Mth 103 TMA SOLUTIONS

MTH103 List of Questions

Q1 Find the equation to the straight line passing through the point of intersection of the lines 5x+6y +1 = 0 and 3x + 2y +5 =0 and perpendicular to the line 3x-5y+11 = 0

 2x + 3y+4 = 0
 5x + 3y+8 = 0
 2x+4y+5 = 0
 3x+ 2y+7 = 0

Q2 Given that vectors P=3i+4j+5k and Q=3i+4j+5k , \[P\dot Q\] is __________
 $$ x^{2}+y^{2}-6x-10y+25=0$$
 $$ x^{2}+y^{2}-5x-16y+34=0$$
 $$ x^{2}+y^{2}-3x+5y+10=0$$
 $$ x^{2}+y^{2}-2x-3y+15=0$$

Q3 The value point A(1, -1) inside the circle is \[x^{2}+y^{2}-3x+4y=12\] ____________
 -17
 -13
 -15
 -12

Q4 Given that ellipse has equation of , the x and y intercepts of the of the equation are ______ and ______
 2, 3
 4, 5
 3,6
 3, 4

Q5 The focus of the parabola whose equation is \[y^{2}+32\] is ________
 F(-8,0)
 F(-6,0)
 F(-4,0)
 F(-5,0)

Q6 Given that ellipse has equation of\[9x^{2}+4y^{2}=36\] , the length of the minor axis is __________
 6
 8
 7
 10

Q7 Given the equation \[9x^{2}-16y^{2}=144\], the intersection on x-axis is ___________
 4
 5
 8
 10

Q8 If a force of 30N acting in the east direction and another force of 40N acting in the north direction. Find the sum of the two vectors is _____________
 50N
 60N
 40N
 100N

Q9 The directrix of the parabola $$ y= -\frac{1}{2} x^{2}$$ is _________
 1/2
 3/5
 1/5
 2/3

Q10 The focus of the parabola $$ y= -\frac{1}{2} x^{2}$$ is _________
 (0, ½)
 (0, 1/3)
 (0, 2/3)
 (0, 3/5)

Q11 If A=i+4j+7k and B=5i-2j+k, find \[A\times B\]
 18i+34j-22k
 22i-15j-10k
 12i+4j-5k
 2i-3j-25k

Q12 Find the equation of the circle with centre (3, 5) and radius 3
 $$ x^{2}+y^{2}-6x-10y+25=0$$
 $$ x^{2}+y^{2}-5x-16y+34=0$$
 $$ x^{2}+y^{2}-3x+5y+10=0$$
 $$ x^{2}+y^{2}-2x-3y+15=0$$

Q13 Find the equation of the tangent at the point (0, 2) to the circle $$x^{2}+y^{2}-4x+2y-8=0$$
 y=2/3 x+2
 y=1/3 x+5
 y=2x+7
 y=3x-1

Q14 Find the standard form of the equation for the parabola with vertex (3, 4) with focus (5, 4).
 \[(y-4)^2=8(x-3)\]
 \[(y+2)^2=8(x+5)\]
 \[(y-3)^2=8(x-5)\]
 \[(y+5)^2=8(x+1)\]

Q15 Given the equation \[9x^2-16y^2=144\]. Find the coordinate of the foci
 \[F_1 (5,0),F_2 (-5,0)\]
 \[F_1 (3,0),F_2 (-3,0)\]
 \[F_1 (2,0),F_2 (-2,0)\]
 F_1 (1,0),F_2 (-1,0)

Q16 Given that ellipse has equation of\ 9x^2+4y^2=36\], Find the length of the major axis
 6
 8
 7
 10

Q17 Find the equation of the parabola having vertex (0, 0), axis along the x-axis and passing through (2, -10)
 \[y^2=4 x\frac{1}{8}\]
 \[y^2=2x \frac{1}{2}\]
 \[y^2=4x\frac{1}{7}\]
 \[y^2=5x\frac{1}{8}\]

Q18 What is the focus in the equation \[y^2=5x\]
 \[(\frac{5}{4}, 0)\]
 \[(\frac{1}{2}, 2)\]
 \[(\frac{1}{4}, 3)\]
 \[(\frac{5}{7}, 5)\]

Q19 What is the equation of the directrix of the parabola whose equation is \[x^2-30y=0\]
 y=-7.5
 y=-5.5
 y=-3.5
 y=-2.5

Q20 Find the equation of the tangent to the circle\[ x^2+y^2-2x+4y+15=0\] at the point (-1, 2)
 2y=x+5
 y=3x+7
 5y=5x+3
 3x+2y=-1

Q21 Find the coordinates of point of contact between the two circles \[x^2+y^2+2x+6y=39\] and \[x^2+y^2-4x-2y+1=0\]
 \[(3\frac{1}{5}, 2\frac{3}{5})\]
 \[(3\frac{3}{5}, 4\frac{1}{5})\]
 \[(2\frac{4}{5}, 4\frac{2}{5})\]
 \[(5\frac{2}{5}, 3\frac{1}{5})\]

Q22 Find the parametric equations of a circle with centre (2, -1) and radius 3
 \[x=2+3 \cos\theta,~~ y=-1+3 \sin⁡\theta\]
 \[x=-1+3 \cos\theta,~~ y=2+3 \sin⁡\theta\]
 \[x=4+9\cos\theta,~~ y=1+9 \sin⁡\theta\]
 \[x=1+9 \cos\theta,~~ y=4+9\sin⁡\theta\]

Q23 Find the points of intersection of the circle x^2+y^2-x-3y=0 with the line y=x-1.
 (1, 0) and (2,1)
 (2, 0) and (0,1)
 (2, 3) and (2, 2)
 (3,1) and (3, 2)

Q24 Find the equation of the circle with circle with centre (-3, 4) which passes through the point (2, 5)
 \[x^{2}+y^{2}+6x-8y=1\]
 \[x^{2}+y^{2}+2x+3y-12=0\]
 \[x^2+y^2+3x-2y-5=0\]
 \[x^2+y^2+3x-5y-6=0\]

Q25 Find the equation of the circle with centre (�??2, 3) and radius 6
 \[x^{2}+y^{2}+4x-6y=23\]
 \[x^{2}+y^{2}+26x-y=3\]
 \[x^{2}+y^{2}+3x-2y=5\]
 \[x^{2}+y^{2}+25x-23y=10\]

Q26 Find the equation of the line through the point (-1, 2) which is perpendicular to y=2x-1
 \[y=-\frac{1}{2}x+\frac{3}{2}\]
 \[y=-\frac{3}{5}x+\frac{7}{3}\]
 \[y=-\frac{2}{3}x+\frac{3}{2}\]
 \[y=-\frac{1}{3}x+\frac{2}{3}\]

Q27 Find the angle between the two lines -3x+4y=8 and -2x-8y-14=0
 \[-14.63^{0}\]
 \[-15.45^{0}\]
 \[-12.37^{0}\]
 \[-24.63^{0}\]

Q28 A straight line has a gradient of 5/3 and it passes through the point (1,3). Find the intercept of the straight line on the y-axis
 -8/3
 -7/5
 -3/4
 -3/4

Q29 Find the sum of vectors \[\bar{AK}, \bar{KL}, \bar{LP}~~and ~~\bar{PQ}\]
 \[\bar{AQ}\]
 \[\bar{AP}\]
 \[\bar{AK}\]
 \[\bar{KP}\]

Q30 A straight line has a gradient of 5/3 and it passes through the point (1,3). Find the equation of the straight line
 3y=5x-8
 2y=6x-2
 y=2x-4
 6y=x-3

Q31 Find the angle of slope of the line joining A(4,3) and B(9,7).
 \[38.66^{0}\]
 \[57.77^{0}\]
 \[55.87^{0}\]
 \[74.53^{0}\]

Q32 If A(3, 6) and B(4, 8) are two points on a line segment. Evaluate the coordinate of the midpoint of AB
 \[(3\frac{1}{2},7)\]
 \[(2,4)\]
 \[(1\frac{1}{2},3)\]
 (5,9)

Q33 Find the distance between the points A(4, 3) and B(6, 5)
 \[2\sqrt(2)\]
 \[\sqrt(3)\]
 \[5\sqrt(5)\]
 \[\sqrt(7)\]

Q34 If a=2i+4j+3k and b=i+5j-2k. Find the vector product a and b
 -23i+7j+6k
 -25i+9j+2k
 -22i+j+10k
 -5i+3j-4k

Q35 Find the magnitude of a components of vector\[\bar{AB}=5i+2j+4k\] expressed in terms of the unit vectors
 \[3\sqrt(5)\]
 \[2\sqrt(5)\]
 \[4\sqrt(5)\]
 \[7\sqrt(5)\]

Q36 Given that\[ Z_1=2i-4j,~~Z_2=2i+6j ~~and~~ Z_3=3i-j\]. Evaluate\[ Z_1-Z_2-Z_3\]
 3i-9j
 -7i+j
 -5i-10j
 -i+6j

Q37 Find the sum of vectors \[\bar{AC}+ \bar{CL}-\bar{ML}\]
 \[\bar{AM}\]
 \[\bar{AL}\]
 \[\bar{AK}\]
 \[\bar{LM}\]

Q38 Evaluate the sum of vectors \[\bar{BC}- \bar{DC}+\bar{DE}+\bar{FE}\]
 \[\bar{BF}\]
 \[\bar{BE}\]
 \[\bar{BD}\]
 \[\bar{DB}\]

Q39 Find the sum of vectors \[\bar{AK}, \bar{KL}, \bar{LP}~~and ~~\bar{PQ}\]
 \[\bar{AQ}\]
 \[\bar{AP}\]
 \[\bar{AK}\]
 \[\bar{KP}\]

Q40 Find the standard equation for parabola whose directrix is the line x=2 and whose focus is the point (-2, 0)
 $$y^{2}= -8x$$
 $$y^{2}= -6x$$
 $$y^{2}= -5x$$
 $$y^{2}= -8x$$

Q41 Find the asymptotes of the hyperbola whose equation is given as \[\frac{x^{2}}{4}-\frac{y^{2}}{9}=1\]
 \[y=\pm\frac{4}{5}x\]
 \[y=\pm\frac{4}{9}x\]
 \[y=\pm\frac{4}{7}x\]
 \[y=\pm\frac{3}{7}x\]

Q42 Find equation of an ellipse whose major axis is vertical, with the center located (-1,3) at the distance between the center and one of the covertices equal to 4, and the distance between the center and one of the vertices equal to 6.
 \[\frac{(x-1)^{2}}{16}-\frac{(y-3)^{2}}{36}=1\]
 \[\frac{(x-1)^{2}}{16}+\frac{(y-3)^{2}}{36}=1\]
 \[\frac{(x+1)^{2}}{16}+\frac{(y+3)^{2}}{36}=1\]
 \[\frac{(x+1)^{2}}{16}+\frac{(y-3)^{2}}{36}=1\]

Q43 Find the equation of the parabola with focus (-1,4) and directrix y=3
 \[y=\frac{1}{2}x^{2}+x+4\]
 \[y=-\frac{1}{2}x^{2}+x+4\]
 \[y=\frac{1}{2}x^{2}-x+4\]
 \[y=\frac{1}{2}x^{2}+x-4\]

Q44 Given the equation \[9x^{2}-16y^{2}=14\], Find the coordinate of the foci
 \[F_{1}(5,0),F_{2}(-5,0)\]
 \[F_{1}(2,0),F_{2}(-2,0)\]
 \[F_{1}(3,0),F_{2}(-3,0)\]
 \[F_{1}(6,0),F_{2}(-5,0)\]

Q45 Given the equation \[9x^{2}-16y^{2}=14\], Find the interception at x
 2
 3
 -4
 5

Q46 Given that ellipse has an equation of \[9x^{2}+4y^{2}=36\]
 \[\frac{x^{2}}{3^{2}}+\frac{y^{2}}{1^{2}}=1\]
 \[\frac{x^{2}}{2^{2}}+\frac{y^{2}}{3^{2}}=1\]
 \[\frac{x^{2}}{4^{2}}+\frac{y^{2}}{3^{2}}=1\]
 \[\frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2}}=1\]

Q47 Find the equation of the parabola having vertex (0,0) axis along the x-axis and pass through (2,-1)
 \[y^{2}=\frac{x}{3}\]
 \[y^{2}=\frac{x}{2}\]
 \[y^{2}=\frac{x}{4}\]
 \[y^{2}=\frac{x}{5}\]

Q48 Find the focus in the equation \[y^{2}=5x\]
 \[F\left(4\frac{1}{4},0\right)\]
 \[F\left(3\frac{1}{4},0\right)\]
 \[F\left(2\frac{1}{4},0\right)\]
 \[F\left(1\frac{1}{4},0\right)\]

Q49 Find the focus of the parabola whose equation is \[y^{2}+32x\]
 F(-5,0)
 F(-8,0)
 F(-7,0)
 F(-6,0)

Q50 Find the directrix of the parabola whose equation is \[x^{2}-30y\]
 \[-7\frac{1}{2}\]
 \[-8\frac{1}{2}\]
 \[6\frac{1}{2}\]
 \[-5\frac{1}{2}\]

Q51 Find the equation of the circle with its center at the origin with points(-3,4) on the circle
 \[x^{2}+y^{2}=25\]
 \[x^{2}-y^{2}=25\]
 \[x^{2}+y^{2}=15\]
 \[x^{2}-y^{2}=15\]

Q52 Find the radius of a circle given by \[x=4+2\cos\theta, y=-3+2\sin\theta\]
 5
 2
 4
 6

Q53 Find the vector product \[a\times b\]. If a = i + 2j - k and b = 2i + 3j + k
 5i - 3j - k
 2i - 4j - k
 3i + j - k
 I - j + 3k

Q54 Find the center of a center of a circle given by \[x=4+2\cos\theta, y=-3+2\sin\theta\]
 (5,3)
 (2,1)
 (4,-3)
 (3,6)

Q55 Find the parametric equations of a circle with centre (2,-1) and radius 3
 \[x=2+3\cos\theta, y=-1+3\sin\theta\]
 \[x=1+2\cos\theta, y=-2+3\sin\theta\]
 \[x=2+5\cos\theta, y=3\sin\theta\]
 \[x=-2+5\cos\theta, y=-3+2\sin\theta\]

Q56 Find the point of the intersection of the center \[x^{2}+y^{2}-3y=0\] with line y=x-1
 (1,0) and (2,3)
 (1,0) and (2,1)
 (0,1) and (1,1)
 (2,3) and (2,2)

Q57 Given the equation of a circle is \[x^{2}+y^{2}+2x-6y-15\]. Find the radius of the circle
 5
 3
 4
 1

Q58 Given the equation of a circle is \[x^{2}+y^{2}+2x-6y-15\]. Find the center of the circle
 (-1,2)
 (2,5)
 (2,5)
 (-1,3)

Q59 Find the center of the circle \[x^{2}+y^{2}+8x + 6y = 0\].
 (-3,-4)
 (-4,-3)
 (3,4)
 (4,3)

Q60 Find the radius of the circle \[x^{2}+y^{2}+8x + 6y = 0\].
 6
 3
 4
 5