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

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.

1. Attempt all questions.
2. Each question carries two marks.

1. Find the value of k, if the circles x² + y² + 4x + 8 = 0 and x² + y² – 16y + k = 0 are orthogonal.

2. If the circle x² + y² – 4x + 6y + a = 0 has radius 4, then find a.

3. Obtain the parametric equations of the circle
(x – 3)² + (y – 4)² = 8².

4. Find the co-ordinates of the points on the parabola y² = 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 \(\int_{-\pi / 2}^{\pi / 2}\)sin²cos⁴x 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.

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

11. Find the pole of 3x + 4y- 45 = 0 with respect to the circle x² + y² – 6x – 8y + 5 = 0.

12. Find the equation of the circle which cuts the circles x² + y² – 4x – 6y + 11 = 0 and x² + y² – 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 x² + 3y² = 3.

15. Find the center, foci, eccentricity, equation of directrices to the hyperbola x² – 4y² = 4.

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

17. Solve the differential equation (1 + x²)\(\frac{d y}{d x}\) + y = \(\mathrm{e}^{\tan ^{-1} x}\)

Section – C

III. Long Answer Type Questions.

1. Attempt any five questions.
2. 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 x² + y² – 6x – 2y + 1 = 0, x² + y² + 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 y² = 4ax and x² = 4by is xa^1/3 + yb^1/3 + a^2/3b^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 = ∫cotⁿ x dx, n being a positive integer, n ≥ 2 and deduce the value of ∫cot⁴ x dx.

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

24. Solve the differential equation (x² + y²) dx = 2xy dy.