AP Inter 2nd Year Maths 2B Model Paper Set 3 with Solutions

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AP Inter 2nd Year Maths 2B Model Paper Set 3 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.

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

Section – A
(10 × 2 = 20)

I. Very Short Answer Type questions.

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

1. Find the equation of the radical axis of the circles
x² + y² – 2x – 4y – 1 = 0
x² + y² – 4x – 6y + 5 = 0.

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

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

4. Find the pole of ax + by + c = 0, (c ≠ 0) with respect to x² + y² = r².

5. Find the equation of tangent to the parabola y² = 16x inclined at an angle 60° with its axis.

6. Evaluate the integral ∫\(\frac{(3 x+1)^2}{2 x}\)dx, x ∈ I ⊂ R / {0}.

7. Evaluate the integral ∫ e^x (sec x + sec x tan x) dx.

8. Evaluate the definite integral \(\int_0^\pi \sqrt{2+2 \cos \theta}\) dθ.

9. Evaluate the definite integral \(\int_0^{\pi / 2}\) sin⁶ x. cos⁴ x dx.

10. Find the general solution of x + y\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 0.

Section – B
(5 × 4 = 20)

II. Short Answer Type questions.

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

11. Find the length of the chord intercepted by the circle
x² + y² – x + 3y – 22 = 0 on the line y = x – 3.

12. If x + y = 3 is the equation of the chord AB of the circle x² + y² – 2x + 4y – 8 = 0, find the equation of the circle having \(\overline{\mathrm{AB}}\) as diameter.

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

14. The tangent and normal to the ellipse x² + 4y² = 4 at a point P(θ) on it meets the major axis in Q and R respectively. If 0 < θ < \(\frac{\pi}{2}\) and QR = 2, then show that θ = cos^-1\(\left(\frac{2}{3}\right)\).

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

16. Find the area enclosed by the curves
y = x² + 1, y = 2x- 2, x = – 1, x = 2.

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

Section – C
(5 × 7 = 35)

III. Long Answer Type questions.

(i) Attempt any five questions.
(ii) Each question carries seven marks.

18. Find the equation of circle passing through each of the three points (3, 4), (3, 2) and (1, 4).

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. Find the equation of the parabola whose axis is parallel to x-axis and which passes through the points (-2, 1), (1, 2) and (- 1, 3).

21. Evaluate the integral \(\int \frac{x+1}{x^2+3 x+12}\) dx.

22. Obtain the reduction formula for ∫tanⁿx dx for an integer n ≥ 2 and deduce the value of ∫tan⁶x dx .

23. Evaluate the integral \(\int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x\).

24. Find the equation of a curve whose gradient is \(\frac{d y}{d x}\) = \(\frac{y}{x}\) – cos²\(\left(\frac{y}{x}\right)\), where x > 0, y > 0 and which passes through the point (1, \(\frac{\pi}{4}\))