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If $f\left(x\right)=\{\begin{array}{cc}ax,& x<2\\ a{x}^2-bx+3,& x\ge 2\end{array}$ is differentiable for all real values of $x,$ then

The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height is

Read the following information carefully to answer the questions that follow. The questions are based on following coding formats:

+ – North
* – South
– East
# – West
$\$$– Either 4 or m
?– Either 5 or 12 m

Conditions given are as:
I. P+?Q
II. P+#R
III. RQ
IV. R#$\$$S
V. T*?S
VI. U#$\$$T
VII. V+$\$$U
VIII. V*#R

Q. If Z*$\$$S and V$\$$Z, what is the maximum area of the quadrilateral formed by joining points Z, V, T V

If $x.a=0,x\times b=c\times b$ then x =

If a is any vector then $(a\times \hat{i}{)}^2+(a\times \hat{j}{)}^2+(a\times \hat{k}{)}^2$ =

Suppose f(x) and g(x) are two continuous functions defined for $0\le x\le 1$.
Given $f\left(x\right)={\int }_0^1{e}^{x+t}.f\left(t\right)dt$ and $g\left(x\right)={\int }_0^1{e}^{x+t}.g\left(t\right)dt+x$.

The value of f(1) equals

Suppose that for a particular economy, investment is equal to 200, government purchases are 150, net taxes (that is lump-sum taxes minus transfer) is 100 and consumption is given by C = 100 + 0.75y

(i) What is the level of equilibrium income?

(ii) Calculate the value of the government expenditure multiplier and the tax multiplier.

(iii) If the government expenditure increases by 200, find the change in equilibrium income.

Let 2x²+y²-3xy=0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

If $A_1,A_2;G_1,G_2$ and $H_1,H_2$ be AM's, GM's and HM's between two quantities, then the value of $\frac{G_1{G}_2}{H_1{H}_2}$ is

Which one of the following is false? Read $\wedge$ as AND , $\vee$ as OR, $\sim$ as NOT, $\to$ as one way implication and $\leftrightarrow$ as two way implication .

If $\overrightarrow{A}$ = $3\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{B}$ = $2\hat{i}+2\hat{j}+4\hat{k}$ then find the value of $|\overrightarrow{A}\times \overrightarrow{B}|$.

Tangent and normal are drawn at $P(16,16)$ on the parabola $y^2=16x$, which intersect the axis of the parabola at $A$ and $B$, respectively. If $C$ is the centre of the circle through the points $P,A$ and $B$ and $\angle CPB=\theta$, then a value of $\tan \theta$ is

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two digit number (ii) a perfect square number (iii) a number divisible by 5.

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at ₹100 and ₹120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

If $(\frac{1}{1+2i}+\frac{3}{1-i})\left(\frac{3-2i}{1+3i}\right)$ is reducible to $a+ib$, then values of $a$ and $b$ are

Question 1(ii)

Check whether the following are quadratic equations:

$\left(ii\right)x^2-2x=(-2)(3-x)$

An angle between the plane, $x+y+z=5$ and the line of intersection of the planes, $3x+4y+z-1=0$ and $5x+8y+2z+14=0$, is :

Which of the following binary operation is not commutative?

If $si{n}^{-1}x=y$, then which one of the following is correct?

a) $0\le y\le \pi$
b) $-\frac{\pi }{2}\le y\le \frac{\pi }{2}$
c) $0<y<\pi$
d) $-\frac{\pi }{2}<y<\frac{\pi }{2}$

Twelve balls are distributed among three boxes. The probability that the first box will contain three balls, is

Find the equation of the hyperbola satisfying the give conditions: Foci, passing through (2, 3)

The equation of the bisector of the acute angle between $4x+3y-6=0$ and $5x+12y+9=0$ is

Value of $\underset{n\to \infty }{lim}\frac{S_1{S}_n+S_2{S}_{n-1}+...+S_n{S}_1}{S_1^2+S_2^2+...+S_n^2}$ is

Prove that:

$cot22{\frac{1}{2}}^{\circ }=\sqrt{2}+1$

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set