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$Let S be the set of all integer solutions, (x, y, z), of the system of equations x - 2 y + 5 z = 0 - 2 x + 4 y + z = 0 - 7 x + 14 y + 9 z = 0 Such that 15 < x^{2} + y^{2} + z^{2} < 150 . Then, the number of elements in the set S is equal to .$

Multiple Choice Questions

722.

$Let f (x) = |x - 2| and g (x) = f (f (x)), x \in [0, 4] . Then \int_{0}^{3} (g (x) - f (x)) d x = ?$

0
$\frac{1}{2}$
$\frac{3}{2}$
1

723.

$The integral \int {(\frac{x}{xsin (x) + \cos (x)})}^{2} dx = ? (Where Cis a constant of integration) :$

$\tan (x) + \frac{xsec (x)}{xsin (x) + \cos (x)} + C$
$\tan (x) - \frac{xsec (x)}{xsin (x) + \cos (x)} + C$
$\sec (x) + \frac{xtan (x)}{xsin (x) + \cos (x)} + C$
$\sec (x) - \frac{xsec (x)}{xsin (x) + \cos (x)} + C$

724.

Let [t] denote the greatest integer $\leq$ t. Then the equation in x, [x]² + 2[x + 2] – 7 = 0 has :

no integral solution
infinitely many solutions
exactly two solutions
exactly four integral solutions

725.
$Let f (x) = \int \frac{\sqrt{x}}{{(1 + x)}^{2}} dx (x \geq 0) . Then f (3) - f (1) = ?$

$- \frac{π}{6} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

$- \frac{π}{12} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

$\frac{π}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

$\frac{π}{6} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

$\frac{π}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

$f (x) = \int \frac{\sqrt{x}}{{(1 + x)}^{2}} dx Let x = \tan^{2} (θ) dx = 2 \tan (θ) \sec^{2} (θ) dθ f (x) = \int \frac{\tan (θ)}{1 + \tan^{2} (θ)} . 2 \tan (θ) \sec^{2} (θ) dθ f (x) = \int \frac{\tan (θ)}{\sec^{4} (θ)} . 2 \tan (θ) \sec^{2} (θ) dθ f (x) = \int 2 \tan^{2} (θ) . \cos^{2} (θ) dθ f (x) = \int 2 \sin^{2} (θ) dθ f (x) = \int (1 - \cos (2 θ)) dθ f (x) = θ - \frac{\sin (2 θ)}{2} = C = θ - \frac{\tan (θ)}{1 + \tan^{2} (θ)} + C f (x) = \tan^{- 1} (\sqrt{x}) - \frac{\sqrt{x}}{1 + x} + C now f (3) - f (1) = \tan^{- 1} (\sqrt{3}) - \frac{\sqrt{3}}{1 + 3} - \tan^{- 1} (\sqrt{1}) + \frac{1}{1 + 1} = \frac{π}{3} - \frac{π}{4} + \frac{1}{2} - \frac{\sqrt{3}}{4} = \frac{π}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

726.

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

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

Short Answer Type

727.

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

728.

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

$- \frac{1}{9}$
$\frac{9}{2}$
$- \frac{1}{18}$
$\frac{7}{18}$

Short Answer Type

729.

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

730.

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