# Areas; Differential Equations

1. If 2 f(x) = f'(x) and f(0) = 3 then f(2) =
1) 3        2) 2            3) 3e4               4) 4e3

2. Solution of the equation (xy4 + y) dx - xdy = 0 is
1) 4x4y3 + 3x3 = cy3            2) 3x3y4 + 4x3 = cx3
3) 3x4y3 + 4x3 = cy3          4) 2x4y3 + x3 = cy4

3. The population p(t) at a time t of certain mouse species satisfies the differential equ-ation dp(t)/dt  = 0.5 p(t) - 450. If p(0) = 850 dt
then the time at which the population becomes zero is.
1) 2 log18            2) log 9        3) 1/2  log 18       4) 1/2  log 9

4. The radius of a circle having minimum area, which touches the curve y = 4 - x2 and the lines y = |x| is .....
1) 2 (√2 - 1)           2) 4 (√2 - 1)            3) 4(√-2 + 1)            4) 2(√2 + 1)

5. Area of the region in the first quadrant enclosed by the x - axis, the line y = x and the circle x2 + y2 = 32
1) 4π                2) 8π           3) 16π             4) 32π

6. The area bounded by y2 = 4ax and y = mx is a2/3 square units then m = .....
1) 1          2) 2                3) 3              4) 4

7. Area bounded y = {x} where {.} fractional part and the lines x = ±1
1) 1          2) 2              3) 3               4) 4

8. The degree of the differential equation

1) 1         2) 2         3) 3       4) not defined

9. A curve passes through the point (4, 2) and at any point (x, y) on it. The product of its slope and the ordinate is equal to abscissa of the curve is ....
1) Parabola              2) Circle             3) Ellipse              4) Hyperbola

10. Integrating factor (I.F.) of the differential equation (y2 - x) dy/dx  = 1 is
1) ex            2) ey          3) e-x              4) e-y

11. Let y = f(x) is a solution of differential equation ey dy/dx - ey = ex and f(0) = 0 then f(1) =
1) log2               2) 3 + log2                 3) log3              4) 1 + log2

12. At present a firm is manufacturing 2000 items. It is estimated that the rate of production p with respect to additional
number of workers x is given by dp/dx = (100 - 12 √x). If the firm employs 25 more workers, then the new level of production of items is

1) 2500              2) 3000           3) 3500            4) 4500

Key: 1-3; 2-3; 3-1; 4-2; 5-1; 6-2; 7-1; 8-1; 9-4; 10-2; 11-4; 12-3.

Conics

1. A movable parabola touches the x - axis and y - axis at (1 0) and (0 1). Then locus of the focus of parabola is
1) 2x2 + 2y2 + 2x + 2y + 1 = 0             2) 2x2 + 2y2 - 2x - 2y + 1 = 0
3) x2 + y2 =1             4) x2 + y2 - x - y + 1 = 0

2. Radius of the largest circle which passes through the focus of the parabola y2 = 4x and contained in it is
1) 4           2) 8              3) 16            4) 1

3. If S is the focus of the parabola y2 = 8x, P is a point on the parabola. The normal at P meets the axis in G. If SPG is an equilateral triangle then P =
1) (6  4√3 )           2) (2   4)     3) (4   4√2)    4) (3   2√3)

4. If y + 3 = m1(x + 2), y + 3 = m2(x + 2) are two tangents the parabola y2 = 8x then angle between two tangents is
1) π/6         2) π/4         3) π/3          4) π/2

5. A ray of light along the line x = 3 is reflected at the ellipse x2/25 + y2/16 = 1 The slope of the reflected ray is
1) 4/15         2) 8/15            3) 15/8           4) 15/4

6. Let Pi and Pi ' be the feet the perpendicular drawn from S, S' on a tangent Ti to an ellipse whose length of semi major axis is 20.

7. The slopes of common tangents to the

1) ±1         2) ±2          3) ± √2           4) ±3

8. The graph represented by the equation x = sin2t and y = 2cost is
1) a part of a sine graph             2) hyperbola
3) a portion of a parabola        4) a parabola

9. The value of k such that the vertex of y = x2 + 2kx + 13 is 4 units above the x- axis is
1) ±1       2) ±2           3) ±3            4) ±4

10. If S, S' are two foci of an ellipse 16x2 + 25y2 = 400 and PSQ is a focal chord such that SP = 16 then S′Q = ....
1) 20               2) 70/9        3) 74/9           4) 74/11

11. If the equation (10x - 5)2 + (10y - 4)2 = λ2(3x + 4y -1)2 represents a hyperbola then λ lies in the interval
1) (-2  2)          2) (-∞ -2) ∪ (2  ∞)           3) (2  ∞)             4) (0  2)

Numerical Value Type

12. The line y = mx + c (m > 0) tangent to y2 = 8 (x + 2) then the minimum value of c is .....

13. If e1, e2, e3 are the eccentricities of a parabola (P) ellipse (E) and hyperbola (H) so that e12 + e22 + e32 = 46/9, e12 - e22 + e32 = 44/9 and eccentrity of conjugate hyperbola (H) is 2/K  then the value of K =

14. If foci of the ellipse x2/25 + y2/16 = 1 and the hyperbola x2/4 + y2/k = 1 coincide then k =

15. If  S, S' are foci of an ellipse of major axis of length 10 units and P is any point on the ellipse such that perimeter of triangle PSS' is 15. Then eccentricity of ellipse is....

16. Equations of directrix and latusrectum of a parabola are 3x - 4y + 17 = 0 and 3x - 4y + 2 = 0 then the length of latusrectum =

17. Number of focal chords of the parabola y2 = 9x whose length is less than 9 is...

18. If F1 and F2 are the  feet of perpendiculars from   foci   S1 and  S2 of  an  ellipse x2/5 + y2/3 = 1 on the tangent  at any point P on the ellipse then (S1 F1 ) (S2 F2) =....

19. If 4x2 + y2 = 1 then maximum value of 12x2 - 3y2 + 16xy is

20. Radius of director circle of a rectangular hyperbola is

Key: 1-2; 2-1; 3-1; 4-4; 5-2; 6-3; 7-1; 8-3; 9-3; 10-3; 11-2; 12-4; 13-1.73; 14-5; 15-0.5; 16-6; 17-0; 18-3;  19-5; 20-0.

Application of Differentiation

1. A person of height 2 mts starts from a lamp post of height 5 mts and walks away at the rate of 6 km per hour. The rate at which his shadow increases is
1) 2 kmph           2) 4 kmph              3) 6 kmph             4) 8 kmph

2. The curve y − exy + x = 0 a vertical tangent at the point
1)  (1 1)              2) (0  1)           3) (1  0)          4) at no point

3.  The condition that the two curves x = y2, xy = k cut orthogonally is
1) k2 = 1            2) 2k2 = 1             3) 4k2 = 1             4) 8k2 = 1

4. If the length of subtangent is 9, length of subnormal is 4 at a point (x y) on y = f(x), then ordinate of the point is
1) ±4          2) ±6            3) ±8            4) ±3

5.  If  a + b + c = 0 then the quadratic equation 3ax2 + 2bx + c = 0 has atleast one root in the interval
1) (0  1)     2) (1   2)    3) (2 3)      4) (-1  0)

6. Through the point (2 3) a straight line is drown making positive intercept on the coordinate axes. The area of the triangle thus formed is least, then the ratio of the intercepts on line x and y axes is
1) 1 : 2            2) 2 : 3            3) 3 : 4           4) 1 : 4

Answers: 1-2,  2-3,  3-4,  4-2,  5-1,  6-2.

Direction Cosines and Direction Ratios;  Plane

1. if  α, β, γ are respectively the acute  angle made by any line with the coordinates axes, then
1) α + β + γ = 90°                2) α + β + γ = 360°
3) 0° < α + β + γ <  270°        4) 0° < α + β + γ < 180°

2. If the angle between the lines whose

possible value of a,b  are
1) -1, 4                2) 4, 2                3) 4, 1               4)  -4, -2

3. The harmonic conjugate of (2, 3, 4) with respect to the line joining points (3,  -2,   2) and (6,  -17, -4) is
1) (0,  0,  0)          2) (1/2 1/3 1/4)          3) (1  1  1)             4) (18/5, -5, 4/5)

4. The equation ax + by + cz + d = 0 (d≠0) does not represents a plane if
1) a2 + b2 + c2 = 0             2) a2 + b2 + c2 ≠ 0
3) a + b - c = 0          4) a + b + c = 0

5. The number of line perpendicular to x−axis lying in yz plane is
1) 0           2) 1             3) 3            4) ∞

6. The angle between a line and normal to a plane is 30°. The angle between line and plane is
1) π/3            2) π/2            3) π/5               4) π/4

7. The planes x = ±3, y = ±4, z = ±6 form a
1) cube             2) Rectangular parallelepiped
3) Tetrahedron         4) Parallelepiped with equal edges

8. The lengths of projection of a line segment on the coordinate planes are 3, 4, 5 then length of the segment is

9. Let a line makes an angle θ with x, z axes and β with y-axes so that √3 sinθ = sinβ then cos2θ =

10. Image of  (1, 2, 3) with respect to a plane ax + by + cz + d = 0 is (-7/3, -4/3, -1/3) then a + b + c + d =

Answers: 1-3,  2-3,  3-4,  4-1,  5-4,  6-1,  7-2, 8-5,  9-0.60,  10-2.

Vector Algebra

Posted Date : 07-07-2021

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గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.