SHARE AND DIVIDEND
RAW- 1
1) Mr. Mehta invested Rs 26000 in 15% Rs 100 shares quoted at a premium of 30%. Find
a) The number of shares bought by Mr. Mehta.
b) Mr. Mehta 's income from the investment.
c) Percentage return on his investment.
Mr. Mehta sold these shares when they were quoted at a premium of 50% and invested the proceeds in 10% Rs 50 shares quoted at a discount of 20%. Find Mr Mehta's income now.
2) Rs 1080 are invested in 8% Rs 10 shares of a company quoted at Rs 12. Find
a) the number of shares bought.
b) dividend due on the shares.
3) An investor buys 50 shares of a company, when Rs 100 shares of the company are quoted at Rs 140. Find the
a) amount of money invested to purchase these shares.
b) rate of interest earned on the investment.
4)
RATIO & PROPORTION
RAW- 1
RATIO
1) A metre scale is cut in two pieces in the ratio 3:2. Find the length of each piece.
2) Ram left Rs 33000 to his three sons to be divided in the ratio 3:4:5. Find the share of each.
3) Two numbers are in the ratio 13:9. Their difference is 56. Find the numbers.
4) Two numbers are in the ratio 7:9. The first is added to twice the second, the result is 100. Find the numbers.
5) A bank has rupee coins and fifty paise coins in the ratio 2:3. The total value of the coins is Rs 24.50. find the number of each type of coin.
6) Ages of Arun and Beena are in the ratio 4:5. Fourteen years ago the ratio of their ages was 2:3. How old are Arun and Beena at present?
7) The ratio of two numbers is 4:3. If 2 is added to the first and 6 is substracted from the second, the ratio becomes 5:3. Find both numbers.
8) On adding 1 to each of two numbers, the ratio of the resulting numbers is 2:5. If 1 is substracted from each, the resulting ratio is 1:3. Find the numbers.
9) The speed of two trains are in the ratio 4:7. They leave in opposite direction from a place at the same time. At the end of 7 hours, the total distance travelled by them is 154 km. Find the speed of each train.
10) There are 25 consecutive positive integers. The ratio of the first to the last integers is 3:7. Find the first integer.
PROPORTION
RAW -1
1) Given a,b,c,d are in continued proportion, show that:
a) a : b - c= cd - d²
b) 5a+ 6d : 87a - 7d = 5a³+ 6b³: 8a³- 7b³.
c) (a+ b + c)²/(a²+ b²+ c²)= (a+ b + c)/(a - b + c).
2) If (8b- 7a)/(8d - 7c)= (8b+ 7a)/(8d +7c), show a: b= c: d.
3) Given a: b= c: d , show that
a) 3a - 5b : 3c - 5d = 3a + 5b : 3c + 5d.
b) (a + c) : (b + d)= √(a²+ c²): √(b²+ d²).
4) Find x, given that the work by (x -3) men in (2x +1) days and the work done by (2x +1) men in (x +4) days is in the ratio 3:10.
5) A vessel contains water and milk in the ratio 1:4. Two litres of the mixture is removed and two litres of water are poured in the vessel. If the ratio of water to milk now is 13:12, find
a) The total amount of the mixture in the vessel
b) the amount of milk originally the vessel.
FACTOR THEOREM
RAW-1
1) Factorise the following:
a) x³+ 6x²+ 11x +6
b) x³+ 2x² - x -2.
c) x³ - 7x²+ 4x + 12
d) 2x³+ 3x² - x - 4
2) Find the value of the constants a and b if (x -2) and (x +3) are factors of the expression x³+ ax²+ bx -12.
INEQUATION
Raw-1
1) Solve the inequation of the following:
a) 30- 2(4x -1)> - 8; x ∈ positive integer.
b) 2(x +1)+ 3(2x -1)> 15; x ∈ {2,3,4,5}
c) 2(x +1)+ 3(x -3) ≤ 2; x ∈ positive integer
d) 4(x +3) - 5(x + 2)> 0 ; x ∈ positive integer
2) Given 1/2≤ (x +1) - (x -1)/2 < 5/2 ; x ∈ {integers}. Simplify the Inequality and list the elements of the solution set.
3) Solve 1< 2x +3 < 5, x ∈ R.
Graph the solution set on the number line.
4) Solve 8 < 5(x +1) -2 ≤ 18, x ∈ R.
Graph the solution set on the number line.
5) Solve 20/3 > (2/3) (8 -x) ≥ -2, x ∈ R.
Graph the solution set on the number line.
6) Given A= {x: -2< x ≤ 2, x ∈ R.}
B={x : -1≤ x < 4, x ∈ R.}
a) Graph sets A and B on number line
b) Write the sets
i) A U B
ii) A∩ B
Graph these on number lines.
7) Solve: 5/2< x - 1/3 ≤ 10/3 over real numbers. Graph the solution set.
8) Solve: -4/3≤ 2(x/4+ 1) - 4/3 < 5/6; x ∈ R.. graph the solution set.
QUADRATIC EQUATION
Raw -1
1) 2x²+ 5x -3=0.
2) x²+ 4x - 21 =0.
3) 3x² - 2x = 1.
4) x² - 4x x -32=0.
5) x² - 6xx = 4.
6) 2x²= 1+ 3x.
PROBLEM ON QUADRATIC EQUATIONS
Raw-1
1) A number is 1 less than twice the second number. The sum of the squares of the numbers is 65. Find the numbers.
2) The product of two positive consecutive even integers is 224. Find the numbers.
3) The length of a rectangle is 10cm more than its width. It's area is 119 cm². Find the length and the width of the rectangle.
4) The diagonal of a rectangle is 2cm longer than the length of the rectangle and 9cm longer than its width. Find the length of the diagonal of the rectangle.
5) In a right angled triangle, the hypotenuse is 8cm longer than one of the other sides and only 1cm longer than the third side. Find the length of the hypotenuse.
6) 200 m² of carpenting is needed to cover floors of two rooms. The width of one room is 3m less than its length. The width of the other room is 2m more than the length of the first and its length is twice the length of the first. Find the dimensions of each room.
7) The sum of the areas of two squares is 325 cm². The side of one square is 5cm longer than the side of the smaller square. Find the side of each square.
8) Travelling at 10 kmph less than usual speed, a bus takes 2 hours more to travel 400 km, than what it takes, if it travels at usual speed. Find the usual speed of the day bus.
9) A number of pupils of a class, share the cost of a Rs 300 gift, which they want to present to their teacher. If 5 more students had joined the plan, each student would have had to contribute Rs 2 less. How many pupils were there in the plan originally.
10) Flying against a 30 kmph wind, a plane takes 10 hrs more to fly 3600 km, than what it would take flying with the same wind ? Find the plane's rate of flying in still air.
11) The length of a rectangle is 3cm more than its width. If the length is decreased by 1cm and the width is increased by 3cm, the area is 3 times of the original rectangle. Find the dimensions of the original rectangle.
12) The denominator of a fraction is 1 more than its numerator. The sum of the fraction and its reciprocal is 5/2. Find the fraction.
13) A train at a speed 12 kmph faster than a bus, travels 260 km in 5/2 hour less than time than the bus. Find the speed of the train and the speed of the day bus.
14) A carpet 3m by 6m is enlarged by the addition of a border of uniform width all round the carpet. Find the width of the border, if the area of the carpet after adding the border to it becomes 20/9 times the area of the original carpet.
15) Two positive numbers are in the ratio 3:2, the difference of their squares is 45. Find the numbers.
MATRICES
Raw-1
1) If A= 2 1
3 4
5 6
a) state the order of A.
b) Write A', the transpose of the matrix A.
c) State the order of A'.
2) If A= x+3 & B= 7
y+2 4
5 x
With the relation A= B, then find x,y,z
3) Find the value of x, y and z in the following:
a) x 1 = 2 1
1 y 1 0
b) 5 2 x = y 2 1
c) A= 0 x & A'= 0 1
1 2 4 2
d) A= 3 2 & B= 3 0
x 1 2 1
e) (x 1 + (2 y = (4 0
0 2) z 3) 3 5)
4) Find the Matrix A from the following:
a) If X= -2 1 & Y= 1 -2
3 0 3 1 with the relation A+ X= Y
b) X= 1 4 & Y= 2 2
0 -2 4 0 with the relation 2X+ 3Y= 2A
5) If A= 2 -1 & B= 1 6
1 2 0 2
Find AB and BA
6) Show that the matrix A²- 2A = 3I, where I is unit Matrix
A= 1 2
2 1
7) Solve for matrix X
If A=2 -1 & B= -7 2
0 5 6 11
With the relation A - 3X = B.
SECTION FORMULA
Raw -1
1) Find the cordinates of the midpoint of the coordinates (2,4p) and (2p,-6).
2) Find the coordinates of the day point dividing the line segment joining the following pairs of points in the given ratios:
a) (1,3) and (6,8) 2:3
b) (-3,1) and (3,-2) 2:1
3) The midpoint of line segment joining (1,4) and (a,2b) is (-3,2). find the values of a and b.
4) The mid-point of the line segment joining (a,b) and (2a + 3b) is (-2,1) . Find the value of and b.
5) In what ratio does the x-axis divide the line segment joining (2,3) and (3,-3)? Find the coordinates of the point of the intersection.
6) In what ratio does the y-axis divide the line segment joining (-2,3) and (5,4)? Find the coordinates of the point of intersection.
7) The coordinates of the midpoint of the three sides of a triangle are (4,-2),(0,0) and (1,-3). Find the coordinates of the vertices of the triangle.
8) Given P(-3,2) and Q(4,5), the join of PQ is intersected by y-axis at R. QM is the perpendicular from Q to the x-axis. N Is the midpoint of RM.
a) find the ratio PR/RQ
b) Find the coordinates of R.
c) Write the coordinates of M.
d) Find the coordinates of N.
9) M(0,4) is the midpoint of AB. Given A is (-2,3).
a) find the coordinates of B. B is joined to O, the origin. P divides OB in the ratio 3:1. Q divides OA in the ratio 1:3.
b) Write coordinates of P and Q.
10) Given O(0,0), Q(1,2), S(-3,0) Q divides OP in the ratio 2:3.
a) Write coordinate of P.
SOQL is a parallelogram
b) Write coordinates of L.
c) Find the ratio in which LP is divided by y-axis.
11) Given P(-1,4), Q(5,-2). PQ intersects y-axis in L and x-axis in M.
a) Write the coordinates of L and M.
b) Write the coordinates of N, the midpoint of LM.
12) Given P(-1,-2). PR= 5 units. PR|| case. PQ is divided at (0,0) in the ratio 1:4. OPRS is a parallelogram.
a) Write coordinate of Q and S.
b) Given T(0,3), write coordinate of U, the midpoint of TS.
EQUATION OF STRAIGHT LINE
Raw-1
1) Write the equation to the line
a) with slope= -1/3 and y-intercept= -3
b) Passing through (-2,-1) and having slope= -2.
c) Passing through (2,2) and y-intercept= 4
d) Passing through (1,-3) and (-2,-1).
e) Passing through (-1,3) and parallel to the line 2x + 3y= 17.
f) Passing through (2,-5) and perpendicular to the line 3x - 4y= 7.
g) Passing through (1,4) and intersecting the line x -11= 2y.
h) Passing through (2,-3) and bisecting the line segment joining (1,4) and (-3,2).
i) Passing through (-5,3) and parallel to the y-axis.
j) Passing through (2,-4) and parallel to the x-axis.
2) A(2,3), B(5,-1) and C(-4,-3) and the vertices of a triangle.
Write equation to the
a) median through C.
b) altitude through B.
c) right bisector of the side AC of the triangle.
3) Find the value of p if
a) lines 2x + 3p = y and px - y = 1 are parallel.
b) lines 3x = 2y -10 and py - x = - 10 are perpendicular to each other.
4) A(-2,2), B(1,4) and C(3, b) are collinear.
a) Find the value of b.
b) Find the equation to the line CD, given that CD is perpendicular to line AB.
c) Write coordinates of P, where CD interesects the x-axis.
d) Write equation of line PQ, given that PQ is parallel to AC.
5) Line CD interesects the x-axis at R and the y-axis at S.
Given C is (-6,-1), OS= 5 and D is (3, t) ROSQ is a rectangle.
a) write equation to the line CD.
b) Write the coordinates of R.
c) Find the value of t.
d) Write the coordinates of Q.
e) Write the equation of line through Q, parallel to CD.
f) Write the equation of line through D, perpendicular to CD.
g) Find the coordinates of P, where the lines in (e) and (f) interesect.
TRIGONOMETRY
Raw-1
a) Find the value of following:
a) 4 sin²60+ 3 tan²30 - 8 sin45 cos45.
b) sin45 cos45 + sin²30 + tan²60.
c) 4/tan²60 + 1/cos²30 - sin²45.
d) 4 cos²60+ 4 tan²45 - sin²30.
e) (cos90+ sin30 - sin45)(sin 0 + cos60+ sin45)
f) (sin90+ sin45 cos45 - tan30)(4 sin²30+ cos60+ 1/tan60).
2) Given cosA= 1/3, A is an acute angle, find tan²A.
3) Given 7 tanθ = 24, θ is an acute angle. Find tan²A.
4) If 5 tanθ = 4, find the value of (5 sinθ - 3 cosθ)/(5 sinθ + 2 cosθ).
5) Given 5 sinθ= 3, θ is an acute angle
Evaluate: (cosθ - 1/tanθ)/2/tanθ.
6) Given 13 tanθ -12= 0, 0< θ<90, find the value of
(sinθ + cosθ)/(Cosθ - sinθ).
7) If tanθ = p/q, is an acute angle?
Find the value of
(cosθ + sinθ)/(Cosθ - sinθ).
8) Use A= B= 30, to verify
(A+ B)= SunA cosB + cosA sinB.
9) Use A= 60 , B= 30 and formula
cos(A+ B)= cosA cosB - sinA sinB to prove cos90= 0.
10) Use A= 30 to verify
Sin2A = 2sinA cosA.
11) Two opposite angles of a rhombus are 60. Find the lengths of the diagonals of the rhombus, if each side of the rhombus measure 10cm.
12) Triangle PQR is right angled at Q.
Angle PRQ= 30 and PQ= 12cm
Angle QSR= 90 and PQ= 12cm
Calculate
a) QR
b) PS
c) PR
13) Given CD= 20m.
Calculate AB and BD.
14) Given PQ= 50m.
Find RS
15) Given CD= 30m
Calculate
a) BC
b) AB
c) BD
16) In the figure,
AD is perpendicular to BC. Given
TanB= 2/3 and tanC= 5/4, find AD, if BC= 11.5cm
Booster - 2
1) Write as T ratios of angles less than 45°
a) sin71
b) cos63
c) sin72
d) cos68
e) tan73
2) Evaluate:
a) sin²25+ sin²65
b) sec50 sin40 - cos40 cosec50
c) cosec²67 - tan²23
d) sin35 sin55 - cos35 cos55
e) 2 tan80/cot10 + cot80/tan10
f) sin²x + sin²(90- x).
3) Given 2 sinA -1= 0
a) find A, in degree
b) value of sin3A.
c) verify that sin3A = 3 sinA - 4 sin³A.
IDENTITY
RAW- 1
4) Prove
a) sin²x/(1- cosx)= 1+ cosx.
b) cos²x/(1+ sinx)= 1- sinx
c) tanx + cotx = 1/sinx cosx = secx cosecx
d) (secx + tanx)(sec x - tan x)= sin²x + cos²x.
e) cosec²x sinx cosx = cotx.
f) (1+ tan²x)/(1+ cot²x)= tan²x.
g) (tan²x - sec²x)/(cot²x - cosec²x)= 1.
h) √(cosecx + cotx) √(cosecx - cotx)=1.
i) (1- cotx)/(1+ cotx) = (sinx - cosx)/(sinx + cosx).
j) √{(1- sinx)/(1+ sinx)}= secx - tanx.
θ
θ θ
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