Trigonometry One Mark Online Test — Class 10 Mathematics

Trigonometry One Mark Online Test (10th Standard Mathematics)

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The value of \(\sin^2 \theta + \frac{1}{1 + \tan^2 \theta}\) is equal to

A.\(\tan^2 \theta\) B.1 C.\(\cot^2 \theta\) D.0

\(\tan \theta \csc^2 \theta - \tan \theta\) is equal to

A.\(\sec \theta\) B.\(\cot^2 \theta\) C.\(\sin \theta\) D.\(\cot \theta\)

If \((\sin \alpha + \csc \alpha)^2 + (\cos \alpha + \sec \alpha)^2 = k + \tan^2 \alpha + \cot^2 \alpha\), then the value of \(k\) is equal to

A.9 B.7 C.5 D.3

If \(\sin \theta + \cos \theta = a\) and \(\sec \theta + \csc \theta = b\), then the value of \(b(a^2 - 1)\) is equal to

A.\(2a\) B.\(3a\) C.0 D.\(2ab\)

If \(5x = \sec \theta\) and \(\frac{5}{y} = \tan \theta\), then \(x^2 - \frac{1}{y^2}\) is equal to

A.25 B.\(\frac{1}{25}\) C.5 D.1

If \(\sin \theta = \cos \theta\), then \(2\tan^2 \theta + \sin^2 \theta - 1\) is equal to

A.\(-\frac{3}{2}\) B.\(\frac{3}{2}\) C.\(\frac{2}{3}\) D.\(-\frac{2}{3}\)

If \(x = a\tan \theta\) and \(y = b\sec \theta\), then

A.\(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\) B.\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) C.\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) D.\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0\)

\((1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta)\) is equal to

A.0 B.1 C.2 D.-1

\(a\cot \theta + b\csc \theta = p\) and \(b\cot \theta + a\csc \theta = q\), then \(p^2 - q^2\) is equal to

A.\(a^2 - b^2\) B.\(b^2 - a^2\) C.\(a^2 + b^2\) D.\(b - a\)

Q10

If the ratio of the height of a tower and the length of its shadow is \(\sqrt{3} : 1\), then the angle of elevation of the sun has measure

A.\(45^\circ\) B.\(30^\circ\) C.\(90^\circ\) D.\(60^\circ\)

Q11

An electric pole subtends an angle of \(30^\circ\) at a point on the same level as its foot. At a second point \(b\) metres above the first, the depression of the foot of the pole is \(60^\circ\). The height of the pole (in metres) is equal to

A.\(b\sqrt{3}\) B.\(\frac{b}{3}\) C.\(\frac{b}{2}\) D.\(\frac{b}{\sqrt{3}}\)

Q12

A tower is \(60\text{ m}\) high. Its shadow reduces by \(x\text{ metres}\) when the angle of elevation of the sun increases from \(30^\circ\) to \(45^\circ\), then \(x\) is equal to

A.\(41.92\text{ m}\) B.\(43.92\text{ m}\) C.\(43\text{ m}\) D.\(45.6\text{ m}\)

Q13

The angles of depression of the top and bottom of a \(20\text{ m}\) tall building from the top of a multistoried building are \(30^\circ\) and \(60^\circ\) respectively. The height of the multistoried building and the distance between the two buildings (in metres) are

A.\(20,\ 10\sqrt{3}\) B.\(30,\ 5\sqrt{3}\) C.\(20,\ 10\) D.\(30,\ 10\sqrt{3}\)

Q14

Two persons are standing \(x\text{ metres}\) apart from each other and the height of the first person is double that of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the shorter person (in metres) is

A.\(2x\) B.\(\frac{x}{2\sqrt{2}}\) C.\(\frac{x}{2}\) D.\(2\sqrt{2}x\)

Q15

The angle of elevation of a cloud from a point \(h\) metres above a lake is \(\beta\). The angle of depression of its reflection in the lake is \(45^\circ\). The height of location of the cloud from the lake is

A.\(\frac{h(1 + \tan \beta)}{1 - \tan \beta}\) B.\(\frac{h(1 - \tan \beta)}{1 + \tan \beta}\) C.\(h\tan(45^\circ - \beta)\) D.none of these

Q16

If \(x = a \\sin \\theta\) and \(y = b \\cos \\theta\), then the value of \(b^2 x^2 + a^2 y^2\) is

A.\(a^2 b^2\) B.\(a^2 + b^2\) C.\(\\frac{a^2}{b^2}\) D.1

Q17

If \(\\tan \\theta + \\cot \\theta = 2\), then the value of \(\\tan^2 \\theta + \\cot^2 \\theta\) is

A.4 B.2 C.1 D.0

Q18

The value of \((1 + \\cot^2 \\theta)(1 - \\cos \\theta)(1 + \\cos \\theta)\) is

A.1 B.-1 C.\(\\sin^2 \\theta\) D.\(\\cos^2 \\theta\)

Q19

If \(\sin \theta = \cos \theta\) for \(0^\circ \le \theta \le 90^\circ\), then the value of \(2\tan^2 \theta + \sin^2 \theta - 1\) is

A.\(\\frac{3}{2}\) B.\(\\frac{1}{2}\) C.1 D.2

Q20

If \(\\sec \\theta + \\tan \\theta = x\), then the value of \(\\sec \\theta - \\tan \\theta\) is

A.\(x\) B.\(\\frac{1}{x}\) C.\(x^2\) D.1 - x

Q21

If \(\\cos \\theta = \\frac{3}{5}\), then the value of \(\\frac{\\sin \\theta - \\cot \\theta}{2 \\tan \\theta}\) is

A.\(\\frac{3}{160}\) B.\(\\frac{3}{80}\) C.\(\\frac{1}{160}\) D.\(\\frac{1}{80}\)

Q22

An observer finds that the angle of elevation of the top of a tower is \(30^\circ\). If the observer moves \(20\) m closer to the tower, the angle of elevation becomes \(60^\circ\). The height of the tower is

A.\(10\\sqrt{3}\) m B.\(20\\sqrt{3}\) m C.\(10\) m D.\(15\) m

Q23

The angle of elevation of the top of a building from a point on the ground, which is \(30\) m away from the foot of the building, is \(45^\circ\). The height of the building is

A.\(30\) m B.\(15\) m C.\(30\\sqrt{3}\) m D.\(\\frac{30}{\\sqrt{3}}\) m

Q24

If a pole of height \(6\) m casts a shadow of length \(2\\sqrt{3}\) m on the ground, then the angle of elevation of the sun is

A.\(30^\circ\) B.\(45^\circ\) C.\(60^\circ\) D.\(90^\circ\)

Q25

If the angle of depression of an object on the ground from the top of a \(75\) m high tower is \(30^\circ\), then the distance of the object from the foot of the tower is

A.\(75\\sqrt{3}\) m B.\(25\\sqrt{3}\) m C.\(75\) m D.\(150\) m

Q26

If \(\\cos \\theta + \\sec \\theta = 2.5\), then the value of \(\\cos^2 \\theta + \\sec^2 \\theta\) is

A.4.25 B.6.25 C.5.25 D.2.25

Q27

The expression \(\\frac{\\sin^3 \\theta + \\cos^3 \\theta}{\\sin \\theta + \\cos \\theta} + \\sin \\theta \\cos \\theta\) simplifies to

A.1 B.2 C.1 + 2\\(\sin\) \\(\theta\) \\(\cos\) \\(\theta\) D.\(\\sin^2 \\theta \\cos^2 \\theta\)

Q28

If \(\\tan \\theta = \\frac{a}{b}\), then the value of \(\\frac{a \\sin \\theta - b \\cos \\theta}{a \\sin \\theta + b \\cos \\theta}\) is

A.\(\\frac{a^2 - b^2}{a^2 + b^2}\) B.\(\\frac{a^2 + b^2}{a^2 - b^2}\) C.\(\\frac{a - b}{a + b}\) D.1

Q29

A ladder leaning against a vertical wall makes an angle of \(60^\circ\) with the ground. If the foot of the ladder is \(3.5\) m away from the wall, then the length of the ladder is

A.\(7\) m B.\(3.5\\sqrt{3}\) m C.\(14\) m D.\(5\) m

Q30

If \(3 \\sin \\theta + 4 \\cos \\theta = 5\), then the value of \(3 \\cos \\theta - 4 \\sin \\theta\) is

A.0 B.1 C.5 D.-5

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About this Trigonometry test

This free online practice test covers Trigonometry from the 10th Standard Mathematics (Samacheer Kalvi) syllabus. Choose the number of questions and an optional time limit, then answer and submit — everything is checked in your browser, with the correct answers and a worked explanation shown at the end. For the full solutions to every question in this set, see the solved MCQs page.

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