Access to a variety of AP Inter 2nd Year Maths 2B Model Papers and AP Inter 2nd Year Maths 2B Question Paper March 2023 allows students to familiarize themselves with different question patterns.

## 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 :

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

1. Find the value of ‘a’ if 2x^{2} + ay^{2} – 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 x^{2} + y^{2} = 35.

3. Find the equation of the radical axis of the given circles x^{2} + y^{2} – 5x + 6y + 12 = 0, x^{2} + y^{2} + 6x – 4y – 14 = 0.

4. Find the value of ‘k’ if the line 2y = 5x + k is a tangent to the parabola y^{2} = 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 \(\int_0^{\pi / 2}\)sin^{6}x∙cos^{4}xdx

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:

- Attempt ANY FIVE questions.
- Each question carries FOUR marks.

11. If a point P is moving such that the length of tangents drawn from P to the circles x^{2} + y^{2} – 2x + 4y – 20 = 0 and x^{2} + y^{2} + 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

x^{2} + y^{2} + 3x + 5y + 4 = 0 and x^{2} + y^{2} + 5x + 3y + 4 = 0.

13. Find the eccentricity, coordinates of foci, length of latus rectum and equations of directrices of the following ellipse 3x^{2} + y^{2} – 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 e^{4} + e^{2} = 1. (e is the eccentricity of the ellipse)

15. Find the equation of the tangents to the hyperbola 3x^{2} – 4y^{2} = 12 which are

- parallel
- 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} + x^{2}e^{-y}

Section – C

(5 × 7 = 35)

III. Long answer type questions :

- Attempt ANY FIVE questions.
- 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 x^{2} + y^{2} – 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} = ∫cos^{n}x dx for an integer n ≥ 2.

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

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