Friday, November 22, 2024

MCQ Mathematics Chapter 9: STRAIGHT LINES, HS 1st year

 

MCQs:

1. Who is credited with systematically introducing the study of geometry using algebra?
A. Isaac Newton
B. René Descartes
C. Euclid
D. Pythagoras
Answer: B. René Descartes

2. What is the formula for finding the distance between two points P(x1,y1)P(x_1, y_1) and Q(x2,y2)Q(x_2, y_2)?
A. (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
B. (x2−x1)+(y2−y1)(x_2 - x_1) + (y_2 - y_1)
C. (x2−x1)+(y2−y1)2\frac{(x_2 - x_1) + (y_2 - y_1)}{2}
D. (x1+x2,y1+y2)(x_1 + x_2, y_1 + y_2)
Answer: A. (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

3. What is the slope (mm) of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2)?
A. y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}
B. x2−x1y2−y1\frac{x_2 - x_1}{y_2 - y_1}
C. (x2−x1)⋅(y2−y1)(x_2 - x_1) \cdot (y_2 - y_1)
D. (y2+y1)/(x2+x1)(y_2 + y_1)/(x_2 + x_1)
Answer: A. y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}

4. What is the equation of a line in slope-intercept form?
A. x=my+cx = my + c
B. y=mx+cy = mx + c
C. y−y1=m(x−x1)y - y_1 = m(x - x_1)
D. ax+by+c=0ax + by + c = 0
Answer: B. y=mx+cy = mx + c

5. If the slope of one line is m1m_1 and the slope of another line is m2m_2, the lines are perpendicular if:
A. m1=m2m_1 = m_2
B. m1⋅m2=−1m_1 \cdot m_2 = -1
C. m1+m2=1m_1 + m_2 = 1
D. m1−m2=0m_1 - m_2 = 0
Answer: B. m1⋅m2=−1m_1 \cdot m_2 = -1

6. The perpendicular distance (dd) of a point (x1,y1)(x_1, y_1) from the line Ax+By+C=0Ax + By + C = 0 is given by:
A. ∣Ax1+By1+C∣A2+B2\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
B. Ax1+By1+CA2+B2\frac{Ax_1 + By_1 + C}{A^2 + B^2}
C. ∣Ax1+By1+C∣|Ax_1 + By_1 + C|
D. Ax12+By12+C2\sqrt{Ax_1^2 + By_1^2 + C^2}
Answer: A. ∣Ax1+By1+C∣A2+B2\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}

7. The equation of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:
A. y−y1=m(x−x1)y - y_1 = m(x - x_1)
B. (x−x1)/(x2−x1)=(y−y1)/(y2−y1)(x - x_1)/(x_2 - x_1) = (y - y_1)/(y_2 - y_1)
C. y=mx+cy = mx + c
D. (y2−y1)/(x2−x1)=m(y_2 - y_1)/(x_2 - x_1) = m
Answer: B. (x−x1)/(x2−x1)=(y−y1)/(y2−y1)(x - x_1)/(x_2 - x_1) = (y - y_1)/(y_2 - y_1)

8. A line with slope mm and intercept cc on the y-axis has the equation:
A. y=mx+cy = mx + c
B. y=c−mxy = c - mx
C. mx=y−cmx = y - c
D. y+mx=cy + mx = c
Answer: A. y=mx+cy = mx + c

9. The angle of inclination of a vertical line is:
A. 0∘0^\circ
B. 90∘90^\circ
C. 45∘45^\circ
D. Undefined
Answer: B. 90∘90^\circ

10. If the slope of a line is zero, the line is:
A. Vertical
B. Horizontal
C. Parallel to y-axis
D. At 45∘45^\circ to x-axis
Answer: B. Horizontal

11. The condition for two lines to be parallel is:
A. Their slopes are equal
B. Their slopes are negative reciprocals
C. The product of their slopes is 1
D. The sum of their slopes is zero
Answer: A. Their slopes are equal

12. The midpoint of a line segment joining points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:
A. (x2−x1,y2−y1)(x_2 - x_1, y_2 - y_1)
B. (x1+x22,y1+y22)(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})
C. (x1+x2,y1+y2)(x_1 + x_2, y_1 + y_2)
D. (x2−x12,y2−y12)(\frac{x_2 - x_1}{2}, \frac{y_2 - y_1}{2})
Answer: B. (x1+x22,y1+y22)(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})

13. The equation of a horizontal line at a distance aa from the x-axis is:
A. x=ax = a
B. y=ay = a
C. y+a=0y + a = 0
D. x−a=0x - a = 0
Answer: B. y=ay = a

14. If the area of a triangle is zero, the three points are:
A. Equidistant
B. Non-collinear
C. Collinear
D. At right angles
Answer: C. Collinear

15. A line passing through the origin and making an angle θ\theta with the x-axis has the equation:
A. y=mx+cy = mx + c
B. y=m(x−d)y = m(x - d)
C. y=mxy = mx
D. x=my+cx = my + c
Answer: C. y=mxy = mx

16. The general form of the equation of a line is:
A. Ax+By+C=0Ax + By + C = 0
B. y=mx+cy = mx + c
C. y−y1=m(x−x1)y - y_1 = m(x - x_1)
D. x+y+z=0x + y + z = 0
Answer: A. Ax+By+C=0Ax + By + C = 0

17. The perpendicular distance between two parallel lines Ax+By+C1=0Ax + By + C_1 = 0 and Ax+By+C2=0Ax + By + C_2 = 0 is:
A. ∣C1+C2∣A2+B2\frac{|C_1 + C_2|}{\sqrt{A^2 + B^2}}
B. ∣C1−C2∣A2+B2\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}
C. ∣C1∣+∣C2∣A2+B2\frac{|C_1| + |C_2|}{A^2 + B^2}
D. C1−C2A2+B2\frac{C_1 - C_2}{A^2 + B^2}
Answer: B. ∣C1−C2∣A2+B2\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}

18. The condition for a point (x1,y1)(x_1, y_1) to lie on a line Ax+By+C=0Ax + By + C = 0 is:
A. Ax1+By1+C=0Ax_1 + By_1 + C = 0
B. Ax1−By1+C=0Ax_1 - By_1 + C = 0
C. Ax1+By1−C=0Ax_1 + By_1 - C = 0
D. Ax1−By1−C=0Ax_1 - By_1 - C = 0
Answer: A. Ax1+By1+C=0Ax_1 + By_1 + C = 0

19. The slope of the line perpendicular to the line x−3y=5x - 3y = 5 is:
A. 13\frac{1}{3}
B. 33
C. −3-3
D. −13-\frac{1}{3}
Answer: B. 33

20. The x-intercept of the line 3x+4y−12=03x + 4y - 12 = 0 is:
A. 4
B. −4-4
C. −3-3
D. 3
Answer: D. 3

21. The y-intercept of the line x+2y−6=0x + 2y - 6 = 0 is:
A. 3
B. 6
C. −6-6
D. −3-3
Answer: A. 3

22. If the slopes of two lines are m1m_1 and m2m_2, the angle θ\theta between them is given by:
A. tan⁡θ=m1+m21−m1m2\tan\theta = \frac{m_1 + m_2}{1 - m_1m_2}
B. tan⁡θ=m2−m11+m1m2\tan\theta = \frac{m_2 - m_1}{1 + m_1m_2}
C. tan⁡θ=m1m2m2−m1\tan\theta = \frac{m_1m_2}{m_2 - m_1}
D. tan⁡θ=m1−m21+m1m2\tan\theta = \frac{m_1 - m_2}{1 + m_1m_2}
Answer: B. tan⁡θ=m2−m11+m1m2\tan\theta = \frac{m_2 - m_1}{1 + m_1m_2}

23. Which of the following is the slope-intercept form of a line?
A. y=mx+cy = mx + c
B. y−y1=m(x−x1)y - y_1 = m(x - x_1)
C. Ax+By+C=0Ax + By + C = 0
D. x+y+z=0x + y + z = 0
Answer: A. y=mx+cy = mx + c

24. The condition for two lines to be perpendicular is:
A. m1=m2m_1 = m_2
B. m1⋅m2=−1m_1 \cdot m_2 = -1
C. m1+m2=0m_1 + m_2 = 0
D. m1−m2=1m_1 - m_2 = 1
Answer: B. m1⋅m2=−1m_1 \cdot m_2 = -1

25. The equation of a vertical line at x=ax = a is:
A. x=ax = a
B. y=ay = a
C. x−a=0x - a = 0
D. Both A and C
Answer: D. Both A and C

26. The equation of a line parallel to the x-axis passing through (–3, 4) is:
A. x=−3x = -3
B. y=4y = 4
C. x+y=1x + y = 1
D. x−y=−7x - y = -7
Answer: B. y=4y = 4

27. The area of a triangle formed by points (x1,y1),(x2,y2),(x3,y3)(x_1, y_1), (x_2, y_2), (x_3, y_3) is given by:
A. 12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
B. 12∣x1x2+x2x3+x3x1∣\frac{1}{2} \left| x_1x_2 + x_2x_3 + x_3x_1 \right|
C. 12∣y1(y2−y3)+y2(y3−y1)+y3(y1−y2)∣\frac{1}{2} \left| y_1(y_2 - y_3) + y_2(y_3 - y_1) + y_3(y_1 - y_2) \right|
D. ∣x1+x2+x3∣|x_1 + x_2 + x_3|
Answer: A. 12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|

28. The slope of the line 2x−y+4=02x - y + 4 = 0 is:
A. 22
B. −2-2
C. 12\frac{1}{2}
D. −12-\frac{1}{2}
Answer: A. 22

29. The condition for collinearity of three points (x1,y1),(x2,y2),(x3,y3)(x_1, y_1), (x_2, y_2), (x_3, y_3) is:
A. Area of triangle=0\text{Area of triangle} = 0
B. (y2−y1)(x3−x2)=(y3−y2)(x2−x1)(y_2 - y_1)(x_3 - x_2) = (y_3 - y_2)(x_2 - x_1)
C. Both A and B
D. None of the above
Answer: C. Both A and B

30. If a line passes through points (1, 2) and (3, 6), its slope is:
A. 22
B. −2-2
C. 23\frac{2}{3}
D. 32\frac{3}{2}
Answer: A. 22

31. The equation of a line making equal intercepts on the axes is:
A. x+y=0x + y = 0
B. x−y=0x - y = 0
C. x+y=cx + y = c
D. x−y=cx - y = c
Answer: C. x+y=cx + y = c

32. The length of the perpendicular drawn from point (h,k)(h, k) to a line Ax+By+C=0Ax + By + C = 0 is:
A. ∣Ah+Bk+C∣A2+B2\frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}}
B. Ah2+Bk2+C2\sqrt{Ah^2 + Bk^2 + C^2}
C. ∣Ah+Bk+C∣|Ah + Bk + C|
D. Ah+Bk+CA2+B2\frac{Ah + Bk + C}{\sqrt{A^2 + B^2}}
Answer: A. ∣Ah+Bk+C∣A2+B2\frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}}

33. The slope of the line passing through (0,0)(0, 0) and (1,1)(1, 1) is:
A. 11
B. −1-1
C. 00
D. Undefined
Answer: A. 11

34. For a line with slope mm and y-intercept cc, the equation is:
A. y=mx+cy = mx + c
B. y=m(x−c)y = m(x - c)
C. y−mx=cy - mx = c
D. x+y=cx + y = c
Answer: A. y=mx+cy = mx + c

35. A vertical line has:
A. Zero slope
B. Undefined slope
C. Positive slope
D. Negative slope
Answer: B. Undefined slope

36. The x-intercept of a line is found by:
A. Substituting x=0x = 0 in the equation
B. Substituting y=0y = 0 in the equation
C. Solving for x+y=0x + y = 0
D. Solving x−y=0x - y = 0
Answer: B. Substituting y=0y = 0 in the equation

37. The equation of a line passing through (x1,y1)(x_1, y_1) and having slope mm is:
A. y−y1=m(x−x1)y - y_1 = m(x - x_1)
B. y−y1=m(x+x1)y - y_1 = m(x + x_1)
C. x−x1=m(y−y1)x - x_1 = m(y - y_1)
D. x+x1=m(y+y1)x + x_1 = m(y + y_1)
Answer: A. y−y1=m(x−x1)y - y_1 = m(x - x_1)

38. The general equation of a line with intercepts aa and bb is:
A. xa+yb=1\frac{x}{a} + \frac{y}{b} = 1
B. ya+xb=1\frac{y}{a} + \frac{x}{b} = 1
C. a+b=1a + b = 1
D. ax+by=1ax + by = 1
Answer: A. xa+yb=1\frac{x}{a} + \frac{y}{b} = 1

39. If the slope of one line is 1/31/3, the slope of a line perpendicular to it is:
A. 33
B. −3-3
C. 13\frac{1}{3}
D. −13-\frac{1}{3}
Answer: B. −3-3

40. The slope of a line parallel to 2x−3y=52x - 3y = 5 is:
A. 2/32/3
B. −2/3-2/3
C. 3/23/2
D. −3/2-3/2
Answer: A. 2/32/3

41. A point is equidistant from two parallel lines if:
A. It lies on their perpendicular bisector
B. It is the midpoint of the segment joining the lines
C. It lies midway between them
D. All of the above
Answer: D. All of the above

42. If a line has inclination θ\theta, the slope is:
A. sin⁡θ\sin \theta
B. tan⁡θ\tan \theta
C. cos⁡θ\cos \theta
D. sec⁡θ\sec \theta
Answer: B. tan⁡θ\tan \theta

43. If the line y=mx+cy = mx + c passes through the origin, what is the value of cc?
A. mm
B. 00
C. 11
D. Undefined
Answer: B. 00

44. The equation of a line with undefined slope is:
A. x=kx = k
B. y=mx+cy = mx + c
C. y=ky = k
D. x+y=kx + y = k
Answer: A. x=kx = k

45. What is the angle of inclination of the x-axis?
A. 0∘0^\circ
B. 90∘90^\circ
C. 180∘180^\circ
D. Undefined
Answer: A. 0∘0^\circ

46. Two lines with slopes m1m_1 and m2m_2 are parallel if:
A. m1=m2m_1 = m_2
B. m1⋅m2=−1m_1 \cdot m_2 = -1
C. m1+m2=0m_1 + m_2 = 0
D. m1≠m2m_1 \neq m_2
Answer: A. m1=m2m_1 = m_2

47. The distance of the line 2x−y+3=02x - y + 3 = 0 from the origin is:
A. 35\frac{3}{\sqrt{5}}
B. 25\frac{2}{\sqrt{5}}
C. 15\frac{1}{\sqrt{5}}
D. 55\frac{5}{\sqrt{5}}
Answer: A. 35\frac{3}{\sqrt{5}}

48. The general form of a line making equal intercepts on the axes is:
A. x+y=cx + y = c
B. x−y=0x - y = 0
C. x+y=0x + y = 0
D. y−x=cy - x = c
Answer: A. x+y=cx + y = c

49. The point equidistant from (2,3)(2, 3) and (4,7)(4, 7) is:
A. (3,5)(3, 5)
B. (6,10)(6, 10)
C. (4,5)(4, 5)
D. (3,7)(3, 7)
Answer: A. (3,5)(3, 5)

50. A line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) has the equation:
A. y−y1=y2−y1x2−x1(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)
B. y−y1=x2−x1y2−y1(x−x1)y - y_1 = \frac{x_2 - x_1}{y_2 - y_1}(x - x_1)
C. y−y1=x−x2y - y_1 = x - x_2
D. y−y1=y2−y1y - y_1 = y_2 - y_1
Answer: A. y−y1=y2−y1x2−x1(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)

51. If the slope of a line is −1-1, what is its angle of inclination?
A. 45∘45^\circ
B. 90∘90^\circ
C. 135∘135^\circ
D. 180∘180^\circ
Answer: C. 135∘135^\circ

52. The equation of the line passing through (1,2)(1, 2) with slope 33 is:
A. y−2=3(x−1)y - 2 = 3(x - 1)
B. y−1=3(x−2)y - 1 = 3(x - 2)
C. y=3x+2y = 3x + 2
D. y+2=3x−1y + 2 = 3x - 1
Answer: A. y−2=3(x−1)y - 2 = 3(x - 1)

53. If a line has slope 22 and y-intercept −3-3, its equation is:
A. y=2x−3y = 2x - 3
B. y=2x+3y = 2x + 3
C. y−3=2(x+3)y - 3 = 2(x + 3)
D. y+2x=−3y + 2x = -3
Answer: A. y=2x−3y = 2x - 3

54. A point (h,k)(h, k) lies on the line 3x+4y−5=03x + 4y - 5 = 0 if:
A. 3h+4k−5=03h + 4k - 5 = 0
B. 3h+4k+5=03h + 4k + 5 = 0
C. 3h−4k+5=03h - 4k + 5 = 0
D. −3h+4k−5=0-3h + 4k - 5 = 0
Answer: A. 3h+4k−5=03h + 4k - 5 = 0

55. A line perpendicular to y=3x+7y = 3x + 7 has slope:
A. −13-\frac{1}{3}
B. 33
C. 11
D. −3-3
Answer: A. −13-\frac{1}{3}

56. The intercept form of a line is:
A. xa+yb=1\frac{x}{a} + \frac{y}{b} = 1
B. ya+xb=1\frac{y}{a} + \frac{x}{b} = 1
C. x+y=cx + y = c
D. x−y=ax - y = a
Answer: A. xa+yb=1\frac{x}{a} + \frac{y}{b} = 1

57. If a line has slope 44, its angle of inclination is:
A. 76∘76^\circ
B. tan⁡−1(4)\tan^{-1}(4)
C. 45∘45^\circ
D. 90∘90^\circ
Answer: B. tan⁡−1(4)\tan^{-1}(4)

58. A horizontal line has slope:
A. 00
B. 11
C. Undefined
D. Negative
Answer: A. 00

59. The distance between two points (1,2)(1, 2) and (4,6)(4, 6) is:
A. 55
B. 44
C. 25\sqrt{25}
D. Both A and C
Answer: D. Both A and C

60. If a line passes through the points (2,3)(2, 3) and (5,7)(5, 7), its slope is:
A. 4/34/3
B. 11
C. −1-1
D. 3/43/4
Answer: A. 4/34/3

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