10 Question MCQ (including Assertion)

Question 1 of 10

The acute angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by which formula?

A \( \tan \theta = \left|\frac{m_1 + m_2}{1 - m_1 m_2}\right| \) B \( \tan \theta = \left|\frac{1 + m_1 m_2}{m_2 - m_1}\right| \) C \( \tan \theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right| \) D \( \tan \theta = \left|\frac{m_2 - m_1}{1 - m_1 m_2}\right| \)

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Question 2 of 10

Assertion (A): The slope of a line making an angle of \( 90^\circ \) with the positive direction of the x-axis is not defined.

Reason (R): The slope of a line with inclination \( \theta \) is given by \( m = \tan \theta \), and \( \tan 90^\circ \) is not defined.

A Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C Assertion (A) is true, but Reason (R) is false. D Assertion (A) is false, but Reason (R) is true.

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Question 3 of 10

Assertion: A line having inclination \(120^\circ\) has a positive slope.
Reason: The slope is \(m=\tan120^\circ=-\sqrt{3}\).

A Both Assertion and Reason are true, and Reason is the correct explanation of Assertion. B Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion. C Assertion is true, but Reason is false. D Assertion is false, but Reason is true.

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Question 4 of 10

For the triangle with vertices \(A(2,3)\), \(B(4,-1)\) and \(C(1,2)\), the equation and length of the altitude from \(A\) are

A \(x+y-5=0\), length \(2\sqrt2\) B \(x-y-1=0\), length \(\sqrt2\) C \(x+y-5=0\), length \(\sqrt2\) D \(x-y+1=0\), length \(\sqrt2\)

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Question 5 of 10

A line makes intercepts \(-3\) and \(2\) on the \(x\)-axis and \(y\)-axis, respectively. Its equation is

A \(2x+3y-6=0\) B \(3x-2y+6=0\) C \(2x-3y+6=0\) D \(2x-3y-6=0\)

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Question 6 of 10

What is the slope of a line that makes an inclination of \( 60^\circ \) with the positive direction of the x-axis?

A \( 1/\sqrt{3} \) B \( \sqrt{3} \) C \( 1 \) D Not defined

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Question 7 of 10

Assertion (A): The points \( (1, 5) \), \( (2, 3) \), and \( (-2, -11) \) are collinear.

Reason (R): Three points A, B, and C are collinear if and only if the slope of AB is equal to the slope of BC.

A Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C Assertion (A) is true, but Reason (R) is false. D Assertion (A) is false, but Reason (R) is true.

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Question 8 of 10

Write the equation of the line passing through the origin \( (0, 0) \) with a slope of \( m \).

A \( y = x \) B \( y = m \) C \( x = my \) D \( y = mx \)

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Question 9 of 10

What is the equation of a horizontal line situated at a distance of '\( a \)' units above the x-axis?

A \( x = a \) B \( y = a \) C \( x = -a \) D \( y = -a \)

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Question 10 of 10

Assertion (A): The line \( y = 2x + 3 \) is perpendicular to the line \( x + 2y = 5 \).

Reason (R): Two non-vertical lines are perpendicular if \( m_1 + m_2 = 0 \).

A Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C Assertion (A) is true, but Reason (R) is false. D Assertion (A) is false, but Reason (R) is true.

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10 Question MCQ (including Assertion)