# 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 ∫ex (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 $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ = ex-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 In = ∫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.