Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

JEE Mathematics Solved Question Paper 2014

Multiple Choice Questions

71.

The differential equation of the family of parabolas with vertex at (0, - 1) and having axis along the Y-axis is

yy' + 2xy + 1 = 0
xy' + y + 1 = 0
xy' - 2y - 2
xy' - y - 1 = 0

72.

$The solution of x \frac{dy}{dx} = y + e^{\frac{y}{x}} with y (1) = 0 is$

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

73.

The solution of cos(y) + (xsin(y) - 1)dy/dx = 0 is,

$xsec (y) = \tan (y) + C$
$\tan (y) - \sec (y) = Cx$
$\tan (y) + \sec (y) = Cx$
$xsec (y) + \tan (y) = C$

74.

Three non-zero non-collinear vectors $\hat{a}, \hat{b} and \hat{c}$ are such that $\hat{a} + 3 \hat{b}$ is collinear with $\hat{c}$ , while $\hat{c}$ is $3 \hat{b} + 2 \hat{c}$ collinear with a. Then $\hat{a} + 3 \hat{b} + 2 \hat{c}$ equals

0
$2 \hat{a}$
$3 \hat{b}$
$4 \hat{c}$

75.

If $\hat{a}, \hat{b} and \hat{c}$ are non-coplanar vectors and if $\hat{d}$ is such that $\hat{d} = \frac{1}{x} (\hat{a} + \hat{b} + \hat{c})$ and $\hat{d} = \frac{1}{y} (\hat{b} + \hat{c} + \hat{d})$ where x and y are non-zero real numbers, then $\frac{1}{xy} (\hat{a} + \hat{b} + \hat{c} + \hat{d})$ equals to

76.

The angle between the lines $\hat{r} = (2 \hat{i} - 3 \hat{j} + \hat{k}) + λ (\hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\hat{r} = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (\hat{i} + 2 \hat{j} - 3 \hat{k})$ is

$\frac{π}{2}$
$\cos^{- 1} (\frac{9}{\sqrt{91}})$
$\cos^{- 1} (\frac{7}{\sqrt{84}})$
$\frac{π}{3}$

77.
If a, b and c are vectors with magnitudes 2, 3 and 4 respectively, then the best upper bound of ${|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2}$ among the given values is

93

97

87

90

$Given, |a| = 2, |b| = 3 and |c| = 4 ∴ {|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2} = a^{2} + b^{2} - 2 ab + b^{2} + c^{2} - 2 bc + c^{2} + a^{2} - 2 ca = 2 (a^{2} + b^{2} + c^{2}) - 2 (ab + bc + ca) \Rightarrow 2 (ab + bc + ca) = 2 (a^{2} + b^{2} + c^{2}) - ({|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2})$

$We know that, {(a + b + c)}^{2} \geq 0 ∴ a^{2} + b^{2} + c^{2} + 2 (ab + bc + ca) \geq 0 \Rightarrow a^{2} + b^{2} + c^{2} + 2 (a^{2} + b^{2} + c^{2}) - ({|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2}) \geq 0 \Rightarrow {|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2} \leq 3 (a^{2} + b^{2} + c^{2}) \Rightarrow {|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2} \leq 3 (2^{2} + 3^{2} + 4^{2}) \Rightarrow {|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2} \leq 3 (4 + 9 + 16) \Rightarrow {|\hat{a} - \hat{b}|}^{2} + {|\hat{b} - \hat{c}|}^{2} + {|\hat{c} - \hat{a}|}^{2} \leq 87$

78.

If x, y and z are non-zero real numbers and $\hat{a} = x \hat{i} + 2 \hat{j}$ , $\hat{b} = y \hat{j} + 3 \hat{k}$ and $\hat{c} = x \hat{i} + y \hat{j} + z \hat{k}$ are such that $\hat{a} \times \hat{b} = z \hat{i} - 3 \hat{j} + \hat{k}$ , then $[\hat{a} \hat{b} \hat{c}]$ equals to