If c o t θ = x y ,then prove that x cos θ - y sin θ x cos θ + y sin θ = x 2 - y 2 x 2 + y 2

Given that, c o t θ = x y cos θ sin θ = x y y cos θ = x sin θ sin θ = y cos θ x In x cos θ - y sin θ x cos θ + y sin θ = x 2 - y 2 x 2 + y 2 L . H . S = x cos θ - y sin θ x cos θ + y sin θ x cos θ - y sin θ x cos θ + y sin θ = x cos θ - y × y cos θ x x cos θ + y × y cos θ x = x 2 cos θ - y 2 cos θ x 2 cos θ + y 2 cos θ = cos θ x 2 - y 2 cos θ x 2 + y 2 cos θ x 2 - y 2 cos θ x 2 + y 2 = x 2 - y 2 x 2 + y 2 Hence, L . H . S = R . H . S x cos θ - y sin θ x cos θ + y sin θ = x 2 - y 2 x 2 + y 2 PROVED

If c o t θ = x y ,then prove that x cos θ - y sin θ x cos θ + y sin θ = x 2 - y 2 x 2 + y 2

Given that, c o t θ = x y cos θ sin θ = x y y cos θ = x sin θ sin θ = y cos θ x In x cos θ - y sin θ x cos θ + y sin θ = x 2 - y 2 x 2 + y 2 L . H . S = x cos θ - y sin θ x cos θ + y sin θ x cos θ - y sin θ x cos θ + y sin θ = x cos θ - y × y cos θ x x cos θ + y × y cos θ x = x 2 cos θ - y 2 cos θ x 2 cos θ + y 2 cos θ = cos θ x 2 - y 2 cos θ x 2 + y 2 cos θ x 2 - y 2 cos θ x 2 + y 2 = x 2 - y 2 x 2 + y 2 Hence, L . H . S = R . H . S x cos θ - y sin θ x cos θ + y sin θ = x 2 - y 2 x 2 + y 2 PROVED

If ∠ P + ∠ Q = 90 ° ,then show that sin P cos Q - sin P cos Q = cos P

We need to prove that, sin P cos Q - sin P cos Q = cos P Given that, ∠ P + ∠ Q = 90 ° ∠ Q = 90 ° - ∠ P In, sin P cos Q - sin P cos Q = sin P cos 90 ° - P - sin P cos 90 ° - P ∵ cos 90 ° - θ = sin θ sin P cos 90 ° - P - sin P cos 90 ° - P = sin P sin P - sin P sin P sin P sin P - sin P sin P = 1 - sin 2 P = cos 2 P = cos P ∵ 1 - sin 2 θ = cos 2 θ Therefore, sin P cos Q - sin P cos Q = cos P Proved

For what values of θ 0 ° ≤ θ ≤ 90 ° the equation 2 cos 2 θ - 3 cos θ + 1 = 0 will be satisfied.

Given that, 2 cos 2 θ - 3 cos θ + 1 = 0 ,where θ 0 ° ≤ θ ≤ 90 ° Let, cos θ = x 2 x 2 - 3 x + 1 = 0 2 x 2 - 2 x - x + 1 = 0 2 x x - 1 - 1 x - 1 = 0 2 x - 1 x - 1 = 0 Thus, x = 1 2 or x = 1 If x = 1 2 cos θ = 1 2 , ⇒ θ = 60 ° And for cos θ = 1 , ⇒ θ = 0 ° Value of θ is 0 ° and 60 °

In an art competition the marks obtained are given below. From the marks, find the mean. Marks 0 or greater than 0 10 or greater than 10 20 or greater than 20 30 or greater than 30 40 or greater than 40 50 or greater than 50 Number of competitors 40 36 22 11 2 0

Given that in an art competition the marks obtained are given below. Marks 0 or greater than 0 10 or greater than 10 20 or greater than 20 30 or greater than 30 40 or greater than 40 50 or greater than 50 Number of competitors 40 36 22 11 2 0 Class interval Mid-point x i Frequency f i cumulative frequency c . f x i f i 0 - 10 5 4 40 20 10 - 20 15 14 36 210 20 - 30 25 11 22 275 30 - 40 35 9 11 315 40 - 50 45 2 2 90 ∑ f i = 40 ∑ x i f i = 910 We know that Arithmetic mean = ∑ x i f i ∑ f i Thus, 910 40 = 22 . 75 Hence, the mean from the given frequency distribution table is 22 . 75 .

MEDIUM

10th West Bengal Board

IMPORTANT

Earn 100

If $c o t θ = \frac{x}{y}$ ,then prove that $\frac{x \cos θ - y \sin θ}{x \cos θ + y \sin θ} = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$

Important Questions on Third Summative Test

MEDIUM

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 12

∠ P + ∠ Q = 90 °

,then show that

\sqrt{\frac{\sin P}{\cos Q} - \sin P \cos Q} = \cos P

MEDIUM

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 12

For what values of

θ (0 ° \leq θ \leq 90 °)

the equation

2 \cos^{2} θ - 3 \cos θ + 1 = 0

will be satisfied.

HARD

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 13

The external radius of a hollow sphere made of

1 c m

thick lead sheet is

6 c m

. By melting the sphere, a solid cylindrical rod of

2 c m

radius is formed. What is the length of the rod ?

MEDIUM

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 13

A heap of wheat is in the form of right circular cone of diameter

9 m

and height

3.5 m

. What is the volume of total wheat. How much polythene in

m^{2}

is required to just cover the heap of wheat?(take

π = 3.14, \sqrt{130} = 11.4

)

HARD

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 13

The upper surface of a metallic sphere is cut out in such a way that the curved surface of the new sphere is half the previous one. What will be the ratio of the volume of the cut-out portion and the remaining portion of the sphere ?

HARD

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 14

The angle of depression of the top and foot of a lamp post as seen from the roof of a house

11

metre high are

30 °

and

60 °

. What is the height of the lamp post ?

HARD

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 14

From a point on a horizontal line lying in the same plane with the chimney if

50

metres proceeded towards the chimney, the angle of elevation of the top becomes

60 °

from

30 °

. What is the height of the chimney ?

HARD

10th West Bengal Board

IMPORTANT

SCHOOL MATHEMATICS Class 10>Chapter 35 - Third Summative Test>THIRD SUMMATIVE TEST>Q 15

In an art competition the marks obtained are given below. From the marks, find the mean.

Marks	$0$ or greater than $0$	$10$ or greater than $10$	$20$ or greater than $20$	$30$ or greater than $30$	$40$ or greater than $40$	$50$ or greater than $50$
Number of competitors	$40$	$36$	$22$	$11$	$2$	$0$

If cotθ=xy,then prove that xcosθ-ysinθxcosθ+ysinθ=x2-y2x2+y2

Important Questions on Third Summative Test

Learn and Prepare for any exam you want !

If $c o t θ = \frac{x}{y}$ ,then prove that $\frac{x \cos θ - y \sin θ}{x \cos θ + y \sin θ} = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$