Untitled Document

Question Bank No: 1

1. The coordinates of the point on the parabola $y^{2}$ = 8x, which is at minimum distance from the circle $x^{2}$ (y + ${6)}^{2}$ = 1 are

a)		(2, -4)
b)		(18, -12)
c)		(2, 4)
d)		None of these

2. Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are

a)		(10, 10)
b)		(12, 8)
c)		(15, 5)
d)		None of these

3. The function f(x) = $x^{5}$ $-$ 5 $x^{4}$ + 5 $x^{3}$ $-$ 1 has

a)		one minima and two maxima
b)		two minima and one maxima
c)		two minima and two maxima
d)		one minima and one maxima

4. If f is an increasing function and g is a decreasing function on an interval I such that fog exists, then

a)		fog is an increasing function on I
b)		fog is a decreasing function on I
c)		fog is neither increasing nor decreasing on I
d)		None of these

5. If f(x) = $x^{3}$ $-$ a $x^{2}$ + bx + 5 $\sin^{2}$ x is increasing on R, then a and b satisfy

a)		$a^{2}$ $-$ 3b $-$ 15 > 0
b)		$a^{2}$ $-$ 3b + 15 < 0
c)		$a^{2}$ $-$ 3b + 15 > 0
d)		a > 0, b > 0

6. The vlaue of b for which the function f(x) = sin x $-$ bx + c is decreasing for x $\in$ R is given by

a)		B < 1
b)		b $\geq$ 1
c)		b > 1
d)		b $\leq$ 1

7. If 4a + 2b + c = 0 then the equation 3a $x^{2}$ + 2bx + c = 0 has at least one real root lying between

a)		0 and 1
b)		1 and 2
c)		0 and 2
d)		None of these

8. If x $\in$ [-1, 1], then the minimum value of f(x) = $x^{2}$ + x + 1 is

a)		- $\frac{3}{4}$
b)		1
c)		3
d)		None of these

9. If f(x) = cos $π$ x + 10x + 3 $x^{2}$ + $x^{3} -$ 2 $\leq$ x $\leq$ 3, then absolute minimum value of f(x) is

a)		0
b)		- 15
c)		3 $-$ 2 $π$
d)		None of these

10. If 2a + 3b + 6c = 0, then at least one root of the equation a $x^{2}$ + bx + c = 0 lies in the interval

a)		(0, 1)
b)		(1, 2)
c)		(2, 3)
d)		None of these

11. The abscissa changes at a faster rate than the ordinate on the curve $x^{3}$ = 12y. Then x lies in the interval

a)		(-2, 2)
b)		(-1, 1)
c)		(0, 2)
d)		None of these

12. The curve x + y = $e^{xy}$ has a tangent parallel to y-axis at the point

a)		(0, 1)
b)		(1, 0)
c)		(1, 1)
d)		None of these

13. The acceleration of a moving particle whose space-time equation is given by s = 3 $t^{2} + 5 t - 15$ is

a)		0
b)		3
c)		5
d)		6

14. The angle of intersection of the two curves xy = $a^{2}$ and $x^{2}$ + $y^{2}$ = 2 $a^{2}$ is

a)		$\frac{π}{6}$
b)		$\frac{π}{3}$
c)		$\frac{π}{4}$
d)		None of these

15. At (0, 0) the curve $y^{2}$ = $x^{3}$ + $x^{2}$

a)		Touches x-axis
b)		bisects the angle between the axes
c)		makes an angle of $60^{o}$ with ox
d)		None of these

16. The tangent curve to the curve y = $x^{2}$ + 3x will pass through the point (0, -9) if it is drawn at the point

a)		(0, 1)
b)		(-3, 0)
c)		(-4, 4)
d)		(1, 4)

17. The tangent to the curve y = 2 $x^{2}$ $-$ x + 1 is parallel to the line y = 3x + 9 at the point

a)		(2, 3)
b)		(2, -1)
c)		(2, 1)
d)		(1, 2)

18. If the tangent at P(1, 1) on $y^{2}$ = ${x (}^{2} - {x)}^{2}$ meets the curve again at Q, then Q is

a)		(2, 2)
b)		(-1, -2)
c)		$\frac{3}{4}$ , $\frac{9}{8}$
d)		None of these

19. The angle at which the circle $x^{2}$ + $y^{2}$ = 16 can be seen from the point (8, 0) is

a)		$\frac{π}{2}$
b)		$\frac{π}{4}$
c)		$\frac{π}{3}$
d)		$\frac{π}{6}$

20. The two curves $x^{3} - 3$ x $y^{2}$ + 5 = 0 and 3 $x^{2}$ y $-$ $y^{3}$ $-$ 7 = 0

a)		Cut at right angles
b)		touch each other
c)		cut at an angle $\frac{π}{4}$
d)		cut at an angle $\frac{π}{3}$

21. If f(x) = $\frac{1}{x + 1} - \log (1 + x) x > 0, then f is$

a)		an increasing function
b)		a decreasing function
c)		both increasing and decreasing
d)		none of these

22. The maximum value of Sin x+Cos x is

a)		1
b)		2
c)		$\sqrt{2}$
d)		$\frac{1}{\sqrt{2}}$

23. The equation to 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

24. If f(x) =x+ $\frac{1}{x}, x > 0 then its greatest value is$

a)		-2
b)		0
c)		3
d)		none of these

25. The point on the curve y=6x- $x^{2} where the tangent is parallel to x - axis is$

a)		(0,0)
b)		(2,8)
c)		(6,0)
d)		(3,9)

26. f(x)=sin x- cos x -Kx+b decreases for all real values is given by

a)		K<1
b)		K $\geq 1$
c)		K $\geq \sqrt{2}$
d)		K< $\sqrt{2}$

27. The function f(x) = $\begin{matrix} Σ^{5} \\ n = 1 \end{matrix}$ ${(x - n)}^{2} assumes$ minimum value for x given by

a)		5
b)		$\frac{5}{2}$
c)		3
d)		2

28. The curve ${(\frac{x}{a})}^{n} + {(\frac{y}{b})}^{n} = 2 touches the straight line \frac{x}{a} + \frac{y}{b} = 2 at the point (a, b)$

a)		for n=3
b)		for n=2
c)		for any value of n
d)		for no value of n

29. f(x)=1+ $[\cos x] x, in 0 < x \leq \frac{π}{2}$

a)		is continious in $[0, \frac{π}{2}]$
b)		has a maximum value 2
c)		has a minimum value 0
d)		is not differentiable at x= $\frac{π}{2}$

30. The maximum value of $\frac{logx}{x} in (2, \propto) is$

a)		1
b)		$\frac{2}{e}$
c)		e
d)		$\frac{1}{e}$

31. Let f(x)= $\frac{logx}{x} + \log 51, then f (x) is$

a)		increasing for x>e
b)		decreasing for x>e
c)		decreasing for 2<x<e
d)		$f^{1} (x) = 0 for x = {2 e}^{3}$

32. The line $\frac{x}{a} + \frac{y}{b} = 1 touches the curve y = {be}^{\frac{- x}{a}} at the point$

a)		(a, $\frac{b}{a})$
b)		(-a,ba)
c)		(a, $\frac{a}{b})$
d)		none of these

33. The normal at the point (1,1) on the curve 2y = 3- $x^{2} is$

a)		x+y=0
b)		x+y+1=0
c)		x-y+1=0
d)		x-y=0

34. The tangent to a given curve perpendicular to x-axis if

a)		$\frac{dy}{dx} = 0$
b)		$\frac{dy}{dx} = 1$
c)		$\frac{dx}{dy} = 0$
d)		$\frac{dx}{dy} = 1$

35. For the curve x= $t^{2} - 1, y = t^{2} - t tangent is parallel to x - axis where .$

a)		t=o
b)		t= $\frac{1}{\sqrt{3}}$
c)		t= $\frac{1}{2}$
d)		$\frac{- 1}{\sqrt{3}}$