A man running a race course notes that the sum of the distances from the two flag posts from him is always 10 meter and the distance between the flag posts is 8 meter. The equation of posts traced by the man is
1.x²/9 + y²/5 = 1
2. x²/9 + y2 /25 = 1
3.x²/5 + y²/9 = 1
4. x²/25 + y²/9 = 1
A rod of length 12 CM moves with its and always touching the co-ordinate Axes. Then the equation of the locus of a point P on the road which is 3 cm from the end in contact with the x-axis is
1. x²/81 + y²/9 = 1
2.x²/9 + y²/81 = 1
3.x²/169 + y²/9 = 1
4.x²/9 + y²/169 = 1
A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
1.2√3(i – j + k)
2.2√3(-i + j + k)
3. 2√3(i + j – k)
4. 2√3(i + j + k)
At what point of the parabola x² = 9y is the abscissa three times that of ordinate
1.(1, 1)
2.(3, 1)
3.(-3, 1)
4.(-3, -3)
Equation of the line passing through (0, 0) and slope m is
1. y = mx + c
2.x = my + c
3.y = mx
4. x = my
If (a, b) is the mid point of a chord passing through the vertex of the parabola y² = 4x, then
1.a = 2b
2.2a = b
3. a² = 2b
4.2a = b²
If 2/3, k, 5/8 are in AP then the value of k is
1.31/24
2. 31/48
3.24/31
4. 48/31
If a is the A.M. of b and c and G1 and G2 are two GM between them then the sum of their cubes is
1.abc
2. 2abc
3.3abc
4.4abc
If a parabolic reflector is 20 cm in diameter and 5 cm deep then the focus of parabolic reflector is
1.(0 0)
2.(0, 5)
3. (5, 0)
4.(5, 5)
If f(x) = x × sin(1/x), x ≠ 0, then Limx→0 f(x) is
1.1
2.0
3.-1
4.does not exist
If α, β, γ are the angles made by a half ray of a line respectively with positive directions of X-axis Y-axis and Z-axis, then sin² α + sin² β + sin² γ =
1.1
2.0
3.-1
4.None of these
If P(x, y, z) is a point on the line segment joining Q(2, 2, 4) and R(3, 5, 6) such that the projections of OP on the axes are 13/5, 19/5, 26/5 respectively, then P divides QR in the ration
1.1 : 2
2.3 : 2
3.2 : 3
4.1 : 3
If the end points of a diagonal of a square are (1, -2, 3) and (2, -3, 5) then the length of the side of square is
1.√3 unit
2. 2√3 unit
3.3√3 unit
4.4√3 unit
If the length of the tangent from the origin to the circle centered at (2, 3) is 2 then the equation of the circle is
1.(x + 2)² + (y – 3)² = 3²
2. (x – 2)² + (y + 3)² = 3²
3.(x – 2)² + (y – 3)² = 3²
4.(x + 2)² + (y + 3)² = 3²
If the sum of the first 2n terms of the A.P. 2, 5, 8, ….., is equal to the sum of the first n terms of the A.P. 57, 59, 61, ….., then n equals
1.10
2.11
3.12
4.13
If the sum of the roots of the quadratic equation ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then a/c, b/a, c/b are in
1.A.P.
2.G.P.
3.H.P
4.A.G.P.
If the third term of an A.P. is 7 and its 7 th term is 2 more than three times of its third term, then the sum of its first 20 terms is
1. 228
2.74
3.740
4.1090
If two vertices of a triangle are (3, -2) and (-2, 3) and its orthocenter is (-6, 1) then its third vertex is
1.(5, 3)
2.(-5, 3)
3. (5, -3)
4.(-5, -3)
In a ΔABC, if A is the point (1, 2) and equations of the median through B and C are respectively x + y = 5 and x = 4, then B is
1.(1, 4)
2.(7, – 2)
3. (4, 1)
4.None of these
In a three dimensional space, the equation 3x – 4y = 0 represents
1.a plane containing Y axis
2. a plane containing Z axis
3.a plane containing X axis
4.None of these
In an ellipse, the distance between its foci is 6 and its minor axis is 8 then its eccentricity is
1. 4/5
2.1/√52
3. 3/5
4.1/2
Let Tr be the r th term of an A.P., for r = 1, 2, 3, … If for some positive integers m, n, we have Tm = 1/n and Tn = 1/m, then Tm n equals
1.1/m n
2.1/m + 1/n
3.1
4. 0
Limx→-1 [1 + x + x² + ……….+ x10] is
1.0
2.1
3.-1
4.2
Limx→0 {(ax – bx)/ x} is equal to1. log a
2. log b
3.log (a/b)
4.log (a×b)
Limx→0 log(1 – x) is equals to
1. 0
2.1
3.1/2
4.None of these
Limy→∞ {(x + 6)/(x + 1)}(x+4) equals1. e
2.e³
3.e5
4. e6
The 35th partial sum of the arithmetic sequence with terms an = n/2 + 1
1. 240
2.280
3.330
4.350
The angle between the lines x – 2y = y and y – 2x = 5 is
1. tan-1 (1/4)
2.tan-1 (3/5)
3.tan-1 (5/4)
4. tan-1 (2/3)
The angle between the vectors with direction ratios are 4, -3, 5 and 3, 4, 5 is
1. π/2
2.π/3
3.π/4
4.π/6
The cartesian equation of the line is 3x + 1 = 6y – 2 = 1 – z then its direction ratio are
1.1/3, 1/6, 1
2.-1/3, 1/6, 1
3. 1/3, -1/6, 1
4.1/3, 1/6, -1
The center of the circle 4x² + 4y² – 8x + 12y – 25 = 0 is?
1.(2,-3)
2. (-2,3)
3. (-4,6)
4.(4,-6)
The center of the ellipse (x + y – 2)² /9 + (x – y)² /16 = 1 is
1.(0, 0)
2.(0, 1)
3.(1, 0)
4.(1, 1)
The coordinate of foot of perpendicular drawn from the point A(1, 0, 3) to the join of the point B(4, 7, 1) and C(3, 5, 3) are
1. (5/3, 7/3, 17/3)
2.(5, 7, 17)
3.(5/3, -7/3, 17/3)
4.(5/7, -7/3, -17/3)
The coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ plane is
1. (0, 17/2, 13/2)
2.(0, -17/2, -13/2)
3.(0, 17/2, -13/2)
4.None of these
The derivative of [1+(1/x)] /[1-(1/x)] is
1.1/(x-1)²
2.-1/(x-1)²
3. 2/(x-1)²
4. -2/(x-1)²
The equation of a hyperbola with foci on the x-axis is
1.x²/a² + y²/b² = 1
2.x²/a² – y²/b² = 1
3. x² + y² = (a² + b²)
4. x² – y² = (a² + b²)
The equation of parabola whose focus is (3, 0) and directrix is 3x + 4y = 1 is
1.16x² – 9y² – 24xy – 144x + 8y + 224 = 0
2. 16x² + 9y² – 24xy – 144x + 8y – 224 = 0
3. 16x² + 9y² – 24xy – 144x – 8y + 224 =0
4.16x² + 9y² – 24xy – 144x + 8y + 224 = 0
The equation of parabola with vertex (-2, 1) and focus (-2, 4) is
1. 10y = x² + 4x + 16
2.12y = x² + 4x + 16
3. 12y = x² + 4x
4. 12y = x² + 4x + 8
The equation of parabola with vertex at origin the axis is along x-axis and passing through the point (2, 3) is
1. y² = 9x
2.y² = 9x/2
3.y² = 2x
4.y² = 2x/9
The equation of plane passing through the point i + j + k and parallel to the plane r . (2i – j + 2k) = 5 is
1.r . (2i – j + 2k) = 2
2. r . (2i – j + 2k) = 3
3.r . (2i – j + 2k) = 4
4.r . (2i – j + 2k) = 5
The equation of straight line passing through the point (1, 2) and parallel to the line y = 3x + 1 is
1.y + 2 = x + 1
2. y + 2 = 3 × (x + 1)
3.y – 2 = 3 × (x – 1)
4.y – 2 = x – 1
The equation of the line passing through the point (2, 3) with slope 2 is
1. 2x + y – 1 = 0
2.2x – y + 1 = 0
3.2x – y – 1 = 0
4.2x + y + 1 = 0
The equation of the line through the points (1, 5) and (2, 3) is
1. 2x – y – 7 = 0
2. 2x + y + 7 = 0
3.2x + y – 7 = 0
4.x + 2y – 7 = 0
The equation of the line which cuts off equal and positive intercepts from the axes and passes through the point (α, β) is
1.x + y = α + β
2. x + y = α
3. x + y = β
4.None of these
The equation of the locus of a point equidistant from the point A(1, 3) and B(-2, 1) is
1. 6x – 4y = 5
2.6x + 4y = 5
3. 6x + 4y = 7
4.6x – 4y = 7
The equation of the set of point P, the sum of whose distance from A(4, 0, 0) and B(-4, 0, 0) is equal to 10 is
1.9x² + 25y² + 25z² + 225 = 0
2.9x² + 25y² + 25z² – 225 = 0
3. 9x² + 25y² – 25z² – 225 = 0
4.9x² – 25y² – 25z² – 225 = 0
The expansion of log(1 – x) is
1. x – x²/2 + x³/3 – ……..
2.x + x²/2 + x³/3 + ……..
3. -x + x²/2 – x³/3 + ……..
4. -x – x²/2 – x³/3 – ……..
The first term of a GP is 1. The sum of the third term and fifth term is 90. The common ratio of GP is
1.1
2.2
3.3
4.4
The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
1.(-3, 5, 2)
2.(3, 5, 2)
3. (3, -5, 2)
4. (3, 5, -2)
The length of the perpendicular from the origin to a line is 7 and the line makes an angle of 150 degrees with the positive direction of the y-axis. Then the equation of line is
1. x + y = 14
2. √3y + x = 14
3.√3x + y = 14
4.None of these
The line lx + my + n = 0 will touches the parabola y² = 4ax if
1. ln = am²
2.ln = am
3.ln = a² m²
4.ln = a² m
The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a
1.Straight line
2. Plane
3.Sphere
4.None of these
The locus of a point, whose abscissa and ordinate are always equal is
1. x + y + 1 = 0
2.x – y = 0
3.x + y = 1
4.None of these
The locus of the point from which the tangent to the circles x² + y² – 4 = 0 and x² + y² – 8x + 15 = 0 are equal is given by the equation
1.8x + 19 = 0
2.8x – 19 = 0
3.4x – 19 = 0
4.4x + 19 = 0
The maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is
1.3
2.4
3.5
4.Can not be find
The number of tangents that can be drawn from (1, 2) to x² + y² = 5 is
1. 0
2.1
3.2
4.More than 2
The parametric coordinate of any point of the parabola y² = 4ax is
1. (-at², -2at)
2. (-at², 2at)
3.(a sin²t, -2a sin t)
4.(a sin t, -2a sin t)
The parametric representation (2 + t², 2t + 1) represents
1. a parabola
2. a hyperbola
3.an ellipse
4.a circle
The perpendicular distance from the point (3, -4) to the line 3x – 4y + 10 = 0
1.7
2.8
3.9
4.10
The plane 2x – (1 + a)y + 3az = 0 passes through the intersection of the planes
1.2x – y = 0 and y + 3z = 0
2.2x – y = 0 and y – 3z = 0
3.2x + 3z = 0 and y = 0
4.2x – 3z = 0 and y = 0
The points on the y- axis which are at a distance of 3 units from the point (2, 3, -1) is
1.either (0, -1, 0) or (0, -7, 0)
2. either (0, 1, 0) or (0, 7, 0)
3.either (0, 1, 0) or (0, -7, 0)
4.either (0, -1, 0) or (0, 7, 0)
The radius of the circle 4x² + 4y² – 8x + 12y – 25 = 0 is?
1. √57/4
2. √77/4
3.√77/2
4.√87/4
The slope of the line ax + by + c = 0 is
1. a/b
2. -a/b
3.-c/b
4.c/b
The sum of AP 2, 5, 8, …..up to 50 terms is
1. 3557
2.3775
3.3757
4.3575
The sum of series 1/2! + 1/4! + 1/6! + ….. is
1.e² – 1 / 2
2.(e – 1)² /2 e
3. e² – 1 / 2 e
4.e² – 2 / e
The sum of squares of the distances of a moving point from two fixed points (a, 0) and (-a, 0) is equal to 2c² then the equation of its locus is
1.x² – y² = c² – a²
2.x² – y² = c² + a²
3.x² +c² – y² = a²
4.x² + y² = c² + a²
The sum of two numbers is 13/6 An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Then the number of means inserted is
1.2
2.4
3.6
4.8
The third term of a geometric progression is 4. The product of the first five terms is
1. 43
2.45
3.44
4.None of these
The value of limit Limx→0 {sin (a + x) – sin (a – x)}/x is1. 0
2.1
3.2 cos a
4.2 sin a
The value of Limx→01 (1/x) × sin-1 {2x/(1 + x²) is1.0
2.1
3.2
4.-2
The value of Limx→01 (1/x) × sin-1 {2x/(1 + x²) is1.0
2.1
3.2
4.-2
The value of limy→0 {(x + y) × sec (x + y) – x × sec x}/y is1.x × tan x × sec x
2.x × tan x × sec x + x × sec x
3.tan x × sec x + sec x
4. x × tan x × sec x + sec x
The value of the limit Limx→0 (cos x)cot2 x is
1. 1
2.e
3.e1/2
4. e-1/2
Three planes x + y = 0, y + z = 0, and x + z = 0
1.meet in a line
2.meet in a unique point
3.meet taken two at a time in parallel lines
4.None of these
Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are coincedent if
1. a1/a2 = b1/b2 ≠ c1/c2
2.a1/a2 ≠ b1/b2 = c1/c2
3.a1/a2 ≠ b1/b2 ≠ c1/c2
4.a1/a2 = b1/b2 = c1/c2
Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are parallel if
1. a1/a2 = b1/b2 ≠ c1/c2
2.a1/a2 ≠ b1/b2 = c1/c2
3. a1/a2 ≠ b1/b2 ≠ c1/c2
4. a1/a2 = b1/b2 = c1/c2
Two lines are perpendicular if the product of their slopes is
1.0
2.1
3.-1
4.None of these
What can be said regarding if a line if its slope is negative
1. θ is an acute angle
2.θ is an obtuse angle
3.Either the line is x-axis or it is parallel to the x-axis.
4.None of these
What can be said regarding if a line if its slope is zero
1.θ is an acute angle
2. θ is an obtuse angle
3.Either the line is x-axis or it is parallel to the x-axis.
4.None of these
y-intercept of the line 4x – 3y + 15 = 0 is
1. -15/4
2.15/4
3.-5
4.5