Maths
Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre. Sol:
Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle (in cm) which touches the smaller circle.
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of sides AB and AC.
If PT is a tangent to a circle with center O and PQ is a chord of the circle such that QPT 70, then find the measure of POQ.
In the given figure common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that AB=CD.
In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If TPQ 70 , find the TRQ.
In the given figure, PA and PB are two tangents to the circle with centre O. If APB 50 then what is the measure of OAB.
In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. \ AB + CD = AD + BC Þ 6 + 8 = AD = 9 Þ AD = 5 cm...
In the given figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If PBT 30 , prove that BA : AT = 2 : 1.
In the given figure, a circle with center O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29cm, AD = 23cm, B 90 and DS=5cm then find the radius of the circle.
Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.
PQ is a chord of length 4.8 cm of a circle of radius 3cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BC and DC into which BC is divided by the point of contact D, are of lengths 6cm and 9cm respectively. If the area of sides AB and AC. ABC 54cm2 then find the lengths of
In the given figure, O is the centre of the two concentric circles of radii 4 cm and 6cm respectively. AP and PB are tangents to the outer and inner circle respectively. If PA = 10cm, find the length of PB up to one place of the decimal.
In the given figure, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that point of contact P bisects the base BC.
In the given figure, PA and PB are the tangent segemtns to a circle with centre O. Show that he points A, O, B and P are concyclic.
A circle is inscribed in a ABC touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR=7cm and CR=5cm, find the length of BC.
From an external point P, tangents PA and PB are drawn to a circle with center O. If CD is the tangent to the circle at a point E and PA = 14cm, find the perimeter of PCD
In the given figure, the chord AB of the larger of the two concentric circles, with center O, touches the smaller circle at C. Prove that AC = CB.
In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6cm, BC=7cm and CD=4 cm. Find AD.
In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length of AD, BE and CF.
Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the larger circle which touches the smaller circle.
A point P is 25 cm away from the center of a circle and the length of tangent drawn from P to the circle is 24 cm. Find the radius of the circle.
Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from the center of the circle
In a four-sided field, the length of the longer diagonal is 128 m. The lengths of perpendiculars from the opposite vertices upon this diagonal are 22.7 m and 17.3 m. Find the area of the field.
The adjacent sides of a parallelogram are 36 cm and 27 cm in length. If the distance between the shorter sides is 12 cm, find the distance between the longer sides.
The diagonals of a rhombus are 48 cm and 20 cm long. Find the perimeter of the rhombus.
A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120 m and 44 m. If one of the sides of the || gm is 66 m long, find its corresponding altitude.
Find the area of quadrilateral ABCD in which AB = 42cm, BC = 21 cm,CD = 29 cm, DA = 34 cm and diagram BD = 20cm.
The cost of fencing a square lawn at 14 per metre is 2800. Find the cost of mowing the lawn at ₹ 54 per 100 m2.
The adjacent sides of a ||gm ABCD measure 34 cm and 20 cm and the diagonal AC is 42 cm long. Find the area of the ||gm.
Find the area of a trapezium whose parallel sides are 11 cm and 25 cm long and non- parallel sides are 15 cm and 13 cm.
Find the area of a rhombus each side of which measures 20 cm and one of whose diagonals is 24 cm.
A lawn is in the form of a rectangle whose sides are in the ratio 5:3 and its area is Find the cost of fencing the lawn at ₹ 20 per metre.
Find the area of a triangle whose sides are 42 cm, 34 cm and 20 cm.
Find the area of a rhombus whose diagonals are 48 cm and 20cm long.
The length of the diagonal of a square is 24 cm. Find its area.
The longer side of a rectangular hall is 24 m and the length of its diagonal is 26 m. Find the area of the hall.
Find the area of an isosceles triangle each of whose equal sides is 13 cm and whose base is 24 cm.
Find the area of an equilateral triangle having each side of length 10 cm. (Take
The parallel sides of a trapezium are 9.7cm and 6.3 cm, and the distance between them is 6.5 cm. The area of the trapezium is (a) 104 cm2 (b) 78 cm2 (c) 52 cm2 (d) 65 cm2
The sides of a triangle are in the ratio 12: 14 : 25 and its perimeter is 25.5 cm. The largest side of the triangle is (a) 7 cm (b) 14 cm (c) 12.5 cm (d) 18 cm
In the given figure ABCD is a trapezium in which AB =40 m, BC=15m,CD = 28m, AD= 9 m and CE = AB. Area of trapezium ABCD is
In the given figure ABCD is a quadrilateral in which
Find the area of trapezium whose parallel sides are 11 m and 25 m long, and the nonparallel sides are 15 m and 13 m long.
The shape of the cross section of a canal is a trapezium. If the canal is 10 m wide at the top, 6 m wide at the bottom and the area of its cross section is 640 m2 , find the depth of the canal.
The parallel sides of trapezium are 12 cm and 9cm and the distance between them is 8 cm. Find the area of the trapezium.
The area of rhombus is 480 c m2 , and one of its diagonal measures 48 cm. Find
(i) the length of the other diagonal,
(ii) the length of each of the sides
(iii) its perimeter
The perimeter of a rhombus is 60 cm. If one of its diagonal us 18 cm long, find (i) the length of the other diagonal, and (ii) the area of the rhombus.
Find the area of the rhombus, the length of whose diagonals are 30 cm and 16 cm. Also, find the perimeter of the rhombus.
The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of the parallelogram.
The area of a parallelogram is 392 m2 . If its altitude is twice the corresponding base, determined the base and the altitude.
The adjacent sides of a parallelogram are 32 cm and 24 cm. If the distance between the longer sides is 17.4 cm, find the distance between the shorter sides.
Find the area of a parallelogram with base equal to 25 cm and the corresponding height measuring 16.8 cm.
Sol: Given: Base = 25 cm Height = 16.8 cm \Area of the parallelogram = Base ´ Height = 25cm ´16.8 cm = 420 cm2
Find the area of the quadrilateral ABCD in which in AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diagonal BD = 20 cm.
Find the perimeter and area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, ACB 90 and AC = 15 cm.
Find the area of the quadrilateral ABCD in which AD = 24 cm, BAD 90 and BCD is an equilateral triangle having each side equal to 26 cm. Also, find the perimeter of the quadrilateral. Sol:
In the given figure ABCD is quadrilateral in which diagonal BD = 24 cm, AL BD and CM BD such that AL = 9cm and CM = 12 cm. Calculate the area of the quadrilateral.
The cost of fencing a square lawn at ₹ 14 per meter is ₹ 28000. Find the cost of mowing the lawn at ₹ 54per100 m2
The cost of harvesting a square field at ₹ 900 per hectare is ₹ 8100. Find the cost of putting a fence around it at ₹ 18 per meter.
The area of a square filed is 8 hectares. How long would a man take to cross it diagonally by walking at the rate of 4 km per hour?
Find the area and perimeter of a square plot of land whose diagonal is 24 m long.
The cost of painting the four walls of a room 12 m long at ₹ 30 per m2 is ₹ 7560 per m2 and the cost of covering the floor with the mat at ₹ dimensions of the room.
The dimensions of a room are 14 m x 10 m x 6.5 m There are two doors and 4 windows in the room. Each door measures 2.5 m x 1.2 m and each window measures 1.5 m x 1 m. Find the cost of painting the four walls of the room at ₹ 35 per m2 .
A 80 m by 64 m rectangular lawn has two roads, each 5 m wide, running through its middle, one parallel to its length and the other parallel to its breadth. Find the cost of gravelling the reads at ₹ 40 per m2 .
A carpet is laid on floor of a room 8 m by 5 m. There is border of constant width all around the carpet. If the area of the border is 12 m2
A room 4.9 m long and 3.5 m board is covered with carpet, leaving an uncovered margin of 25 cm all around the room. If the breadth of the carpet is 80 cm, find its cost at ₹ 80 per metre.
The length and breadth of a rectangular garden are in the ratio 9:5. A path 3.5 m wide, running all around inside it has an area of 1911m2 . Find the dimensions of the garden.
A footpath of uniform width runs all around the inside of a rectangular field 54m long and 35 m wide. If the area of the path is 420 m2 , find the width of the path.
A rectangular plot measure 125 m by 78 m. It has gravel path 3 m wide all around on the outside. Find the area of the path and the cost of gravelling it at ₹ 75 per m2
The area of rectangle is 192cm2 and its perimeter is 56 cm. Find the dimensions of the rectangle.
A 36-m-long, 15-m-borad verandah is to be paved with stones, each measuring 6dm by 5 dm. How many stones will be required?
A room is 16 m long and 13.5 m broad. Find the cost of covering its floor with 75-m-wide carpet at ₹ 60 per metre.
A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. The area of the lawn is 3375m2 . Find the cost of fencing the lawn at ₹ 65 per metre.
The area of a rectangular plot is 462m2is length is 28 m. Find its perimeter
The length of a rectangular park is twice its breadth and its perimeter is 840 m. Find the area of the park.
The perimeter of a rectangular plot of land is 80 m and its breadth is 16 m. Find the length and area of the plot.
In the given figure, ABC is an equilateral triangle the length of whose side is equal to 10 cm, and ADC is right-angled at D and BD= 8cm. Find the area of the shaded region
Find the area and perimeter of an isosceles right angled triangle, each of whose equal sides measure 10cm.
Each of the equal sides of an isosceles triangle measure 2 cm more than its height, and the base of the triangle measure 12 cm. Find the area of the triangle.
The base of an isosceles triangle measures 80 cm and its area is 360 cm2. Find the perimeter of the triangle.
Find the length of the hypotenuse of an isosceles right-angled triangle whose area is 200cm2 . Also, find its perimeter
Find the area of a right – angled triangle, the radius of whose, circumference measures 8 cm and the altitude drawn to the hypotenuse measures 6 cm.
The base of a right – angled triangle measures 48 cm and its hypotenuse measures 50 cm. Find the area of the triangle.
If the area of an equilateral triangle is 81
11. If the area of an equilateral triangle is 36 Sol: cm2 , find its perimeter.
The height of an equilateral triangle is 6 cm. Find its area.
Each side of an equilateral triangle is 10 cm. Find (i) the area of the triangle and (ii) the height of the triangle.
The length of the two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 c m2 , find the perimeter of the triangle.
The difference between the sides at the right angles in a right-angled triangle is 7 cm. the area of the triangle is 60 c m2 . Find its perimeter.
The sides of a triangle are in the ratio 5:12:13 and its perimeter is 150 m. Find the area of the triangle.
Find the area of the triangle whose sides are 18 cm, 24 cm and 30 cm. Also find the height corresponding to the smallest side.
Find the areas of the triangle whose sides are 42 cm, 34 cm and 20 cm. Also, find the height corresponding to the longest side.
Find the area of triangle whose base measures 24 cm and the corresponding height measure 14.5 cm.
Draw two concentric circles of radii 4 cm and 6 cm. Construct a tangent to the smaller circle from a point on the larger circle. Measure the length of this tangent.
Draw a circle of radius 3.5 cm. Draw a pair of tangents to this circle which are inclined to each other at an angle of 60 . Write the steps of construction.
Draw a ABC , right-angled at B such that AB = 3 cm and BC = 4cm. Now, Construct a triangle
Construct an isosceles triangle whose base is 9 cm and altitude 5cm. Construct another
Construct a ABC in which BC = 5cm, C 60 and altitude from A equal to 3 cm. Construct
Construct a ABC Sol: in which B= 6.5 cm, AB = 4.5 cm and ABC 60
Draw a line segment AB of length 6.5 cm and divided it in the ratio 4 : 7. Measure each of the two parts.
Draw a line segment AB of length 5.4 cm. Divide it into six equal parts. Write the steps of construction.
Construct a tangent to a circle of radius 4 cm form a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.
Draw a circle of radius 32 cm. Draw a tangent to the circle making an angle 30 with a line passing through the centre.
Write the steps of construction for drawing a pair of tangents to a circle of radius 3 cm , which are inclined to each other at an angle of 60 .
Draw a circle of radius 4.2. Draw a pair of tangents to this circle inclined to each other at an angle of 45
Draw a line segment AB of length 8 cm. Taking A as centre , draw a circle of radius 4 cm and taking B as centre , draw another circle of radius 3 cm. Construct tangents to each circle form the centre of the other circle.
Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the pair of tangents form the point P to the circle
Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle. Construct tangents to the circle at each of the two end points of the diameter AB.
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form its centre.
Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw two tangents to the circle. Also, measure the lengths of the tangents.
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3
Construct an isosceles triangles whose base is 8 cm and altitude 4 cm and then another
To construct a triangle similar to
7. Construct a ABC in which BC = 8 cm, B 45 and C 60 . Construct another
6. Construct a ABC in which AB = 6 cm, A 30and AB 60 . Construct
Sol: Steps of Construction with base AB’ = 8 cm. Step 1: Draw a line segment AB = 6cm. Step 2: At A, draw ÐXAB = 30°. Step 3: At B, draw ÐYBA = 60°. Suppose AX and BY intersect at C. Thus, DABC is...
5. Construct a ABC with BC = 7 cm, B 60 and AB = 6 cm. Construct another triangle
Construct a triangle with sides 5 cm, 6 cm, and 7 cm and then another triangle whose sides
3. Construct a PQR , in which PQ = 6 cm, QR = 7 cm and PR =- 8 cm. Then, construct
Find the probability that a leap year selected at random w ill contain 53 Sundays.
A game consists o f tossing a 1 rupee coin three times, and noting Its outcomes each time. Find the probability o f getting (I) 3 heads, (II) at least 2 tails.
All kings, queens, and aces are removed from a pack o f 52 cards. The remaining cards are well-shuffled and then a card Is drawn from I t Find the probability that the drawn card Is
(I) a black face card,
(II)a red face card.
All red face cards are removed from a pack o f playing cards. The remaining cards are well-shuffled and then a card Is drawn at random from them. Find the probability that the drawn card Is
(I) a red card,
(II) a face card,
(III)a card of clubs.
What Is the probability that an ordinary year has 53 Mondays?
A card Is drawn at random from a well-shuffled pack o f 52 cards. Find the probability that the card was drawn Is neither a red card nor a queen.
5 cards the ten, Jack, queen, king and ace o f diamonds are well shuffled with their faces downward. One card Is then picked up at random. (a) What Is the probability that the drawn card Is the queen? (b) If the queen Is drawn and put aside and a second card Is drawn, find the probability that the second card Is (I) an ace, (II) a queen.
A letter Is chosen at random from the letter o f the word ‘ASSOCIATION’. Find the probability that the chosen letter Is a (I) vowel (II) consonant (III) S
Two dice are rolled once. Find the probability o f getting such numbers on 2 dice whose product Is a perfect square.
A die Is rolled twice. Find the probability that _9_ _ 3 12 4
(I) 5 w ill not come up either time,
(II) 5 w ill come up exactly one time,
(III) 5 w ill come up both the times.
A group consists o f 12 persons, o f which 3 are extremely patient, other 6 are extremely honest and rest are extremely kind. A person from the group Is selected at random. Assuming that each person Is equally likely to be selected, find the probability o f selecting a person who Is
(I) extremely patient,
(II) extremely kind o r honest. Which o f the above values did you prefer more?
A carton consists o f 100 shirts o f which 88 are good and 8 have minor defects. Rohlt, a trader, w ill only accept the shirts which are good. But, Kamal, and another trader w ill only reject the shirts which have major defects. 1 shirt Is drawn at random from the carton. What Is the probability that It Is acceptable to
(I)Rohlt,
(II) Kamal?
A Jar contains 54 marbles, each o f which some are blue, some are green and some are white. The probability o f selecting a blue marble at random Is and the probability o f selecting a green marble at random Is | . How many white marbles does the Jar contain?
A Jar contains 24 marbles. Some o f these are green others are blue. If a marble Is drawn at random from the Jar, the probability that It Is green Is | . Find the number o f blue marbles In the Jar.
A bag contains 18 balls out o f which x balls are red.
(I)If one ball Is drawn at random from the bag, what Is the probability that It Is not red?
(II) If two more red balls are put In the bag, the probability o f drawing a red ball w ill be | times the probability o f drawing a red ball In the firs t case. Find the value o f x.
The probability o f selecting a red ball at random from a Jar that contains only red, blue and orange balls Is j . The probability of selecting a blue ball at random from the same Jar Is j .If the Jar contains 10 orange balls, find the total number o f balls In the Jar.
A piggy bank contains hundred 50-p coins, seventy Rs. 1 coin, fifty Rs. 2 coins and th irty Rs. 5 coins. If It Is equally likely that one of the coins will fall out when the blank Is turned upside down, what Is the probability that the coin(I) will bea R s. 1 coin? (II) will not be a Rs. 5 coin (III) will be 50-p or a Rs. 2 coin?
A box contains 80 discs, which are numbered from 1 to 80. If one disc Is drawn at random from the box, find the probability that It bears a perfect square number.
Tickets numbered 2 ,3 ,4 , 5……………100,101 are placed In a box and mix thoroughly. One ticket Is drawn at random from the box. Find the probability that the number on the ticket Is(III) a number which Is a perfect square (Iv) a prime number less than 40.
Tickets numbered 2 ,3 ,4 , 5……………100,101 are placed In a box and mix thoroughly. One ticket Is drawn at random from the box. Find the probability that the number on the ticket Is
(I)an even number
(II)a number less than 16
Cards marked with numbers 1,3, 5……………..101 are placed In a bag and mixed thoroughly. A card Is drawn at random from the bag. Find the probability that the number on the drawn card Is
(I)less than 19,
(II) a prime number less than 20.
A box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bearsA box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bearsA box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bears
(III) an odd number less than 30,
(Iv) a composite number between 50 and 70.
A box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bears
(I) a 1 digit number,
(II)a number divisible by 5,
Cards bearing numbers 1,3, 5……………..35 are kept In a bag. A card Is drawn at random from the bag. Find the probability o f getting a card bearing
(I)a prime number less than 15,
(II) a number divisible by 3 and 5.
Card numbered 1 to 30 are put In a bag. A card Is drawn at random from the bag. Find the probability that the number on the drawn card Is
(I) not divisible by 3,
(II)a prime number greater than 7,
(III)not a perfect square number.
A box contains cards numbered 3, 5 , 7 , 9 ……..35,37. A card Is drawn at random from the box. Find the probability that the number on the card Is a prime number.
A box contains 25 cards numbers from 1 to 25. A card Is drawn at random from the bag. Find the probability that the number on the drawn card Is(I) divisible by 2 or 3, (II) a prime number.
A card Is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card was drawn Is
(III)either a king or queen
(Iv) neither a king nor the queen.
A card Is drawn at random from a well-shuffled deck o f playing cards. Find the probability that the card was drawn Is
(I)a card of a spade or an Ace
(II)a red king
One card Is drawn from a well-shuffled deck of 52 cards. Find the probability o f getting(v) a Jack o f hearts (vl) a spade.
One card Is drawn from a well-shuffled deck of 52 cards. Find the probability o f getting
(III)a red face card
(Iv) a queen o f black suit
One card Is drawn from a well-shuffled deck of 52 cards. Find the probability o f getting
(I)a king o f red suit
(II) a face card
There are 40 students In a class o f whom 25 are girls and 15 are boys. The class teacher has to select one student as a class representative. He writes the name o f each student on a separate, the card being Identical. Then she puts cards In a bag and stirs them thoroughly. She then draws one card from the bag. What Is the probability that the name written on the card Is the name of
(I) A girl?
(II)A boy?
A bag contains lemon-flavored candles only. Hema takes out 1 candy without looking Into the bag. What Is the probability that she takes out
(I)An orange-flavored candy
(II)A lemon-flavored candy
(I) A lo t o f 20 bulbs contain 4 defective ones. 1 bulb Is drawn at random from the lot. What Is the probability that this bulb Is defective? (II) Suppose the ball drawn In
(I) Is not defective and not replaced. Now, ball Is drawn at random from the rest. What Is the probability that this bulb Is not defective?
A box contains 90 discs which are numbered from 1 to 90 If one disc Is drawn at random from the box, find the probability that It bears
(I) A two-digit number
(II) A perfect square number
(III) A number divisible by 5.
A lot consists o f 144 ballpoint pens o f which 20 are defective and others good. Tanvl will buy a pen If It Is a good but will not buy If It Is defective. The shopkeeper draws 1 pen at random and gives It to her. What Is the probability that
(I) She will buy It,
(II) She will not buy It?
12 defective pens are accidentally mixed with 132 good ones, It Is not possible to Just look at pen and tell whether or not It Is defective. 1 pen Is taken out at random from this lot. Find the probability that the pen taken out Is good one.
A game of chance consists of spinning and arrow which Is equally likely to come to the rest pointing to one of the numbers 1 , 2 ,3 ,4 12 as shown In the figure. What Is the probability that It will point to
(III)A prime number
(Iv) A number which Is a multiple o f 5
A game of chance consists of spinning and arrow which Is equally likely to come to the rest pointing to one of the numbers 1 , 2 ,3 ,4 12 as shown In the figure. What Is the probability that It will point to
(I)6
(II)An even number
Cards marked with numbers 5 to 50 are placed In a box and mixed thoroughly. A card Is drawn from the box at random. Find the probability that the number on the taken out card Is
Two dice are rolled together. Find the probability o f getting such numbers on the two dice whose product Is 12.
Two dice are rolled together. Find the probability o f getting such numbers on two dice whose product Is perfect square.
Solution: When two different dice are thrown, then total number of outcomes = 36. Let E be the event of getting the product of numbers, as a perfect square. These numbers are (1,1), (1,4), (2,2),...
When two dice are tossed together, find the probability that the sum o f the numbers on their tops Is less than 7.
Solution: When two different dice are thrown, the total number of outcomes = 36. Let E be the event of getting the sum of the numbers less than 7. These numbers are (1,1), (1,2), (1,3), (1,4),...
Two different dice are rolled simultaneously. Find the probability that the sum o f the numbers on the two dice Is 10.
Solution: When two different dice are thrown, the total number of outcomes = 36. Let E1 be the event of getting the sum of the numbers on the two dice is 10. These numbers are (4 ,6), (5,5)...
A medicine company has factories at two places, 
and Y. From these places, supply is made to each of its three agencies situated at 
and 
. the monthly requirement of the agencies are respectively 40 packets, 40 packets and 50 packets of medicine, while the production capacity of the factories at and 
are 60 packets and 70 packets respectively. The transportation costs per packet from the factories to the agencies are given as follows.
How many packets from each factory should be transported to each agency so that the cost of transportation is minimum? Also, find the minimum cost. Solution: Let $x$ packets of medicines be...
Graph the solution sets of the following inequations: 
Given $x \geq y-2$ $\Rightarrow \mathrm{y} \leq \mathrm{x}+2$ Consider the equation $y=x+2$ Finding points on the coordinate axes: If $x=0$, the $y$ value is 2 i.e, $y=2$ $\Rightarrow$ the point on...
Graph the solution sets of the following inequations: 
Given $x+2 y>1$ $\begin{array}{l} \Rightarrow 2 y>1-x \\ \Rightarrow y>\frac{1}{2}-\frac{x}{2} \end{array}$ Consider the equation $y=\frac{1}{2}-\frac{x}{2}$ Finding points on the...
Solve 
$\begin{array}{l} x^{2}-4 a x+4 a^{2}-b^{2}=0 \\ \Rightarrow x^{2}-4 a x+(2 a+b)(2 a-b)=0 \\ \Rightarrow x^{2}-[(2 a+b)+(2 a-b)] x+(2 a+b)(2 a-b)=0 \\ \Rightarrow x^{2}-(2 a+b) x-(2 a-b) x+(2 a+b)(2...
Solve 
$\begin{array}{l} x^{2}+5 x-\left(a^{2}+a-6\right)=0 \\ \Rightarrow x^{2}+5 x-(a+3)(a-2)=0 \\ \Rightarrow x^{2}+[(a+3)-(a-2)] x-(a+3)(a-2)=0 \\ \Rightarrow x^{2}+(a+3) x-(a-2) x-(a+3)(a-2)=0 \\...
Solve 
$\begin{array}{l} 4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0 \\ \Rightarrow 4 \sqrt{3} x^{2}+8 x-3 x-2 \sqrt{3}=0 \\ \Rightarrow 4 x(\sqrt{3} x+2)-\sqrt{3}(\sqrt{3} x+2)=0 \\ \Rightarrow(\sqrt{3} x+2)(4...
Solve 
$\begin{array}{l} \sqrt{3} x^{2}+10 x-8 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x^{2}+12 x-2 x-8 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x(x+4 \sqrt{3})-2(x+4 \sqrt{3})=0 \\ \Rightarrow(x+4 \sqrt{3})(\sqrt{3}...
Solve 
$\begin{array}{l} 2 x^{2}+a x-a^{2}=0 \\ \Rightarrow 2 x^{2}+2 a x-a x-a^{2}=0 \\ \Rightarrow 2 x(x+a)-a(x+a)=0 \\ \Rightarrow(x+a)(2 x-a)=0 \\ \Rightarrow x+a=0 \text { or } 2 x-a=0 \\ \Rightarrow...
Find the value of 
so that the quadratic equation 
has equal roots.
It is given that the quadratic equation $x^{2}-4 k x+k=0$ has equal roots. $\begin{array}{l} \therefore D=0 \\ \Rightarrow(-4 k)^{2}-4 \times 1 \times k=0 \\ \Rightarrow 16 k^{2}-4 k=0 \\...
If one root of the quadratic equation 
is reciprocal of the other, find the value of .
Let $\alpha$ and $\beta$ be the roots of the equation $3 x^{2}-10 x+k=0$. $\therefore \alpha=\frac{1}{\beta} \quad$ (Given) $\Rightarrow \alpha \beta=1$ $\Rightarrow \frac{k}{3}=1 \quad \quad$...
If 1 is a root of the equation 
. and 
. then find the value of ab.
It is given that $y=1$ is a root of the equation $a y^{2}+a y+3=0$. $\begin{array}{l} \therefore a \times(1)^{2}+a \times 1+3=0 \\ \Rightarrow a+a+3=0 \\ \Rightarrow 2 a+3=0 \\ \Rightarrow...
If the roots of the quadratic equation 
are equal then find the value of 
.
It is given that the roots of the quadratic equation $2 x^{2}+8 x+k=0$ are equal. $\begin{array}{l} \therefore D=0 \\ \Rightarrow 8^{2}-4 \times 2 \times k=0 \\ \Rightarrow 64-8 k=0 \\ \Rightarrow...
If 
is a solution of the quadratic equation 
. Find the value of .
It is given that $x=\frac{-1}{2}$ is a solution of the quadratic equation $3 x^{2}+2 k x-3=0$ $\begin{array}{l} \therefore 3 \times\left(\frac{-1}{2}\right)^{2}+2 k...
The sum of two natural numbers is 8 and their product is 
, Find the numbers.
Let the required natural numbers be $x$ and $(8-x)$. It is given that the product of the two numbers is $15 .$ $\begin{array}{l} \therefore x(8-x)=15 \\ \Rightarrow 8 x-x^{2}=15 \\ \Rightarrow...
The length of a rectangular field exceeds its breadth by 
and the area of the field is 
. The breadth of the field is
(a) 
(b) 
(c) 
(d) 
Let the breadth of the rectangular field be $x \mathrm{~m}$. $\therefore$ Length of the rectangular field $=(x+8) m$ Area of the rectangular field $=240 \mathrm{~m}^{2}$ $\therefore(x+8) \times...
The sum of a number and its reciprocal is 
. The number is
(a) 
or 
(b) 
or 
(c) 
or 
(d) 
or 6
Answer is (a) $\frac{5}{4}$ or $\frac{4}{5}$ Let the required number be $x$. According to the question: $\begin{array}{l} x+\frac{1}{x}=\frac{41}{20} \\ \Rightarrow \frac{x^{2}+1}{x}=\frac{41}{20}...
For what value of , the equation 
has real roots?
(a) 
(b) 
(c) 
(d)None of these
Answer is (b) $k \geq \frac{-9}{2}$ It is given that the roots of the equation $\left(k x^{2}-6 x-2=0\right)$ are real. $\begin{array}{l} \therefore D \geq 0 \\ \Rightarrow\left(b^{2}-4 a c\right)...
The roots of the equation 
are
(a) real, unequal and rational
(b) real, unequal and irrational
(c) real and equal
(d) imaginary
Answer is (b) real, unequal and irrational $\begin{array}{l} \because D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 2 \times 3 \\ =36-24 \\ =12 \end{array}$ 12 is greater than 0 and it is not a...
The roots of the equation 
are
(a) real, unequal and rational
(b) real, unequal and irrational
(c) real and equal
(d) imaginary
Answer is (d) imaginary $\begin{array}{l} \because D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 2 \times 7 \\ =36-56 \\ =-20<0 \end{array}$ Thus, the roots of the equation are...
If the equation 
has equal roots then value of 
(a) 
(b) 
(c) 
(d) 
Answer is $(\mathrm{d}) \pm \frac{4}{3}$ It is given that the roots of the equation $\left(4 x^{2}-3 k x+1=0\right)$ are equal. $\begin{array}{l} \therefore\left(b^{2}-4 a c\right)=0 \\...
If the equation 
has equal roots then 
?
(a) 0 or 0
(b) 
or 0
(c) 2 or
(d) 0 only
Answer is (c) 2 or $-2$ It is given that the roots of the equation $\left(9 x^{2}+6 k x+4=0\right)$ are equal. $\begin{array}{l} \therefore\left(b^{2}-4 a c\right)=0 \\ \Rightarrow(6 k)^{2}-4 \times...
The roots of the equation 
will be reciprocal each other if
(a) 
(b) 
(c) c=a
(d) none of these
Answer is (c) $\mathrm{c}=\mathrm{a}$ Let the roots of the equation $\left(a x^{2}+b x+c=0\right)$ be $\alpha$ and $\frac{1}{\alpha}$. $\therefore$ Product of the roots $=\alpha \times...
If the sum of the roots of a quadratic equation is 6 and their product is 6 , the equation is
(a) 
(b) 
(c) 
(d)
Answer is (a) $x^{2}-6 x+6=0$ Given: Sum of roots $=6$ Product of roots $=6$ Thus, the equation is: $x^{2}-6 x+6=0$
The roots of a quadratic equation are 5 and . Then, the equation is
(a) 
(b) 
(c) 
(d) 
Answer is (b) $x^{2}-3 x-10=0$, It is given that the roots of the quadratic equation are 5 and $-2$. Then, the equation is: $\begin{array}{l} x^{2}-(5-2) x+5 \times(-2)=0 \\ \Rightarrow x^{2}-3...
If one root of the equation 
is 
then the other root is
(a) 
(b)
(c) 
(d) 3
Answer is (d)3. Given: $3 x^{2}-10 x+3=0$ One root of the equation is $\frac{1}{3}$. Let the other root be $\alpha$. Product of the roots $=\frac{c}{a}$ $\begin{array}{l} \Rightarrow \frac{1}{3}...
If the product of the roots of the equation 
is then the value of is
(a)
(b) 
(c) 
(d) 12
Answer is (c) 8 It is given that the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$ The equation can be rewritten as: $x^{2}-3 x+(k-10)=0$ Product of the roots of a quadratic equation...
If one root of the equation 
is 2 then 
?
(a) 
(b) 
(c) 
(d) 
Answer is (b) $-7$ It is given that one root of the equation $2 x^{2}+a x+6=0$ is 2 . $\begin{array}{l} \therefore 2 \times 2^{2}+a \times 2+6=0 \\ \Rightarrow 2 a+14=0 \\ \Rightarrow a=-7...
Which of the following is a quadratic equation?
(a) 
(b) 
(c) 
(d) None of these
Answer is (b) $x^{3}-x^{2}=(x-1)^{3}$ $\because x^{3}-x^{2}=(x-1)^{3}$ $\Rightarrow x^{3}-x^{2}=x^{3}-3 x^{2}+3 x-1$ $\Rightarrow 2 x^{2}-3 x+1=0$, which is a quadratic equation
Which of the following is a quadratic equation?
(a) 
(b) 
(c) 
(d) 
Answer is (d) $2 x^{2}-5 x=(x-1)^{2}$ A quadratic equation is the equation with degree 2 . $\because 2 x^{2}-5 x=(x-1)^{2}$ $\Rightarrow 2 x^{2}-5 x=x^{2}-2 x+1$ $\Rightarrow 2 x^{2}-5 x-x^{2}+2...
The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 
and exceeds twice the length of the altitude by 
. Find the length of each side of the triangle.
Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...
The area of right -angled triangle is 165 sq meters. Determine its base and altitude if the latter exceeds the former by 7 meters.
Let the base be $x \mathrm{~m}$. Therefore, the altitude will be $(x+7) m$ $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text { Altitude } \\ \therefore...
The area of a right triangle is 
. If the base of the triangle exceeds the altitude by 
, find the dimensions of the triangle.
Let the altitude of the triangle be $x \mathrm{~cm}$ Therefore, the base of the triangle will be $(x+10) \mathrm{cm}$ $\begin{array}{l} \text { Area of triangle }=\frac{1}{2} x(x+10)=600 \\...
The length of a rectangle is thrice as long as the side of a square. The side of the square is 
, more than the width of the rectangle. Their areas being equal, find the dimensions.
Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...
A rectangular filed in long and 
wide. There is a path of uniform width all around it, having an area of 
. Find the width of the path
Let the width of the path be $x \mathrm{~m}$. $\therefore$ Length of the field including the path $=16+x+x=16+2 x$ Breadth of the field including the path $=10+x+x=10+2 x$ Now, (Area of the field...
The length of rectangle is twice its breadth and its areas is 
. Find the dimension of the rectangle.
Let the length and breadth of the rectangle be $2 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 2 x \times x=288 \\ \Rightarrow 2 x^{2}=288 \\...