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

## AP Inter 2nd Year Maths 2B Model Paper Set 2 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.

- Attempt all questions.
- Each question carries two marks.

1. Find the value of k, if the circles x^{2} + y^{2} + 4x + 8 = 0 and x^{2} + y^{2} – 16y + k = 0 are orthogonal.

2. If the circle x^{2} + y^{2} – 4x + 6y + a = 0 has radius 4, then find a.

3. Obtain the parametric equations of the circle

(x – 3)^{2} + (y – 4)^{2} = 8^{2}.

4. Find the co-ordinates of the points on the parabola y^{2} = 8x whose focal distance is 10.

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

6. Evaluate \(\int \frac{d x}{\cosh x+\sinh x}\) on R.

7. Evaluate \(\int \frac{x^8}{1+x^{18}}\) on R.

8. Find \(\int_{-\pi / 2}^{\pi / 2}\)sin^{2}cos^{4}x dx.

9. Evaluate \(\int_0^\pi \sqrt{2+2 \cos \theta}\) dθ.

10. Find the order and degree of the differential equation

\(\left[\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3\right]^{6 / 5}\) = 6y.

Section – B

(5 × 4 = 20)

II. Short Answer Type Questions.

- Attempt any five questions.
- Each question carries four marks.

11. Find the pole of 3x + 4y- 45 = 0 with respect to the circle x^{2} + y^{2} – 6x – 8y + 5 = 0.

12. Find the equation of the circle which cuts the circles x^{2} + y^{2} – 4x – 6y + 11 = 0 and x^{2} + y^{2} – 10x – 4y + 21 = 0 orthogonally and has the diameter along the straight line 2x + 3y = 7.

13. Show that the points of intersection of the perpendicular tangents to an ellipse lie on a circle.

14. Find the value of k if 4x + y + k = 0 is a tangent to the ellipse x^{2} + 3y^{2} = 3.

15. Find the center, foci, eccentricity, equation of directrices to the hyperbola x^{2} – 4y^{2} = 4.

16. Evaluate \(\int_0^{\pi / 2} \frac{d x}{4+5 \cos x}\)

17. Solve the differential equation (1 + x^{2})\(\frac{d y}{d x}\) + y = \(\mathrm{e}^{\tan ^{-1} x}\)

Section – C

III. Long Answer Type Questions.

- Attempt any five questions.
- Each question carries seven marks.

18. Find the equation of a circle which passes through (4, 1), (6, 5) and having the center on 4x + 3y – 24 = 0.

19. Show that the circles x^{2} + y^{2} – 6x – 2y + 1 = 0, x^{2} + y^{2} + 2x – 8y + 13 = 0 touch each other. Find the point of contact and the equation of common tangent at their point of contact.

20. Show that the common tangent to the parabola y^{2} = 4ax and x^{2} = 4by is xa^{1/3} + yb^{1/3} + a^{2/3}b^{2/3} = 0.

21. Evaluate \(\int \frac{2 \sin x+3 \cos x+4}{3 \sin x+4 \cos x+5} d x \text {. }\)

22. Obtain the reduction formula for I_{n} = ∫cot^{n} x dx, n being a positive integer, n ≥ 2 and deduce the value of ∫cot^{4} x dx.

23. Evaluate \(\int_0^{\pi / 4}\)log(1 + tan x)dx.

24. Solve the differential equation (x^{2} + y^{2}) dx = 2xy dy.