#### A railway carriage has a wheel of diameter 3 m and it makes 7200 revolutions in 10 minutes. Then, the speed of the train is ____.

#### The angle of a regular polygon are in A. P. The smallest angle is 2π/3 radians and the common difference is 5^{0}. Then, the number of sides of the polygon is ____.

The sum of the exterior angles of polygon = 360^{0} Sum of n terms of an A. P is S_{n} =n/2(2a+(n-1)d) The great exterior angle is π^{c }/3=60^{0}(linear pair of2π^{c } /3) And the common difference for this A. P. is - 5^{0}. 360^{0 }=n/2(2(60^{0})+(n-1)(-5^{0})) 720 = n(120 - 5n + 5) 720 = 5 n^{2}+ 125n n^{2}- 25n + 144 = 0 (n - 16) (n - 9) = 0 n = 16 or 9 For n = 16, t_{16} = 60 + 15(-5) = -15. We cannot have negative angle in a polygon. Therefore, n = 9

#### If sin A = sin B and cos A = cos B and A ≠ B, then A and B differ by ____.

sin A = sin B A = nπ + (-1)^{n} B,where n is an integer (A = .........-2π + B, -π + B, B, π - B...........) cos A = cos B A = 2mπ ± B, where A = ....-2π ± B, ±B, 2π ± B.........and m is an integer Common solution A = ...-2π ± B, ±B, 2π ± B....... A = 2nπ + B, where n is an integer using the condition A ≠ B A - B = 2nπ Putting n = 1, A - B = 2π

#### The range of y = 2 - cos x is _________________.

The range of cosine function is [-1, 1]. When cos x takes minimum value -1, -cos x = 1. 2 - cosx = 3. Similarly, when cos x takes maximum 1, then -cos x = -1. Then, 2 - cosx = 2 - 1 = 1. Therefore, the range is [1, 3].

#### The angles of a triangle are in the ratio 2: 5: 8. Then, the smallest angle measure in radians is ____.

The sum of the three angles in a triangle is = 180^{0}. And, 180^{0}= Π radians Therefore, 2x + 3x + 5x = π^{c} The smallest angle is (2x/10x)(π) = π^{c} /5

#### The range of the function |3 cos x| is _______________ .

-1 ≤ cos x ≤ 1 -3 ≤ 3 cos x ≤ 3 0 ≤ |3 cos x| ≤ 3 Range of the function is [0, 3]

#### A wheel makes 540 revolutions in 1 minute. The angle swept in 1 second by the wheel, measured in radians is ____.

In 1 revolution, the angle swept by the wheel is 2 Π radians. In 540 revolutions, the angle swept is 1080 Π radians and this is covered in 1 minute. The angle swept in 1 second is 1080 Π/ 60 = 18 Π radians

#### The general solutions of trigonometric equations are infinitely many, since trigonometric functions are ____.

Trigonometric functions are periodic in nature. They attain the same value after a specific For example, sine and cosine functions periods are the same, viz 2π. sin θ, sin 2π + θ, ........have the same value. These functions are called many to one functions.

#### The domain of the function y = tan x is ______________.

The function tan x becomes undefined whenever x is an odd multiple of Π/2. Therefore, those points are to be removed from (-?,?). Hence, the domain is (- ∞, ∞) – {(2n, +, 1)?/ 2}n?Z

#### The angle in radians between the hour hand and minute hand of a clock at half past one is ____.

An hour hand covers 2π radians in 12 hours. In 1 hour, it covers 2π/12 = π/6 radians A minute hand covers 2π radians in 1 hour. In 30 minutes, it covers π radians. The angle between them = π − π/6 = 5 π^{c }/6

#### The measure of an angle in grades of a regular pentagon is ____.

The sum of all exterior angles of a regular polygon is 360^{0}. A pentagon has 5 sides. Being regular, all angles are of same measure. 1 right angle = 90^{0}= 100 grades Sum of 5 exterior angles = 400^{g}, each angle = 80^{g} Then, each interior angle is linear pair of each exterior angle. The interior angle is 200^{g} - 80^{g} = 120^{g}

#### The domain of y = cot x is ____.

cot x = cos x/sin x. The function becomes undefined whenever sin x becomes 0. Therefore, from real numbers, remove those values of x where the function becomes undefined. We know sin x = 0 when x = nΠ where n ε Z. Hence, the domain of y = cot x is (-∞, ∞) – n, n/2:n∈Z.

#### The period of 4 sin 3/2 x is __________.

We know that sine and cosine functions are periodic functions with period 2Π. Therefore, 4sin3/2 x = 4sin(3/2, x, +, 2, π) = 4sin 3(1/2, x,+,2π/3) The period of the function is 2π/3.

#### If t_{n} = cos^{n} θ + sin^{n} θ, then 2t_{6} - 3t_{4}+ 1 is ______________ .

t_{6 }= cos^{6} θ + sin^{6} θ 2t_{6} = 2(cos^{6}θ + sin^{6} θ) = 2 {(cos^{2} θ + sin^{2} θ)(cos^{4} θ - cos^{2} θ sin^{2} θ + sin^{4} θ)} t_{4} = cos^{4} θ + sin^{4 }θ We know that (cos^{2} θ + sin^{2} θ)^{2} = cos^{4} θ + 2sin^{2} θ cos^{2} θ + sin^{4}θ Implies, 1^{2} = cos^{4}θ + 2sin^{2}θ cos^{2}θ + sin^{4}θ Therefore, t_{4 }= cos^{4}θ + sin^{4}θ = 1 - 2sin^{2}θcos^{2}θ Hence, -3t_{4} = -3 (cos^{4} θ + sin^{4 }θ) =- 3(1 - 2sin^{2} θ cos^{2} θ) Therefore, 2t_{6} - 3t_{4} + 1 = 2{(cos^{2} θ + sin^{2} θ)(cos^{4} θ - cos^{2} θ sin^{2} θ + sin^{4} θ)} - 3(1 -2sin^{2}θ cos^{2} θ) + 1 = 2(1) (cos^{2}θ + sin^{4}θ - cos^{2} θ sin^{2} θ) - 3(1 - 2sin^{2} θ cos^{2 } θ) +1 = 2 (1 - 2 sin^{2}θ cos^{2}θ - sin^{2}θcos^{2}θ) - 3(1 – 2 sin^{2}θ cos^{2}θ) + 1 = 2 - 3 + 1 = 0

#### The range of the function 3 + sin x is ________________.

-1 ≤ sin x ≤ 1 - 1 + 3 ≤ 3 + sin x ≤ 1 + 3 2 ≤ sin x ≤ 4 Hence, of the options given, the range of 3 + sin x is [2, 4].

#### The angles of a quadrilateral are in G. P. with common ratio r > 1. The ratio of the larger angle to smaller angle is 8. Then, the second largest angle of the quadrilateral is ____.

Let the angles which are in G. P be a, ar, ar^{2} and ar^{3}. The sum of the angles in a quadrilateral is 360^{0}. a + ar + ar^{2} + ar^{3} = 3600 ar^{3 }/a = 8, since r > 1 ⇒ r^{3} = 8 ⇒ r = 2 a(1 + 2 + 2^{2} + 2^{3}) = 360^{0 } ⇒15a = 360^{0} ⇒ a = 24^{0} The other angles are 48^{0}, 96^{0} and 192^{0} The second largest angle is 96^{0}.

#### The trigonometric expression 2+2+2cos 2θ is equal to ____.

2+2+2cos2θ=2+2+2(2 cos^{2}θ-1) =2+4 cos^{2}θ =2+2cosθ =2(1+cosθ) =4 cos^{2}θ/2 =2cosθ/2

#### A negative angle which lies in the I quadrant from the given four angles is ______________.

When we measure angles, we always start from X - axis and go in the anticlockwise direction for + angles. When we start from X – axis and go in clockwise direction, we get negative angles. Among the given angles, -70^{0} lies in the IV quadrant. Therefore, -330^{0}lies in the I quadrant.

#### In a circle of diameter 20 cm, the length of the chord is 10 cm. Then, the length of the major arc of the chord is ____.

The length of the major arc = Circumference - length of the arc AC = 20π - 60^{0 }/360^{0}20π =5/6 20π =50/3 πcm

#### The maximum value of the function y = 3sin x + 5 cos x in the domain [0, Π] is ______________.

y = 3 sin x + 5 cos x in [0, Π] is maximum when cos x = 1, i.e. when x is 0. Hence, maximum value is 5.

#### The wiper of a car is of length 35 cm. It sweeps an angle of 45^{0} for every one move. The area swept by the wiper in 1 move is ____.

The wiper sweeps a sector of area = θ^{0 }/360^{0}πr^{2}, where r is the length of the wiper. Area = 45^{0}/36^{0 }π(35)^{2} =1225/8πsq.cm

#### The ratio of the radii of two circles is 22: 13. Then, the ratio of the angle subtended by an arc of length l at the centre of each of the circle is ____.

Let the radii of two circles be 22x and 13x respectively and the angle subtended in the first circle be θ_{1} and in the second circle be θ_{2}. {(θ1 /3600 )2π22x}/ {( θ2 /3600 )2π13x = 1 θ_{1}/θ_{2 } = 13/22

#### sin 163° sin 163° + sin 73° sin 107° = ____ .

sin 163° sin 163° + sin 73° sin 107° = sin (180° - 17°) sin (90° + 73°) + sin (90° - 17°) sin (180° - 73°) = sin 17° cos 73° + cos 17° sin 73° = sin (17° + 73°) = sin (90°) = 1

#### The trigonometric ratios that are positive for θ = -8Π/3 are ________________.

- angle implies one should go in the clockwise direction, and -8π/3 = -480^{0} will arrive at the III quadrant, where tan and cot ratios are positive.

#### The domain of y = secant x is _______________ .

y = secant x = 1/cos x The function becomes undefined whenever cos x = 0. Therefore, the domain of the function is domain of cos x - the points where cos x becomes 0. The domain of secant x is (-∞, ∞) - {(2n + 1) Π/2}, where n is an integer.

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