Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

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.

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.

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

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

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

Ans : 18

$Let \vec{a} = x \hat{i} = x \hat{i} + y \hat{j} + z \hat{k} \hat{i} \times (\vec{a} \times \hat{i}) = (\hat{i} . \hat{i}) \vec{a} - (\vec{a} . \hat{i}) \hat{i} = y \hat{j} + z \hat{k} similarly \hat{j} \times (\vec{a} \times \hat{j}) = x \hat{i} + z \hat{k} and \hat{k} \times (\hat{a} \times \hat{k}) = x \hat{i} + y \hat{k} {|\hat{i} \times (a \times i)|}^{2} + {|\hat{j} \times (\vec{a} \times \hat{j})|}^{2} + {|\hat{k} \times (\vec{a} \times \hat{k})|}^{2} {|y \hat{j} + z \hat{k}|}^{2} + {|x \hat{i} + z \hat{k}|}^{2} + {|x \hat{i} + y \hat{j}|}^{2} = 2 {|a|}^{2} = 2 (9) = 18$

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

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

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