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1.Show that diameter of a sphere subtends a right angle at a point on the surface.

2. P represents the variable complex number z. Find the locus of P, if Im (z^{2} ) = 4

3. The tangent at any point of the rectangular hyperbola xy = c^{2} makes intercepts a, b and the normal at the point makes intercepts p, q on the axes. Prove that ap+bq = 0.

4. Find the equations of the tangent and normal to the curve y = x^{2 – x-2 at the point (1, -2). }

5. 20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using (i) Binomial distribution (ii) Poisson distribution [e-^{2} = 0.1353].

6. Prove that cos(A-B) = cosAcosB +sinAsinB

7. Find the vector and Cartesian equation to the plane through the point (-1, 3, 2) and perpendicular to the planes x+2y+2z=5 and 3x+y+2z=8.

8. Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?

9. A kho-kho player in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him if the distance between the poles is 6m.

10. Show that the line x-y+4=0 is a tangent to the ellipse x^{2} +3y^{2} =12. Find the co-ordinates of the point of contact.

11. Find the area of region enclosed by y ^{2} = x and y = x-2.

12. The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value 31 psi and standard deviation 0.2 psi.

(i)What is the probability that the pressure for a randomly selected tyre** (a) between 30.5 and 31.5 psi (b) between 30 and 32 psi. **

(ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi [ Area table: P(0<z<2.5) = 0.4938 ].

13. At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km/hr and ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4.00 p.m?.

14. Derive the formula for the volume of a right circular cone with radius ‘r’ and height ‘h’

15. Prove that (AB)^{-1} = B^{-1}A^{-1}, where A and B are two non-singular matrices.

16. The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to sun and the greatest possible distance between the earth and the sun.

17. In a gambling game a man wins Rs.10 if he gets all heads or all tails and loses Rs. 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.

18. The girder of a railway bridge is in the parabolic form with span 100 ft. and the highest point on the arch is 10 ft, above the bridge. Find the height of the bridge at 10 ft, to the left or right from the midpoint of the bridge.

19. The arch of a bridge is in the shape of a semi –ellipse having a horizontal span of 40 ft and 16 ft high at the centre. How high is the arch, 9 ft from the right or left of the centre.

20. Find the perimeter of the circle with radius a

21.The mean score of 1000 students for an examination is 34 and S.D. is 16

(i) How many candidates can be expected to obtain marks between 30 and 60 assuming the normally of the distribution and

(ii) Determine the limit of the marks of the central 70% of the candidates. [ Area table : P(0<z<0.25) = 0.0987, P(0<z<1.63) = 0.4484 , P(0<z<1.04) = 0.35 ]

22. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line x = 2 y – 5 = 0 and passes through the points (6,0) and (-3,0).

23.Find the intervals of concavity and the points of inflection for the given functions y = 12 x ^{2} – 2x^{3} – x^{4}

24. Find the equations of those tangents to the circle x^{2} + y^{2} = 52 , which are parallel to the straight line 2x + 3y = 6

25. Find the area of the region bounded by x^{2} = 36 y , y – axi, y = 2 and y = 4

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