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Find the values of k for whic the system

2x + ky = 1

3x - 5y = 7

will have (i) a unique solution, and (ii) no solution. Is there a value of k for which the system has infinitely many solutions?

The sum of 4th and 8th terms of an A.P. is 24 and the sum of 6th and 10th terms is 44.Find the A.P.

If you need to construct a triangle with point P as one of its vertices, which is the angle that you need to construct a side of the triangle?

If a circle S(x,y)=0 touches the line x + y = 5 at the point (2, 3) and S(1, 2) = 0, then the radius of such circle is .

Find the value of k such that the line (k - 2) x + (k + 3) y - 5 = 0 is :

(i) perpendicular to the line 2x - y + 7 = 0

(ii) parallel to it.

By investing Rs 45000 in 10% Rs 100 shares, and get Rs 3000 as dividend. Find the market value of the share.

Draw a circle inscribing a regular hexagon of side 5.8 cm. Find the length of the radius.

A card is drawn at random from a well- shuffled deck of 52 cards. What is the probability of getting a black king?
(a) $\frac{1}{13}$ (b) $\frac{1}{26}$ (c) $\frac{2}{39}$ (d) None of these

For what value of k do the equations kx-2y=3 and 3x+y=5 represent two lines intersecting at a unique point?

(a) k=3 (b) k=-3

(c) k=6 (d) all real values except -6

A piggy bank contains hundred 50-p coins, seventy 1 coin, fifty Rs. 2 coins and thirty 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin

(i) will be a 1 coin

(ii) Will not be a 5 coin

(iii) Will be 50 - p or a 2 coin?

Prove that the diagonals of a rectangle bisect each other and are equal.

If 15 cot A = 8, find the values of sin A and sec A.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of ${30}^{\circ }$ with the ground. The distance between the foot of the tree to the point where the top touches the ground is 8m. Find the height of the tree.

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then, construct another triangle whose sides are $\frac{5}{3}$ times the corresponding sides of the given triangle.

A moving boat is observed from the top of a 1550 m high cliff moving away from the cliff. The angle of depression of the boat changes from ${60}^{\circ }$ to ${45}^{\circ }$ in 2 mintues. Find the speed of the boat in m/h.

Find the condition that the point P(x, y) may lie on the line joining (3, 4) and (-5, -6).

A die is thrown once, the probability of getting an even prime number is___

$Themodelofabuildingisconstructedwithscalefactor1:30.\left(i\right)iftheheightofthemodelis80cm,findtheactualheightofthebuildinginmetres.\left(ii\right)Thetheactualvolumeofatankatthetopofthebuildingis27m^3,findthevolumeofthetankonthetopofthemodel$

$\text{In the given figure,}AX:XB=3:5\text{and}XY||BC.$

$\text{If XY = 18 cm, then BC = \_\_\_ .}$

The ratio of the sum of n terms of two A.P.’s is $(7n+1):(4n+27)$. Find the ratio of their $m^{th}$ terms.

Find x and y if $\left(\begin{array}{c}x-8\\ y+2\end{array}\right)=\left(\begin{array}{c}2x+3\\ 3y-6\end{array}\right)$

One of the roots of the equation $(x+\frac{1}{a}{)}^2-b^2=0$ is .

Question 1(d)

Determine if the following are in proportion:

32, 48, 70, 210

Express the trigonometric ratios sin A , sec A and tan A in terms of cot A

Find the largest possible positive integer that will divide 398, 436 and 542 leaving remainder 7, 11, 15 respectively.

Q83. With reference to Priority Sector Lending Certificates (PSLCs), consider the following statements:

1. They are issued to lenders who have made loans to categories eligible under PSL.

2. They are non-tradable certificates.

Which of the above statement(s) is/are not correct?

AB is the longest chord of a circle with centre O and P is a point on it, not coinciding with A or B. Then $\angle APB=$

If the zeroes of $a{x}^2+bx+c$ are equal then

A rectangular paper is folded into a cylinder. The length and breadth of the paper are L and B respectively. The surface area of the cylinder is

If the first term of an AP is 3 and the common difference is 5, the $n^{th}$ term of the AP is .

Which of the following is true about an equilateral triangle?

What will be the centroid of a triangle whose vertices are (2, 4), (6, 4) and (2, 0)?

Aju was standing at the corner of a square field. He started walking along the North-East direction. When he was $10\sqrt{2}$ m away from where he started, he took a left turn and started walking again. Taking the square field as co-ordinate plane, find the equation of Aju’s current path.

Two water taps together can fill a tank in $9\frac{3}{8}$ hours. The tap of larger diameter takes $10$ hours less than the smaller one to fill the tank separately. find the time in which each tap can separately fill the tank.

The volume of a conical tent is 1232 $m^3$ and the area of the base floor is 154 $m^2$. Calculate the :

(i) radius of the floor,

(ii) height of the tent,

(iii) length of the canvas required to cover this conical tent if its width is 2 m.

Prove that 7 root 5 is irrational

If the first term of a GP is 5 and the sum of the first three terms is 31/ 5, find the common ratio.

In a $△ABC,\angle A={60}^{\circ },\angle B={30}^{\circ }$ and $\angle C={90}^{\circ }.$

Then the ratio $AC:BC:AB$ will be equal to

Given that one of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is zero, the product of the other two zeroes is

If one of the zeroes of the polynomial $g\left(x\right)=(k^2+4)x^2+13x+4k$ is reciprocal of the other, then value of k is,

The points A(2,3), B(-5,0) and C(-2,a) are collinear , find the value of 'a'.

If the equation $9x^2+6kx+4=0$ has equal roots, then find the value of k.

If a,b are the zeros of polynomial f(x)=x^2-p(x+1)-c,then (a+1)(b+1)=

From the top of a cliff 92 m high, the angle of depression of a buoy is 20^o. Calculate to the nearest metre, the distance of the buoy from the foot of the cliff.

Note: tan 20 = 2.237, tan 70 = 2.747