Question 1

1. Theside of an equilateral triangle is increasing at the rate of 2 cm/s. Therate at which area increases when the side is 10 is

(a) 10 cm²/s

(b) cm²/s

(c) 10 cm²/s

(d) cm²/s

Accepted Answer

(c)

Answer

(d)

Answer

(a)

Answer

(b)

Question 2

2. Thepoint(s) on the curve y= x²,at which y-coordinate ischanging six times as fast as x-coordinateis/are

(a) (2, 4)

(b) (3, 9)

(c) (3, 9), (9, 3)

(d) (6, 2)

Accepted Answer

(b)

Answer

(d)

Answer

(a)

Answer

(c)

Question 3

3. The equation of the normal to the curve y = sin x at (0, 0) is

(a) x = 0

(b) y = 0

(c) x + y = 0

(d) x – y = 0

Accepted Answer

(c)

Answer

(b)

Answer

(a)

Answer

(d)

Question 4

4. The point on the curve where tangent to the curve y² = x, makes an angle of 45°clockwise with the x-axis is

(a) $\left(-\frac{1}{2},\frac{1}{4}\right)$

(b) $\left(\frac{1}{4},-\frac{1}{2}\right)$

(c) (–2,4)

(d) (4, –2)

Accepted Answer

(b)

Answer

(c)

Answer

(a)

Answer

(d)

Question 5

5. The line y = x + 1 is a tangent to the curve y² = 4x at the point

(a) (–1, 2)

(b) (1, 2)

(c) (1, –2)

(d) (2, 1)

Accepted Answer

(b)

Answer

(a)

Answer

(c)

Answer

(d)

Question 6

6. The curves y = ae^–x and y = be^x are orthogonal if

(a) a = b

(b) a = –b

(c) ab = –1

(d) ab = 1

Accepted Answer

(d)

Answer

(b)

Answer

(a)

Answer

(c)

Question 7

7. If the curves ay + x² = 7 and x³ = y cut orthogonally at (1, 1), then the value of a is (a) 1

(b) 0

(c) –6

(d) 6

Accepted Answer

(d)

Answer

(c)

Answer

(b)

Answer

(a)

Question 8

8. The angle between the curve y²= x and x² = y at (1, 1) is

(a) 60°

(b) tan^{–1 $\left(\frac{4}{3}\right)$}

(c) cot^{–1 $\left(\frac{4}{3}\right)$}

(d) 90°

Accepted Answer

(c)

Answer

(d)

Answer

(b)

Answer

(a)

Question 9

9. The absolute maximum value of y= x³ –3x + 2 in 0 ≤x ≤ 2 is

(a) 4

(b) 6

(c) 2

(d) 0

Accepted Answer

(a)

Answer

(c)

Answer

(b)

Answer

(d)

Question 10

10. The maximum value of xy, subject to x + y = 8 is

(a) 8

(b) 16

(c) 20

(d) 24

Accepted Answer

(b)

Answer

(a)

Answer

(d)

Answer

(c)

Question 11

11. A spherical balloon has a variable diameter ^{$\frac{3}{2}$}(2x + 1). The rate of change of its volume with respect to x is

(a) ^{$\frac{27\pi}{8}$}(2x + 1)²

(b) ^{$\frac{9}{4}$}π(2x + 1)³

(c) ^{$\frac{9\pi}{16}$}(2x + 1)³

(d) π(2x+ 1)²

Accepted Answer

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Question 12

12. If the curves 4x = y² and 4xy = k cut at right angles, then

(a) k² = 64

(b) k² = 512

(c) k² = 4

(d) k² = 256

Accepted Answer

(b)

Answer

(a)

Answer

(d)

Answer

(c)

Question 13

13. Approximate value of^{$\frac{3}{2}$} is

(a) 3.074

(b)2.926

(c) 5

(d) none of these

Accepted Answer

(b)

Answer

(d)

Answer

(c)

Answer

(a)

Question 14

14. For the function y = x³ + 21, the value of x, when y increases 75 times as fast as x, is

(a) ± 3

(b) ± 5

(c) ± 5

(d)none of these

Accepted Answer

(c)

Answer

(b)

Answer

(d)

Answer

(a)

Question 15

15. Equation of tangent to the curve y= 1 – e^x^/2at the point of intersection with the y-axis is

(a) x + 2y = 1

(b) 2x + y = 0

(c) x – y = 2

(d) none of these

Accepted Answer

(a)

Answer

(c)

Answer

(d)

Answer

(b)

Question 16

16. Given function f(x) = x²e^–x, then ‘f’ increases in the interval

(a) (– ∞, ∞)

(b) (–2, 0)

(c) (2, ∞)

(d) (0, 2)

Accepted Answer

(d)

Answer

(c)

Answer

(a)

Answer

(b)

Question 17

17. Which of the following function is decreasing on ^{$\left(0,\frac{\pi}{2}\right)$}?

(a) cos x

(b) – cos 2x

(c) cos 3x

(d) tan x

Accepted Answer

(a)

Answer

(c)

Answer

(b)

Answer

(d)

Question 18

18. The maximum value of is

(a) e

(b) e^e

(c) $\frac{1}{e^e}$

(d) $\left(\frac{4}{3}\right)$

Accepted Answer

(c)

Answer

(b)

Answer

(a)

Answer

(d)

Question 19

19. The height of the cylinder of greatest volume which can be inscribed in a right circular cone, of height h, is

(a) $\frac{h}{2}$

(b) $\frac{h}{4}$

(c) $\frac{h}{3}$

(d) none of these

Accepted Answer

(c)

Answer

(d)

Answer

(a)

Answer

(b)

Question 20

20. A vector equally inclined to axes is

(a) $\hat{i}+\hat{j}+\hat{k}$

(b) $\hat{i}-\hat{j}+\hat{k}$

(c) $\hat{i}-\hat{j}-\hat{k}$

(d) $\hat{i}+\hat{j}-\hat{k}$

Accepted Answer

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Question 21

21. The position vector of a point which divides the join of points with position vectors $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ in the ratio 1 : 2 internally is

(a) $\frac{\overrightarrow{\mathrm{3a}}+\overrightarrow{\mathrm{2b}}}{3}$

(b) $\frac{27\pi}{8}$

(c) $\frac{\overrightarrow{\mathrm{5a}}-\overrightarrow{\mathrm{b}}}{3}$

(d) $\frac{9}{4}$

Accepted Answer

(d)

Answer

(a)

Answer

(c)

Answer

(b)

Question 22

22. A vector in the direction of vector $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ that has magnitude 15 is

(a) $\hat{i}+\hat{j}+\hat{k}$

(b) $15\hat{i}-30\hat{j}+30\hat{k}$

(c) $\hat{i}-2\hat{j}+15\hat{k}$

(d) $5\hat{i}-10\hat{j}+10\hat{k}$

Accepted Answer

(d)

Answer

(c)

Answer

(a)

Answer

(b)

Question 23

23. If ^{∣ $\frac{3}{2}$ ∣} = 4 and –3 ≤ λ ≤ 2 then the range ^{I $\frac{h}{2}$ I} of is

(a) [0, 8]

(b)[–12, 8]

(c) [0,12]

(d) [8, 12]

Accepted Answer

(c)

Answer

(a)

Answer

(d)

Answer

(b)

Question 24

24. If $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ are unit vectors, then what is the angle between $\overrightarrow{\mathrm{a}}$ and $15\hat{i}-30\hat{j}+30\hat{k}$ $\frac{\overrightarrow{\mathrm{3a}}+\overrightarrow{\mathrm{2b}}}{3}$ for to be a unit vector?

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Accepted Answer

(a)

Answer

(d)

Answer

(c)

Answer

(b)

Question 25

25. If for non zero vectors $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ , $\overrightarrow{\mathrm{a}}$ × $15\hat{i}-30\hat{j}+30\hat{k}$ is a unit vector and I $\hat{i}-\hat{j}+\hat{k}$ I =I $\frac{h}{4}$ I = $\frac{27\pi}{8}$ , then angle θ between vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is

(a) $\frac{\pi}{2}$

(b) $\frac{\pi}{3}$

(c) $\frac{\pi}{6}$

(d) $-\frac{\pi}{2}$

Accepted Answer

(c)

Answer

(b)

Answer

(d)

Answer

(a)

Question 26

26. The area of a parallelogram whose one diagonal is $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and one side is $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ is

(a) $\frac{\overrightarrow{\mathrm{3a}}+\overrightarrow{\mathrm{2b}}}{3}$

(b) $\frac{27\pi}{8}$ sq units

(c) $\hat{i}+\hat{j}-\hat{k}$ sq units

(d) 6 sq units

Accepted Answer

(b)

Answer

(a)

Answer

(d)

Answer

(c)

Question 27

27. If ^{I $\frac{3}{2}$ I} = 8, ^{I $\frac{h}{2}$ I} = 3 and ^{I $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}$ I}= 12 then the value of ^{I $\left(\frac{1}{4},-\frac{1}{2}\right)$ I} is

(a) 12

(b)12

(c) 6

(d)4

Accepted Answer

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Question 28

28. If vectors $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ are coplanar, then value of λ is

(a) 4

(b) 0

(c) –3

(d) 2

Accepted Answer

(b)

Answer

(c)

Answer

(a)

Answer

(d)

Question 29

29. Position vectors of points A and B are $\overrightarrow{\mathrm{a}}$ + $15\hat{i}-30\hat{j}+30\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ - $\frac{h}{4}$ . Then $\frac{27\pi}{8}$ equal to

(a) 3^{$\left(\frac{4}{3}\right)$}

(b) –^{$\overrightarrow{\mathrm{a}}$} + 2^{$\overrightarrow{\mathrm{b}}$}

(c)^{$\overrightarrow{\mathrm{a}}$} – 2^{$\overrightarrow{\mathrm{b}}$}

(d) none of these

Accepted Answer

(c)

Answer

(d)

Answer

(b)

Answer

(a)

Question 30

30. Given vector ^{$\left(0,\frac{\pi}{2}\right)$}, then –2^{$\hat{i}+\hat{j}+\hat{k}$} is a vector whose

(a) magnitude is twice that of ^{$\left(-\frac{1}{2},\frac{1}{4}\right)$} and direction is same as that of ^{$\left(\frac{1}{4},-\frac{1}{2}\right)$}

(b) magnitude is twice that of ^{$\overrightarrow{\mathrm{a}}$} and direction is opposite to that of ^{$\frac{9}{4}$}

(c) magnitude is same as that of ^{$\overrightarrow{\mathrm{a}}$} and direction is opposite to that of ^{$\overrightarrow{\mathrm{a}}$}

(d) none of these

Accepted Answer

(b)

Answer

(d)

Answer

(c)

Answer

(a)

Question 31

31. If $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ are non-zero vectors, such that $\overrightarrow{\mathrm{a}}$ · $15\hat{i}-30\hat{j}+30\hat{k}$ = 0, then

(a) $5\hat{i}-10\hat{j}+10\hat{k}$ is parallel to $\left(\frac{1}{4},-\frac{1}{2}\right)$

(b) $\left(\frac{4}{3}\right)$ and $\overrightarrow{\mathrm{b}}$ are collinear

(c) $\frac{\pi}{3}$ is perpendicular to $\overrightarrow{\mathrm{b}}$

(d) none of these

Accepted Answer

(c)

Answer

(b)

Answer

(d)

Answer

(a)

Question 32

32. If $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ then

(a) $\frac{h}{2}$

(b) $\left(-\frac{1}{2},\frac{1}{4}\right)$

(c) $\overrightarrow{\mathrm{b}}$ are non-coplanar

(d)none of these

Accepted Answer

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Question 33

33. Area of parallelogram, whose diagonals are along vectors $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ is

(a) $\frac{\overrightarrow{\mathrm{3a}}+\overrightarrow{\mathrm{2b}}}{3}$

(b) $\hat{i}-\hat{j}-\hat{k}$

(c) $\frac{1}{2}\sqrt{29}$

(d) none of these

Accepted Answer

(c)

Answer

(a)

Answer

(b)

Answer

(d)

Question 34

34. Unit vectors along vector $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ are

(a) $\frac{h}{2}$

(b) $\left(-\frac{1}{2},\frac{1}{4}\right)$

(c) $\overrightarrow{\mathrm{b}}$

(d) none of these

Accepted Answer

(c)

Answer

(b)

Answer

(d)

Answer

(a)

Question 35

35. The position vectors of opposite vertices of a parallelogram are $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ . Then position vector of the point of intersection of diagonals is

(a) $\frac{\overrightarrow{\mathrm{3a}}+\overrightarrow{\mathrm{2b}}}{3}$

(b) $\hat{i}-\hat{j}-\hat{k}$

(c) $\frac{1}{2}\sqrt{29}$

(d) none of these

Accepted Answer

(c)

Answer

(b)

Answer

(d)

Answer

(a)

Question 36

36. If for non-zero vectors $\overrightarrow{\mathrm{a\ }}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{2a\ }}-\overrightarrow{\mathrm{b}}$ , then

(a) $\frac{\overrightarrow{\mathrm{3a}}+\overrightarrow{\mathrm{2b}}}{3}$ are parallel to each other

(b) $\hat{i}-\hat{j}-\hat{k}$ are perpendicular to each other

(c) $\frac{1}{2}\sqrt{29}$

(d) none of these

Accepted Answer

(b)

Answer

(d)

Answer

(a)

Answer

(c)

Question 37

37. If ^{I $\frac{3}{2}$ I =I $\hat{i}+\hat{j}+\hat{k}$ I}, then

(a) $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}$ is parallel to $5\hat{i}-10\hat{j}+10\hat{k}$

(b) $\overrightarrow{\mathrm{b}}$ is perpendicular to $\frac{1}{2}\sqrt{29}$

(c)I $\overrightarrow{\mathrm{a}}$ I=I $\left(\frac{4}{3}\right)$ I

(d) $\overrightarrow{\mathrm{a}}$ = $\overrightarrow{\mathrm{a}}$

Accepted Answer

(b)

Answer

(a)

Answer

(d)

Answer

(c)

Question 38

38. If ^{I $\frac{3}{2}$ I}= 5,^{I $\frac{h}{2}$ I}= 13 and ^{I $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}$ I}= 25, then $\frac{27\pi}{8}$ · $\overrightarrow{\mathrm{a}}$ is equal to

(a) 12

(b) 5

(c) 13

(d) 60

Accepted Answer

(d)

Answer

(b)

Answer

(a)

Answer

(c)

Question 39

39. The distance of point (2, 5, 7) from the x-axis is

(a) 2

(b) $\hat{i}-2\hat{j}+15\hat{k}$

(c) $5\hat{i}-10\hat{j}+10\hat{k}$

(d) $\overrightarrow{\mathrm{b}}$

Accepted Answer

(b)

Answer

(d)

Answer

(a)

Answer

(c)

Question 40

40. P is a point on the line segment joining the points (3, 5, –1) and (6, 3, –2). If y-coordinate of point P is 2, then its x-coordinate will be

(a) 2

(b) $\frac{h}{3}$

(c) $\frac{15}{2}$

(d) –5

Accepted Answer

(c)

Answer

(d)

Answer

(a)

Answer

(b)

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