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The relation R defined on the set of natural numbers as {(a, b) : a is greater than b by 3}, is given by

Question 1 (iv)
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

$4u^2+8u$

Find the equation of a straight line which cuts an intercept of length $4$ units in the negative direction of y-axis and is parallel to the line joining the points $(3,-2)$ and $(1,4)$

The points (-1,3), (0,3), (3,3) and (6,3) lie on which line? [1 MARK]

The square of( $x+\frac{5}{2}$) is $\frac{25}{4}$.What is $x$?

Dylan drew 1 heart, 1 star, and 26 circles. What is the ratio of circles to hearts? What is the ratio of hearts to star? [2 MARKS]

If the sum of first 7 terms of an AP is 49 and that of last 17 terms is 289, find the sum of its first n terms.

A tent is in the form of a cylinder surmounted by a cone. The slant height of the cone is 10 m, and height of the cylinder is 8 m. If the diameter of the cylinder is 12 m, then find the total surface area of the tent.

In right-angled triangle ABC, $\angle A={45}^{\circ }$, The ratio of AB : AC : BC will be equal to

The equation $x^2+bx+c=0$ would have real roots if ____.

The sum of the first 1000 positive integers is ___.

Find the value of $\frac{1-ta{n}^2{30}^{\circ }}{1+ta{n}^2{30}^{\circ }}.$

If one root of the equation 2$x^2$ + ax + 6 = 0 is 3, then find the value of a

A parachute is tied to a stone (A). Initially a man was there on parachute when it makes ${60}^{\circ }$ with the horizontal plane. After sometime the man ties the rope to himself and then jumps off from the parachute. At some point C man makes ${30}^{\circ }$ with the horizontal plane. From B, an imaginary line is drawn perpendicular to the ground so that it cuts AC (in the fig) in D. Distance BD is ___ m (Round off to nearest integer)

Find Equation of a straight line which passes through the points given by position vectors $\overline{a}and\overline{b}$ .

A fourth degree polynomial is called

If $\sin \theta =\frac{24}{25}$ and $\theta$ lies in the second quadrant, then $\sec \theta +\tan \theta =$

Find the sum and the product of the roots of the following equations 8$x^2$ – 3x + 6 = 0

The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB.

In $\Delta ABC,AB=BC=6cm$. A circle is drawn with center at B and radius 2 cm. What is the scale factor of the triangle formed by the point B and the points where the circle intersects $△ABC$?

Which one could be a solution for the equation 3x + 4y = 2?

The perpendicupar from A on side BC of a triangle ABC intersects bc at d such that db=3cd prove that 2qb square= 2ac square + bc square.

In the figure given below, the radius of the circle is 42 cm. The angle in the sector is ${60}^{\circ }$​. What is the area of the major segment?

The quadrilateral formed by joining the mid-points of consecutive sides of a rectangle ABCD, taken in order, is a rhombus.

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

What is the area of the minor segment of a circle of radius 14 cm , when the angle of the corresponding sector is ${60}^{\circ }$

In $\Delta ABC$, the bisector of $\angle B$ meets AC at D. A line PQ || AC meets AB, BC and BD at P, Q and R respectively.

Show that PR $\times$ BQ = QR $\times$ BP.

A river of 5 $\text{m}$ width and 2 $\text{m}$ depth flows with a speed of 20 $\text{km/hr}$. Water from this river is used to irrigate a field of area $x{\text{m}}^2$. The field is filled with water to a height of 0.5 $\text{m}$. If it takes 30 minutes to irrigate the field, then find $x$.

ABCD is a square. F is the mid-point of AB. BE is one third of BC. li the area of $\Delta FBE=108c{m}^2$, find the length of AC.

In the figure given below AB = 7 cm BC = 8 cm CA = 5 cm. What are the radii of the circles A, B and C?

The length of the longest pole that can be kept in a room $(12m\times 9m\times 8m)$ is

(a) 29 m (b) 21 m (c) 19 m (d) 17 m

The area of an equilateral ΔABC is 17320.5 $c{m}^2$. A circle is drawn taking the vertex of the triangle as centre. The radius of the circle is half the length of the side of triangle. Find the area of the shaded region (in $c{m}^2$) . (π = 3.14 , $\sqrt{3}$ =1.73205)

If k, (2k- 1) and (2k + 1) are the three successive terms of an AP, find the value of k.

The area of a rhombus is 480 $c{m}^2$ and the length of one of its diagonals is 20 cm. The length of each side of the rhombus is

(a) 24 cm (b) 30 cm (c) 26 cm (d) 28 cm

In a hospital, the ages of diabetic patients were recorded as follows. Find the median age.
$\begin{array}{cccccc}\text{Age (in years)}& 0-15& 15-30& 30-45& 45-60& 60-75\\ \text{Numbers of patients}& 5& 20& 40& 50& 25\end{array}$

If 3 and -3 are two zeros of the polynomial $(x^4+x^3-11x^2-9x+18)$, find all the zeros of the given polynomial.

Explain why $3\times 5\times 7+7$ is a composite number.

There are 2 cubes A and B each of volume $Xc{m}^3$. Cube B is cut into smaller cubes each of volume $Yc{m}^3$. The surface area and volumes of the resulting cubes are compared

Statement (S1): The ratio of volumes of A and B(after cutting) = 1:1

Statement (S2): The ratio of surface areas of A and B(after cutting) = 1:1

Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x)=3x+1,x=$\frac{-1}{3}$

(ii) p(x)=5x−$\pi$,x=$\frac{4}{5}$

(iii) p(x)=$x^2$-1,x=1,−1

(iv) p(x)=(x+1)(x−2),x=−1,2

(v) p(x)=$x^2$,x=0

PQ is a post of given height a, and AB is a tower at some distance. If $\alpha$ and $\beta$ are the angles of elevation of B, the top of the tower, at P and Q respectively. Find the height of the tower and its distance from the post.

The length of an arc of a sector of angle ${\theta }^{\circ }$ of a circle with radius R is

(a) $\frac{2\pi R\theta }{180}$

(b) $\frac{2\pi R\theta }{360}$

(c) $\frac{\pi R^2\theta }{180}$

(d) $\frac{\pi R^2\theta }{360}$