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If $x$ = $\sqrt{a}tan\alpha$ and $y$ = $\sqrt{a}sec\alpha$. The value of $y^2$ – $x^2$ is ___.

Is 302 is the term of 3,8,13....

Find the equation of the perpendicular dropped from the point (-1, 2) onto the line joining the points (1, 4) and (2, 3).

Which of the following fraction is terminating decimal?

The shaded region consists of a chord and the arcs of the circle. Which of the following options is correct?

Find the value of $\frac{P}{Q}$ if $(a,a),(-a,-a)$ and $(Pa,Qa)$ are the vertices of an equilateral triangle.

If the sum of first 4 terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms.

OR

Find the sum of the first 30 positive integers divisible by 6.

The square root of a negative number is not a real number. Why?

Solve the following pair of linear equations

(a-b)x + (a+b)y = a² - 2ab + b²

(a+b)(x+y) = a² + b²

In the centre of a rectangular lawn of dimensions $50m\times 40m$, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 $m^2$ [see figure]. Find the length and breadth of the pond.

Find the largest number which leaves the remainder 4 and 6 respectively when it divides 220 and 186

Find the value of 'a' for which the quadratic equation $x^2-2ax+2a-1=0$ has equal roots.

If the second and fifth term of a G.P. is $3$ and $81$ respectively, then the first term of this G.P. is

Find the ratio in which the point P (x, 2) divides the line segment joining the points A (12, 5) and B (4, -3). Also find the value of X

In ∆PQR, right angled at Q and ∠P = ∠R, then sin P cos R + cos P sin R is

When x² - 2x + k divides the polynomial x⁴ - 6x³ + 16x² - 25x + 10, the remainder is (x + a) . The value of a is _______

If Karan stands at a point which is 2m from A and 4m from B, the angles of elevation of the top of the poles at points A and B are complimentary. What is the shortest distance between the topmost points of the poles, if the height of the pole at B is twice the height of the pole at A?

An equilateral triangle is drawn with its vertices if we put a dot in it without looking into the picture, find the probability of the dot being outside the triangle?

Question 2 (iv)
Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
(iv) $2x^2+x+4=0$

The line $ax+by=ab$, will meet the x- axis at _____.

In the figure, if $PQ\parallel RS,\angle MXQ={135}^{\circ }$ and $\angle MYR={40}^{\circ }$, find $\angle XMY$. [4 MARKS]

Indians, Egyptians, Europeans and Australians reside in a village of area 200$k{m}^2$. If the area is equally divided among the four civilizations, what is the probability that a particular person lands on the land occupied by Indians or Europeans?

If polynomial= $2x^3+3x^2-2x$ , Divisor= $x-1$ and quotient = $2x^2+5x+3$, what is the remainder ?

What values of $x$ satisfy the given equation
$x^2-3x-10=0$ ?

Ram has a semicircular disc. He rotates it about its diameter by 360 degrees. When he rotates the disc, a volume of air in his room gets swept.The name of the object/shape that exactly occupies this volume is

A number consisting of two digits is seven times the sum of its digits.When 27 is subtracted from the number,the digits are reversed.Find the number.

Which of the following is reduced form of the equation $(2x-1{)}^2$ = $x^2$ + 2?

A horse is tied to a pole with 28 m long string. Find the area where the horse can graze. (Take $\pi =\frac{22}{7}$)

In Fig. ABCD is a trapezium with $AB\parallel DC$, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. Circles of equal radii 7 cm with centres A, B, C and D have been drawn. Then, find the area of the shaded region of the figure. (Use $\pi =\frac{22}{7}$)

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are ${30}^{\circ }$ and ${45}^{\circ }$, respectively. If the bridge is at a height of 6 m from the banks, find the width of the river.

A circular field has a perimeter of 650 m. A square plot having its vertices on the circumference of the field is marked in the field. Calculate the area of the square plot.

An observer 1.5 m tall is 28.5 m away from a tower and the angle of elevation of the top of the tower from the eye of the observer is ${45}^{\circ }$. The height of the tower is

(a) 27 m

(b) 30 m

(c) 28.5 m

(d) none of these

Prove the following trigonometric identities:

$\frac{(1+ta{n}^2\theta )cot\theta }{cose{c}^2\theta }=tan\theta$

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, -2 and -24 respectively.

Solve for x and y:

$\frac{1}{2x}+\frac{1}{3y}=2,\frac{1}{3x}+\frac{1}{2y}=\frac{13}{6}(x\ne 0,y\ne 0).$

In Fig. 14.58,from a cuboidal solid metallic block,of dimensions $15cm\times 10cm\times 5cm$,a cylindrical hole of diameter 7 cm is drilled out.Find the surface area of the remaining block.

Surface areas and volumes

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$x^2+5x-(a^2+a-6)=0$

The horizontal distance between two poles is 15 m. The angle of depression of the top of the first pole as seen from the top of the second pole is ${30}^{\circ }$. If the height of the second pole is 24 m, find the height of the first pole. $(\sqrt{3}=1.732)$