schoolservicecommission_math_1st mocktest_hpg

8/14/2019 schoolservicecommission_math_1st mocktest_hpg

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1. Prove that the remainder when cube of any integer is divisible by 9 is 0, 1 or 8.

2. If| cos(x + iy)| = 1 where x, y are real, prove that cos 2x + cosh2y = 2.

3. Let R be a ring with unity 1, and a R. If there exists a unique b R such that ab = 1, prove

that ba = 1 and a is a unit.

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4. Find the linear mapping T : R3 R2 which maps the basis vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) of

R3 to the vectors (1, 1), (2, 3), (3, 2) respectively.

5. Ifx , y , z are positive real number such that x + y + z = 1 find the minimum value of

(x + 1 + 1x

)2 + (y + 1 + 1y

)2 + (z + 1 + 1z

)2.

6. If a, am and an are generators of a finite cyclic group, then prove that m and n are relatively

prime.

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7. If f be a differentiable homogeneous function of x, y and z of degree n and if x = u2 v2,

y = v2 w2 and z = w2 u2, show that f is also a homogeneous function of u, v and w of

degree 2n.

8. Test the convergency of the series: 1 + (23

)2 + ( 35

)3 + ( 47

)4 + ( 59

)5 + + .

9. A function f is differentiable on [0, 2] and f(0) = 0, f(1) = 2, f(2) = 1. Prove that f(c) = 0 for

some c (0, 2).

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10. Iff(x, y) =

xy2

x2+y4if (x, y) = (0, 0)

0 if (x, y) = (0, 0), show that f is not continuous at (0, 0).

11. Solve: x2 d2y

dx2+ 7x dy

dx+ 5y = x5.

12. Solve: (3x2y4 + 2xy)dx + (2x3y3 x2)dy = 0.

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13. Prove that the family of parabolas y2 = 4a(x + a) is self orthogonal.

14. If the tangent at a point P of a conic meets the directrix at Q, prove that P SQ = 2

, S being

corresponding focus.

15. If the pole of the normal at P on a parabola lies on the normal at Q, show that the pole of the

normal at Q lies on the normal at P.

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16. Find the surface generated by the lines which intersect the lines y = mx, z = c; y = mx,

z = c and x-axis.

17. A plane passing through a fixed point (a,b,c) cuts the axes in A , B , C . Show that the locus of

the centre of the sphere OABC is ax

+ by

+ cz

= 2.

18. Show that the equation of the plane which contains the straight line r = + t and is

perpendicular to the plane r = a is [r ] = 0.

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19. Ifa ,b , c ,

d are coplanar vectors, show that (a

b ) (c

d ) =

0

20. A particle starts at rest from the origin under a force ( per unit mass ) 4t + 1 and 6t parallel to

the x and y axes respectively at time t. Find the path of the particle.

21. A particle moves from rest in a straight line under an attractive force ( distance )2 per unit

mass to a fixed point on the line. Show that if the initial distance from the centre of force be

2a, then the distance will be a after a time (2

+ 1)a3

.

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25. Show that the mean and variance of a Poisson variate are equal.

26. If the probability of a man hitting a target is 0.25 then how many times a he should fire so that

the probability of his hitting the target at least once is greater than 2764

?

27. One bag contains 5 white and 4 black balls. Another bag contains 7 white and 9 black balls. A

ball is transferred from the first bag to the second bag and then a ball is drawn from the second.

Find the probability that the drawn ball is black.

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28. Find the value of k and the marginal density functions fX(x) and fY(y) if the joint density

function f of X and Y is given by f(x, y) =

k(x + y) , 0 < x < 1, 0 < y < 1

0 , otherwise

29. Evaluate10 (x2+x)dx by Simpsons one-third rule, taking 4 ordinates, correct up to 3 significant

figures.

30. Use Lagranges interpolation formula to find the value of f(0) from the following table:

x 1 2 2 4

f(x) 1 9 11 69

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Space for rough work

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schoolservicecommission_math_1st mocktest_hpg

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