1. The algebraic sum of the intercepts made by the plane x + 3y - 4z = 6 on the coordinate axes is
1) 13 2) 16 3) 
4) 
Ans: (3)
Explanation:
2. The distance of origin from the image of (1, 2, 3) in the plane x - y + z = 5 is
1) 
2) 
3) 
4) 
Ans: (3)
3. The equation of a circle that touches X - axis at a distance 3 from origin and makes an intercept 
on Y - axis is (1, 2, 3)
1) x2 + y2 - 6x ± 8y + 9 = 0 2) x2 + y2 ± 6x - 8y - 9 = 0
3) x2 + y2 + 6x ± 8y - 9 = 0 4) x2 + y2 ± 6x + 8y + 9 = 0
Ans: (1)
Explanation: x2 + 2gx + c ≡ (x - 3)2 = x2 - 6x + 9
2g = - 6, c = 9
f2 - 9 = 7, f = ± 4
... Circle = x2 + y2 - 6x ± 8y + 9 = 0
4. The points of intersection of the ellipse b2x2 + a2y2 = a2b2 and the line lx + my + n = 0 are joined to origin. The lines thus formed are coincident if
1) a2l2 + b2m2 = n2 2) a2m2 + b2l2 = n2
3) l2a2 - m2b2 = n2 4) m2a2 - l2b2 = n2
Ans: (1)
5. Tangents drawn to the parabola y2 = 4ax at the points P and Q meet at a point R, on the line y - 2x = a, a > 0. If the line segment PQ passes through the focus of the parabola, then its length is:
1) 2a 2) 3a 3) 5a 4) 6a
Ans: (3)
6. Tangents are drawn to the circle x2 + y2 = 9 from points which lie on the line 4x - 5y = 20. The equation to the locus of mid - points of the chords of contact of tangents is given as
m(x2 + y2) - nx + py = 0. Then m + n + p =
1) 20 2) 29 3) 36 4) 45
Ans: (2)
Explanation: P(x1, y1) is a point of which chord of contact is S1 = 0
xx1 + yy1 = 9 
(1)
P(x1, y1) lies on 4x - 5y = 20 4x1 - 5y1 = 20 (2)
If (h, k) is any variable mid - point, then S1 = S11 implies x(h) + y(k) = h2 + k2 (3)
7. Tangents which are parallel to the line y = 2x - 1 are drawn to the hyperbola 
= 1. If (a, b), (c, d) are the points of contact, then a + b + c + d =
1) 
2) 0 3) 2 4) -2
Ans: (2)
Explanation: The tangent parallel to y = 2x - 1 meets the hyperbola in first and third quadrants.
Hence (a, b) = (a, b) and (c, d) = (-a, b)
... a + b + c + d = 0
8. Let PQ be the common chord of the parabola y2 = 8x and the circle x2 + y2 - 2x - 4y = 0. If S is the focus of the parabola, then the area of the triangle PQS in square units is:
1) 2 2) 4 3) 8 4) 12
Ans: (2)
Explanation: y2 = 4(2)x ... a = 2
P = (at2, 2at) P = (2t2, 4t). This lies on the circle also.
... 4t4 + 16t2 - 4t2 - 16t = 0 t4 + 3t2 - 4t = 0
... t = 0 or 1 P = (0, 0), Q = (2, 4), S = (2, 0)
9. A circle passing through (-1, 0) and touching the Y-axis at (0, 2) also passes through:
1) 
2) 
3) 
4) 
Ans: (4)
Explanation: y2 + 2fy + c ≡ (y - 2)2 = y2 - 4y + 4
2f = -4; c = 4
x2 + y2 + 2gx + 2fy + c = 0 passes through (-1, 0) implies
-2g + c = -1 -2g = -5 ... 2g = 5
... Circle: x2 + y2 + 5x - 4y + 4 = 0 is satisfied only by (-4, 0).
10. Let P be any point on the parabola y2 = 4x, dividing the line segment joining (0, 0) and (x, y) in the ratio 1 : 3. The equation to the locus of P is
1) y2 = x 2) y2 = 2x 3) x2 = y 4) x2 = 2y
Ans: (1)
Explanation: P = (h, k); O = (0, 0); A = (x, y); 1 : 3 ratio
(h, k) = 
implying 4h = x, 4k = y
y2 = 4x 16k2 = 16h (or) k2 = h
... Equation to locus of P is y2 = x.
11. Normals drawn to the parabola y2 = 4x pass through the point (9, 6). Their equations are:
1) y = x - 3; y = 2x - 12; y = 33 - 3x 2) y = x + 3; y = 2x + 12; y = 33 + 3x
3) y = x - 3; y = x + 12; y = x - 33 4) y = 3x - 2; y = 3x - 12; y = 3x - 11
Ans: (1)
Explanation: Simple method: (9, 6) satisfies (1). So Answer is (1)
Actual Method: y + xt = 2at + at3
y + xt = 2(1)t + (1)t3
(9, 6) 
6 + 9t = 2t + t3
t3 - 7t - 6 = 0
t = -1; -2, +3
... t = -1 y - x = -3 (or) y = x - 3
t = -2 y - 2x = -12 (or) y = 2x - 12
t = 3 y + 3x = 33 (or) y = 33 - 3x
12. Normal drawn to the hyperbola 
at the point (6, 3) meets X - axis at (9, 0). If 'e' is the eccentricity, then 6e2 =
1) 3 2) 6 3) 9 4) 12
Ans: (3)
13. The eccentricity of the hyperbola 
is a reciprocal to the eccentricity of the ellipse 
and the hyperbola passes through the focus of the ellipse. Then which one of the following is correct?
1) Equation of hyperbola is x2 - 3y2 = 3 2) Equation of hyperbola is 2x2 - 3y2 = 6
3) Focus of hyperbola is (2, 0) 4) Focus of hyperbola is (
, 0)
Ans: (1), (3)
14. The equations of the common tangents to y2 = 4ax and (x + a)2 + y2 = a2 are
Ans: (1)
15. Two straight lines with slopes 3 and 6 intersect one another at the point (30, 40). The difference between their y - intercepts is
1) 30 2) 50 3) 90 4) 140
Ans: (3)
Explanation: y - 40 = 3(x - 30) y = 3x - 50 (one line)
y - 40 = 6(x - 30) y = 6x - 140 (second line)
... Magnitude of difference between the y - intercepts = 90.
16. If the area of the auxiliary circle of the ellipse b2x2 + a2y2 = a2b2, a > b, is two times the area of the ellipse itself, then eccentricity of the ellipse is:
Ans: (2)
Explanation: Auxiliary circle of the ellipse 
(a > b) is x2 + y2 = a2
... Area = π a2
(1)
Area of ellipse = π ab (2)
Given π a2 = 2. π. a. b
a = 2b
17. If the distance between the foci and the distance between the directrices of a hyperbola 
are in the ratio 3 : 2, then a : b =
1) 2 : 1 2) 1 : 
3) : 1 4) 1 : 2
Ans: (3)
18. Let LL' be a latus rectum and S' be the other focus of a hyperbola b2x2 - a2y2 = a2b2. If LL'S' forms an equilateral triangle, the eccentricity of the hyperbola is
1) 
2) 
3) 
4) 
Ans: (2)
19. Two tangents are drawn to a circle x2 + y2 = 16 form a point A on X - axis. If the tangents meet Y - axis at the points P and Q, then the minimum value of AP2 + AQ2 =
1) 32 2) 48 3) 64 4) 128
Ans: (4)
Explanation:
OA2 + OP2 = 16 (sec2 θ + cosec2 θ) = OA2 + OQ2
Minimum value of OA2 + OP2 = 16(1 + 1)2 = 64
AP2 + AQ2 = OA2 + OP2 + OA2 + OQ2
= 2(OA2 + OP2)
... Minimum value of AP2 + AQ2 = 2 . minimum value of (OA2 + OP2)
= 2(64)
= 128
