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Describe the graphical representation of investment multiplier.

OR

If a change in investment of Rs. 100 crores is required to bring a change in income by Rs. 1000 crores, calculate MPC and MPS.

If $y=(1+x)(1+x^2)(1+x^4),then\frac{dy}{dx}atx=1$

The solution of D.E $y^2\frac{dy}{dx}+y^2+1=0$ is

The unit vector in the direction of the resultant of vectors $\overrightarrow{a}=2\hat{i}+2\hat{j}-5\hat{k}and\overrightarrow{b}=2\hat{i}+\hat{j}+3\hat{k}$ is

The number of solutions of $\sqrt{3x^2+x+5}=x-3$ is

Combine these two statements using, 'if and only if'

p:If a rectangle is a square, then all its four sides are equal.

q:If all the four sides of a rectangle are equal, then the rectangle is a square.

If (a - b), (b - c), (c - a) are in G.P. then prove that $(a+b+c{)}^2=3(ab+bc+ca)$

The maximum value of the function $f\left(x\right)=3x^3-18x^2+27x-40$ on the set $S=\{x\in R:x^2+30\le 11x\}$ is :

Let $f_k\left(x\right)=\frac{1}{k}({\sin }^{k}x+{\cos }^{k}x)$ for $k=1,2,3,\dots \dots$ Then for all $x\in R$, the value of $f_4\left(x\right)-f_6\left(x\right)$ is equal to:

For any two sets A and B, prove that

(i) $B\subset A\cup B$

(ii) $A\cap B\subset A$

(iii) $A\subset B\Rightarrow A\cap B=A$

If rth term in the expansion of ${(2x^2-\frac{1}{x})}^{12}$ is without x, then r is equal to

The geometric mean between - 9 and - 16 is

If a $3$-digit number is randomly chosen, what is the probability that either the number itself or some permutation of the number (which is a $3$-digit number) is divisible by $4$ and $5$?

The expression $\sin A(1+\tan A)+\cos A(1+\cot A)$ is equivalent to

The angle between the pair of tangents of the parabola $y^2+12x=0$ which are normal to $x^2+y^2-6x-7y-4=0$ is

A man wants to cross a river (from point A to B) 500 m wide. Rowing speed of the man relative to water is 3 km/h and river flows at the speed of 2 km/hr. If man’s walking speed on the shore is 5 km/hr, then at what angle with line AB he should start rowing in order to reach the directly opposite point on the other bank in shortest time.

Let r be a root of the equation $x^2$+ 2x + 6 = 0. The value of (r + 2) (r + 3) (r + 4) (r + 5) is equal to.

Let $g$ be the inverse function of $f$ and $f^{\prime }\left(x\right)=\frac{x^{10}}{1+x^2}.$ If $g\left(2\right)=a,$ then $g^{\prime }\left(2\right)$ is equal to

The vertices of a $\Delta ABC$ are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that
$\frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4}$. Calculate the ratio of the area of the $\Delta ADE$ and $\Delta ABC$.

If $a,b,c$ are three consecutive positive integers and $\log (1+ac)=2k,$ then the value of $k$ is

If $f\text{'}\text{'}\left(x\right)>0,\forall x\in R,f\text{'}\left(3\right)=0$ and $g\left(x\right)=f({\tan }^{2}x-2\tan x+4),0<x<\frac{\pi }{2}$, then $g\left(x\right)$ is increasing in

If $y\left(x\right)$ satisfies equations $(1+x^2)\frac{dy}{dx}+2xy-4x^2=0$ and $y\left(0\right)=0$, then $y\left(1\right)$ is

Let $z_1$ and $z_2$ be two imaginary roots of $z^2+pz+q=0$, where $p$ and $q$ are real. The points $z_1,z_2$ and origin form an equilateral triangle if

Find the equation of a straight line through the point of intersection of the lines 4x-3y=0 and 2x-5y+3=0 and parallel to 4x+5y+6=0.

If $1,\omega ,{\omega }^2,\cdots \cdots {\omega }^{n-1}$ are the $n^{\text{th}}$ roots of unity and $z_1$ and $z_2$ are any two complex numbers, then $\sum _{k=0}^{n-1}|z_1+{\omega }^k{z}_2{|}^2$ is equal to

The number of ways of selecting two squares on a chess board such that they have a side in common is

If $x=r\sin A\sin B,y=r\cos A\sin B,z=r\cos B,$ then $x^2+y^2+z^2$ is equal to

Find the value of the following:

$ta{n}^{-1}\left[2cos\left(2si{n}^{-1}\frac{1}{2}\right)\right]$

Integrate the following functions.
$\int \frac{3x}{1+2x^4}dx.$

If $ta{n}^{-1}x+ta{n}^{-1}y=\frac{4\pi }{5},thenco{t}^{-1}x+co{t}^{-1}y$ equals to

(a) $\frac{\pi }{5}$ (b) $\frac{2\pi }{5}$ (c) $\frac{3\pi }{5}$ (d) $\pi$

The point on the curve $x^2=2y$ which is nearest to the point (0, 5) is

(A) $(2\sqrt{2},4$) (B) $(2\sqrt{2},0$) (C) (0, 0) (D) (2, 2)

Form the differential equation representing the family of curves given by $(x-a{)}^2+2y^2=a^2,$ where a is an arbitrary constant.

Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment?

Which of the following function(s) not defined at $x=0$ has/have irremovable discontinuity at $x=0$?

Let $I=\underset{\pi /4}{\overset{\pi /3}{\int }}\frac{\sin x}{x}dx$. Then

The expression $-5x^2+4x+3$, has

Mean of a given Data set is

If the sum of reciprocal of intercepts made by a line on the coordinate axes is $k$, then the line passes always through the point

The general solutions for the equation $\cos 4x=\frac{\sqrt{5}+1}{4}$ is

Please explain the graph of trignometric function given as.

f(x)= sin 3x

The equation of the auxiliary circle of the hyperbola $\frac{x^2}{64}-\frac{y^2}{36}=1$ is

complete the series

7,11,17,25,35,47

harikiran purchased a house in Rs. 15000 and paid Rs. 5000 at once. Rest money he promised to pay in annual instalment of Rs. 1000 with 10% per annum interest. How much money is to be paid by him

If $\sqrt{(1-x^6)}+\sqrt{(1-y^6)}=a(x^3-y^3)$ and $\frac{dy}{dx}=f(x,y)\sqrt{\left(\frac{1-y^6}{1-x^6}\right)}$, then

The sum of 50 terms of $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+....$ is