Find the value of k if the points \[A\,\,\left( k+3,\text{k}-1 \right),\] \[B\,\,\left( 2k-3,k-3 \right)\] and \[C\,\,\left( 5k+1,2k \right)\] are collinear.
Let \[D\,\,\left( 3,-2 \right),\]\[E\,\,\left( -3,\,\,1 \right)\] and \[F\,\,\left( 4,-\,3 \right)\] be the midpoints of the sides BC, CA and AB respectively of\[\Delta \,ABC\]. Find the coordinates of the vertex A of the \[\Delta \]ABC.
The line joining the points (2, 1) and \[\left( 5,-\,8 \right)\] is trisected at the points P and Q. If point P lies on the line \[2x-y+k=0,\] find the value of k.
A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.
The proportion of the sheet area that remains after punching is:
A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.
Find the area of the part of the circle (round punch) falling outside the square sheet:
A tomb is constructed whose lower part is cylindrical and upper part is hemispherical. The radius of both the parts is 5.6 meters and the height of the cylindrical part is 6.3 meters. Find the inner curved surface area of the tomb.
If the coefficient of z in the quadratic equation\[{{z}^{2}}+az+b=0\] is taken as 18 in place of 12, then its roots were found to be \[-\,16\] and\[-\,2\]. The roots of the original equation are:
Two circles intersect at M and N. IMNK and MNLJ are the quadrilaterals inscribed in the two circles as shown in the figure given below. If\[\angle I={{95}^{o}}\] and \[\angle k={{65}^{o}},\] then the values of x and y are respectively:
If \[x=\operatorname{cosec}\theta -\sin \theta \] and \[y=\sec \theta -\cos \theta ,\]and, then the value of \[{{x}^{2}}{{y}^{2}}({{x}^{2}}\,+\,{{y}^{2}}\,+3)\] is
At the foot of a mountain the elevation of its summit is \[45{}^\circ \]. After ascending 600 meter towards the mountain, upon an inclination of\[30{}^\circ ,\] the elevation changes to \[60{}^\circ \]. Find the height of the mountain.
If the polynomial\[g\,(m)=6{{m}^{4}}+8{{m}^{3}}+17{{m}^{2}}+21m+7\] is divisible by another polynomial\[h\,(m)=3{{m}^{2}}+4m+1\]gives the remainder \[qm+a,\]then find the value of q and a.
Smith lives at place A and works at place B. He travels to his office by train. There are two trains from his home to his office. The first train travels at the speed of 5 km/h more than its usual speed, while the second train travels at the speed of 4 km/h slower than its usual speed. The first train takes 3 hours less for the same journey and the second train takes 3 hours more for the same journey. Find the distance between the places A and B, if the usual speed of both the train is same.
Robert speaks truth in 60% of cases and Mary speaks truth in 90% of cases. If both of them are asked to give a statement, then the probability that they contradict each other?s statements is:
There are two friends Jack and Anderson. Jack said to Anderson if you give me hundred of your shares, then I will be twice as rich as you are. At this Anderson replied that if you give me ten of your, then I will be six times as rich as you are. The share of Jack and Anderson are respectively.
Find the value of m if the sum of the products of the three roots of the equation\[(m+1)\,\,{{y}^{4}}+25{{y}^{3}}+45{{y}^{2}}-(2m\,-3)\,\,y+9=0\] is\[\frac{7}{4}\].
Thomas have some cards of different colours. There are some particular number of cards of each colour. The number of differently colour cards are in A.P. If he has cards of seven different colours in the order of VIBGYOR such that the number of cards of third colour is four times the number of cards of first colour. If the number of sixth colour cards is 17, then the number of fifth colour cards is:
The initial salaries of Sonu and Monu are Rs. 16500 and Rs. 18500 respectively. If their annual increment will be Rs. 2000 and Rs. 1800 each respectively, then in which year will Sonu start earning more salary than Monu?
A crow is sitting on the top of a house. The house is 40 m high. The angle of elevation of the crow as seen from a point on the ground is \[45{}^\circ \]. The crow flies away horizontally and remains at a constant height. After 3 seconds the angle of elevation of the crow from the point of observation becomes\[30{}^\circ \]. Find the speed of the crow.
Four numbers are in arithmetic progression. The sum of first and last term is 8 and the product of both middle terms is 15. The least number of the series is
If the polynomial \[{{x}^{4}}-6{{x}^{3}}+16{{x}^{2}}-25x+10\] is divided by another polynomial \[{{x}^{2}}-2x+k\], the remainder comes out to be \[x+a\]. Find the values of k and a.
In a row of boys facing the North, A is sixteenth from the left end and C is sixteenth from the right end. B, who is fourth to the right of A, is fifth to the left of C in the row. How many boys are there in the row?
Direction: In given question, three statements are given followed by four conclusions I, II, III and IV. You have to consider the statements to be true even if they seem to be at variance from commonly known facts. You are to decide which of the given conclusions, if any, follow from the given statements.
Directions: In each of the following questions a number series is given. After the series below it in the next line a number is given and is followed by (a), (b), (c), (d) and (e). You have to complete the series starting with the given number following the pattern of the given series. Then answer the questions given below it.
\[\begin{matrix} 140, & 68, & 36, & 16, & 10, & 3 \\ 284, & (A) & (B) & (C) & (D) & (E) \\ \end{matrix}\] Which number will come in place of (B)?
Directions: In each of the following questions a number series is given. After the series below it in the next line a number is given and is followed by (a), (b), (c), (d) and (e). You have to complete the series starting with the given number following the pattern of the given series. Then answer the questions given below it.
\[\begin{matrix} 25, & 194, & 73, & 154, & 105, & {} \\ 14, & (A) & (B) & (C) & (D) & {} \\ \end{matrix}\] Which number will come in place of (D)?
Directions: In each of the following questions a number series is given. After the series below it in the next line a number is given and is followed by (a), (b), (c), (d) and (e). You have to complete the series starting with the given number following the pattern of the given series. Then answer the questions given below it.
\[\begin{matrix} 6, & 9, & 18, & 45, & 135 \\ 20, & (\text{A}) & (B) & (C) & (D) \\ \end{matrix}\] Which number will come in place of (C)?
From a square sheet of paper of side 21 cm, four regions each in the form of a quadrant of circles each of radius 10.5 cm are removed as shown in the adjoining figure. Find the area of the shaded region.
Out of a number of Saras birds, \[\frac{1}{4}\]th the number are moving about in lotus plants, \[\frac{1}{9}\]th coupled (along) with \[\frac{1}{4}\]th as well as 7 times the square root of the number move on a hill and 56 birds remain in Vakula trees. What is the total number of birds?
The mean of the following frequency distribution is 53. But the frequencies \[{{f}_{1}}\]and \[{{f}_{2}}\]in the classes 20 - 40 and 60 - 80 are missing. Find the missing frequencies.
The figure shows the region enclosed by a cardboard of length 30 m and breadth 18 m. The enclosed region so formed is a square where the grass are grown for the cattle to feed them. Find the length of the sides of the square region.
A regular hexagonal shape is made on the design with the help of coins of equal size as shown in figure. If the first figure has two coins along each side of the design and second figure has three coins along each side of the figure, then find the number of coins required to make a figure of regular hexagon using 33 coins along each side of the hexagon.
If three times the larger of two numbers is divided by the smaller, we get the quotient 6 and remainder 6. If five times the smaller is divided by the larger we get the quotient 2 and remainder 3. Find the numbers.