Linear Functions

Q. The domain of $f\left(x\right)=\sqrt{(x^2-3x-10)\left[\ln \right(x-3){]}^2}$ is

Q. f(x)=sqrt(-x^(2)+4x-3)+sqrt(sin(pi)/(2)(sin(pi)/(2)(x-1))) Find range.

Q.

Prove that $\frac{1-\cos A+\cos B-\cos (A+B)}{1+\cos A-\cos B-\cos (A+B)}=\tan \frac{A}{2}\cot \left(\frac{B}{2}\right)$.

Q. Which of the following graphs represent a linear function (over its whole domain)?

1. Only $\left(1\right)$ and $\left(2\right)$
2. Only $\left(1\right),\left(2\right)$ and $\left(3\right)$
3. $\left(1\right),\left(2\right),\left(3\right)$ and $\left(4\right)$
4. Only $\left(2\right)$

Q.

Prove that:

sin(π/7)*sin(2π/7)*sin(3π/7)=(√7)/8

Q. If $f:Z\to Z$ be a linear function such that $f\left(2\right)+f\left(1\right)=4,f\left(2\right)-2f\left(1\right)=-1$, then $f\left(3\right)$ is

1. $1$
2. $3$
3. $2$
4. $\frac{2}{3}$

Q.

If (sin A)/(sin B) = √3/2 , (cos A)/(cos B) = √5/2, then find the values of tan A and tan B.

Q.

Evaluate $2\sin \left(\frac{\pi }{12}\right)$

Q. Let $R$ be the set of real numbers. If $f:R\to R$ is a function defined by $f\left(x\right)=x^4$, then $f$ is

1. an injective function but not a surjective function
2. a surjective but not an injective function
3. a bijective function
4. Neither an injective nor a surjective function

Q.

Show that the function f: R_* → R_* defined byis one-one and onto, where R_* is the set of all non-zero real numbers. Is the result true, if the domain R_* is replaced by N with co-domain being same as R_*?

Q. The value of integral $\underset{-1}{\overset{2}{\int }}\left[\frac{\left[x\right]}{1+x^2}\right]dx,$ where $[\cdot ]$denotes the greatest integer function, is equal to

1. $-2$
2. $-1$
3. $0$
4. $1$

Q. How to find out the equation of shm from F=-kx that is X=A sin(wt +fi)

Q. If $f:[1,\infty )\to B$ defined by the function $f\left(x\right)=x^2-2x+6$ is a surjection, then $B$ is equal to

1. $[1,\infty )$
2. $[5,\infty )$
3. $[6,\infty )$
4. $[2,\infty )$

Q. Let $f:R\to Z$ be a continuous function such that $f\left(4\right)=4$. Then the value of $f\left(5\right)$ is

Q. Which of the following graphs represent a linear function (over its whole domain)?

1. Only $\left(1\right),\left(2\right)$ and $\left(3\right)$
2. Only $\left(1\right)$ and $\left(2\right)$
3. $\left(1\right),\left(2\right),\left(3\right)$ and $\left(4\right)$
4. Only $\left(2\right)$

Q. The function $f\left(x\right)=\frac{9x}{5}+32$ is the formula to convert $x^{\circ }C$ to Fahrenheit units. Then at what ${}^{\circ }C$ both Celsius and Fahrenheit is same?

1. ${32}^{\circ }C$
2. $-{32}^{\circ }C$
3. ${40}^{\circ }C$
4. $-{40}^{\circ }C$

Q. Let $N$ be the set of natural numbers. Suppose $f:N\to N$ is a function satisfying the following conditions :
$\left(a\right)f(m+n)=f\left(m\right)+f\left(n\right)\left(b\right)f\left(2\right)=2$
Then the value of $\frac{1}{7}\sum _{k=1}^{20}f\left(k\right)$ is

1. $30$
2. $20$
3. $40$
4. $35$

Q.

If f(x)=cos(log x), then prove that

f(1/x).f(1/y)-1/2(f(x/y)+f(xy))=0

Q. The number of integral value(s) of $a$ for which function $f\left(x\right)=\{\begin{array}{cc}\frac{x^2-5x+6}{x-2},& x<2\\ a,& x=2\\ x^2+6,& x>2\end{array}$
is strictly increasing at $x=2$, is equal to

Q.

If tanA/2=±√1-e/1+e*tanB/2

So prove

CosB = cosA-e/1-ecosA

Q. Integrate cosx/[ (1-sinx) (2-sinx) ] with respect to x

Q. $\int \frac{1}{\cos \left(x+a\right)\cos \left(x+b\right)}\mathrm{d}x$

Q. ntIn any triangle ABC, prove the following: asin[A/2].sin[(B-C)/2] + bsin[B/2].sin[(C-A)/2] + csin[C/2].sin[(A-B)/2] = 0n

Q. If $f:R\to R$ defined by $f\left(x\right)=\{\begin{array}{c}2x+5,x>0\\ 3x-2,x\le 0\end{array}$ then $f$ is

1. many-one function
2. one-one function
3. onto function
4. into function

Q.

Evaluate tan^-1(a - b)/(1+ab)+ tan^-1(b - c)/(1+bc)+ tan^-1(c - a)/(1+cb) ; ab, bc, ac>-1

1. -1

2. π

3. 2π

4. 0

Q. 20. If X is the angle between unit Vector a and b, then sin(X/2) equals

Q. If $f:Z\to Z$ be a linear function such that $f\left(2\right)+f\left(1\right)=4,f\left(2\right)-2f\left(1\right)=-1$, then $f\left(3\right)$ is

1. $1$
2. $3$
3. $2$
4. $\frac{2}{3}$

Q. †an^{-1}(x/\{a+\sqrt{a^2+x^2}\

Q. lim to 0+ sin(root)/root(sin)=

Q. The function defined by $f\left(x\right)=max\{x^2,(x-1{)}^2,2x(1-x)\}$ for $0<x<1$

1. is differentiable for all $x$.
2. is differentiable for all $x$ except at one point.
3. is differentiable for all $x$ except at two points.
4. is not differentiable at more than two points.