If \(\begin{vmatrix}2a & x_1 &y_1 \\ 2b & x_2 & y_2 \\ 2c & x_3 &y_3 \end{vmatrix}={abc\over 2}\neq 0,\) then the area of the triangle whose vertices are \(\begin{pmatrix} {x_1\over a}, {y_1\over a} \end{pmatrix}\), \(\begin{pmatrix} {x_2\over b}, {y_2\over b} \end{pmatrix}\), \(\begin{pmatrix} {x_3\over c}, {y_3\over c} \end{pmatrix}\) is

(a)

\({1\over 4}\)

(b)

\({1\over 4} abc\)

(c)

\({1\over 8}\)

(d)

\({1\over 8}abc\)

The value of the determinant of A = \(\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is\)

(a)

-2abc

(b)

abc

(c)

0

(d)

a² + b² + c²

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

(a)

3

(b)

\(1\over3\)

(c)

6

(d)

\(1\over6\)

Let f :\(R \rightarrow R\) be defined by \(f(x)= \begin{cases}x & x \text { is irrational } \\ 1-x & x \text { is rational }\end{cases}\)  then f is

(a)

discontinuous at x = \({1\over 2}\)

(b)

continuous at  x = \({1\over 2}\)

(c)

continuous everywhere

(d)

discontinuous everywhere

\(\lim _{ x\rightarrow 1 }{ \frac { { x }^{ m }-1 }{ { x }^{ n }-1 } } is\)

(a)

mn

(b)

m+n

(c)

m-n

(d)

\(\frac { m }{ n } \)

The function \(f\left( x \right) =\tan { x } \) is discontinuous on the set

(a)

\(\left\{ n\pi :\quad n\in z \right\} \)

(b)

\(\left\{ 2n\pi :\quad n\in z \right\} \)

(c)

\(\left\{ (2n+1)\frac { \pi }{ 2 } ,\quad n\in z \right\} \)

(d)

\(\left\{ n\frac { \pi }{ 2 } ,\quad n\in z \right\} \)

A function f(x) is said to be continuous at x=a if  \(\lim _{ x\rightarrow a }{ f\left( x \right) } \)is equal to

(a)

\(f\left( a \right) \)

(b)

\(f\left( -a \right) \)

(c)

\(2f\left( a \right) \)

(d)

\(f\left( \frac { 1 }{ a } \right) \)

If the derivative of (ax - 5)e^3x at x = 0 is -13, then the value of a is

(a)

8

(b)

-2

(c)

5

(d)

2

\(\text { If } f(x)= \begin{cases}x-5 & \text { if } x \leq 1 \\ 4 x^2-9 & \text { if } 1<x<2 \\ 3 x+4 & \text { if } x \geq 2\end{cases}\), then the right hand derivative of f(x) at x = 2 is

(a)

0

(b)

2

(c)

3

(d)

4

Choose the correct or the most suitable answer from the given four alternatives.
\(If\sin { (x+y)= } \log { (x+y) } \) then \(\frac { dy }{ dx } \) is _____

(a)

2

(b)

-2

(c)

1

(d)

-1

Choose the correct or the most suitable answer from the given four alternatives.
If, \(y=a+b{ x }^{ 2 }\) where a, b are arbitrary constants, then ____

(a)

\(\frac { d^{ 2 }y }{ d{ x }^{ 2 } } =2xy\)

(b)

\(x\frac { d^{ 2 }y }{ d{ x }^{ 2 } } ={ y }_{ 1 }\)

(c)

\(x\frac { d^{ 2 }y }{ d{ x }^{ 2 } } -\frac { dy }{ dx } +y=0\)

(d)

\(x\frac { d^{ 2 }y }{ d{ x }^{ 2 } } =2xy\)

Assertion (A) : f (x) =\(\begin{cases} \begin{matrix} x+1, & x<2 \end{matrix} \\ \begin{matrix} 2x-1, & x\ge 2 \end{matrix} \end{cases}\) then f'(2) does not exist.
Reason (R) : f(x) is not continuous at 2.

(a)

Both A and it are true and R is the correct explanation of A

(b)

Both A and R are true but R is not the correct explantion of A

(c)

A is true R is false

(d)

A is false R is true

If \(\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c\), then the value of k is

(a)

log 3

(b)

-log 3

(c)

\(-{1\over log3}\)

(d)

\({1\over log3}\)

\(\int \sin ^3 x d x\) is

(a)

\({-3\over 4}cos \ x-{cos \ 3x\over 12}+c\)

(b)

\({3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(c)

\({-3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(d)

\({-3\over 4}sin \ x-{sin \ 3x\over 12}+c\)

\(\int \frac{x^2+\cos ^2 x}{x^2+1} \operatorname{cosec}^2 x d x\) is

(a)

cot x + sin ^-1x + c

(b)

-cot x + tan^-1x + c

(c)

-tan x + cot^-1x + c

(d)

-cot x - tan^-1x + c

\(\int { \frac { 1 }{ 9x^{ 2 }-4 } } \) dx = ________+c.

(a)

log\(\left| \frac { 3x-2 }{ 3x+2 } \right| \)

(b)

\(\frac { 1 }{ 12 } \)log\(\left| \frac { 3x-2 }{ 3x+2 } \right| \)

(c)

12log\(\left| \frac { 3x-2 }{ 3x+2 } \right| \)

(d)

\(\frac { 1 }{ 12 } log\left| \frac { 3x+2 }{ 3x-2 } \right| \)

Match List - I with List II.

	List - I		List - II
i	\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log(tan\quad x)dx } \)	a	\(\frac { 16 }{ 35 } \)
ii	\(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 10 }dx } \)	b	\(\frac { 120 }{ 46 } \)
iii	\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ si{ n }^{ 7 }xdx } \)	c	\(\frac { 1 }{ 132 } \)
iv	\(\int _{ 0 }^{ \infty }{ { x }^{ 5 }{ e }^{ -4x }dx } \)	d	0

The Correct match is

(a)

i	ii	iii	iv
d	c	a	b

(b)

i	ii	iii	iv
d	c	b	a

(c)

i	ii	iii	iv
b	d	c	a

(d)

i	ii	iii	iv
b	c	a	d

Two items are chosen from a lot containing twelve items of which four are defective, then the probability that at least one of the item is defective

(a)

\({19\over 33}\)

(b)

\({17\over 33}\)

(c)

\({23\over 33}\)

(d)

\({13\over 33}\)

Three integers are chosen at random from the first 20 integers. The probability that their product is even is

(a)

\(\frac { 2 }{ 19 } \)

(b)

\(\frac { 3 }{ 19 } \)

(c)

\(\frac { 17 }{ 19 } \)

(d)

\(\frac { 4 }{ 19 } \)

If P(A\(\cup \)B) = 0.8 and P(A\(\cap \)B) = 0.3 then \(P(\bar { A } )+P(\bar { B } )\) =

(a)

0.3

(b)

0.5

(c)

0.7

(d)

0.9

If A and B are two events such that P(A) = \(\frac { 4 }{ 5 } \) and \(P(A\cap B)=\frac { 7 }{ 10 } \) then P(B/A) = 

(a)

\(\frac { 1 }{ 10 } \)

(b)

\(\frac { 1 }{ 8 } \)

(c)

\(\frac { 7 }{ 8 } \)

(d)

\(\frac { 17 }{ 20 } \)

If P(B)=\(\frac { 3 }{ 5 } \), P(A/B)=\(\frac { 1 }{ 2 } \) and P(AUB)=\(\frac { 4 }{ 5 } \), then P(B/\(\bar { A } \)) =

(a)

\(\frac { 1 }{ 5 } \)

(b)

\(\frac { 3 }{ 10 } \)

(c)

\(\frac { 1 }{ 2 } \)

(d)

\(\frac { 3 }{ 5 } \)

A flash light has 8 batteries out of which 3 are dead. If 2 batteries are selected without replacement and tested, the probability that both are dead is

(a)

\(\frac { 3 }{ 28 } \)

(b)

\(\frac { 1 }{ 14 } \)

(c)

\(\frac { 9 }{ 64 } \)

(d)

\(\frac { 33 }{ 56 } \)

Choose the incorrect pair :

(a)

P(A) + P (B) -2P(A∩B) - Exactly one of them occur

(b)

P(A∩B) - Simultaneous occurrence of A and B

(c)

peA) + P (B) - P(A∩B) - Occurrence of either A or B or both

(d)

1 - P(A∪B) - Occurrence of only A

Choose the correct statement

(a)

Permutation and Combination are equal

(b)

Permutation is greater than combination

(c)

Permutation is lesser than combination

(d)

Permutation and combination are unrelated