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NTSE Maths Volume II Chapter 17 to 22

Illustrative Examples

19 Qs

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Find the volume of a right circular cylinder of height 7 cm and radius of the base 2 cm.

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How many cubic meters of earth must be dug out to make a well of dimeter 5 m and depth 14 m?

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Find the lateral surface area and the total surface area of a cylinder with base radius 7 m and height 9 m

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Find the total surface area of a hollow cylinder of internal radius 3 cm thickness 1 cm and height 14 cm

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The sum of the radius and the height of a solid cylinder is 37 cm If the total surface area of the solid is 1628 cm find the circumference of the base and the volume of the solid

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A roll of newspaper of length 7000 m and thickness 0.022 cm is rolled into a solid cylinder Find its radius

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The radius and height of a cylinder are in the ratio 5:7 and its volume is 550 cm Find its radius 

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The curved surface area of a cone is 670 cm and its radius is 15cm Find its slant height and total surface area

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A conical cup has a height of 21 cm and the diameter of its base is 18 cm Find the amount of water it can hold

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A conical cup cm high has a circular base of diameter cm The cup is full of water which is now poured into a cylindrical vessel of circular base whose diameter is cm What will be the height of water in the vessel

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What is the total surface area of a cone with slant height m and radius of base m?

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The circumference of the base of a m high conical tent is m. Calculate the length of canvas used in making the tent, if the width of the canvas is m.

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The radius of the base and the height of a right circular cone are cm and cm respectively Find the total surface area of the cone

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The radius of a sphere is 9 cm It is melted and drawn into a wire of diameter 2 mm Find the lenght of the wire in meters

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A metallic sphere of radius cm is melted and then recast into small cones each of radius cm and height cm Find the number of cones thus formed

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Find the volume of the sphere whose diameter is 30 cm

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Find the surface area of the sphere whose radius is .

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The inner radius of a hemispherical steel bowl is cm If the thickness of the bowl is m find the volume of steel used in making the bowl

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The diameter of acopper sphere is cm The sphere is melted and is drawn into a long wire of uniform circular cross-section If the length of the wire is m find its diameter

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Multiple choice questions

30 Qs

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The height of a right circular cylinder is 10 cm and the radius of the base is 7 cm Then the difference between the total base area and the curved surface area is

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The circumference of the base of  the of a cylinder is 12 m and its height is  meters. The volume of the cylinder is

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Two cylinders have their radii in the ratio 4:5 and their heights in the ratio 5:6. Then, the ratio of their volumes is

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A well 2 m in radius and 14 m deep is to be dug. What is the cost of digging the well at Rs. 1.20 per cubic meter?

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Earth dug out on making a circular tank of radius 7 m is spread all round the tank uniformly to a width of 1 m to form an embankment of height 3.5 m. Calculate the depth of the tank.

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The sum of the radius and the height of a solid cylinder is 37 cm  If the total surface area of the solid is  find the circumference of the base

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An iron pipe is cm long and its experior radius is cm If the thickness of the pipe is cm and iron weights  the weight of the pipe is

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The diameters of two cylinders are in the ratio 2:3 Find the ratio of their heights if their volume is the same

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The radius of a roller is 0.7 m and it is 2 m long How much area will it cover in 10 revolutions?

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A cube of copper of edge 11 cm is melted and formed into a cylindrical wire of diameter 0.5 cm What length of wire will be obtained from the cube?

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A right circular cone is 84 cm high The radius of the base is 350 m Find the curved surface area

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Given that the volume of a cone is  and the area of its base is  Its height is

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The total surface area of a cone is . If its slant height is four times the radius of the cone, the diameter of the cone is:

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The radius and the slant height of a cone are in the ratio and the curved surface area is . Find its radius.

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From a solid cylinder whose height is 8 cm and radius is 6 cm a conical cavity of height 8 cm and base radius 6 cm is hollowed out Find the volume of the remaining solid

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A right circular cone of vertical height has volume . Its curved surface area is:

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A right triangle with sides 5 cm 12 cm and 13 cm is revolved about the side 12 cm  Find the volume of the cone thus formed

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The radius of a conical tent is and the slant height is . Find the length of canvas required to make the tent it the width of canvas is .

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Two cones have their heights in the ratio 1:3 and their radii in the ratio 3:1 Find the ratio of their volumes

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The ratio of the volumes of a cylinder and a cone having equal radii and equal heights is

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The surface area of a sphere is . Its volume is

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The volume of a solid hemisphere of radius 2 cm is

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The radii of two spheres are in the ratio 3:5 The ratio of their volumes is

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If the radius of a sphere is doubled what is the ratio of the volume of the first sphere to that of the second?

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If the numerical value of surface area of a sphere is equal to its volume, then the radius of the sphere is:

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If the sum of the radii of two spheres is 2 km and their volumes are in the ratio 64:27 then the ratio of their radii is

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How many spherical bullets can be made out of a solid cube of lead whose edge measures 44 cm if each bullet has radius 2 cm?

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200 wooden balls each of diameter 70 mm are to be painted Find the cost of painting these balls at 10 paise/

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The largest sphere is carved out of a cube of edge 14 cm Find the volume of the sphere

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A solid metal sphere is cut through the center into two equal parts. Find the total surface area of each part if the radius of the sphere is .

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Multiple choice questions

28 Qs

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In any continuous class interval table (a-b)

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Consider the class intervals 1-10, 11-20, 21-30, etc Here what is the class boundary of class interval 11-20?

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The mean of  is M. If every , 2,....,50, is replaced by /50 the mean will be

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Find the mode of 0,0,2,2,3,3,3,4,5,5,5,5,6,6,7,8

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The midvalue of the class interval (a-b) is

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The mode of the series 2,3,1,2,5,3,2,2,3,5 is

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The mean of  is M. When =1,2,....,10, is replaced by , the mean is . Then =

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The median of 9, 5, 7, 11, 13, 3 is

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Which of the following is true?

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In a bar chart:

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In any pie chart the sum of the central angles is

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If the number of boys and the number of girls in a class are 45% and 55% respectively and it is represented in a pie chart what will be the central angles of the portions representing boys and girls respectively?

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The mean of the data is:

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Find the mean of the following frequency distribution.
x10153025422111
f5264382

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The weights in kilogram of members in a school boxing team are . If the average is , then is:

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The mode of some observations is 4 and the median is 3 Then mean is

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The median of 1, 3, 6, 8, 4, 2, 7, 10 is

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The geometric mean of 4,5,6,7,4 is

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A letter is chosen at random from the letters of the English alphabet The probability that it is not a vowel is

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A box has tokens numbered 3 to 100. If a token is taken out at random the chance that the number is divisible by 7 is

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A pair of dice is thrown once The probability that the sum of the outcomes is less than 11 is

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A bag contains six red four green and eight white balls If a ball is picked at random the probability that it is not white is

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There are 50 marbles of 3 colors: blue yellow and black The probability of picking up a blue marble is 3/10 and that of picking up a yellow marble is 1/2 The probability of picking up a black ball is 

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A box has 20 balls of which x are red When 4 more red balls are added and then a ball is picked up the probability is 1/2 The value of x is

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A bag contains some green white and pink beads The probability of taking out one green bead is 1/3 and that of picking up one pink bead is 1/4 If it is known that the box has 10 white beads how many beads were in the box initially?

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Out of 132 screws in a pack 12 screws are known to be defective If one screw is picked up at random the probability that it is a good screw is

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A bag contains five yellow balls and some white balls If the probability of picking a white ball is twice that of picking a yellow ball how many white balls are there?

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A coin is tossed three times If all the outcomes are identical then I win What is the chance that I lose?

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Multiple choice questions

70 Qs

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The reflection of (5,2) on the x-axis has coordinates

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The reflection of (-6,-3) on the y-axis has coordinates

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Two adjacent vertices of a rhombus are (-1,0) and (-3,4) The other two vertices are

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A point P(4,3) is reflected on the x-axis Its image Q is reflected on the y-axis to get R and R is again reflected on the x-axis If S is the image of R then the length of the diagonal of rectangle PQRS is

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The vertices of a triangle AOD are (4,0) (0,0) (0,3) If the triangle is rotated clockwise about its circumcenter till AD returns to its original position then the vertices of the triangle in its new positions are

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The reflection of a point P(-3,4) on the y-axis is Q and the reflection of Q on the x-axis is R Then PR=

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A point P(-3,2) reflected about the line l shown in the diagram goes over to Q If Q is reflected again about the x-axis and it goes over to R then the coordinates of R are

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The points (4,0), (0,4), (-4,0), and (0, -4) form

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The points and are the vertices of:

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The points (-2,10), (-2,2), and (6,2) are the vertices of

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The distance between (a,-b) and (-a, -b) is given by

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If the distance between the origin and (x,3)is 5 then x=

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The triangle formed by the points (3,5), (6,9), and (2,6) is

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The triangle formed by the points (2,4), (2,1) and (6,1) is:

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The points (3,7), (6,5), and (15,-1)

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The radius of the circle with center (0,0) and which passes through (-6,8) is

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The diameter of the circle with center (1,2) and which passes through (-4,6) is

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The points (-2,0), (3,0), (3,5) and (-2,5) are the vertices of a:

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The center of a circle is at the origin and the radius is 10 units The point (6,8) lies

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If the point (x,y) is equidistant from (a,0) and (2a,a) then

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The vertices of a rhombus are (-5,0), (0,8), (5,0), and (0,-8) The product of the lengths of its diagonals is

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An isosceles triangle has vertices at (4,0), (-4,0), and (0,8) The length of the equal sides is

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The vertices P, Q, R, and S of a parallelogram are at (3,-5), (-5,-4), (7,10) and (15,9) respectively The length of the diagonal PR is

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If the point (P,0) is equidistant from (3,2) and (5,3) then P=

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The center A of a circle is (3,8) and its radius is 5 The length of the tangent from B(-6,8) is 

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The circumcenter of the triangle with vertices (9,3),(7,-1) and (-1,3) is (4,3) Circumradius is

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If A(x,0), B(-4,6), and C(14, -2) form an isosceles triangle with AB=AC, then x=

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If P(1,4), Q(9,-2), and R(5,1) are collinear then

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If the points (1,1) (2,3) and (5,-1) form a right triangle, then the hypotenuse is of length

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The length of the diagonal of rectangle ABCD formed by A(2,-2), B(8,4), C(5,7) and D(-1,1) is

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The midpoint of the line segment with endpoints and is

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The line segment with endpoints (-9,7) and (6,4) is trisected at P and Q The coordinates of Q are

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If the midpoint of PQ is (7,10) and Q is (5,8) then the coordinates of P are

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A(4,0) and B(5,0) are two points If AB is produced to C such that AB=BC then the coordinates of C are

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The ratio in which the line joining of (-2,5) and (1,-9) is divided by the x-axis is

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The ratio in which the joining of (-3,2) and (5,6) is divided by the y-axis is

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The centroid of the triangle with vertices (2,6), (-5,6) and (9,3) is

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A(2,6) and B(1,7) are two vertices of a triangle ABC and the centroid is (5,7) The coordinates of C are

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In the diagram M(2,3) is the midpoint of AB The coordinates of A and B are respectively

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In the diagram P(-4,-2) divides AB in the ratio 2:1 The coordinates of A and B are respectively

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The circumcenter of the triangle with vertices at (0,0), (8,0) and (0,-6) is at

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A triangle has vertices A(1,-1) B(2,4) and C(6,0) The length of the median from A is

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The opposite vertices of a rectangle ABCD are A(-5,4) and C(5,-4) The area of the rectangle is

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The opposite vertices of a rectangle OABC are and If is a point on the other diagonal AC then the ratio in which P divides AC is

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A line segement AB is divided into four equal parts at P, Q, R in that order If A is (8,12) and B is (12,16) then R has coordinates

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A line segment AB is produced to C such that AC=2AB. If the coordinates of A and B are (6,8) and (8,6) respectively then the coordinates of C are

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In a triangle ABC, vertex A is (6,8) and the midpoint of BC is (3,2) Then the coordinates of the centroid G are

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A and B are the centres of two circles that just touch each other at P If A is (4,1), B is (2,2) and the radii of the circles are 2 and 3 respectively then P has coordinates

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A(2,4), B(6,-2), and C(0,6) are the vertices of a triangle ABC. If L and M are, respectively, the midpoints of AB and AC, then LM=

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P(4,-5) and Q(5,8) are the endpoints of a line segment PQ. The ratio in which PQ is divided by the y-axis is

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The area of the triangle whose vertices are (2,-3), (3,2) and (-2,5) is

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AD is the median of triangle ABC with vertices A(-3,2), B(5,-2) and C(1,3) The area of triangle ABD is

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L,M, and N are the midpoints of the sides BC CA and AB respectively of triangle ABC If the vertices are A(3,-4), B(5,-2), and C(1,3) the area of  is

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In the diagram the coordinates of P,Q, and R are (3,6),(-1,3) and (2,-1) respectively The area of  is

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If the area of the triangle formed by (-2,5), (x,-3) and (3,2) is 14 square units then x=

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If the area of the triangle formed by the points (-2,3), (4,-5), and (-3,y) is 10 square units then y=

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If the points (1,3), (x,2) and (4,6) are collinear then x=

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If the points (4,y) (6,4), and (-1,-3) are collinear then y=

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If the points (2,1), (3,-2) and (a,b) are collinear then

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The condition for the points (x,y), (-2,2) and (3,1) to be collinear is

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If (a,b) (c,d) and (a-c, b-d) are collinear then

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The midpoints of the sides of triangle ABC are (-1,-2),(6,1), and (3,5) The area of  is 

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The area of the quadrilateral with vertices (0,0), (4,0), (4,2), and (0,2) is

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The area of the quadrilateral formed by the points (0,0), (1,0), (1,4) and (0,2) is

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If G is the centroid of the triangle with vertices A(-1,6), B(7,1) and C(3,5) then the area of triangle GAB is

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The area of the triangle formed by the points (a,b+c), (b,c+a) and (c,a+b) is

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The point (x,y) lies on the line joining (3,4) and (-5,-6) if

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If the points (a,0), (0,b), and (1,1) are collinear then

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The area of the parallelogram with vertices (0,0), (2,3), (-2,3) and (-4,0) is

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The area of the rhombus formed by the points (3,0),(0,4),(-3,0) and (0,-4) is

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Multiple choice questions

13 Qs

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The angle of elevation at the top of a tower from a point 10 m from the foot of the tower is   The height of the tower is 

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the angle of depression of  a boat from the top of a cliff 300 m high is   The distance of the boat from the foot of the cliff is 

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The length of the shadow of a pole is  times its height  The elevation of the sum must be 

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the shadow cast by a tower is 30 m long when the elevation of sum is  If the elevation of sum is   then the length of the shadow is 

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The elevation at the top of a building under construction at a point 120 m from the base is  How much higher should the building be raised so that the elevation becomes ?

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The angles  of depression at the top and foot of a tower as seen from a hill 45 m high are  and  respectivelyThe height of the tower is 

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The angles of elevation at the top of a lamp-post 4m tall from a point on the ground is How far should the observer walk so that the elevation gets doubled?

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A kite is flying with the string inclined at  to the horizontal   If the string is straight and 50 m long the height at which the kite is flying is 

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A flagstaff stands on the top of a building From a point on the ground 15 away the elevations at the top of the building and the flagstaff are  and   respectively the length of the flagstaff is 

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A man in a boat rowed away from a cliff m high It takes minutes to change the elevation at the top of the cliff from  to  The speed of the boat is 

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The angles of elevation of a church from two points at distance  and  from the floor of the church on the side are complementary The height of the church is

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From the top of a tower 60 m tall the elevation at the top of a hill is found to be equal to the depression at the foot of the hill The height of the hill is

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There are two poles on a horizontal plane From the midpoint of the line joining their feet the tops of the poles appear at angles of elevation of  and  If the height of the first pole is 100 m  then the height of the second is

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Multiple choice questions

40 Qs

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If  is acute and  then  

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If  is acute and  then 

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If  is acute and  then 

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If  and  is acute then 

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If  where  is an acute angle then

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If  and  then 

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If  and  then 

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If  and  then 

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If  and  then 

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If  and  then 

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If  then 

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If  and  are in GP  then 

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If  then 

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If  then 

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If  then

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The value of  is

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The value of  is

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The value of  is

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The value of 

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A pole 15 m long rests against a vertical wall at an angle of  with the ground How high up the wall does the pole reach?

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If  then 

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If  is acute and  then 

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A kite flying at a height of 75 m from the ground is attached to a straight string  Which is inclined at an angle of  to the ground  The length of the string is

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In the diagram,  and AC = 8 cm If  and  then BC =

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In the diagram PQRS is a square of side 3 cm and  Then the length of TR is approximately

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In the diagram,  represents a flag pole on top of a building. From the information given in the figure, the approximate length of the flag pole is

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The value of 

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If  then x =

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Multiple choice questions

15 Qs

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If  then x+2y=

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If A= and B=  then 2A-B=

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If A=, B=,
C=, then 2A+3B-C=

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If +=,then (x,y)=

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If A+= then A=

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If -= then the values of x and y respectively are 

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If A+B=  and A-B= then A is 

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If A=   B=  then 2A+B =

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If P= and Q= then P-Q=

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IF A= And B= then A+B=

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IF A=  B=and A+B=I then the values of x and y respectively  are 

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If and , then A+B=

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If +=, then

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The values of x satisfying the equation
-= are

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The value of x satisfying the equation 2+=+are

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