# AP Inter 1st Year Maths 1B Question Paper April 2022

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## AP Inter 1st Year Maths 1B Question Paper April 2022

Time : 3 Hours
Max. Marks : 75

Section – A
(10 × 2 = 20)

I. Very Short Answer Type Questions.

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

1. Transform the equation 3x + 4y + 12 = 0 into

1. Slope-intercept form
2. intercept form

2. Find the value of p, if the straight lines x + P = 0, y + 2 = 0 and 3x + 2y + 5 = 0 are concurrent.

3. If (3, 2, -1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a tetrahedron, find the fourth vertex.

4. Find the equation of the plane whose intercepts on X, Y, Z- axes are 1, 2, 4 respectively.

5. Compute $$\lim _{x \rightarrow 0} \frac{e^x-1}{\sqrt{1+x}-1}$$

6. Compute $$\lim _{x \rightarrow \infty} \frac{8|x|+3 x}{3|x|-2 x}$$

7. If f(x) = sin (log x), (x > 0) find f(x).

8. Find the derivative of cos (log x + ex).

9. Find the slope of the normal to the curve x = a cos3θ, y = a sin3θ at θ = $$\frac{\pi}{4}$$.

10. Find the intervals on which f(x) = x2 – 3x + 8 is increasing or decreasing.

Section – B

II. Short answer type questions :

1. Attempt any five questions.
2. Each question carries four marks.

11. If the distance from P to the points (2, 3) and (2, -3) are in the ratio 2 : 3, then find the equation of the locus of P.

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

13. A straight line with slope 1 passes through Q (-3, 5) and meets the straight line x + y – 6 = 0 at P, Find the distance PQ.

14. If f given by

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

15. Find $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ for the function

16. Find the angle between the curve 2y = $$e^{\frac{-x}{2}}$$ and Y-axis.

17. Show that $$\frac{x}{1+x}$$ < In (1 + x)< x, ∀ x > 0.

Section – C
(5 × 7 = 35)

III. Long answer type questions :

1. Attempt any five questions.
2. Each question carries seven marks.

18. If p and q are the lengths of the perpendiculars from the origin to the straight lines x secα + y cosecα = a and x cosα – y sinα = a cos2α, prove that 4p2 + q2 = a2.

19. Show that the pairs of straight lines 6x2 – 5xy – 6y2 = 0 and 6x2 – 5xy – 6y2 + x + 5y – 1 = 0 form a square.

20. Find the values of k if the lines joining the origin to the points of intersection of the curve 2x2 – 2xy + 3y2 + 2x – y – 1 = 0 and the line x + 2y = k are mutually perpendicular.

21. Find the direction cosines of two lines which are connected by the relations l + m + n = 0 and mn – 2nl – 2lm = 0.

22. If xy = yy then

23. Show that the tangent at P(x1, y1) on the curve
= $$\sqrt{x}+\sqrt{y}$$ = $$\sqrt{a}$$ is
$$y y_1{ }^{\frac{-1}{2}}$$ = $$a^{\frac{1}{2}}$$

24. The profit function P(x) of a company, selling x items per day is given by P(x) = (150 – x) x – 1600. Find the number of items that the company should sell for maximum profit. Also find the maximum profit.