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If $f\left(x\right)=\left|\begin{array}{ccc}1& 3\cos x& 1\\ \sin x& 1& 3\cos x\\ 1& \sin x& 1\end{array}\right|,$ then the maximum value of $f\left(x\right)$ is

Metal spheres, each of radius 2 cm, are packed into a rectangular box of internal dimension $16cm\times 8cm\times 8cm$ when 16 spheres are packed the box is filled with preservative liquid. Find the volume of this liquid .

Find the number of terms in each of the following A.P.

I. 7, 13, 19, …, 205

II.

Find the distance of each of the following points from the origin:

(i) A(5, -12) (ii) B (-5, 5) (iii) C (-4,-6).

$(\frac{1+tan\theta }{1+cot\theta }{)}^2=$

Which are the two most popular types of accommodations?

Two pillars of equal heights stand on either side of a roadway, which is 150 m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are ${60}^{\circ }and{30}^{\circ }$, find the height of the pillars and the position of the point.

Samuel has $x$ chocolates with him from which Marie took $3$. The correct algebraic expression for the chocolates left with Samuel is

Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of one pole is ${60}^{\circ }$ and the angle of depression from the top of another pole at P is ${30}^{\circ }$. Find the height of each pole and distances of the point P from the poles.

What is the probability of getting a four when a die is rollled once?

If $2\left[\begin{array}{cc}x& 5\\ 7& y-3\end{array}\right]+\left[\begin{array}{cc}3& -4\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}7& 6\\ 15& 14\end{array}\right],$ then $(x,y)$ is

If p,q & r are the zeroes of a cubic polynomial $a{x}^3+b{x}^2+cx+d$, then what will be p+q+r?

The sum of four consecutive number in an AP is 32 and the ratio of the product of the 1st and the term of the product of two middle term is 7:15 find the number.

Find the sum of first $24$ terms of the A.P. whose $n^{th}$ term is given by $a_n=3+2n$

Find the missing terms in the AP______, 3,_______, -5.

From a solid cylinder of height 2.8 cm and diameter 4.2 cm. A conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. [Take $\pi =\frac{22}{7}$]

Express the following in the form, where p and q are integers and q ≠ 0.

(i) (ii) (iii)

From a point P on the ground, the angleof elevation of a tower 200m high is 45°. The point P is ___m from the foot of the tower.

If $\Delta$ABC and $\Delta$PQR are two similar triangles shown in the figure. AM and PN are the medians on $\Delta$ABC and $\Delta$PQR respectively. The ratio of areas of $\Delta$ABC and $\Delta$PQR is $9:25.$ If $AM=PO=5cm,$ the value of $3\left(ON\right)$ is ___ $cm$

In the given figure, ABCD is a parallelogram with x : y = 1 : 1 and OE || AD. Then, (area of $\Delta ADC$ + area of $\Delta BOC$) : area of $\Delta ODC$ is

A man sitting at a height of 20m on a tall tree on a small island in the middle of a river observes two poles directly opposites to each other on the two banks of the river and in line with the foot of tree. If the angles of depression of the feet of the poles from a point at which the man is sitting on the tree on either side of the river are ${60}^{\circ }$ and ${30}^{\circ }$ respectively. Find the width of the river.