If |x+3| ≥10 then

1. x ∊ (-13,7]

2. x ∊ [-13,7)

3. x ∊ (-∞,-13] ᴗ [7,∞)

4. x ∊ (-∞,-13] ᴗ [7,∞)

\(\underset{ x\rightarrow 2 }{lim}{ \frac { 2{ x }^{ 2 }+x+1 }{ x+2 } } \) is equal to

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

2. 2

3. \(\frac { 11 }{ 4 } \)

4. 0

\(\underset{x \rightarrow 3}{lim}\left\lfloor x \right\rfloor =\)

1. 2

2. 3

3. does not exist

4. 0

5C₁+5C₂+5C₃+5C₄+5C₅ is equal to

1. 30

2. 31

3. 32

4. 33

The angle between the lines \((x^2+y^2)sin^2\alpha=(xcos\alpha-y\beta)^2\)

1. \(\alpha\)

2. \(2\alpha\)

3. \(\alpha+\beta\)

4. None

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

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

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

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

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

What must be the matrix X, if 2x+\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

1. \(\begin{bmatrix} 1& 3 \\ 2 &-1 \end{bmatrix}\)

2. \(\begin{bmatrix} 1& -3 \\ 2 &-1 \end{bmatrix}\)

3. \(\begin{bmatrix} 2& 6 \\ 4 &-2 \end{bmatrix}\)

4. \(\begin{bmatrix} 2& -6 \\ 4 &-2 \end{bmatrix}\)

The equation of the bisectors of the angle between the co-ordinate axes are

1. x+y=0

2. x-y=0

3. x\(\pm\)y=0

4. x=0

The value of the series\(\quad \frac { 1 }{ 2 } +\frac { 7 }{ 4 } +\frac { 13 }{ 8 } +\frac { 19 }{ 6 } +\).....is

1. 14

2. 7

3. 4

4. 6

\(\underset{\alpha \rightarrow {\pi/4}}{lim}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}\) is

1. \(\sqrt{2}\)

2. \(1\over \sqrt{2}\)

3. 1

4. 2

Let X = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (1, 3), (2, 2), (3, 3), (2, 1), (3, 1), (1, 4),(4, 1)}. Then R is

1. reflexive

2. symmetric

3. transitive

4. equivalence

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

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

1. 1

2. -1

3. 0

4. 2

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

The length of perpendicular from the origin to a line is 12 and the line makes an angle of 120° with the positive direction of y-axis. then the equation of line is

1. \(x+y\sqrt 3=24\)

2. \(x+y=12\sqrt 2\)

3. x+y=24

4. \(x+y=12\sqrt 3\)

Given A={5,6,7,8}. Which one of the following is incorrect?

1. Ø⊆A

2. A⊆A

3. {7,8,9}⊆A

4. {5}⊆A

The product of first n odd natural numbers equals

1. ²ⁿC_n\(\times\)ⁿP_n

2. \({ \left( \frac { 1 }{ 2 } \right) }^{ n }\)²ⁿC_n\(\times\)ⁿP_n

3. \({ \left( \frac { 1 }{ 4 } \right) }^{ n }\times \)²ⁿC_n\(\times\)²ⁿP_n

4. ⁿC_n \(\times\)ⁿP_n

Let \(\alpha\) and \(\beta\) are the roots of a quadratic equation px² + qx + r = 0 then

1. \(\alpha +\beta=-{p\over r}\)

2. \(\alpha\beta={p\over r}\)

3. \(\alpha +\beta={-q\over p}\)

4. \(\alpha \beta =r\)

The product of r consecutive positive integers is divisible by

1. r!

2. r!+1

3. (r+1)

4. none of these

A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are\({3\over4},{1\over2},{5\over 8}\). The probability that the target is hit by A or B but not by C is

1. \({21\over64}\)

2. \({7\over32}\)

3. \({9\over64}\)

4. \({7\over8}\)

\({d\over dx}({2\over \pi}sin \ x^o)\)is

1. \({\pi\over 180}cos \ x^o\)

2. \({1\over 90} cos \ x^o\)

3. \({\pi\over 90}cos \ x^o\)

4. \({2\over \pi}cos \ x^o\)

The projection of \(\overrightarrow { b } \) on \(\overrightarrow { a } \) is

1. \(\left( \frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { b } | } \right) \overrightarrow { b } \)

2. \(\frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { b } | } \)

3. \(\frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { a } | } \)

4. \(\left( \frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { a } | } \right) \)

\(\underset{x\rightarrow o}{lim}{8^x-4^x-2^x+1^x\over x^2}=\)

1. 2 log 2

2. 2( log2)²

3. log 2

4. 3 log 2

It is given that the events A and B are such that \(P(A)={1\over4},P(A/B)={1\over2}\) and \(P(B/A)={2\over3}.\) Then P(B) is

1. \({1\over 6}\)

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

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

4. \({1\over 2}\)

In a \(\triangle\)ABC, \(\hat{A}\) = 60°, \(\hat{C}\) = 30°, b = 2\(\sqrt{3}\) ,c = 2 then a is

1. 0

2. 1

3. 4

4. 2

cos35⁰+cos85⁰+cos155⁰=

1. 0

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

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

4. cos 275⁰

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,\(\infty\))

Choose the incorrect statement

 

1. log 1 to any base is zero

2. \(\frac{d}{dx}\) (e^x)=e^x

3. Inverse function of log x is \(\frac{1}{x}\)

4. |x| is not differentiable at x =0

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

1. zero 

2. straight

3. co-terminal

4. standard

In aΔABC, if (i) \(sin\frac { A }{ 2 } sin\frac { B }{ 2 } sin\frac { C }{ 2 } \) (ii) sinA sinB sinC>0

1. Both (i) and (ii) are true

2. Only (i) is true

3. Only (ii) is true

4. Neither (i) nor (ii) is true

If the lines represented by the equations 6x²+41xy-7y²=0 make angles and with x-axis then \(\alpha\tan\beta=\)

1. \(-\frac{6}{7}\)

2. \(+\frac{6}{7}\)

3. \(-\frac{7}{6}\)

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

If cos \(\alpha\)=\(\frac{3}{5}\) and cos \(\beta=\frac{5}{13}\), then

1. cos\((\alpha+\beta)=\frac{33}{65}\)

2. \(sin(\alpha+\beta)=\frac{56}{65}\)

3. \(sin^2\frac{(\alpha-\beta)}{2}=\frac{4}{65}\)

4. \(cos(\alpha-\beta)=\frac{66}{65}\)

The differential coefficient of log₁₀ x with respect to log_x10 is

1. 1

2. -(log₁₀ x)²

3. (log_x 10)²

4. \(x^2\over100\)

The number of relations on a set containing 3 elements is

1. 9

2. 81

3. 512

4. 1024

If a^2-a C₂=a^2-a C₄ then the value of 'a' is

1. 2

2. 3

3. 4

4. 5

If A is a square matrix, then which of the following is not symmetric?

1. A+ A^T

2. AA^T

3. A^T A

4. A− A^T

If ABCD is a parallelogram, then \(\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}\) is equal to 

1. \(2(\overrightarrow{AB}+\overrightarrow{AD})\)

2. \(4\overrightarrow{AC}\)

3. \(4\overrightarrow{BD}\)

4. \(\overrightarrow{0}\)

When h² = ab, the angle between the pair of straight lines ax² + 2hxy + by² = 0 is

1. \(\frac\pi4\)

2. \(\frac\pi3\)

3. \(\frac\pi6\)

4. 0^o

Which one of the following statements is false?

1. The image of a point \((\alpha\beta)\) about x-axis \((\alpha,-\beta)\)

2. The image of the line ax+by+c=0 about x-axis is ax-by+c=0

3. The image of a point \((\alpha,\beta)\) about y-axis \((-\alpha,\beta)\)

4. The image of the line ax+by+c=0 about y-axis is ax-by+c=0

If sin\(\theta\) + cos\(\theta\) = 1 then sin⁶\(\theta\) + cos⁶\(\theta\) is

1. 1

2. 0

3. -1

4. 2

Integrate the function with respect to x
\(\left( { x }^{ 2 }+1 \right) \sqrt { x+1 } \)

1. Find the n^th term of the series 3 - 6 + 9 -12 + ...

2. A girl is reading a book having 446 pages and she has already finished reading 271 pages. She wants to finish reading this book within a week. What is the minimum number of pages she should read per day to complete reading the book within a week?

Examine the continuity of the following:\(|x-2|\over |x+1|\)

Find seven numbers A₁, A₂, ... , A₇ so that the sequence 4, A₁, A₂, ... , A₇, 7 is in arithmetic progression and also 4 numbers G₁, G₂, G₃, G₄ so that the sequence 12, G₁, G₂, G₃, G₄, is in geometric progression.

If the function f:R➝R be given by f(x)=x²+2 and g(x)=2x, find fog and gof

Show that the function is \(f\left( x \right) =\begin{cases} \frac { \sin { x } }{ x } +\cos { x,\quad x\neq 0 } \\ 2,\quad \quad \quad x=0 \end{cases}\) continuous at x =0.

Consider the matrix A\(\alpha\)=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)

Show that \(A _{\alpha}A_{\beta}=A_{(\alpha + \beta)}\) .

There are 3 types of toy car and 2 types of toy train are available in a shop. Find the number of ways a baby can buy a toy car and a toy train?

Find the area of a triangle ABC in which ㄥA=60⁰, b=4 cm and c=\(\sqrt { 3 } \) cm.

If log₂x + log₄x + log₁₆x=\(\frac{7}{2}\), find the value of x.

Evaluate:\(cos^{-1}(-{1\over\sqrt{2}})\)

Simplify: \(\frac { 7! }{ 2! } \).

Simplify by rationalising the denominator \(\frac { 7+\sqrt { 6 } }{ 3-\sqrt { 2 } } \)

In a \(\triangle ABC,\) a = 3, b = 5 and c = 7. Find the values of cos A, cos B and cos C.

If p(A) denotes the power set of A, then find \(n(P(P(P(\phi)))).\)

If k > 0,then solve the inequation |x| ≤ K

If x = 1 is one root of two equation. x³ - 6x + 11x - 6 = 0 find the other roots.

Find sin (x - y) given that sin x = \(\frac{8}{17}\) with\(0 and cos y = \(-\frac{24}{25}\) with\(\pi

Show that the relation R on R defined as R={(a,b):a≤b} is reflexive and transitive but not symmetric.

Integrate the function with respect to x
\(\sqrt { (2x+1)^{ 2 }+9 } \)

Given a = 8, b = 9, c = 10, find all the angles.

If A is a 3 × 4 matrix and B is a matrix such that both A^TB and BA^T are defined, what is the order of the matrix B?

If \(sin\theta=\frac{3}{5}\) and the angle θ is in the second quadrant, then find the values of other five trigonometric functions.

By taking suitable sets A, B, C, verify the following results:
A \(\times\) (B\(\cup \)C)=(A\(\times\)B) \(\cup \) (A\(\times\)C)

If \(\cos { \left( \alpha -\beta \right) } +\cos { \left( \beta -\gamma \right) } +\cos { \left( \gamma -\alpha \right) } =\frac { -3 }{ 2 } \) then prove that \(\cos { \alpha } +\cos { \beta } +\cos { \gamma } =\sin { \alpha } +\sin { \beta } +\sin { \gamma } =0\)

Integrate the following with respect to x\(sin 2x\over a^2+b^2 sin^2x\)

Evaluate the following limits :
\(\underset{x\rightarrow2}{lim}{{1\over x}-{1\over2}\over x-2}\)

Evaluate the following integrals\(\int e^x(sin \ x+cos \ x)dx\)

Separate the equations 5x² + 6xy +y² = 0.

Derive cosine formula using the law of sines in a \(\triangle ABC.\)

Evaluate \(\int { \frac { \left( { a }^{ x }+{ b }^{ x } \right) ^{ 2 } }{ { a }^{ x }{ b }^{ x } } } \)dx

A straight line L with negative slope passes through the point (9, 4) cuts the positive coordinate axes at the points P and Q. As L varies, find the minimum value \(|OP|+|OQ|\) of , where O is the origin.

Find the number of 4-digt numbers that can be formed using the digits 1,2,3,4,5 if no digit is repeated. How many of these will be even?

Prove that \({{cos\ 2\ x}\over{1+sin\ 2x}}=tan\left( {{\pi}\over{4}}-x \right)\)

Expand \({\left( 2x-{1\over 2x} \right)}^{4}.\)

What are the points on x-axis whose perpendicular distance from the line 4x + 3y =12 is 4?

Prove that \(\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-{ e }^{ -x } }{ x } } =2\)

Find the principal value of tan^-1\(\sqrt { 3 } \)

Let A and B be two points with position vectors 2\(\overrightarrow{a}\)+ 4\(\overrightarrow{b}\) and 2\(\overrightarrow{a}\) −8\(\overrightarrow{b}\). Find the position vectors of the points which divide the line segment joining A and B in the ratio 1:3 internally and externally.

If n(A\(\cap\)B)=3 and n(A\(\cup\)B) = 10 then find n(P(A\(\Delta \)B))

Using the mathematical induction, show that for any natural number n,\(\frac { 1 }{ 1.2 } +\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } +...+\frac { 1 }{ n(n+1) } =\frac { n }{ n+1 } \).

1. If \(y=\sqrt { \frac { 1+{ e }^{ x } }{ 1-{ e }^{ x } } } ,\) show that \(\frac { dy }{ dx } =\frac { { e }^{ x } }{ (1-{ e }^{ x })\sqrt { 1-{ e }^{ 2x } } } \)

2. Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
   Show that A ∩ (B ∩ C) = (A ∩ B) ∩ C

Show that 4x² + 4xy + y² - 6x - 3y - 4 = 0 represents a pair of parallel lines.

Express each of the following as a product.
cos 35^o - cos 75^o

Show that \(\begin{vmatrix} 2bc-a^2 & c^2 &b^2 \\ c^2 &2ca-b^2 &a^2 \\ b^2 & a^2 &2ab-c^2 \end{vmatrix}=\begin{vmatrix} a &b &c \\ b & c & a \\ c &a &b \end{vmatrix}^2\) .

Solve the equation \(8+9\sqrt{(3x-1)(x-2)}=3x^2-7x.\)

Let f and g be the two functions from R to R defined by f(x) = 3x - 4 and g(x) = x²+ 3. Find gof and fog.

If A is a square matrix such that A²=I, then find the simplified value of (A-I)³+(A+I)³-7A.

If \(sin x=\frac{4}{5}\) (in I quadrant) and \(cos\ y=\frac{-12}{13}\) (in II quadrant), then find (i) sin (x - y), (ii) cos (x -y).

Find the Co-efficient of x⁶ and the co -efficient of x² in \(\left( { x }^{ 2 }-\frac { 1 }{ { x }^{ 3 } } \right) ^{ 6 }\)

Evaluate \(\int { \frac { dx }{ tanx+cotx+secx+cosecx } } \)

If y = tan^-1\(({1+x\over 1-x}),find \ y'\)