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.

Continuity and Differentiability

Multiple Choice Questions

1.
$If z = \frac{y}{x} [\sin (\frac{x}{y}) + \cos (1 + \frac{y}{x})], then x \frac{\partial z}{\partial x} is equal to$

$y \frac{\partial z}{\partial y}$

$- y \frac{\partial x}{\partial y}$

$2 y \frac{\partial z}{\partial y}$

$2 y \frac{\partial z}{\partial x}$

$- y \frac{\partial x}{\partial y}$

$We have, z = (\frac{y}{x}) [\sin (\frac{x}{y}) + \cos (1 + (\frac{y}{x}))] \frac{\partial z}{\partial x} = (\frac{y}{x}) [\cos (\frac{x}{y}) (\frac{1}{y}) - \sin (1 + (\frac{y}{x})) (\frac{- y}{x^{2}})] + (\frac{- y}{x^{2}}) [\sin (\frac{x}{y}) + \cos (1 + (\frac{y}{x}))]$

$x \frac{\partial z}{\partial x} = y [(\frac{1}{y}) \cos (\frac{x}{y}) + (\frac{y}{x^{2}}) \sin (1 + (\frac{y}{x}))] - (\frac{y}{x}) [\sin (\frac{x}{y}) + \cos (1 + (\frac{y}{x}))] \Rightarrow x \frac{\partial z}{\partial x} = \cos (\frac{x}{y}) + (\frac{y^{2}}{x^{2}}) \sin (1 + (\frac{y}{x})) - z . . . (i) Similarly, y \frac{\partial z}{\partial y} = \cos (\frac{x}{y}) - (\frac{y^{2}}{x^{2}}) \sin (1 + (\frac{y}{x})) + z . . . (ii) On adding Eqs . (i) and (ii) x \frac{\partial z}{\partial x} = - y \frac{\partial z}{\partial y}$

limit as straight n rightwards arrow infinity of space open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis left parenthesis straight n plus 2 right parenthesis....3 straight n over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 divided by straight n end exponent

is equal to

18/e⁴
27/e²
9/e²
9/e²

303 Views

The area (in sq. units) of the region $open curly brackets left parenthesis straight x comma straight y right parenthesis colon straight y squared space greater or equal than space 2 straight x space and space straight x squared space plus straight y squared space less or equal than 4 straight x comma space straight x space greater or equal than 0 comma space straight y greater or equal than 0 close curly brackets$ is

$straight pi minus 4 over 3$
$straight pi minus 8 over 3$
$straight pi minus fraction numerator 4 square root of 2 over denominator 3 end fraction$
$straight pi minus fraction numerator 4 square root of 2 over denominator 3 end fraction$

299 Views

If a curve y=f(x) passes through the point (1, −1) and satisfies the differential equation, y(1+xy) dx=x dy, then f(-1/2) is equal to

-2/5
-4/5
2/5
2/5

424 Views

If f and ga re differentiable functions in (0,1) satisfying f(0) =2= g(1), g(0) = 0 and f(1) = 6, then for some c ε] 0,1[

2f'(c) = g'(c)
2f'(c) = 3g'(c)
f'(c) = g'(c)
f'(c) = g'(c)

149 Views

The population p(t) at time t of a certain mouse species satisfies the differential equation $fraction numerator dp space left parenthesis straight t right parenthesis over denominator dt end fraction space equals space 0.5 space left parenthesis straight t right parenthesis space minus 450$ . if p (0) = 850, then the time at which the population becomes zero is

2 log 18
log 9
$1 half space log space 18$
$1 half space log space 18$

493 Views

Consider the function f(x) = |x – 2| + |x – 5|, x ∈ R.
Statement 1: f′(4) = 0
Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).

Statement 1 is false, statement 2 is true
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

154 Views

fraction numerator straight d squared straight x over denominator dy squared end fraction

equal to

$open parentheses fraction numerator straight d squared straight x over denominator dy squared end fraction close parentheses to the power of negative 1 end exponent$
$negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses to the power of negative 1 end exponent open parentheses dy over dx close parentheses to the power of negative 3 end exponent$
$open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 2 end exponent$
$open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 2 end exponent$