Quantitative Aptitude - Differential and Integral Calculus
Exam Duration: 45 Mins Total Questions : 30
The derivative of \({x^3\over 2}\) (x > 0) is-
- (a)
\(2{x^2\over 3}\)
- (b)
\(3{x^2\over 2}\)
- (c)
52x/5
- (d)
55x/2
Differentiate y W.r.t. x when y = (x2 - 2x)(x2 + 1)
- (a)
4x3 +6x2 - 2x + 2
- (b)
4x3 - 6x + 2
- (c)
4x3 - 6x2 + 2x - 2
- (d)
None of these
The gradient of the curve y = 2x3 - 5x2 - 3x at x = 0 is
- (a)
3
- (b)
-3
- (c)
1/3
- (d)
-1
If Y = \(e^{\sqrt{2x}}{dy\over dx}\) is calculated as
- (a)
\({e^{\sqrt{2x}}\over \sqrt{2x}}\)
- (b)
\(e^{\sqrt{2x}}\)
- (c)
\({e^{\sqrt{2x}}\over \sqrt{2x}}\)
- (d)
None of these
If y = x1/2 then \({dy\over dx}\) is
- (a)
(-1/2)x-3/2
- (b)
(1/2)x-3/2
- (c)
(1/2)x3/2
- (d)
None
Find \({dy\over dx}\) for x2y2 + 3xy + y = 0
- (a)
\({(2xy-y)\over (x-2x)}\)
- (b)
\(-{(2xy^2+3y)\over (2x^2y+3x+1)}\)
- (c)
\({x^2y^2-2y\over 2xy}\)
- (d)
\(-{(2x^2y-3y)\over (x^2y+3x)}\)
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 \({dy\over dx}\) is
- (a)
\(-{(ax+hy+g)\over (hx+by+f)}\)
- (b)
\({(ax+hy+g)\over (hx+by+f)}\)
- (c)
\({(ac-hy+g)\over (hx-by+f)}\)
- (d)
\({h(ax-y+g)\over (x-by+f)}\)
If x2ey + 4 log x = 0 then \({dy\over dx}\) is
- (a)
\({e^y2x^2+4+8x}\over x^3e^y\)
- (b)
\({e^y2x^2-4}\over x^3e^y\)
- (c)
\({-e^y2x^2-4}\over x^3e^y\)
- (d)
None of these
The derivative of \({3-5x}\over 3+5x\) is
- (a)
30(3 + 5x)-2
- (b)
1/(3 + 5x)2
- (c)
\(-{30\over (3+5x)^2}\)
- (d)
None of these
If \(y={{x^{1/2}(5-2x)^{2/3}}\over (4-3x)^{3/4}(7-4x)^{4/5}}\) then the value of \({dy/dx\over y}\) is
- (a)
\({1\over 2x}-{4\over 3(5-2x)}+{9\over 4(4-3x)}+{16\over 5(7-4x)}\)
- (b)
\({1\over 2x}-{1\over 4(5-2x)}+{4\over 9(4+3x)}+{16\over (7+4x)}\)
- (c)
\({1\over x}-{3\over 4(5-2x)}+{4\over 9(4-3x)}+{16\over 5(7-4x)}\)
- (d)
None
If y =\(\left( X+\sqrt { { X }^{ 2 }-1 } \right) \) then the value of (X2-1)\(\left( \frac { dY }{ dX } \right) ^{ 2 }\)-m2 y2
- (a)
-1
- (b)
1
- (c)
0
- (d)
None of these
If y=2x2+3x+10then\(\frac{dY}{dX}\)at(0,0)is
- (a)
10
- (b)
0
- (c)
3
- (d)
None of these
Given x = at2; y = 2at\(\frac{dY}{dX}\)is
- (a)
t
- (b)
-1/t
- (c)
1/t
- (d)
None of these
Let f(y)=xx3then f'(y) is ______
- (a)
x3[x2 + 3x.log x]
- (b)
xX3[X2 +3x2.logx]
- (c)
XX3[x2 - 3x.log x]
- (d)
None of these
If Y = xa + aX + XX+ aa then the value of \(\frac{1}{Y}\times\frac{dY}{dX}\)is
- (a)
x-2(1-logx)
- (b)
x2(1-logx)
- (c)
x2(1+logx)
- (d)
None
If f(x) = x4 then 3rd order derivative of f(x) when x =3 is-
- (a)
72
- (b)
108
- (c)
27
- (d)
81
The gradient of the curve y=-2x3 + 3x + 5 at x = 2 is-
- (a)
-20
- (b)
27
- (c)
-16
- (d)
-21
Evaluate result of \(\int{(x^2-1)^2dx}\) is
- (a)
\({x^5\over5}-{2\over3}x^3+x+k\)
- (b)
\({x^5\over5}-{2\over3}x^3+k\)
- (c)
2x
- (d)
None of these.
At x=3,y=(x-2)6(x-3)5 is
- (a)
A maxima
- (b)
A minima
- (c)
A point of inflexion
- (d)
None
Y = x3 - 3x2 + 3x + 7 has
- (a)
A maxima
- (b)
A minima
- (c)
Both maxima and minima
- (d)
None
If the pattern of total revenue (y) for x number of units sold is y = 40,00,000-(x-2,000)2
Find what number of units sold maximizes total revenue.
- (a)
2000
- (b)
4000
- (c)
6000
- (d)
None
Evaluate \(\int{e^{3x}+e^{-3x}\over e^x}dx\)
- (a)
\({e^{3x}\over3}-{1\over 2x}+c\)
- (b)
\({e^{2x}\over2}-{1\over 4e^{4x}}+c\)
- (c)
\({e^{3x}\over2}-{1\over 3e^{2x}}+c\)
- (d)
\(-{e^{2x}\over2}-{1\over 3e^{2x}}+c\)
Find \(\int {3^x dx}\)
- (a)
loge3+c
- (b)
\({e^x\over 3}log 3+c\)
- (c)
\({3x\over log_e3}+c\)
- (d)
3x+c
Integrate \((ax+{b\over x^3}+{c\over x^7})x^2\)
- (a)
\({1\over4}ax^4+blogx-{1\over4}cx^{-4}+k\)
- (b)
\(4ax^4+blogx-{1\over4}cx^{-4}+k\)
- (c)
\({1\over4}ax^4+blogx+{1\over4}cx^{-4}+k\)
- (d)
None
Evaluate \(\int _{ 0 }^{ 1 }{ ({ 2x }^{ 2 }-{ x }^{ 3 }) } \)dx and the value is
- (a)
4/3+k
- (b)
5/12
- (c)
-4/3
- (d)
None of these
Evaluate \(\int{x^3+5x^2-3\over(x+2)}dx\)
- (a)
\({x^3\over 3}+{2x^2\over 5}+4x+6log(x+3)+c\)
- (b)
\({x^3\over 5}+{7x^2\over 2}-5x-9log(x-8)+c\)
- (c)
\({x^3\over 2}-{7x^2\over 2}-6x-9log(x-4)+c\)
- (d)
\({x^3\over 3}+{3x^2\over 2}-6x+9log(x+2)+c\)
Integrate \((x^2+2)^{-3}x^3\)
- (a)
\(-{2x^2+3\over 2(x^2+2)^2}\)
- (b)
\({1\over2}{(2x^2+3)\over (x^2+1)^2}\)
- (c)
\(-{1\over4}{(2x^2+1)\over x^2+1}\)
- (d)
\({1\over4}{(2x^2+1)\over x^2+1}\)
\(\int \sqrt{x^2+a^2}\)dx is equal to
- (a)
\({x\over 2}\sqrt{x^2+a^2}+{a^2\over2}log|x^2+\sqrt{x^2+a^2}|\)
- (b)
\({x\over 2}\sqrt{x^2-a^2}+{a^2\over2}log|x^2-\sqrt{x^2-a^2}|\)
- (c)
\({x\over 2}\sqrt{x^2-a^2}-{a^2\over2}log|x^2+\sqrt{x^2+a^2}|\)
- (d)
None of these
Evaluate \(\int x^3e^x dx\)
- (a)
(x3 -3x2 +6x -6)eX +c
- (b)
(x3+3x2 +6x -6)eX +c
- (c)
(x3 -3x2 -6x -6)eX +c
- (d)
(x3 +3x2 +6x+6)eX +c
Integrate \({xe^x\over (x+1)^2}\)
- (a)
\(e^x\over x+1\)
- (b)
\(e^x\over (x+1)^2\)
- (c)
\(xe^x\over x+1\)
- (d)
None