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৪৯তম বিসিএস ⎯ গণিত [৫৫১]

পরীক্ষা৪৯তম বিসিএস ⎯ গণিত [৫৫১]তারিখতারিখ অনির্ধারিতসময়30 minutes২৯ বৈধ · ১ অসম্পূর্ণ
মোট প্রশ্ন৩০
সিলেবাস
Exam - 3 Topics: 1. Pairs of straight lines. Transformation of coordinates. 2. General equation of the second degree, Reduction to standard forms, Conics in general. 3. Planes and straight lines in three dimensions, shortest distance between two straight lines. 4. Vector algebra with applications to geometry. [Source: Class - 3 and Relevant Books]
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

৪৯তম বিসিএস ⎯ গণিত [৫৫১]

৪৯তম বিসিএস ⎯ গণিত [৫৫১] · তারিখ অনির্ধারিত · ৩০ প্রশ্ন

১.
What is the angle between straight lines represented by the equation x2+ 2hxy - y2= 0?
  1. ক
    π/4
  2. খ
    π/2
  3. গ
    0
  4. ঘ
    none
ব্যাখ্যা
Here 
Comparing x2 + 2hxy - y2 = 0 to ax2 + 2hxy + by2 = 0 we get a = 1 and b = -1
​and a + b = 0 so lines are perpendicular, so angle is π/2
২.
Equation of the bisectors of the angle between the straight lines represent by ax2+ 2hxy - by2= 0
    ব্যাখ্যা
    we know 
    Bisector of the angles between the pair of straight lines ax2+2hxy+by2=0 is
    ​Here b=- b, so the equation of bisector becomes or
    ৩.
    What is the condition that ax2+ 2hxy + by2+ 2gx + 2fy + c = 0 represents a pair of straight lines
    1. ক
      abc + 2fgh + af2- bg2- ch2= 0
    2. খ
      abc + 2fgh + af2+ bg2+ ch2= 0
    3. গ
      - abc - 2fgh + af2+ bg2+ ch2= 0
    4. ঘ
      abc - 2fgh + af2+ bg2+ ch2= 0
    ব্যাখ্যা
    The equation represents a pair of straight lines if and only if:
    abc+2fgh-af2-bg2-ch2=0
    ​or, -abc-2fgh+af2+bg2+ch2=0 [multiplying both sides by -1]
    ৪.
    If the direction of axes is turned through an angle 45° and the origin remain unchanged then find the transformed coordinates of (√2, 2√2)
    1. ক
      (3, 1)
    2. খ
      (1, 3)
    3. গ
      (0, √2)
    4. ঘ
      (√2, 0)
    ব্যাখ্যা
    If the trans formed coordinates are (x1, y1) or (√2, 2√2)
    ​We know x1=xcosθ+ysinθ=√2cos45°+2√2sin45°=3
    ​y1=-xsinθ+ycosθ=-√2sing45°+2√2cos45°=1
    ​(x1, y1)=(3, 1)
    ৫.
    The equation of lines bisecting the angles between the coordinate axes is:
    1. ক
      y = x and y = - x
    2. খ
      x = 0 and y = 0
    3. গ
      x + y = 0 and x - y = 0
    4. ঘ
      x = y = 1
    অনির্ধারিত
    ব্যাখ্যা
    সঠিক উত্তর: ক) y = x and y = - x এবং গ) x + y = 0 and x - y = 0
    ​অপশনে দ্বৈত উত্তর থাকায় প্রশ্নটি বাতিল করা হলো। 
    ​------------------- 
    ​
    ​Lines y=±x bisect the right angles formed by x=0(y-axis) and y=0 (x-axis). 
    ​Since the angle between the axes is ±90°, bisector makes angle θ=45° and passes through origin
    ​So equation of bisectors y=tan(±45°) x or, y=±x
    ৬.
    The equation 3x2+ 10xy + 3y2= 0 represents:
    1. ক
      Real and distinct lines
    2. খ
      Coincident lines
    3. গ
      Imaginary lines
    4. ঘ
      Perpendicular lines
    ব্যাখ্যা
    h2-ab=25-9=16>0, so real and distinct.
    ৭.
    The direction cosines of coordinate axes are:
    1. ক
      (1, 0, 0), (0, 1, 0), (0, 0, 1)
    2. খ
      (1, 1, 0), (1, 0, 1), (0, 1, 1)
    3. গ
      (0, 0, 0) for all
    4. ঘ
      None of these
    ব্যাখ্যা
    Direction cosine of x-axis is (cos0, cos90, cos90)=(1, 0, 0)
    Direction cosine of y-axis is (cos90, cos0, cos90)=(0, 1, 0)
    Direction cosine of z-axis is (cos90, cos90, cos0)=(0, 0, 1)
    ৮.
    The sum of slopes of lines represented by ax2+ 2hxy + by2= 0 is:
    1. ক
      - 2h/b
    2. খ
      2h/a
    3. গ
      - 2h/a
    4. ঘ
      2h/b
    ব্যাখ্যা
    If slope of two lines are m1, m2. 
    Then let (y-m1x)(y-m2x)=0, or, y2-(m1+m2)xy+m1m2x2=0  
    And, ax2+2hxy+by2=0 or, y2+(2h/b)xy(a/b)x2=0   
    Comparing we get, m1+m2=-(2h/b)
    ৯.
    Equation x2+ y2= 0 represents:
    1. ক
      Point circle
    2. খ
      Circle with radius 0
    3. গ
      Both A and B
    4. ঘ
      None
    ব্যাখ্যা
    Explanation: Point circle is a circle  with radius 0.
    ১০.
    For ellipse (x2/b2) + (y2/a2) = 1, and b > a eccentricity is:
    1. গ
      b/a
    2. ঘ
      None
    ব্যাখ্যা
    Formula for ellipse eccentricity
    ১১.
    Find the nature of 6x2+ 5xy - 6y2- 4x + 7y + 11=0
    1. ক
      Circle 
    2. খ
      Ellipse
    3. গ
      Hyperbola
    4. ঘ
      Parabola
    ব্যাখ্যা
    ১২.
    If in ax2+ 2hxy + by2+ 2gx + 2fy + c = 0, h2< ab, the conic represents: 
    1. ক
      Ellipse
    2. খ
      Parabola
    3. গ
      Hyperbola
    4. ঘ
      Pair of straight lines
    ব্যাখ্যা
    If h2 < ab → Ellipse
    ১৩.
    Conjugate hyperbola of (x2/a2) - (y2/b2) = 1 is:
    1. ক
      (y2/b2) - (x2/a2) = 1
    2. খ
      (x2/b2) - (y2/a2) = 1
    3. গ
      y2 - x2 = 1
    4. ঘ
      None
    ব্যাখ্যা
    Swap terms to get conjugate hyperbola.
    ১৪.
    (x2/4) + (y2/9) = 1 Find the eccentricity?
    1. ক
      √5/3
    2. খ
      3/5
    3. গ
      3/√5
    4. ঘ
      none
    ব্যাখ্যা
    Compare with (x2/a2) + (y2/b2) = 1 we get a = 2, b=3 and a<b
    ​
    ১৫.
    The conic with eccentricity e=1 is:
    1. ক
      Ellipse
    2. খ
      Parabola
    3. গ
      Circle
    4. ঘ
      Hyperbola
    ব্যাখ্যা
    Parabola has eccentricity = 1.
    ১৬.
    Conic represented by x2+ 2xy + y2= 0 is:
    1. ক
      Ellipse
    2. খ
      Hyperbola
    3. গ
      Pair of lines
    4. ঘ
      Circle
    ব্যাখ্যা
    Factorization (x+y)2=0. Repeated line.
    ১৭.
    Equation of a plane passing through origin and perpendicular to vector (2, - 1, 3) is:
    1. ক
      2x - y + 3z = 0
    2. খ
      2x + y - 3z = 0
    3. গ
      x - 2y + 3z = 0
    4. ঘ
      2x - y - 3z = 0
    ব্যাখ্যা
    Equation: ax+by+cz=0, where (a, b, c) is normal.
    ​Substituting: 2x-y+3z=0.
    ১৮.
    Distance of point (2, - 1, 3) from plane x - 2y + 2z = 4 is:
    1. ক
      2
    2. খ
      3
    3. গ
      4
    4. ঘ
      5
    ব্যাখ্যা
    ১৯.
    If a line has direction ratios (2, - 1, 2), then one set of direction cosines is:
    1. ঘ
      All of these
    ব্যাখ্যা
    ২০.
    Equation of a line passing through (1, - 2, 3) and parallel to vector (2, 1, - 1) is:
    1. ঘ
      None
    ব্যাখ্যা
    ২১.
    Condition for two lines to be perpendicular with direction ratio (l1, m1, n1) and (l2, m2, n2) is:
    1. ক
      l1l2+ m1m2+ n1n2= 0
    2. গ
      l1m2 - l2m1= 0
    3. ঘ
      None
    ব্যাখ্যা
    Condition for two lines to be perpendicular with direction ratio is- l1,l2+m1,m2+n1n2=0
    ২২.
    Shortest distance between skew lines is always:
    1. ক
      Along line perpendicular to both lines
    2. খ
      Along bisector
    3. গ
      In plane containing them
    4. ঘ
      None
    ব্যাখ্যা
    The shortest distance between two skew lines in 3D space is always the length of the line segment that is perpendicular to both of the skew lines.
    ২৩.
    The distance between parallel planes x + y - z = 5 and x + y - z = 7 is:
    1. ক
      2/3
    2. খ
      3/2
    3. গ
      2/√3
    4. ঘ
      None
    ব্যাখ্যা
    Distance between two parallel planes
    ২৪.
    If equation of two lines are and and direction cosine of the line of shortest distance is find the shortest distance?
    1. ক
      1/√6
    2. খ
      2/√6
    3. গ
      0
    4. ঘ
      1
    ব্যাখ্যা
    ২৫.
    The angle θ between two vectors a and b is given by:
    1. ঘ
      None
    ব্যাখ্যা
    The dot product formula gives
    ২৬.
    Which one of the following is true?
      ব্যাখ্যা
      From vector triple product we get
      ​
      ২৭.
      If a = 2i + 6j - 3k and b = i + 4j + 8k. Find the projection of a along b?
      1. ক
        9/2
      2. খ
        9/4
      3. গ
        2/9
      4. ঘ
        4/9
      ব্যাখ্যা
      Projection of a long b is
      ২৮.
      If a line is perpendicular to a plane, its direction vector is:
      1. ক
        Parallel to the plane's normal
      2. খ
        Perpendicular to plane's normal
      3. গ
        Zero vector
      4. ঘ
        None
      ব্যাখ্যা
      Line ⊥ Plane ⇒ direction vector ∥ plane normal.
      ২৯.
      What is perpendicular unit vector of two vectors a, b
      1. ঘ
        None
      ব্যাখ্যা
      Perpendicular unit vector of two vectors a, b is =
      ৩০.
      If two sides of a triangle are i - 4j - k and - 2i - j + k. What is the area of the triangle?
      1. ক
        √(107)/2
      2. খ
        √107
      3. গ
        107
      4. ঘ
        None
      ব্যাখ্যা