Access to a variety of TS Inter 1st Year Maths 1B Model Papers and TS Inter 1st Year Maths 1B Question Paper May 2023 allows students to familiarize themselves with different question patterns.

## TS Inter 1st Year Maths 1B 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 Marks)

I. Very short answer type questions :

- Attempt all the questions.
- Each question carries two marks.

1. Find the slope of the straight line passing through the points (3, 4), (7, -6).

2. Transform the following straight line equation into normal form 3x + 4y = 5.

3. Find the centroid of the tetrahedron whose vertices are (2,3, -4), (-3, 3, -2),

(-1, 4, 2), (3, 5, 1)

4. Write the equation of the plane 4x – 4y + 2z + 5 = 0 in the intercept form.

5. Compute \(\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{4}{x^2-4}\right)\)

6. Compute \(\lim _{x \rightarrow 0} \frac{\mathrm{e}^{7 x}-1}{x}\) = 7.

7. If y = log (sin (log x)), find \(\frac{d y}{d x}\).

8. Find the derivative of sin^{-1}(3x – 4x^{3}).

9. Find the slope of the tangent to the curve y = 5x^{2} at (-1, 5).

10. Find dy and ∆y of y = f(x) = x^{2} + x at x = 10 when ∆x = 0.1.

Section – B (5 × 4 = 20)

II. Short answer type questions.

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

11. The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0). Find the equation of the locus of its third vertex.

12. When the axes are rotated through an angle \(\frac{\pi}{6}\), find the transformed equation of x^{2} + 2\(\sqrt{3}\) xy – y^{2} = 2a^{2}.

13. If Q(h, k) is the foot of the perpendicular from (P(x_{1}, y_{1}) on the straight line ax + by + c = 0 then prove that

\(\frac{\mathrm{h}-x_1}{\mathrm{a}}\) = \(\frac{k-y_1}{b}\) = –\(\frac{\left(a x_1+b y_1+c\right)}{a^2+b^2}\)

14. Compute \(\lim _{x \rightarrow 0} \frac{1-\cos 2 m x}{\sin ^2 n x}\) (m, n ∈ Z).

15. Find the derivative of the function tan 2x from the first principle.

16. If the increase in the side of a square is 2%, then find the approximate percentage of increase in its area.

17. Find the lengths of subtangent, subnormal at a point ‘t’ on the curve

x = a (cos t + t sin t), y = a (sin t – t cos t).

Section – C (5 × 7 = 35)

III. Long answer type questions.

- Answer ANY FIVE questions.
- Each question carries SEVEN marks.

18. Find the orthocentre of the triangle with the following vertices (-2, -1), (6, -1) and (2, 5).

19. If the equation ax^{2} + 2hxy + by^{2} = 0 represent a pair of straight lines and ‘0’ is the angle between the lines then prove that cosθ = \(\frac{|a+b|}{\sqrt{(a-b)^2+4 h^2}}\)

20. Find the values of k, if the lines joining the origin to the points of intersection of the curve 2x^{2} – 2xy + 3y^{2} + 2x – y – 1 = 0 and the line x + 2y = k are mutually perpendicular.

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

22. If \(\sqrt{1-x^2}+\sqrt{1-y^2}\) = a(x – y) then show that \(\frac{d y}{d x}\) = \(\sqrt{\frac{1-y^2}{1-x^2}}\)

23. Show that the curves y^{2} = 4 (x + 1) and y^{2} = 36 (9 – x) intersect orthogonally.

24. The volume of a cube is increasing at a rate of 9 cubic centimeters per second. How fast is the surface area increasing when the length of the edge is 10 centimetres ?