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Integrals

Multiple Choice Questions

1011.

$Let f be the twice differential function on (1, 6) . If f (2) = 8, f' (2) = 5, f' (x) \geq 1 and f'' (x) \geq 4, for all x \in (1, 6), then :$

$f (5) + f' (5) \geq 28$
$f' (5) + f'' (5) \leq 20$
$f (5) \leq 10$
$f (5) + f' (5) \leq 26$

Short Answer Type

1012.

If the system of equations

x – 2y + 3z = 9

2x + y + z = b

x – 7y + az = 24,

has infinitely many solutions, then a – b is equal to :

Multiple Choice Questions

1013.
$The integral \int_{\frac{π}{6}}^{\frac{π}{3}} \tan^{3} (x) . \sin^{2} (3 x) (2 \sec^{2} (x) \sin^{2} (3 x) + 3 \tan (x) \sin (6 x)) dx = ?$

$- \frac{1}{9}$

$\frac{9}{2}$

$- \frac{1}{18}$

$\frac{7}{18}$

$- \frac{1}{18}$

$\int_{\frac{π}{6}}^{\frac{π}{3}} \tan^{3} (x) . \sin^{2} (3 x) (2 \sec^{2} (x) \sin^{2} (3 x) + 3 \tan (x) \sin (6 x)) dx \int_{\frac{π}{6}}^{\frac{π}{3}} (\frac{\frac{d}{dx} \tan^{4} (x)}{2} \sin^{4} (3 x) + \tan^{4} (x) . \frac{\frac{d}{dx} \sin^{4} (3 x)}{2}) = \frac{1}{2} \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{d}{dx} (\tan^{4} (x) \sin^{4} (3 x)) dx = \frac{1}{2} {[\tan^{4} (x) \sin^{4} (3 x)]}_{\frac{π}{6}}^{\frac{π}{3}} = \frac{1}{2} [{(3)}^{2} \times 0 - \frac{1}{{(\sqrt{3})}^{4}}] = - \frac{1}{2} \times \frac{1}{9} = - \frac{1}{18}$

Short Answer Type

1014.

$Let {x} and [x] denote the fractional part of x and the greatest integer r \leq x respectively of a real number x . if \int_{0}^{n} \{x\} d x and 10 (n^{2} - n), (n \in N, n > 1) are three consecutive terms of a G . P . then n is equal to . . . . . .$

1015.

$If \vec{a} = 2 \hat{i} + \hat{j} + 2 \hat{k}, then the value of {|\hat{i} \times (\vec{a} \times \hat{i})|}^{2} + {|\hat{j} (\vec{a} \times \hat{j})|}^{2} + {|\hat{k} \times (\vec{a} \times \hat{k})|}^{2} = ?$

Multiple Choice Questions

1016.

$If \int (e^{2 x} + 2 e^{x} - e^{- x} - 1) e^{(e^{x} + e^{- x})} dx = g (x) e^{(e^{x} + e^{- x})} + C, where c is a constant of integration, then g (0) is$

1017.

The value of $\int_{\frac{- π}{2}}^{\frac{π}{2}} \frac{1}{1 + e^{\sin (x)}} d x is :$

$\frac{π}{4}$
$\frac{π}{2}$
$\frac{3 π}{2}$
$π$

1018.

$If \int \frac{\cos (θ)}{5 + 7 \sin (θ) - 2 \cos^{2} (θ)} dθ = {Alog}_{e} |B (θ)| + C, where C is a constant of integration, then \frac{B (θ)}{A} can be :$

$\frac{2 \sin (θ) + 1}{5 (\sin (θ) + 3)}$
$\frac{5 (2 \sin (θ) + 1)}{\sin (θ) + 3}$
$\frac{2 \sin (θ) + 1}{\sin (θ) + 3}$
$\frac{5 (\sin (θ) + 3)}{2 \sin (θ) + 1}$

1019.

The general solution of the differential equation

$\sqrt{1 + x^{2} + y^{2} + x^{2} y^{2}} + xy \frac{dy}{dx} = 0 (where C is constant of integration)$

$\sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} - 1}{\sqrt{1 + x^{2} + 1}}) + C$
$\sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} + 1}{\sqrt{1 + x^{2}} - 1}) + C$
$\sqrt{1 + y^{2}} - \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} - 1}{\sqrt{1 + x^{2} + 1}}) + C$
$\sqrt{1 + y^{2}} - \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} + 1}{\sqrt{1 + x^{2} - 1}}) + C$