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## AP Inter 2nd Year Maths 2B Question Paper April 2022

Time : 3 Hours

Max. Marks : 75

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

Section – A (10 × 2 = 20)

I. Very short answer type questions :

- Attempt ALL questions.
- Each question carries TWO marks.

1. Find the equation of circle whose extremities of a diameter are (1, 2) and (4, 5).

2. Locate the position of the point (2, 4) with respect to the circle x^{2} + y^{2} – 4x – 6y + 11 = 0.

3. Find the equation of the radical axis of the circles

S ≡ x^{2} + y^{2} – 5x + 6y + 12 = 0

S_{1} ≡ x^{2} + y^{2} + 6x – 4y – 14 = 0

4. Find the vertex and focus x^{2} – 6x – 6y + 6 = 0.

5. If e, e_{1} are the eccentricities of a hyperbola and its conjugate hyperbola PT. \(\frac{1}{e^2}\) + \(\frac{1}{\mathrm{e}_1^2}\) = 1

6. Evaluate ∫sec^{2} x cosec^{2}xdx .

7. Evaluate ∫x log x dx on (0, ∞].

8. Evaluate ∫\(\int_0^\pi \sqrt{2+2 \cos \theta} d \theta\)

9. Evaluate \(\int_0^{\frac{\pi}{2}}\)cos^{7} x sin^{2} x dx.

10. Find the general solution of \(\frac{d y}{d x}\) = e^{x+y}

Section – B

II. Short Answer Type Questions.

- Answer any FIVE questions.
- Each Question carries FOUR marks.

11. Find the equations of the tangents to the circle x^{2} + y^{2} – 4x + 6y – 12 = 0 which are parallel to x + y – 8 = 0.

12. Find the equation of the circle passing through the points of intersection of the circles

x^{2} + y^{2} – 8x – 6y + 21 = 0

x^{2} + y^{2} – 2x – 15 = 0 and (1, 2)

13. Find the length of major axis, minor axis, latus rectum, eccentricity, coordinates of centre, foci and the equation of directrices of the ellipse x^{2} + 2y^{2} – 4x + 12y + 14 = 0.

14. Find the equation of the ellipse with focus at (1, -1), e = \(\frac{2}{3}\) and directrix as x + y + 2 = 0.

15. Find the centre, foci, eccentricity, equation of the directrices, length of the latus rectum of the hyperbola 16y^{2} – 9x^{2} = 144.

16. Evaluate \(\int_0^{\frac{\pi}{2}} \frac{\cos ^{\frac{5}{2}} x}{\sin ^{\frac{5}{2}} x+\cos ^{\frac{5}{2}}}\)dx.

17. Solve (1 + x^{2})\(\frac{d y}{d x}\) + 2xy – 4x^{2} = 0

Section – C

III. Long Answer Type Questions.

- Answer ANY FIVE questions,
- Each Question carries SEVEN marks.

18. If (2, 0), (0, 1), (4, 5) and (0, c) are concyclic then find c.

19. Show that x + y + 1 = 0 touches the circle x^{2} + y^{2} – 3x + 7y + 14 = 0 and find its point of contact.

20. Define parabola and derive its equation in standard form.

21. Evaluate \(\int \frac{1}{1+\sin x+\cos x} d x\).

22. Obtain Reduction formula for ∫tan^{x} dx and also evaluate ∫tan^{4}xdx.

23. Evaluate \(\int_0^1 \frac{\log (1+x)}{1+x^2}\)dx.

24. Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{x^2+y^2}{2 x^2}\).