AP Inter 1st Year Maths 1B Question Paper March 2023

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AP Inter 1st Year Maths 1B Question Paper March 2023

Time : 3 Hours
Max. Marks : 75

Section – A
(10 × 2 = 20)

I. Very Short Answer Type Questions.

2. Each question carries two marks.

1. Find the distance between parallel lines 5x – 3y – 4 = 0 and 10x – 6y – 9 = 0.

2. Show that the points A(3, 2, -4), B(5, 4, -6) and C(9, 8, -10) are collinear.

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

4. If the increase in side of square is 4%, then find the approximate percentage of increase in the area of square.

5. Transform the equation $$\sqrt{3}$$x + y + 10 = 0 into

1. slope- intercept form
2. normal form.

6. Find the second order derivative of y = tan-1$$\left(\frac{2 x}{1-x^2}\right)$$.

7. Find the derivative of e2x.log (3x + 4) (x > $$\frac{-4}{3}$$)

8. Verify Rolle’s theorem for the function f(x) = x2 – 1 on [-1, 1]

9. Compute $$\lim _{x \rightarrow 2^{+}}$$([x] + x) and $$\lim _{x \rightarrow 2}$$ ([x] + x).

Section – B
(5 × 4 = 20)

2. Each question carries FOUR marks.

11. If f, given by

is a continuous function on R, then find values of k.

12. Find the derivative of ‘sec 3x’ from the first principle.

13. The volume of a cube is increasing at the rate of 8 cm3/sec. How fast is the surface area increasing when the length of an edge is 12 cm ?

14. Find the lengths of normal and sub-normal at a point on the curve
y = $$\frac{a}{2}\left(e^{x / a}+e^{-x / a}\right)$$.

15. Find the equation of locus of ‘P’, if the ratio of distances from P to A(5, -4) and B(7, 6) is 2 : 3.

16. When the axes are rotated through an angle, find the transformed equation of 3x2 + 10xy + 3y2 = 0.

17. Find the point on the straight line 3x + y + 4 = 0 which is equidistant from the points (-5, 6) and (3, 2).

Section – C
(5 × 7 = 35)

2. Each question carries SEVEN marks.

18. Find the orthocentre of triangle formed by the lines x + 2y = 0, 4x + 3y – 5 = 0 and 3x + y = 0.

19. Find the angle between the lines joining the origin to the points of intersection of the curve x2 + 2xy + y2 + 2x + 2y – 5 = 0 and the line 3x – y + 1 = 0.

20. Show that the area of triangle formed by the lines ax2 + 2hxy + by2 = 0 and lx + my + n = 0 is $$\left|\frac{n^2 \sqrt{h^2-a b}}{a m^2-2 h l m+b l^2}\right|$$

21. Find the angle between the lines whose direction cosines are given by the equations 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.

22. If xy + yx = ab, then show that $$\frac{d y}{d x}$$ = –$$\left(\frac{y \cdot x^{y-1}+y^x \cdot \log y}{x^y \log x+x \cdot y^{x-1}}\right)$$.

23. If the tangent at any point on the curve x2/3 + y2/3 = a2/3 intersects the co-ordinate axes in A and B, then show that the length AB is a constant.

24. The profit function P(x) of a company selling x items per day is given by P(x) = (150 – x) x – 1000. Find the number of items that the company should manufacture to get maximum profit. Also find the maximum profit.