If A=\(\begin{bmatrix} 1& 2 &2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}\) is a matrix satisfying the equation AA^T = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

1. (2, - 1)

2. (- 2, 1)

3. (2, 1)

4. (- 2, - 1)

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) = 

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

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

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

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

The function f:[0,2π]➝[-1,1] defined by f(x)=sin x is

1. one-to-one

2. on to

3. bijection

4. cannot be defined

Number of solutions of the equation tan x+sec x=2 cos x lying in the interval [0,2π] is

1. 0

2. 1

3. 2

4. 3

The general solution of cosec_\(\theta\) = -2 is

1. \(2n\pi +(-1)^n({\pi\over 6})\)

2. _{\(n\pi +(-1)^n({-\pi\over 6})\)}

3. \(2n\pi \pm({\pi\over 6})\)

4. \(-{\pi\over 6}+n\pi\)

The co-ordinates of the foot of the perpendicular drawn from the point (2,3) to the line 3x-y+4=0 is

1. \((\frac{1}{10},\frac{37}{10})\)

2. \((\frac{-1}{10},-\frac{37}{10})\)

3. \((\frac{-1}{10},\frac{37}{10})\)

4. \((\frac{37}{10},\frac{-1}{10})\)

\({d\over dx}(e^{x+5log \ x})\)is

1. e^x.x⁴(x+5)

2. e^x.x(x+5)

3. e^x\(+{5\over x}\)

4. e^x\(-{5\over x}\)

If x₁,x₂,x₃ as well as y₁,y₂,y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂,y₂), (x₃,y₃ ) are

1. vertices of an equilateral triangle

2. vertices of a right angled triangle

3. vertices of a right angled isosceles triangle

4. collinear

If \(\left\lfloor . \right\rfloor \) denotes the greatest integer less than or equal to the real number under consideration and −1\(\le\)x < 0, 0 \(\le\) y <1, 1\(\le\) z <2, then the value of the determinant \(\begin{vmatrix} \left\lfloor x \right\rfloor +1& \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor +1& \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor +1\end{vmatrix}\) is

1. \(\left\lfloor z \right\rfloor \)

2. \(\left\lfloor y \right\rfloor \)

3. \(\left\lfloor x \right\rfloor \)

4. \(\left\lfloor x \right\rfloor \)+1

\(\underset{\theta\rightarrow0}{lim}{Sin\sqrt{\theta}\over \sqrt{sin \theta}} \)

1. 1

2. -1

3. 0

4. 2

A line perpendicular to the line 5x - y = 0 forms a triangle with the coordinate axes. If the area of the triangle is 5 sq. units, then its equation is

1. \(x+5y\pm5\sqrt2=0\)

2. \(x-5y\pm5\sqrt2=0\)

3. \(5x+y\pm5\sqrt2=0\)

4. \(5x-y\pm5\sqrt2=0\)

For what values of x is the rate of increase of x³ - 2x² + 3x + 8 is twice the rate of increase of x?

1. \(\left( -\frac { 1 }{ 3 } ,-3 \right) \)

2. \(\left( \frac { 1 }{ 3 } ,3 \right) \)

3. \(\left( -\frac { 1 }{ 3 } ,3 \right) \)

4. \(\left( \frac { 1 }{ 3 } ,1 \right) \)

There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

1. 10²

2. 1023

3. 2¹⁰

4. 10!

If -3x+17 < -13 then

1. x ∈ (10,∞)

2. x ∈ [10,∞)

3. x ∈ (-∞,10]

4. x ∈ [10,10)

If f(x)=x(-1)^{\(\left\lfloor 1\over x \right\rfloor \)},\(x\le0\),then the value of \(\underset{x\rightarrow 0}{lim}f(x)\) is equal to

1. -1

2. 0

3. 2

4. 4

The condition that the equation ax² + bx + c = 0 may have one root is the double the other is:

1. 2b² = 9ac

2. b²= ac

3. b² = 4ac

4. 9b² = 2ac

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

1. 5

2. 7

3. 26

4. 10

If P(A) = 0.4, P(B) = 0.3 and P(AUB) = 0.5, then \(P(\bar { B } \cap A)\)

1. \(\frac { 2 }{ 3 } \)

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

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

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

\(\underset{x \rightarrow 0}{lim}{e^{tan \ x}-e^x\over tan x-x}=\)

1. 1

2. e

3. \({1\over2}\)

4. 0

The number of five digit telephone numbers having at least one of their digits repeated is

1. 90000

2. 10000

3. 30240

4. 69760

Which one of the following is false?

1. A⋂(BΔ\C) = (A ⋂ B)Δ(A ∩ C)

2. A∩(B - C) = (A ∩B) \ (A∩C)

3. (A U B), = A' ∩ B'

4. (A \ B) U B = A ⋂ B

If the arcs of same lengths in two circles sustend central angles 30° and 40° find the ratio of their radii

1. 3:4

2. 4:3

3. 7:12

4. none of these

The middle term in the expansion of  is \((x- \frac{2}{x})^{12}\) is

1. 12C₆

2. 12C₆2⁶

3. 12C₇

4. 12C₆2⁷

If cos28⁰+sin28⁰=k³, then cos 17⁰ is equal to

1. \(\frac { { k }^{ 3 } }{ \sqrt { 2 } } \)

2. -\(\frac { { k }^{ 3 } }{ \sqrt { 2 } } \)

3. ±\(\frac { { k }^{ 3 } }{ \sqrt { 2 } } \)

4. -\(\frac { { k }^{ 3 } }{ \sqrt { 3 } } \)

If \(y={(1-x)^2\over x^2}\) ,then \({dy \over dx}\) is 

1. 8

2. 1

3. 4

4. 5

The ratio of the coefficient of x ¹⁵ to the term independent of x in \([x^2+(\frac{2}{x})]^{15}\) is

1. 1:16

2. 1:8

3. 1:32

4. 1:64

\(\frac{1}{360}\) of a complete rotation clockwise is

1. -1°

2. -360°

3. -90°

4. 1°

There are 15 points in a plane of which exactly 8 are collinear. The number of straight lines obtained by joining these points is

1. 105

2. 28

3. 77

4. 78

If |x-2|>5, then x belongs to

 

1. (\(\infty\),-2]∪[5,\(\infty\)]

2. (-\(\infty\),-3]∪[7,\(\infty\))

3. (-\(\infty\),-3)∪(7,\(\infty\))

4. (\(\infty\),-2]∪(5,\(\infty\))

The equation of the straight line joining the origin to the point of intersection of y-x+7=0 and y+2x-2=0 is

1. 3x+4y=0

2. 3x-4y=0

3. 4x-3y=0

4. 4x+3y=0

The remainder when 38¹⁵ is divided by 13 is

1. 12

2. 1

3. 11

4. 5

The locus of a point which moves such that it maintains equal distance from the fixed point is a

1. straight line

2. line bisector

3. circle

4. angle bisector

If \(\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix}=\begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}\), then the value of x+y is

1. 5

2. 6

3. 4

4. 3

If \(\underset{x \rightarrow 0}{lim}{sin \ px\over tan \ 3x}=4\) , then the value of p is

1. 6

2. 9

3. 12

4. 4

If tanx =\({1\over 7}\) ,tan y = \({1\over 3}\) then x + y is:

1. \({\pi\over 4}\)

2. \({\pi\over 3}\)

3. \({\pi\over 2}\)

4. \({\pi}\)

Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter 4 + 2\(\sqrt{2}\) is

1. x+y+2=0

2. x+y-2=0

3. \(x+y-\sqrt{2}=0\)

4. \(x+y+\sqrt{2}=0\)

1 - 2x + 3x² - 4x³ + ..., Ixl< 1 is:

1. (1-x)^-2

2. (1+x)^-2

3. (1-x)²

4. (1+x)²

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to

1. 1

2. 3

3. 23

4. 21

Choose the incorrect pair

 

1. \(\underset { x\rightarrow \infty }{ lim } \frac { sin\quad x }{ x } =0\)

2. \(\underset { x\rightarrow \frac { \pi }{ 2 } }{ lim } \frac { 2x-\pi }{ cos\quad x } =-2\)

3. \(\underset { x\rightarrow \infty }{ lim } \frac { { a }^{ x }-{ b }^{ x } }{ x } =log\left( \frac { a }{ b } \right) \)

4. \(\frac { { a }^{ x }-1 }{ x } =log\ a>0\)

Number of sides of a polygon having 44 diagonals is

1. 4

2. 4!

3. 11

4. 22

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

1. 3

2. \(1\over3\)

3. 6

4. \(1\over6\)

The term without x in \({ \left( 2x-\frac { 1 }{ 2{ x }^{ 2 } } \right) }^{ 12 }\) is 

1. 495

2. -495

3. -7920

4. 7920

Find the odd one of out of the following

 

1. tan x

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

3. \(\\ \\ \\ \\ \\ \\ \\ \frac { { x }^{ 2 }+5x+4 }{ { x }^{ 2 }+4x+4 } \)

4. cos x

Let A and B be two events such that P(A) = 0.6, P(B) = 0.2 and P(A/B) = 0.5, Then P(\(\bar { A } /\bar { B } \)) is

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

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

3. \(\frac { 3 }{ 8 } \)

4. \(\frac { 6 }{ 7 } \)

\(\int { sin } \)e^x . d (e^x) = ________+c.

1. cos (e^x)

2. sin(e^x)

3. - cos (e^x)

4. -sin(e^x)

The image of the point (3,8) in the line x+3y=7 is

1. (1,4)

2. (-1,-4)

3. (-4,-1)

4. (1,-4)

Find the odd one out of the following:

 

1. \(\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\)

2. \(\begin{bmatrix} 0 & \frac { -7 }{ 2 } \\ \frac { 7 }{ 2 } & 0 \end{bmatrix}\)

3. \(\begin{bmatrix} 0 & 3.2 \\ -3.2 & 0 \end{bmatrix}\)

4. \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)

Let f:Z➝Z be given by f(x)=\(\begin{cases} \frac { x }{ 2 } \quad if\quad x\quad is\quad even \\ 0\quad if\quad x\quad is\quad odd \end{cases}\) . Then f is

1. one-one but not onto

2. onto but not one-one

3. one-one and onto

4. neither one-one nor onto

The angle between the lines 2x - y + 3 = 0 and x + 2y + 3 = 0 is

1. 90°

2. 60°

3. 45°

4. 30°

If quadratic with real coefficients has no real roots, then its discriminant is ________.

 

1. 0

2. <0

3. >0

4. 1

If cosθ+\(\sqrt{3}\)sinθ=2 and θ∈[0,2π] then θ is

1. \(\frac{\pi}{3}\)

2. \(\frac{5\pi}{3}\)

3. \(\frac{2\pi}{3}\)

4. \(\frac{4\pi}{3}\)

The n^th term of the sequence 1, 2, 4, 7, 11,... is

1. n³+3n²+2n

2. n3-3n²+3n

3. \(\frac { n(n+1)(n+2) }{ 3 } \)

4. \(\frac { { n }^{ 2 }-n+2 }{ 2 } \)

There is a letter lock with 3 rings each marked with 5 letters and do not know the keyword. The total number of attempts can be made to know the keyword is

1. 3⁵

2. 5³

3. 124

4. 5

\(\theta\) is acute angle between the lines x²-xy- 6y² = 0, then \(\frac{2\cos\theta+3\sin\theta}{4\sin\theta+5\cos\theta}\) is

1. 1

2. \(-\frac{1}{9}\)

3. \(\frac{5}{9}\)

4. \(\frac{1}{9}\)

If  \(\frac { kx }{ (x+2)(x-1) } =\frac { 2 }{ x+2 } +\frac { 1 }{ x-2 } \) ,then the value of k is

1. 1

2. 2

3. 3

4. 4

Angle with measures 45° and -315° are __________ angles.

1. zero 

2. straight

3. co-terminal

4. standard

If S is the sample space, P(A) = \(\frac { 1 }{ 3 } \)  P(B) and S = A\(\cup \)B, where A and B are mutually exclusive events then P(A) =

1. \(\frac { 1 }{ 4 } \)

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

3. \(\frac { 3 }{ 4 } \)

4. \(\frac { 3 }{ 8 } \)

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

1. \(\frac { \sin { x } }{ 2y-1 } \)

2. \(\frac { \sin { x } }{ 1-2y } \)

3. \(\frac { \cos { x } }{ 1-2y } \)

4. \(\frac { \cos { x } }{ 2y-1 } \)

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is

1. 45

2. 40

3. 39

4. 38

Solve 3x² + 5x - 2≤0

1. (2,\(\frac{1}{3}\))

2. [2,\(\frac{1}{3}\)]

3. (-2,\(\frac{1}{3}\))

4. (-2,\(\frac{-1}{3}\))

If \(\overrightarrow{BA}=3\hat{i}+2\hat{j}+\hat{k}\) and the position vector of is \(\hat{i}+3\hat{j}-\hat{k}\) ,then the position vector A is

1. \(4\hat{i}+2\hat{j}+\hat{k}\)

2. \(4\hat{i}+5\hat{j}\)

3. \(4\hat{i}\)

4. \(-4\hat{i}\)

The points (k+1,1),(2k+1,3) and (2k+2,2k) are collinear if

1. k=-1

2. \(k=\frac{1}{2}\)

3. k=3

4. k=2

If the points (2k,k)(k,2k) and (k,k)enclose a triangle of area 18 sq units, then the centroid of the triangle is

1. (8,8)

2. (4,4)

3. (3,3)

4. (2,2)

If A and B are any two finite sets having m and n elements respectively then the cardinality of the power set of A x B is

1. 2^m

2. 2ⁿ

3. mn

4. 2^mn

Choose the correct statement

 

1. Matrix addition is not associative

2. Matrix addition is not commutative

3. Matrix multiplication is associative

4. Matrix multiplication is commutative

Assertion (A): A plumber will be paid according to following schemes: In first scheme he will be paid Rs.500 plus rs.70 her hour and in the second scheme he will paid rs.120 per hour. If he work for less than 10 hours the first scheme give better wages.
Reason (R): In the first scheme he will get a fixed Rs.500 whereas longer than 10 hours second scheme will pay more.

 

1. Both A and R are true and R is the correct explanation of A

2. Both A and R are true and R is not a correct explanation of A

3. A is true but R is false

4. A is false but R is true

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.

 

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

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

3. A is true R is false

4. A is false R is true

\(\underset{x \rightarrow \infty}{lim}{a^x-b^x\over x}=\)

1. log ab

2. log\(({a\over b})\)

3. log\(({b\over a})\)

4. \({a\over b}\)

There are three events A, B, and C of which one and only one can happen. If the odds are 7 to 4 against A and 5 to 3 against B, then odds against C is

1. 23: 65

2. 65: 23

3. 23: 88

4. 88: 23

\(\int {1\over x\sqrt{(log \ x)^2+5}}dx\)is

1. \(log|x+\sqrt{x^2-5}|+c\)

2. \(log|logx+\sqrt{logx-5}|+c\)

3. \(log|logx+\sqrt{(logx)^2-5}|+c\)

4. \(log|logx-\sqrt{(logx)^2-5}|+c\)