Access to a variety of AP Inter 2nd Year Maths 2B Model Papers Set 1 allows students to familiarize themselves with different question patterns.

## 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.

- Answer all questions.
- 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 x^{2} + y^{2} + 2ky = 0 is 1, then find k.

3. Find the angle between the circles given by the equations

x^{2} + y^{2} – 12x – 6y + 41 = 0 and x^{2} + y^{2} + 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) e^{x} dx.

8. Evaluate \(\int_0^\pi\)sin^{3}x cos^{3}x 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

II. Short Answer Type questions.

- Attempt any five questions.
- 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 x^{2} + y^{2} – 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

x^{2} + y^{2} – 6x + 3y + 5 = 0 and x^{2} + y^{2} – x – 7y = 0 orthogonally.

13. Find the eccentricity, co-ordinates of foci, length of latus rectum and equations of directrix of the following ellipse

9x^{2} + 16y^{2} – 36x + 32y – 92 = 0.

14. Find the equation of the tangents to 9x^{2} + 16y^{2} = 144 which makes equal intercepts on the co-ordinate axes.

15. Find the equations of tangents to the hyperbola 3x^{2} – 4y^{2} = 12 which are

- Parallel and
- 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

III. Long Answer type questions.

- Attempt any five questions.
- 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

x^{2} + y^{2} + 22x – 4y – 100 = 0 and x^{2} + y^{2} – 22x + 4y + 100 = 0.

20. If y_{1}, y_{2}, y_{3} are the y-co-ordinates of the vertices of the triangle inscribed in the parabola y^{2} = 4ax, then show that the area of the triangle \(\frac{1}{8 a}\)|(y_{1} – y_{2}) (y_{2} – y_{3}) (y_{3} – y_{1}) | square units.

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

22. Obtain the reduction formula for I_{n} = ∫sec^{n}xdx, n being a positive integer n ≥ 2 and hence deduce the value of ∫sec^{5}x dx.

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

24. Solve (1 + y^{2}) dx = (tan^{-1} y – x) dy.