AP Inter 2nd Year Maths 2B Question Paper March 2023

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AP Inter 2nd Year Maths 2B Question Paper March 2023

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 :

1. Attempt ALL questions.
2. Each question carries TWO marks.

1. Find the value of ‘a’ if 2x² + ay² – 3x + 2y – 1 = 0 represents a circle and also find its radius.

2. Find the value of ‘k’ if the points (1, 3) and (2, k) are conjugate with respect to the circle x² + y² = 35.

3. Find the equation of the radical axis of the given circles x² + y² – 5x + 6y + 12 = 0, x² + y² + 6x – 4y – 14 = 0.

4. Find the value of ‘k’ if the line 2y = 5x + k is a tangent to the parabola y² = 6x.

5. If the eccentricity of a hyperbola is 5/4, then find the eccentricity of its conjugate hyperbola.

6. Evaluate ∫\(\sqrt{1-\cos 2 x}\) dx

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

8. Evaluate \(\int_0^2|1-x| d x\)

9. Evaluate \(\int_0^{\pi / 2}\)sin⁶x∙cos⁴xdx

10. Find the order of the differential equation of the family of all circles with their centres at the origin.

Section – B (5 × 4 = 20)

II. Short answer type questions:

1. Attempt ANY FIVE questions.
2. Each question carries FOUR marks.

11. If a point P is moving such that the length of tangents drawn from P to the circles x² + y² – 2x + 4y – 20 = 0 and x² + y² + 2x – 8y + 1 =0 are in the ratio 2 : 1 them find the equation of the locus of P.

12. Find the equation and length of the common chord of the two circles
x² + y² + 3x + 5y + 4 = 0 and x² + y² + 5x + 3y + 4 = 0.

13. Find the eccentricity, coordinates of foci, length of latus rectum and equations of directrices of the following ellipse 3x² + y² – 6x – 2y – 5 = 0.

14. If the normal at one end of a latus rectum of the ellipse \(\frac{x^2}{\mathrm{a}^2}\) + \(\frac{y^2}{b^2}\) = 1 passes through one end of the minor axis, then show that e⁴ + e² = 1. (e is the eccentricity of the ellipse)

15. Find the equation of the tangents to the hyperbola 3x² – 4y² = 12 which are

1. parallel
2. perpendicular to the line y = x – 7.

16. Evaluate \(\int_0^{\pi / 2} \frac{1}{4+5 \cos x} d x\)

17. Solve the following differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) = e^x-y + x²e^-y

Section – C
(5 × 7 = 35)

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 centre on 4x + y – 16 = 0

19. Find the equation of the circle which touches the circle x² + y² – 2x – 4y – 20 = 0 externally at (5, 5) with radius 5.

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

21. Evaluate : \(\int \frac{\cos x+3 \sin x+7}{\cos x+\sin x+1}\)dx

22. Obtain reduction formula for I_n = ∫cosⁿx dx for an integer n ≥ 2.

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

24. Solve (1 + x²)\(\frac{\mathrm{d} y}{\mathrm{~d} x}\) + 2xy – 4x² = 0.