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

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

Time : 3 Hours
Max. Marks : 75

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 circle with center C = (-1, 2) and radius r = 5.

2. Find the equation of the normal at P = (3, 5) of the circle
S ≡ x² + y² – 10x – 2y + 6 = 0.

3. Find the equation of the radical axis of the following circles :
x² + y² – 3x – 4y +5 = 0, 3(x² + y²) – 7x + 8y – 11 = 0.

4. Find the coordinates of the points on the parabola y² = 8x whose focal distance is 10.

5. If 3x – 4y + k = 0 is a tangent to x² – 4y² = 5, find the value of k.

6. Evaluate ∫\(\sqrt{1-\cos 2 x}\) dx on I ⊂ [2nπ, (2n + 1) π], n ∈ Z.

7. Evaluate \(\int \frac{x^8}{1+x^{18}}\) dx or R.

8. Evaluate \(\int_2^3 \frac{2 x}{1+x^2}\)dx.

9. Find \(\int_0^{\pi / 2}\) cos⁸x dx.

10. Find the order and degree of
\(\left[\frac{\mathrm{d}^3 y}{\mathrm{~d} x^3}\right]^2\) – 3\(\left[\frac{\mathrm{d} y}{\mathrm{~d} x}\right]^2\) – e^x = 4

Section – B

II. Short Answer Type Questions.

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

11. Find the length of the chord formed by x² + y² = a² on the line x cos a + y sin a = R

12. Find the equation of the circle which cuts orthogonally the circle x² + y² – 4x + 2y – 7 = 0 and having the center at (2, 3).

13. Find the eccentricity, coordinates of foci, length of latus rec¬tum and equations of directrices of the following ellipse :
9x² + 16y² – 36x + 32y – 92 = 0.

14. Find the condition for the line lx + my + n = 0 to be a tangent to the ellipse \(\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}\) = 1.

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

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

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

17. Solve the differential equation
\(\frac{d y}{d x}\) = \(\frac{x y+y}{x y+x}\)

Section – C
(5 × 7 = 35)

III. Long Answer Type Questions.

1. Answer ANY FIVE questions.
2. Each question carries SEVEN marks.

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

19. Find the equation of circle with center (-2, 3) cutting a chord length 2 units on 3x + 4y + 4 = 0.

20. Derive the equation of the parabola in standard form.

21. Evaluate \(\int \frac{d x}{5+4 \cos 2 x}\)

22. Evaluate \(\int \frac{d x}{x(x+1)(x+2)}\)

23. Evaluate \(\int_{\pi / 6}^{\pi / 3} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\)dx

24. Solve the differential equation
(x² + y²) dy = 2xy dx.