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JEE Mathematics Solved Question Paper 2020

Multiple Choice Questions

11.
$For a suitably chosen real constant a, let a function, f : R - \{- a\} \to R be defined by f (x) = \frac{a - x}{a + x} . Further suppose that for any real number x \neq 0 and f (x) \neq - a, (fof) (x) = x . Then f (- \frac{1}{2}) = ?$

3

- 3

$\frac{1}{3}$

$- \frac{1}{3}$

$fof (x) = \frac{a - f (x)}{a + f (x)} = x \Rightarrow \frac{a - ax}{1 + x} = f (x) \Rightarrow \frac{a (1 - x)}{1 + x} = \frac{a - x}{a + x} \Rightarrow a = 1 so f (x) = \frac{1 - x}{1 + x} f (- \frac{1}{2}) = 3$

12.

Let f : R $\to$ R be a function defined by f(x) = max {x, x²}. Let S denote the set of all points in R, where f is not differentiable. Then:

$ϕ (an empty set)$
$\{1\}$
$\{0\}$
$\{0, 1\}$

13.

For all twice differentiable functions f : R $\to$ R, with f(0) = f(1) = f'(0) = 0

f''(x) = 0, for some x $\in$ (0, 1)
$f'' (x) = 0, at every point x \in (0, 1)$
f''(0) = 0
$f'' (x) \neq 0, at every point x \in (0, 1)$

14.

The area (in sq. units) of the region enclosed by the curves y = x² – 1 and y = 1 – x² is equal to :

$\frac{8}{3}$
$\frac{7}{2}$
$\frac{4}{3}$
$\frac{16}{3}$

15.

$The probabilities of three events A, B and C are given P (A) = 0.6, P (B) = 0.4 and P (C) = 0.5 . If P (A \cup B) = 0.8, P (A \cap C) = 0.3, P (A \cap B \cap C) = 0.2, P (B \cap C) = β and P (A \cup B \cup C) = α, where 0.85 \leq x \leq 0.95, then β lies in the interval :$

$[0.36, 0.40]$
$[0.35, 0.36]$
$[0.25, 0.35]$
$[0.20, 0.25]$

16.

$The integral \int_{1}^{2} e^{x} . x^{2} (2 + \log_{e} (x)) d x = ?$

e(2e - 1)
e(4e + 1)
4e² - 1
e(4e - 1)

17.

The common difference of the A.P. b₁,b₂,....,bm is 2 more than common difference of A.P. a₁,a₂,...,a_n. If a₄₀= –159, a₁₀₀= – 399 and b₁₀₀= a₇₀, then b₁is equal to :

127
81
- 127
- 81

18.

$If y = (\frac{2}{π} x - 1) \csc (x) is the solution of the differential equation, \frac{dy}{dx} + p (x) y = \frac{2}{π} \csc (x), 0 < x < \frac{π}{2} then the function p (x) is equal to :$

$\tan (x)$
$\csc (x)$
$\cot (x)$
$\sec (x)$

19.

Let z = x + iy be a non-zero complex number such that z² = i |z|², where i = $\sqrt{- 1}$ , then z lies on the:

real axis
line y = x
line y = - x
imaginary axis

20.

A plane P meets the coordinate axes at A, B and C respectively. The centroid of $∆$ ABC is given to be (1, 1, 2). Then the equation of the line through this centroid and perpendicular to the plane P is: