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Mathematics All India Set 1 2011-2012 CBSE Class 10 Question Paper Solution

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Mathematics [All India Set 1]
Marks: 80Date: 2011-2012 March

[1]1

Find the roots of the quadratic equation `2x^2-x-6=0` 

 

A.`-2,3/2`

B.`2,-3/2`

C.`-2,-3/2`

D.`2,3/2`

Concept: Quadratic Equations Examples and Solutions
Chapter: [2.03] Quadratic Equations
[1]2

If the nth term of an AP is (2n + 1) then find the sum of its first three terms

A. 6n + 3

B. 15

C. 12

D. 21

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
[1]3

From a point Q, 13 cm away from the centre of a circle, the length of tangent PQ to the circle is 12 cm. The radius of the circle (in cm) is 

 

 

A. 25

B. `sqrt313`

C. 5

D. 1

Concept: Tangent to a Circle
Chapter: [3.01] Circles
[1]4

In Figure 1, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC

= 4 cm, then the length of AP (in cm) is

A. 7.5

B. 15

C. 10

D. 9

Concept: Tangent to a Circle
Chapter: [3.01] Circles
[1]5

The circumference of a circle is 22 cm. The area of its quadrant (in cm2) is

 

A.`77/2`

B.`77/4`

C.`77/8`

D.`77/16`

Concept: Circles Examples and Solutions
Chapter: [3.01] Circles
[1]6

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is

A. 1 : 2

B. 1 : 4

C. 1 : 6

D. 1 : 8

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[1]7

A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is

A. 45°

B. 30°

C. 60°

D. 90°

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[1]8

The Distance of the point (−3, 4) from the x-axis is

A. 3

B. −3

C. 4

D. 5

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[1]9

In Figure 2, P (5, −3) and Q (3, y) are the points of trisection of the line segment joining A (7, −2) and B (1, −5). Then y equals

A. 2

B. 4

C. −4

D. `-5/2`

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
[1]10

Cards bearing numbers 2, 3, 4, ..., 11 are kept in a bag. A card is drawn at random from the bag. The probability of getting a card with a prime number is

A. `1/2`

B. `2/5`

C. `3/10`

D. `5/9`

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
[2]11

Find the value of p for which the roots of the equation px (x − 2) + 6 = 0, are equal.

Concept: Quadratic Equations
Chapter: [2.03] Quadratic Equations
[2]12

How many two-digits numbers are divisible by 3?

 

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[2]13

In Figure 3, a right triangle ABC, circumscribes a circle of radius r. If AB and BC are of lengths of 8 cm and 6 cm respectively, find the value of r.

Concept: Tangent to a Circle
Chapter: [3.01] Circles
[2]14

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[2]15

In Figure 4, ABCD is a square of side 4 cm. A quadrant of a circle of radius 1 cm is drawn at each vertex of the square and a circle of diameter 2 cm is also drawn. Find the area of the shaded region. (Use π = 3.14)

Concept: Areas of Combinations of Plane Figures
Chapter: [7.01] Areas Related to Circles
[2]16

A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm.Find the number of cones so formed. 

Concept: Surface Areas and Volumes Examples and Solutions
Chapter: [7.02] Surface Areas and Volumes
[2]17

Find the value of k, if the point P (2, 4) is equidistant from the points A(5, k) and B (k, 7).

Concept: Pair of Linear Equations in Two Variables
Chapter: [2.01] Pair of Linear Equations in Two Variables
[2]18

A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability of getting

(i) a red king.

(ii) a queen or a jack.

Concept: Probability - A Theoretical Approach
Chapter: [5.01] Probability
[2]19

Solve the following quadratic equation for x:

x2 − 4ax − b2 + 4a2 = 0

Concept: Solutions of Quadratic Equations by Factorization
Chapter: [2.03] Quadratic Equations
[3]20

Find the sum of all multiples of 7 lying between 500 and 900.

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
[3]21

Draw a triangle ABC with BC = 7 cm, ∠ B = 45° and ∠C = 60°. Then construct another triangle, whose sides are `3/5` times the corresponding sides of ΔABC.

Concept: Constructions Examples and Solutions
Chapter: [3.03] Constructions
[3]22

In Figure 5, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR =12 cm. Find the lengths of QM, RN and PL ?

Concept: Areas Related to Circles Examples and Solutions
Chapter: [7.01] Areas Related to Circles
[3]23

In the given figure, O is the centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90°. Find the area of the shaded region.

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle
Chapter: [7.01] Areas Related to Circles
[3]24

A hemispherical bowl of internal radius 9 cm is full of water. Its contents are emptied in a cylindrical vessel of internal radius 6 cm. Find the height of water in the cylindrical vessel.

Concept: Surface Areas and Volumes Examples and Solutions
Chapter: [7.02] Surface Areas and Volumes
[3]25

The angles of depression of the top and bottom of a tower as seen from the top of a 60  `sqrt(3)` m high cliff are 45° and 60° respectively. Find the height of the tower.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[3]26

Find the coordinates of a point P, which lies on the line segment joining the points A (−2, −2), and B (2, −4), such that `AP=3/7 AB`.

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
[3]27

If the points A (xy), B (3, 6) and C (−3, 4) are collinear, show that x − 3y + 15 = 0.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[3]28

All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn from it. Find the probability that the drawn card is (i) a black face card. (ii) a red card.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
[4]29 | attempt any one of the following
[4]29.1

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by `1/15`
. Find the fraction

Concept: Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
Chapter: [2.03] Quadratic Equations
[4]29.2

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.

Concept: Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
Chapter: [2.03] Quadratic Equations
[4]30

Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[4]31

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[4]32 | attempt any one of the following
[4]32.1

A hemispherical tank, full of water, is emptied by a pipe at the rate of `25/7`litres per sec.

How much time will it take to empty half the tank if the diameter of the base of the tank is 3 m?

Concept: Volume of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[4]32.2

A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.  [use π=22/7]

Concept: Frustum of a Cone
Chapter: [7.02] Surface Areas and Volumes
[4]33

A military tent of height 8.25 m is in the form of a right circular cylinder of base diameter 30 m and height 5.5 m surmounted by a right circular cone of same base radius. Find the length of the canvas used in making the tent, if the breadth of the canvas is 1.5 m.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4]34

The angles of elevation and depression of the top and bottom of a light-house from the top of a 60 m high building are 30° and 60° respectively. Find (i) the difference between the heights of the light-house and the building. (ii) the distance between the light-house and the building.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances

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