Prove that: tanθ 1 - cotθ + cotθ 1 - tanθ = 1 + secθ cosec θ

To prove: tanθ 1 - cotθ + cotθ 1 - tanθ = 1 + secθ cosec θ Consider LHS, we have ⇒ tanθ 1 - cotθ + cotθ 1 - tanθ = sinθ cosθ 1 - cosθ sinθ + cosθ sinθ 1 - sinθ cosθ = sinθ × sinθ cosθ sinθ - cosθ + cosθ × cosθ sinθ cosθ - sinθ = sin 2 θ cosθ sinθ - cosθ - cos 2 θ sinθ sinθ - cosθ = sin 3 θ - cos 3 θ cosθ sinθ - cosθ sinθ = sinθ - cosθ sin 2 θ + cos 2 θ + sinθcosθ sinθcosθ sinθ - cosθ = 1 + sinθcosθ sinθcosθ = 1 sinθcosθ + sinθcosθ sinθcosθ = 1 sinθ 1 cosθ + 1 = cosecθ . secθ + 1 = 1 + secθ . cosecθ = RHS Hence, proved.

Radhika has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2 . 5 cm and is full of water. Metallic spherical balls each of diameter 0 . 5 cm are put in the vessel due to which 2 5 th of the water in the vessel flows out. Find out how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?

Given that height of the conical vessel = 11 cm , radius of top = 2 . 5 cm . As conical vessel is full of water, so volume of cone will be equal to the volume of water. ⇒ Volume of water = Volume of cone = 1 3 π r 2 h = 1 3 × π × ( 2 . 5 ) 2 × 11 cm 3 ⇒ Volume of water flows out of cone = 2 5 of total volume of water = 2 5 × 1 3 × π × ( 2 . 5 ) 2 × 11 cm 3 Diameter of metallic spherical ball = 0 . 5 cm ∴ Radius of metallic spherical ball = 1 2 × 0 . 5 cm = 0 . 25 cm Volume of 1 metallic spherical ball = 4 3 π r 3 = 4 3 × π × ( 0 . 25 ) 3 cm 3 Let number of balls put in the vessel be n . Total volume of n balls = n × 4 3 × π × ( 0 . 25 ) 3 cm 3 ∴ Volume of water flows out = Volume of n balls ⇒ 2 5 × 1 3 × π × ( 2 . 5 ) 2 × 11 = n × 4 3 × π × ( 0 . 25 ) 3 ⇒ n = 2 × π × 2 . 5 2 × 11 5 × 3 × 3 4 × π × 0 . 25 3 ⇒ n = 2 . 5 × 2 . 5 × 11 10 × 0 . 25 × 0 . 25 × 0 . 25 ⇒ n = 2 . 5 × 2 . 5 × 11 × 10 × 10 × 10 10 × 2 . 5 × 2 . 5 × 2 . 5 ⇒ n = 1100 2 . 5 ∴ n = 440 Values showed are: Conservation of water and natural resources.

The following distribution gives the weights of 60 students of a class. Find the mean and mode weights of the students. Weight(in kg) Number of students ( f i ) 40 ‐ 44 4 44 ‐ 48 6 48 ‐ 52 10 52 ‐ 56 14 56 ‐ 60 10 60 ‐ 64 8 64 ‐ 68 6 68 ‐ 72 2

We have to make a frequency distribution table by adding two more column as class-marks and f i x i . Class-marks x i can be obtained by using the formula: Upper limit + lower limit 2 . Weight(in kg) Number of students ( f i ) Class-marks x i f i x i 40 ‐ 44 4 42 168 44 ‐ 48 6 46 276 48 ‐ 52 10 50 500 52 ‐ 56 14 54 756 56 ‐ 60 10 58 580 60 ‐ 64 8 62 496 64 ‐ 68 6 66 396 68 ‐ 72 2 70 140 Total ∑ i = 1 8 f i = 60 ∑ i = 1 8 f i x i = 3312 Using the direct mean formula: Mean x ¯ = ∑ i = 1 8 f i x i ∑ i = 1 8 f i = 3312 60 = 55 . 2 Hence, mean weight of the students is 55 . 2 kg . Since the class 52 - 56 has the maximum frequency 14 . ⇒ 52 - 56 is the modal class. ∴ l = 52 , h = 4 , f 0 = 10 , f 1 = 14 , f 2 = 10 . Using the formula: Mode = l + f 1 - f 0 2 f 1 - f 0 - f 2 × h = 52 + 14 - 10 2 14 - 10 - 10 × 4 = 52 + 16 8 ∴ Mode = 54 . Hence, the modal weight of students is 54 kg .

MEDIUM

10th CBSE

IMPORTANT

Earn 100

An aeroplane when flying at a height of $4000 m$ from the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two aeroplanes from the same point on the ground are $60 °$ and $45 °$ respectively. Find the vertical distance between the two aeroplanes at that instant in $m$ corrected to one decimal place taking $\sqrt{3} = 1.73$ .

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Important Questions on Model Test Paper

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 25

If two sides and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then the two triangles are similar.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 25

Two poles of height

a

metres and

b

metres are

p

metres apart. Prove that the height of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is given by

\frac{ab}{a + b}

metres.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 26

Draw a circle of radius

3 cm

. Take two points

P

and

Q

on either side of the extended diameter at a distance of

8 cm

and

5 cm

respectively from the centre. Draw tangents to the circle from the points

P

and

Q

. Write steps of construction.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 27

Prove that:

\frac{tanθ}{1 - cotθ} + \frac{cotθ}{1 - tanθ} = 1 + secθ cosec θ

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 27

Given that

\tan (A + B) = \frac{tanA + tanB}{1 - tanAtanB}

, find the values of

\tan 75 °

and

\tan 90 °

by taking suitable values of

A

and

B

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 28

Two circles touch internally. The sum of their areas is

116 π {cm}^{2}

and distance between their centres is

6 cm

. Find the radii of the circles.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 29

Radhika has a vessel, of the form of an inverted cone, open at the top, of height

11 cm

and radius of top as

2.5 cm

and is full of water. Metallic spherical balls each of diameter

0.5 cm

are put in the vessel due to which

{\frac{2}{5}}^{th}

of the water in the vessel flows out. Find out how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?

EASY

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 30

The following distribution gives the weights of

60

students of a class. Find the mean and mode weights of the students.

Weight(in kg)	Number of students $(f_{i})$
$40 ‐ 44$	$4$
$44 ‐ 48$	$6$
$48 ‐ 52$	$10$
$52 ‐ 56$	$14$
$56 ‐ 60$	$10$
$60 ‐ 64$	$8$
$64 ‐ 68$	$6$
$68 ‐ 72$	$2$

Important Questions on Model Test Paper

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