Derivative

Q.

If $∆=\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 2x-3& 3x-4& 4x-5\\ 3x-5& 5x-8& 10x-17\end{array}\right|=A{x}^3+B{x}^2+Cx+D$, then $B+C$ is equal to

1. $1$

2. $-1$

3. $-3$

4. $9$

Q. Differentiate the following w.r.t. x :

Q.

What Is the limit As $x$ approaches infinity of $\cos \left(x\right)$ $?$

Q. Let $f:N\to Y$ be a function defined as $f\left(x\right)=4x+3$, where $Y=\{y\in N:y=4x+3\text{for some}x\in N\}$. Show that $f$ is invertible. Find the inverse.

Q. Let f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x - 7.

(i) For what real numbers $x,f\left(x\right)=g\left(x\right)?$

(ii) For what real numbers x, f(x) < g(x)

Q.

If $f\left(x\right)=x,-1\le x\le 1$, then function $f\left(x\right)$ is

1. increasing

2. decreasing

3. stationary

4. discontinuous

Q.

For what minimum value of n is [(1+I)/(1-i)]ⁿ real?

Q.

Divide $15$pearls between Sonu and Monu so that they get $20\%$and $80\%$ of them respectively.

Q. Evaluate
(1+i) /(1-i)

Q.

The value of k for which f (x) = kx³ - 9 kx² + 9x + 3 is increasing on R.is...

1. 

2. 

3. 

4. 

Q. If f(x ^ 2) = 4x ^ 6 + 3x ^ 4 , then f(5) is equal to

Q.

Let A = {$x:xϵR,x>4$} and B = {$xϵR:x<5$}. Then $A\cap B$ =

1. (4, 5]

2. (4, 5)

3. [4, 5)

4. [4, 5]

Q. Let $g\left(x\right)$ be a polynomial of degree one and $f\left(x\right)$ be defined by
$f\left(x\right)=\{\begin{array}{cc}g\left(x\right),& x\le 0\\ {\left[\frac{1+x}{2+x}\right]}^{1/x},& x>0\end{array}$
Let $f\left(x\right)$ be a continuous function satisfying $f^{\prime }\left(1\right)=f(-1).$ Then $f(-2)$ is equal to

1. $0$
2. $-2$
3. $\frac{2}{9}-\frac{4}{3}\log \left(\frac{2}{3}\right)$
4. $\frac{1}{6}-\log \left(\frac{2}{3}\right)$

Q. 2. 2x 3y sin y

Q. If with prove that

Q.

The maximum value of the function f(x) = -|x + 2| + 3 is :

1. - 2

2. 3

3. 2

4. ∞

Q. DIFFERENTIATE THE FOLLOWING (x^2+3x)^4

Q. If $f\left(x\right)=x^n$

and f (1) = 15, then the value of n is

Q.

If f(x)=x-1/x+1 than proved that f(2x)=3f(x)+1/f(x)+3

Q. If $x^m{y}^n=(x+y{)}^{m+n}$ then $\frac{dy}{dx}$ is equal to

1. $\frac{x+y}{xy}$
2. $xy$
3. $\frac{y}{x}$
4. $0$

Q. This section contains 1 Assertion-Reason type question, which has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct
.
A : The tangent at x = 1 to the curve y = x³ – x² – x + 2 again meets the curve at x = –2.
R : When a equation of a tangent solved with the curve, repeated roots are obtained at point of tangency.

इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (A), (B), (C) तथा (D) दिये गये हैं, जिनमें से केवल एक सही है।

A : x = 1 पर वक्र y = x³ – x² – x + 2 की स्‍पर्श रेखा, वक्र से पुन: x = –2 पर मिलती है।
R : जब एक स्‍पर्श रेखा के समीकरण को वक्र के साथ हल किया जाता है, तब स्‍पर्शिता के बिंदु पर पुनरावर्त मूल पाये जाते हैं।

1. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
2. (A) is true but (R) is false
   (A) सही है लेकिन (R) गलत है
3. (A) is false but (R) is true
   (A) गलत है लेकिन (R) सही है
4. Both (A) and (R) are true and (R) is the correct explanation of (A)
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है

Q. Let $g\left(x\right)$ be a polynomial of degree one and $f\left(x\right)$ be defined by
$f\left(x\right)=\{\begin{array}{cc}g\left(x\right),& x\le 0\\ {\left[\frac{1+x}{2+x}\right]}^{1/x},& x>0\end{array}$
Let $f\left(x\right)$ be a continuous function satisfying $f^{\prime }\left(1\right)=f(-1).$ Then $f(-2)$ is equal to

1. $0$
2. $-2$
3. $\frac{2}{9}-\frac{4}{3}\log \left(\frac{2}{3}\right)$
4. $\frac{1}{6}-\log \left(\frac{2}{3}\right)$

Q. Mark the correct alternative in each of the following:
Let $f\left(x\right)=x-\left[x\right],xϵR,$ then $f^{\prime }\left(\frac{1}{2}\right)$ is

1. $\frac{3}{2}$
   
2. 0
   
3. 1
   
4. -1

Q. Let $g\left(x\right)$ be a polynomial of degree one and $f\left(x\right)$ be defined by
$f\left(x\right)=\{\begin{array}{cc}g\left(x\right),& x\le 0\\ {\left[\frac{1+x}{2+x}\right]}^{1/x},& x>0\end{array}$
Let $f\left(x\right)$ be a continuous function satisfying $f^{\prime }\left(1\right)=f(-1).$ Then $f(-2)$ is equal to

1. $0$
2. $-2$
3. $\frac{2}{9}-\frac{4}{3}\log \left(\frac{2}{3}\right)$
4. $\frac{1}{6}-\log \left(\frac{2}{3}\right)$

Q. If tany= 2^x/1+2^2x+1 then dy/dx at x=0

Q. If $x^y.y^x=16$ then $\frac{dy}{dx}$ at (2, 2) is

1. $1$
2. $0$
3. $-1$
4. none of these

Q.

Which of the following statements is true

1. A continuous function is an increasing function

2. An increasing function is continuous

3. A differentiable function is continuous

4. A continuous function is differentiable

Q. If f : R → R, g : R → are given by f(x) = (x + 1)² and g(x) = x² + 1, then write the value of fog (−3).

Q. If $y=\sqrt{\frac{1-\cos 2x}{1+\cos 2x}},x\in \left(0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right)$, then $\frac{dy}{dx}$ is equal to

1. $se{c}^{2}x,ifx\in \left(0,\frac{\pi }{2}\right)$
2. $-se{c}^{2}x,ifx\in \left(\frac{\pi }{2},\pi \right)$
3. $se{c}^{2}x,ifx\in \left(\frac{\pi }{2},\pi \right)$
4. $-se{c}^{2}x,ifx\in \left(0,\frac{\pi }{2}\right)$

Q. What is the solution of DE $\left(\frac{y}{x}+\sin y\right)\mathrm{dx}+\left(\log x+x\cos y\right)\mathrm{dy}=0$

1. y sin x + x log y =k
2. y log x + x sin y = k
3. log x + sin y = k
4. log y + sin x = k