AP Inter 2nd Year Maths 2B Model Paper Set 1 with Solutions

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AP Inter 2nd Year Maths 2B Model Paper Set 1 with Solutions

Time : 3 Hours
Max. Marks : 75

Note : This question paper consists of three sections A, B and C.

Section – A

I. Very Short Answer type questions.

2. Each question carries two marks.

1. Find the equation of the circle passing through (- 2, 3) having the centre at (0, 0).

2. If the length of the tangent from (5, 4) to the circle x2 + y2 + 2ky = 0 is 1, then find k.

3. Find the angle between the circles given by the equations
x2 + y2 – 12x – 6y + 41 = 0 and x2 + y2 + 4x + 6y – 59 = 0.

4. Find the equation of parabola whose focus is S (1, -7) and vertex is A (1, -2).

5. If the eccentricity of hyperbola is $$\frac{5}{4}$$, then find the eccentricity of its conjugate hyperbola.

6. Evaluate $$\int\left(x+\frac{4}{1+x^2}\right)$$dx

7. Evaluate ∫(tan x + log sec x) ex dx.

8. Evaluate $$\int_0^\pi$$sin3x cos3x dx.

9. Evaluate $$\int_0^4 \frac{x^2}{1+x} d x$$.

10. Find the order and degree of the equation $$\left[\frac{d^2 y}{d x^2}-\left(\frac{d y}{d x}\right)^3\right]^{\frac{6}{5}}$$ = 6y

Section – B

1. Attempt any five questions.
2. Each question carries four marks.

11. Find the value of k, if kx + 3y – 1 = 0, 2x + y + 5 = 0 are conjugate lines with respect to the circle x2 + y2 – 2x – 4y – 4 = 0.

12. Find the equation of the circle which passes through the point (0, -3) and intersects the circles given by the equations
x2 + y2 – 6x + 3y + 5 = 0 and x2 + y2 – x – 7y = 0 orthogonally.

13. Find the eccentricity, co-ordinates of foci, length of latus rectum and equations of directrix of the following ellipse
9x2 + 16y2 – 36x + 32y – 92 = 0.

14. Find the equation of the tangents to 9x2 + 16y2 = 144 which makes equal intercepts on the co-ordinate axes.

15. Find the equations of tangents to the hyperbola 3x2 – 4y2 = 12 which are

1. Parallel and
2. Perpendicular to line y = x – 7.

16. Evaluate $$\int_0^{\pi / 4} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x$$

17. Solve $$\frac{\mathrm{dy}}{\mathrm{dx}}$$ – x tan (y – x) = 1.

Section – C

1. Attempt any five questions.
2. Each question carries seven marks.

18. Show that (1, 2), (3, -4), (5, -6) and (19, 8) four points are concyclic.

19. Find the direct common tangents of the circles
x2 + y2 + 22x – 4y – 100 = 0 and x2 + y2 – 22x + 4y + 100 = 0.

20. If y1, y2, y3 are the y-co-ordinates of the vertices of the triangle inscribed in the parabola y2 = 4ax, then show that the area of the triangle $$\frac{1}{8 a}$$|(y1 – y2) (y2 – y3) (y3 – y1) | square units.

21. Evaluate ∫(3x – 2)$$\sqrt{2 x^2-x+1}$$ dx

22. Obtain the reduction formula for In = ∫secnxdx, n being a positive integer n ≥ 2 and hence deduce the value of ∫sec5x dx.

23. Evaluate $$\int_0^{\pi / 2} \frac{\sin ^2 x}{\cos x+\sin x} d x$$

24. Solve (1 + y2) dx = (tan-1 y – x) dy.