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If the sum of first n even natural numbers is equal to k times the sum of the first n odd natural numbers, then k =

Find k, if the equation kx²-2√5x+4=0 has equal roots.

Find the value for k which each of the following systems of equations has a unique solution:

$2x+3y-5=0,kx-6y-8=0.$

A container, open at the top, is in the form of a frustum of a cone of height 24 cm with radii of its lower and upper circular ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container at the rate of Rs 21 per litre.

Points $P(1,3)$ and $Q(-5,-5)$ are two points in the coordinate plane. The X-coordinate of the point equidistant from the points $P$ and $Q$ and lying on the line $y=-1$ is

(i) $(se{c}^2\theta -1)co{t}^2\theta =1$ (ii) $(se{c}^2\theta -1)(cose{c}^2\theta -1)=1$

(iii) $(1-co{s}^2\theta )se{c}^2\theta =ta{n}^2\theta$

Which of following is not a solution of 3a + b = 12?

The areas of two similar triangles ABC and MR are in the ratio 9 : 16. If BC 4.5 cm find the length of QR.

Show that x = -1 is a zero of quadratic equation $x^2$– 2x – 3 = 0

Solve the following quadratic equation by factorization.
$\frac{1}{x}-\frac{1}{x-2}=3,x\ne 0,2$

A pearachutist is descending vertically and makes angles of elevation of ${45}^{\circ }and{60}^{\circ }$ at two observing points 100 m apart from each other on the left side of himself. Find the maximum height from which he falls and the distance of the point where he falls on the ground from the just observation point.

Prove the following trigonometric identities:

$(cosec\theta +sin\theta )(cosec\theta -sin\theta )=co{t}^2\theta +co{s}^2\theta$

In making 1000 revolutions, a wheel covers 88 km. The diameter of the wheel is

(a) 140 (b) 150 (c) 160 (d) 166

If PA and PB are two tangents to a circle with centre O such that $\angle$ AOB = ${110}^o$ then $\angle$ APB is equal to

(a) ${55}^o$ (b) ${60}^o$ (c) ${70}^o$ (d) ${90}^o$

Prove the following trigonometric identities:

$\frac{cosecA}{cosecA-1}+\frac{cosecA}{cosecA+1}=2se{c}^{2}A$

Using the formula, $tan2A=\frac{2tanA}{1-ta{n}^2{A}^{\prime }}$ find the value of $tan{60}^{\circ },$ it being given that $tan{30}^{\circ }=\frac{1}{\sqrt{3}}.$

sec ${70}^o$ sin ${20}^o$ + cos ${20}^o$ cosec ${70}^o$ = ?

(a) 0 (b) 1 (c) -1 (d) 2

For what value of k(k>0) is the area of the triangle with vertices (-2,5), (k, -4) and (2k+1,10) equal to 53 square units ?

A survey regarding the heights (in cm) of 50 girls of a class was conducted and the following data was obtained:
$\begin{array}{ccccccc}\text{Height (in cm)}& 120-130& 130-140& 140-150& 150-160& 160-170& \text{Total}\\ \text{Number of girls}& 2& 8& 12& 20& 8& 50\end{array}$
Find the mean, median and mode of the above data.

Solve each of the following systems of equations graphically:

$2x+3y-4=0,3x-y+5=0.$

which of the following about remainder theorem and factor theorem is true?

A number when divided by 61 gives 27 a quotient and 32 as remainder. Find the number.

Solve each of the following quadratic equations:
$3^{(x+2)}+3^{-x}=10$

The speed of a boat in still water is 8km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.

The area of the rectangle is 192 $c{m}^2$ and its perimeter is 56 cm.Find the dimensions of the rectangle.

Let there be an A.P. with first term 'a', common difference 'd'. If $a_n$ denotes its nth term and $S_n$ the sum of first n terms, find.
(i) n and $S_a$, if a=5, d=3 and $a_n$=50.
(ii) n and a, if $a_n$=4, d=2 and $S_n$=-14.
(iii) d, if a=3, n=8 and $S_n$=192.
(iv)a, if $a_n$=28, $S_n$=144 and n=9.
(v) n and d, if a=8, $a_n$=62 and $S_n$=210.
(vi) n and $a_n$, if a = 2, d=8 and $S_n$=90.
(vii) k, if $S_n=3n^2$+5n and $a_k$=164.
(viii) $S_{22}$, if d =22 and $a_{22}$ =149

Find the third vertex of a $△ABC$ if two of its vertices are B(-3, 1) and C(0, -2), and its centroid is at the origin.

The base of a right-angled triangle measures 48 cm and its hypotenuse measures 50 cm. Find the area of the triangle.

A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by $\frac{1}{15}$. Find the fraction.

Assertion : In rolling a die, event A = {1, 3, 5} and event B = {2, 4} are mutually exclusive events.
Reason : In a sample space two events are mutually exclusive if they do not occur at the same time.

Solve, using cross-multiplication

$0.4x-1.5y=6.50.3x+0.2y=0.9$

Question 14

$\frac{3}{7}$ of $\frac{2}{5}$ is equal to

(a) $\frac{5}{12}$
(b) $\frac{5}{35}$
(c) $\frac{1}{35}$
(d) $\frac{6}{35}$

The equation of the line through origin with slope = 1

In the given figure, PQ is a transverse common tangent to two circles with centers A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.

The sum of the first six terms of an AP is 42.the ratio of its 10th term to its 13th term is 1:3 calculate the first and the thirteenth term of an AP

n2 - n is divisible by 2 for every postive integer n.

Find the equation of the line perpendicular to the line x- 7y + 5 = 0

and having x intercept 3.

A man deposited Rs. 10000 in a bank at the rate of $5\%$ simple interest anually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.

Triangle ABC is similar to triangle PQR. If AD and PM are altitudes of the two triangles, then $\frac{AB}{PQ}=\frac{AD}{PM}$

8th term of an AP is40. Calculate the sum of first 15terms

In the following passage, there are some numbered blanks. Fill in the blanks by selecting the most appropriate word for each blank from the given options.

At markets or at county fairs in the old days, the customer had to be on guard against a dishonest trader. A house wife, for example, wanting (1) ___ buy a live piglet might be (2) ___ a discount if she bought (3) ___ packed one, tied up in a small sack (4) ___ a poke. Anyone who agreed to (5) ___ a pig in a poke was naturally (6) ___ a risk. The pig might be ill (7) ___ even dead: Or it might turn (8) ___ to be not a piglet at all.Which of the following words will be most appropriate for blank (1)?

Write whether the square of any positive integer can be of the form 3m+2 , where m is a natural number . Justify your answer ?

The HCF of 65 and 117 is expressible in the form 65m-117. Find the value of m. Also find the LCM of 65 and 117 using prime factorization method.

A $12\%$ stock yielding $10\%$ is quoted at: