1

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Question 3

Question 4

Question 8

Question 9

Question 11



Question 18

Question 28



= 892.57 m2

Question 29



Height at which the jet is flying = 2598 mts

Question 3

3.In an A.P., the sum of its first n terms is n2+2n. Find its 18th term.Sol:

GivenReplacing n by (n 1), we get

Now

From (1) and (2)

To get AP, substituting n = 1, 2, 3,. . respectively in (3), we get

.(from (3))

Hence, AP is 3, 5, 7, 9,

Now 18th term =Hence, 18th term = 37. 7.A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find the

probability that the ball drawn is (a) white. (b) neither red nor white.

Sol: Number of balls in the box = 5 + 4 + 7 = 16

n(S) = 16

(i) Let A be the event of drawing a white ball

n(A) = 7

Probability of drawing a white ball =(ii) Let B be the event of getting a ball which is neither red nor white. The selection can be made from green balls.

n(B) = 4

Probability of drawing neither white nor red ball



. Question 10

10.The first term, common difference and last term of an A.P are 12, 6 and 252 respectively. Find the sum of allterms of this A.P

Sol:

Let the first term, a = 12, Common difference, d= 6 Last term, L = 252

Hence, the sum of all terms of A.P is 5412.

Question 13

13.A toy in the form of a cone mounted on a hemisphere with same radius. The diameter of the base of the

conical portion is 7 cm and the total height of the toy is 14.5 cm. Find the volume of the toy. (use π = )

Sol:

Given AB = 7 cm

Total height = h = 14.5 cm Volume of the toy = Volume of the cone + volume of the hemi sphere

=

=



=

=

=

=

Question 23

23. A hemispherical bowl of internal diameter 36 cm is full of some liquid. This liquid is to be filled in cylindricalbottles of radius 3 cm and height 6 cm. Find the number of bottles needed to empty the bowl.

Sol:

Given 2r = 36

⇒ r = 18

Volume of the hemi-sphere =

Now volume of the cylinder = r 2h

The number of cylinders that can be filled with cylinder

= 72

Hence the number of bottles to empty the bowl = 72 bottles

(OR)

Water flows out through a circular pipe whose internal radius is 1 cm, at the rate of 80 cm/second into an emptycylindrical tank, the radius of whose base is 40 cm. By how much will the level of water rise in the tank in half anhour ? Sol: Rate of flow of water = 80 cm/sec Radius of cylindrical PIPE = r = 1 cm

Radius of cylindrical Tank = R = 40 cm Time = 30 minutes = 30 × 60 = 1800 sec Length of the pipe = h = 80 × 30 × 60 = 144000 Let the height of the water level raised = H

Volume of water flows from a pipe = Volume of water in a cylindrical tank

Hence, the level of water rise up to a height of 90 cm

Question 24



24. A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from apoint A on the ground is 60° and the angle of depression of point A from the top of the tower is 45°. Find the

height of the tower. (Take = 1.732)

Sol: Let BC = height of the Tower = x

CD = height of the pole = 5 m Angle of elevation = ∠BAD = 600

Angle of depression = ∠CAB = 450

Let AB = yConsider ⊿BAD,

-----------(1)

Consider⊿CAB

-------------(2)

Substituting (2) in (1)

Rationalising the denominator

'>

Hence, the height of the tower = 6.83 m

Question 7

7.A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find theprobability that the ball drawn is



(a) white. (b) neither red nor white.

Sol: Number of balls in the box = 5 + 4 + 7 = 16

n(S) = 16 (i) Let A be the event of drawing a white ball

n(A) = 7

Probability of drawing a white ball =(ii) Let B be the event of getting a ball which is neither red nor white. The selection can be made from green balls.

n(B) = 4

Probability of drawing neither white nor red ball

.

Question 10

10.The first term, common difference and last term of an A.P are 12, 6 and 252 respectively. Find the sum of allterms of this A.P

Sol: Let the first term, a = 12, Common difference, d= 6 Last term, L = 252


Question 12

12.Draw a circle of radius 4.5 cm. At a point A on it, draw a tangent to the circle without using the centre. Sol:

Steps of Construction:



1. Draw a circle of radius 4.5 cm.

2. Draw any chord AC through the given point A on the circle.3. Take a point B on the circle and join A,B and C

4. Construct ∠CAY = ∠ ABC on the opposite side of chord AC

5. Produced YA to X to get YAX as the required tangent.

Question 13

13.A toy in the form of a cone mounted on a hemisphere with same radius. The diameter of the base of the

conical portion is 7 cm and the total height of the toy is 14.5 cm. Find the volume of the toy. (use π = )

Sol:

Given AB = 7 cm

Total height = h = 14.5 cm Volume of the toy = Volume of the cone + volume of the hemi sphere

=

=

=

=

=

=

Question 15

15.All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is then drawn at

random from the remaining pack. Find the probability of getting (a) a black face card, (b) a queen, (c) a blackcard.

Sol: Total number of cards = 52 Number of Face cards of spades = 3

Remaining cards = n(S) = 49 Let A be the event of drawing (a) a black face card

(b) a queen card

(c) a black card



Question 10

10.The first term, common difference and last term of an A.P are 12, 6 and 252 respectively. Find the sum of all

terms of this A.P Sol: Let the first term, a = 12, Common difference, d= 6 Last term, L = 252


Question 15

15.All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is then drawn atrandom from the remaining pack. Find the probability of getting (a) a black face card, (b) a queen, (c) a blackcard.

Sol: Total number of cards = 52 Number of Face cards of spades = 3 Remaining cards = n(S) = 49 Let A be the event of drawing

(a) a black face card

(b) a queen card

(c) a black card

Question 18

18. If the point C (1, 2) divides internally the line segment AB in the ratio 3 : 4, where the coordinates of A are (2,5), find the coordinates of B.

Sol: Let the coordinates of B be (x, y) Given AC : CB = 3 : 4 (internally)

Co ordinates of C =



Hence, the coordinates of B are ( 5, 2)

Question 4

4. Using the quadratic formula, solve the equation:

Sol: Comparing

We know that,

Question 13

13. A solid is in the form of a right circular cylinder with hemispherical ends. The total height of the solid is 58 cmand the diameter of the cylinder is 28 cm. Find the total surface area of the sold.

Sol:



Radius of the cylindrical portion

Radius of the hemisphere

Length of the cylindrical portion

Curved surface area of the cylindrical part

Curved surface area of hemisphere

As there are two such hemispheres, the curved surface area

Total surface area = Curved surface area of the cylindrical part + Curved surface area of two hemispheres

Question 20

20. A card is drawn at random from a well shuffled deck of playing cards. Find the probability that the carddrawn is (a) a card of spades or an ace(b) a red king

(c) neither a king nor a queen (d) either a king or queen

Sol: Total no. of outcomes 52 (a) There are 13 spades and 4 aces. But 1 ace appears in 13 spade cards

So, favourable outcomes

(b) There are two red kings, one is diamonds and other is hearts

So, favourable outcomes = 2

Required probability(c) Number of king cards or queen cards = 4 + 4 = 8.

Favourable outcomes

Required probability =

(d) Required probability

Question 23

23. A bucket made up of a metal sheet is in the form of a frustum of a cone. Its depth is 24 cm and thediameters of the top and bottom are 30 cm and 10 cm respectively. Find the cost of milk which can completelyfill the bucket at the rate of Rs. 20 per litre and the cost of the metal sheet used, if it costs Rs. 10 per 100 cm2.

Sol:



Given

Curved surface area of the bucket

Total metal sheet required = Curved surface area of the bucket + Area of the base

Area of metal sheet used

Cost of metal sheet used

Volume of bucket (frustum of cone)

Cost of milk = Volume Rate

OR Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is50 m long and 44 m wide. Find the time which the level of water in the tank will rise by 21 cm.

Sol:

Volume of cylinder

RadiusVolume of water flowing through the cylindrical pipe in an hour at the rate of 15 km/hr

Volume of cuboid





Distance of the ship from the hill OR From a window h meters high above the ground in a street, the angles of elevation and depression of the top and

foot of the other house on the opposite side of the street are respectively. Show that the height of the

opposite house is

Sol:

Let W be the window and AB be the house on the opposite side Then, WP is the width of the street Height of the window = h metres = BP Let AP = x metres

In right triangle

----- (i)

In right triangle

(From (i))

Height of the opposite house

Question 20

20.A bag contains red, white and black balls. A ball is drawn at random from the bag. Find the probabilitythat the drawn ball is:(i) Red or white (ii) Not black (ii) Neither white nor black.

Sol:

A bag contains red, white and black balls.

A ball is drawn at random from the bag.

Total number of balls

Probability that the ball drawn is red

Probability that the ball drawn is black

Probability that the ball drawn is white

The probability that the ball drawn is red or white is



The probability that the ball drawn is not black will be

The probability that the ball drawn is neither white nor black is

Question 1

1.Solve the following system of linear equations:

6(ax + by) = 3a + 2b, 6(bx – ay) = 3b – 2a

Sol:

Given system of linear equations can be written as

6ax + 6 by – (3a + 2b) = 0 ……………..(1) 6bx – 6ay – (3b – 2a) = 0 ………………(2)

Solving (1) and (2)

x y 1 6b -(3a + 2b) 6a 6b

6a -(3b – 2a) 6b -6a

Hence, the values of x and y are respectively.

Question 2

2.Using quadratic formula, solve the following quadratic equation for x: p2x2 + (p2 – q2)x – q2 = 0

Sol:



Given quadratic equation is p2x2 + (p2 – q2)x – q2 = 0

Here a = p2, b = p2 – q2, c = -q2

Therefore D = b2 – 4ac

D = (p2 – q2)2 – 4p2(-q2)

= p4 + q4 – 2p2q2 + 4p2q2

= p4 + q4 + 2p2q2

= (p2 + q2)2 ³ 0

\ Roots exist

So the roots of the given equation are

Question 3

3.If (x – 3) (x + 2) is the GCD of the polynomials P(x) = (x2 – 2x – 3) (2x2 + ax – 2) and Q(x) = (x2 + x – 2) (3x2 + bx – 3) find the value of a and b.

Sol:

Since, (x – 3) (x + 2) is a factor of P(x) and Q(x)

P(3) = (9 – 6 – 3) (2(9) + a(3) – 2) = 0

P(-2) = (4 – 2(2) 3) (2(4) + a(2) 2) = 0= (4 + 4 – 3) (8 – 2a – 2) = 0

= 5(6 – 2a) = 0

Þ 6 – 2a = 0

Þ 2a = 6

Þ a = 3

Q(-2) = (4 2 2)(3(4) + b(2) 3) = 0

Q(3) = (9 + 3 2) (3(9) + b(3) - 3) = 0

= (10) (27 + 3b 3) = 0Question 5

5.The 8th term of an arithmetic progression (A.P.) is 37 and its 12 th term is 57. Find the A.P.

Sol:

Let ‘a’ be the first term and ‘d’ be the common difference of the A.P

Given that 8th term of A.P. = 37

Þ a8 = 37Þ a + (8 – 1)d = 37

Þ a + 7d = 37 ………………(1)



12th term of AP = 57

Þ a + (12 – 1)d = 57

Þ a + 11d = 57 ……………(2)

Now, (2) – (1)

Þ 4d = 20

Þ d = 5

Substituting d = 5 in (1), we get

a + 7(5) = 37

Þa = 37 – 35 = 2

\ First term = a = 2, common difference = d = 5

Hence the A.P. is a, a + d, a + 2d, a + 3d, ……

i.e., 2, 7, 12, 17,……….

Question 6

6.Find the sum of the first 25 terms of an A.P. whose nth term is given by t n = 7 – 5n.Sol:

Given tn = 7 – 5nt1 = 7 – 5(1) = 7 – 5 = 2

t2 = 7 – 5(2) = 7 – 10 = 3

\ d = t2 – t1 = 3 - 2 = 5

Now,

= 25(58)

= 1450

Hence, the sum of first 25 terms of A.P. is 1450

(OR) Which term of the arithmetic progression 3, 10, 17, ….. will be 84 more than its 13 th term?Sol:

We have 3, 10, 17, ……

Here a = 3, d = 10 – 3 = 7

Now, let required term be an





= 60

\ Hence the number of bottles to empty the bowl = 60 bottles

Question 19

19.A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that

the ball drawn is (i) white (ii) red (iii) not black (iv) red or white. Sol:

Let ‘S’ be the sample space of total number of balls.

n(S) = 4 + 5 + 6 = 15

Let A be the event of drawing

(i) a white ball

n(A) = 6

(ii) a red ball

n(A) = 4

(iii) not black ball Þ a ball can be drawn from both Red and white balls

(iv) Red or white

Question 23

23.If the radii of the circular ends of a bucket, 45 cm high, are 28 cm and 7 cm (as shown in Fig. 4), find the capacity

of the bucket.

Fig.4

Sol:

Given



Height of the bucket (h) = 45 cm

Radius of upper end (r1) = 28 cm

Radius of lower end (r2) = 7 cm

Capacity of the bucket = Volume of the bucket

Volume of the bucket =

= 48510 cm3

(OR)

A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the

remainder is of the curved surface of the whole cone, find the ratio of the line segments into which the altitude of the

cone is divided by the plane.

Sol:

In the figure, the smaller cone, APQ has been cut off through the plane PQ || BC. Let r and R be the radii of the smaller

cone and larger cones respectively. Let l and L be the slant heights of smaller cone and larger cones respectively. In the figure, OQ = r, MC = R, AQ = l, AC = L

Since AOQ AMC

Given that curved surface area of the remainder = of the curved surface area of the whole cone

Curved surface area of smaller cone =

of the curved surface area of the whole cone



Since AOQ and AMC are similar triangles, we have

OM = AM OA

Hence, the required ratio of their heights = 1 : 2

Question 24

24.A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a

hill as 600 and angle of depression of the base of the hill as 450 . Find the distance of the hill from the ship and height of thehill.

Sol:

Let AB = Height of the deck of a ship from the water level = 10 m (given)

CE = Height of the tower = h m

Angle of elevation = ∠CAD = 600

Angle of depression = ∠DAE = 450

Let AD = BE = x

From ⊿ AEB ,

x = 10 m

Hence, the distance between the ship and the tower = 10 m From ⊿ CAD ,



m

Hence, the height of the tower is

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Maths Prev 1