TS Inter 1st Year Maths 1B Question Paper May 2022

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TS Inter 1st Year Maths 1B Question Paper May 2022

Time : 3 Hours
Max. Marks : 75

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

Section – A
(10 × 2 = 20 Marks)

I. Very short answer type questions :

1. Answer all the questions.
2. Each question carries two marks.

1. Find the equation of the line containing the points (2, -3) and (0, -3).

2. Transform the equation 3x + 4y = 5 into intercept form. Find the length of the perpendicular from (-2, -3) to the straight line 5x – 2y + 4 = 0.

3. Find the length of the perpendicular from (-2, -3) to the straight line 5x – 2y + 4 = 0

4. Find the ratio in which the X-axis divide the line segment \(\overline{\mathrm{AB}}\) joining A(2, -3) and B(3, -6).

5. Show that the points A (3, -2, 4), B (1, 1, 1) and C (-1, 4, -2) are collinear.

6. Find x if the distance between (5, -1, 7) and (x, 5, 1) is 9 units.

7. Find the ratio in which the XZ-plane divides the line joining A (-2, 3, 4) and B (1, 2, 3).

8. Find the direction cosines of the normal to the plane x + 2y + 2z – 4 = 0.

9. Find

10. Show that

11. Find

12. Find the derivative of f (x) = e^x. (x² + 1)

13. If y = Log (S in (L og x )), then find \(\frac{d y}{d x}\).

14. Find ∆y and dy for the function y = x² + 3x + 6, x = 10 and ∆x = 0.01.

15. Find the slope of the tangent to the curve y = 3x⁴ – 4x at x = 4.

Section – B
(5 × 4 = 20)

II. Short Answer Type Questions.

1. Answer any FIVE questions.
2. Each Question carries FOUR marks.

16. Find the locus of the third vertex of a right angled triangle, the ends of whose hypotenuse are (4, 0) and (0, 4).

17. A(5, -3), B(3, -2) are two fixed points. Find the equation of the locus of P, so that the area of triangle PAB is 9.

18. Find the equation of locus of a point which is equidistant from the points A(3, -2) and B(0, 4).

19. When the origin is shifted to (-1, 2) by the translation of axes, find the transformed equation of x² + y² + 2x – 4y + 1 = 0.

20. When the axes are rotated through an angle π/6, find the transformed equation of x² + \(2 \sqrt{3}\)xy – y² = 2a².

21. Find the value of F if the lines 3x + 4y = 5, 2x + 3y = 4, Px + 4y = 6 are concurrent.

22. Find the foot of the perpendicular drawn from (4, 1), upon the straight line 3x – 4y + 12 = 0.

23. Find the coordinates of the vertex C of ∆ABC if its centroid is the origin and the vertices A, B are (1, 1, 1) and (-2, 4, 1) respectively.

24. Compute \(\lim _{x \rightarrow \infty} \frac{x^2+5 x+2}{2 x^2-5 x+1}\)

25. Find the derivative of the function Cos ax from the first principles.

26. Find the equation of tangent and normal to the curve xy = 10 at (2, 5)

27. Find two positive integers whose sum is 16 and the sum of whose squares is minimum.

Section – C

III. Long Answer Type Questions.

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

28. Find the circumcenter of the triangle whose vertices are (1, 0), (-1, 2) and (3, 2).

29. Find the orthocenter of the triangle whose vertices are (-2, -1), (6, -1) and (2, 5).

30. Find the equation of the straight line passing through (1, 3) and

i) Parallel to
ii) Perpendicular to the line passing through the points (3, – 5) and (-6, 1).

31. Show that the product of the perpendicular distances from a point (α, β) to the pair of straight lines \(\frac{\left|a \alpha^2+2 h \alpha \beta+b \beta^2\right|}{\sqrt{(a-b)^2+4 h^2}}\)

32. Find the condition for the chord lx + my = 1 of the circle x² + y² = a² (whose center is the origin) to subtend a right angle at the origin.

33. Find the angle between two diagonals of a cube.

34. Find the derivatives of the functions:

i) Log (Tan 5x)
ii) Tan (e^x)

35. Find the derivative of

36. Find the derivatives of the functions:

37. Find the angle between the curves y² = 4x, x² + y² = 5.