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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 :
- Answer all the questions.
- 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) = ex. (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.
- Answer any FIVE questions.
- 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.
- Answer ANY FIVE questions.
- 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 (ex)
35. Find the derivative of
36. Find the derivatives of the functions:
37. Find the angle between the curves y2 = 4x, x2 + y2 = 5.