JEE (Advanced) 2014 Paper-1

determinant of $$(M^2 + MN^2)$$ is 0

there is a $$3 \times 3$$ non-zero matrix UU such that $$(M^2 + MN^2)$$ U is the zero matrix

determinant of $$(M^2 + MN^2) \geq 1$$

for a $$3 \times 3$$ matrix UU, if $$(M^2 + MN^2)$$U equals the zero matrix then U is the zero matrix

$$(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$$ for some $$c \in [0, 1]$$

$$(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$$ for some $$c \in [0, 1]$$

$$(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$$ for some $$c \in [0, 1]$$

$$(f(c))^2 = (g(c))^2$$ for some $$c \in [0, 1]$$

f(x) is monotonically increasing on $$[1, \infty)$$

f(x) is monotonically decreasing on (0, 1)

$$f(x) + f \left(\frac{1}{x}\right) = 0,$$ for all $$x \in (0, \infty)$$

$$f(2^x)$$ is an odd function of x on R

f(x) has three real roots if a > 4

f(x) has only one real root if a > 4

f(x) has three real roots if a < -4

f(x) has three real roots if - 4 < a < 4

g(x) is continuous but not differentiable at a

g(x) is differentiable on R

g(x) is continuous but not differentiable at b

g(x) is continuous and differentiable at either a or b but not both

f(x) is an odd function

f(x) is a one-one function

f(x) is an onto function

f(x) is an even function

$$\sqrt 2$$

$$1$$

$$-1$$

$$-\sqrt 2$$

$$\overrightarrow{b} = (\overrightarrow{b} . \overrightarrow{z})(\overrightarrow{z} - \overrightarrow{x})$$

$$\overrightarrow{a} = (\overrightarrow{a} . \overrightarrow{y})(\overrightarrow{y} - \overrightarrow{z})$$

$$\overrightarrow{a} . \overrightarrow{b} = - (\overrightarrow{a} . \overrightarrow{y}) (\overrightarrow{b} . \overrightarrow{z})$$

$$\overrightarrow{a} = (\overrightarrow{a} . \overrightarrow{y})(\overrightarrow{z} - \overrightarrow{y})$$

radius of S is 8

radius of S is 7

centre of S is (−7, 1)

centre of S is (−8, 1)

the first column of M is the transpose of the second row of M

the second row of M is the transpose of the first column of M

M is a diagonal matrix with nonzero entries in the main diagonal

the product of entries in the main diagonal of M is not the square of an integer