Tips for Improving the Performance of Students

TIPS FOR IMPROVING THE PERFORMANCE OF STUDENTS

General Tips.

From the beginning of the session above average students must be trained in HOTS at least once in a week.
From the beginning of the session average students must be trained in Text book questions at least twice in a week.
Slips test based on one or two concepts with variation in questions can be given periodically.
During remedial classes chapter wise questions based on Board pattern must be worked out by the students.
Two sets of question papers can be given for every formal test to avoid copying.
Answers must be discussed and the same may be worked out in the H.W note book.

10. Test papers must be always in the board pattern.

11. Basic concepts must be recalled before the beginning of

Lesson without discouraging the students.

12. Revision must be given before every test, stressing on the

Board questions.

13. Sure shot questions in 4 marks, 6marks must be drilled.

15. Chapter wise formulae maybe prepared for a quick revision

16. Minimum five board question papers must be worked out.

17. Solving system of linear equations using matrices, LPP,

maxima& minima and area bounded between the curves

questions must be drilled every day during the break before the

maths examination.

18. From each chapter 1 mark board questions can be drilled.

19. To teach integration start with simple questions before starting

Text book questions.

20. Integration formulae can be drilled by categorizing into different form so

as to enable them to recall while solving the problems.

22. The students must be insisted to write the values in the valued

Based questions.

23.To improve quality result, special classes can be arranged for selected students from each school (not more than five) at cluster level on every Sunday, Second Saturday and Holidays from 1st of November.( Special training classes for students belonging to under achievers and Gifted students at cluster level may be organized

RELATIONS AND FUNCTIONS & INVERSE TRIGONOMETRIC FUNCTIONS

ONE MARK QUESTIONS

Give an example of a relation. Which is

Symmetric but neither reflexive nor transitive.

Give an example of a relation. Which is

Transitive but neither reflexive nor symmetric

Give an example of a relation. Which is

Reflexive and symmetric but not transitive

Give an example of a relation. Which is

Reflexive and transitive but not symmetric

Give an example of a relation. Which is

Symmetric and transitive but not reflexive

Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

Find the principal values of the following:

Find the principal values of the following:

Find the principal values of the following:

Find the principal values of the following:

Find the value of the following:

Find the value of the following:

FOUR MARKS

Show that the relation R in R defined as R = {(a, b) : a ≤b}, is reflexive and transitive but not symmetric

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is Symmetric but neither reflexive nor transitive.

Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i)f : R Rdefined by f (x) = 3 – 4x

Let * be a binary operation on the set Q of rational numbers as follows:

Let A = N × N and * be the binary operation on A defined by

(a, b) * (c, d) = (a + c, b + d)

Show that *is commutative and associative. Find the identity element for * on A, if any.

SIX MARKS

MATRICES AND DETERMINANTS

ONE MARK QUESTIONS

1. Write the value of x – y + z from the following equation|

2. If =, write the value of x.

3. Simplify: cos ᴓ + sin ᴓ

4. Find the value of x + y from the following equation|

5. If a is square matrix such that = a then write the value of - 3A

6. For what value of x is the matrix A = a skew symmetric matrix

7. If a is a square matrix and =2 then write the value of where A’ is the transpose

8. If A is a square matrix of order 3 such that = 64 find

9. If 2,then write the value of (x + Y).

10. Express the following matrix as the sum of a symmetric and a skew symmetric matrix and verify the result.

11. If A =’ find .

12. For what value of K the matrix has no inverse.

13. If A is a square matrix of order 3 and | 3A | = k |A|, then write the value of k.

14.What positive value of x makes the following pair of determinants equal

15. What is the value of the following determinant

16. Evaluate

17. If A=.write .in terms of A

18. If =.write the positive value of x.

19. Let A be a square matrix of order 3 x 3 write the value of |2A|,where |A|=4

20. If =.then write the value of x

21. Evaluate

FOUR MARK QUESTIONS

Using properties of determinants show that

=4.

Show that the matrix A= satisfies the equation -5A+71=0.hence find .

Using properties of determinants prove that =.
If xy ≠z and =0 then show that xyz=-1
Solve for x =0
Using properties of determinants show that

=1 +++

Using properties of determinants prove the following:

If a, b, and c are positive and unequal show that the value of the determinant is negative.
Let A= , then show that A2 - 4A +7 I = 0. Using this result , calculate A5 also.
Let A = and I the identity matrix of order2 × 2, show that I + A = ( I - A ) .
Let A , show that (aI + bA )n = an I + n a n – 1 b A, where I is the identity matrix of order 2 and n € N.
Using properties of determinants , show that = (3 .

SIX MARK QUESTIONS

If A= find .using solve the following system of equations

2x-3y+5z=16;3x+2y-4z=-4;x+y-2z=-3.

Use product to solve the system of equations

x - y+2z=1;2y – 3z=1; 3x – 2y+4z= 2.

3.solve for x y z

+ + = 4 ; - + =1; + -

4. If = and B=.find ;

5 A, B and C ARE THREE FRIENDS. During lunch time they went to buy three plates of chhole-bhature. Each one has some amount but not equal to the price of per plate. They decided to buy together. The sum of all the amount is Rs 90. The sum if the money with A and C is added to the twice of the amount of B which resultsin RS 115. A has the amount equal to the sum of the amount of B and C. Find the amount that each one has, using the matrix method.

6. 10 students were selected from a school on the basis of values for giving awards and were divided into three groups. The first group comprises Hard workers, the second group has Honest and Law Abiding students and the third group conatins Vigilant and Obedient students;Double the number of students of the first group added to the number in the second group gives 13;, while the combined strength of first and second group is four times that of the third group. Using matrix method , find the number of students in each group. Apart from the values ,Hard work, Honesty and Respect for Law, Vigilance and Obedience, suggest one more value, which in your opinion the school should consider for awards.

7. Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prizes at the rate of Rs x, Rsy and Rsz respectively per person. The first institution decided to award respectively 4 , 3 and 2 employees with a total prize money of Rs 37000 and the second institution decided to award respectively 5 , 3 and 4 employees with a total prize money ofRs 47000. If all the three prizes per person together amount to Rs 12000, then using matrix method find the value of x, y and z. what values are described in this question?

8. Using elementary transformations, find the inverse of the matrix

CONTINUITY AND DIFFERENTIABILITY

ONE MARK QUESTIONS

If y =cos xo find dy/dx.
Find the point of dis continuity if any for f(x) =1/x+3.
Is the function f(x) = continuous at x=0 given reason.
Differentiate sin x with respect to ex
Find an angle ,which increases twice as fast as its sine.
Find the slope of the normal to the curve x=
For what value of x in [0, ] does the function sin 2x attains the maximum value.
Find the point on the curve 9y2=x3 , where the normal to the curve makes equal intercepts with the axes.
Find the maximum value of x1/x; x>0.
Show that the function f(x) =+7 is decreasing for x € R(x=0).
Let f(x)= , then find f1(0)
Find the points of removable discontinuity of f(x)= 4-x2 for x

4x-x3

If y=(1+x1/4) (1+x1/2) (1-x ¼) find dy/dx.
The volume of the cube is increasing at a constant rate .prove that the increase in surface area varies inversely as the length of the edge of the cube.
A balloon which always remains spherical has a variable diameter (2x+3) Determine the rate of change of volume with respect to ‘x’.
For what value x; f(x)=+; x=0 is strictly increasing?
Find the angle made by the tangents to the curve y= x2-5x+6 at the points (2,0)and (3,0) .
If the radius of a sphere increase by 0.01% find the percentage increases in the volume.
Without using derivative find the maximum and minimum values

f(x)= sin (4x+3)

A Particle move along the curve y=x2-7 find the point on the curve at which y-co-ordinate is changing twice as fast as x-co-ordinate.

FOUR MARK QUESTIONS

Differentiate y = x3with respect to ‘x’.

(7-3x)5 8+5

The function f(x) ={

If f(x)={ ; x show that f(x) is discontinuous at x=0

If the function f(x)={ is continuous at x=1 find the value of a and b

5. If y=ex tan- x prove that (1+x2) y11-2 (1-x+x2)y + (1- x 2)y= 0

6. If xy =ex-y prove that y1=

7. Verify rolle’s theorem for the function f(x) =log (x2+ab) on (a,b)

x (a+b)

8. find a point on the parabola y=(x-4)2 where the tangent is parallel to the chord joining (4,0) and (5,1).

9.find the intervals in which the function f (x)=log (cosx) in 0<x<2π is increasing or

decreasing.

10.find the equations of the tangent to the curve y=√3x-2which is parallel to the line

4x-2y+5=0.

11.Prove that the triangle of maximum area inscribed is a circle must be equilateral triangle.

12.If the line y=x touches the curve y=x2+bx+c at a point (1,1) find the value of b and c.

13.diffrentiate tan-1 (√1+x - √1-x) with respect to ‘x’.

(√1+x + √1-x)

SIX MARK QUESTIONS

A window is in the form of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m.find the dimension of the rectangle that will produce the largest area of the window.
If the length of three sides of a trapezium of other then the base are equal to 10cm each, then find the area of trapezium when it is maximum.
Find the maximum area of an isosceles triangle inscribed in the ellipse x2/25+y2/16=1 with vertex at one end of the major axis.
An open box with a square base is to be made out of a given quantity of cardboard of area c2sq unit’s show that the maximum volume of the box is c3/6√3 cubic units.
Show that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.
A wire of length 28m is where cut in to two pieces one of the two pieces is the mode is to a square and the other intra circle. What should be the length of the pieces so that the combined area of circle and square is minimum?
A tank with rectangular base and rectangular sides operate the top is to be constructed so that its depth 2m and volume 8 m3 if building of tank cast Rs70 per sq.mt for the base and Rs.45 per sq. mts for sides what is the cast of least expensive tank?
A point on the hypotenuse of a right triangle is at a distance a and b from the sides. Show that the minimum length of the hypotenuse is (a2/3+b2/3)3/2.

INTEGRATION AND APPLICATION OF INTEGRATIONS

ONE MARK QUESTIONS

Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Given . Write the value of f(x).
Evaluate
Evaluate
Evaluate
Write the value of
If , find the value of a.

FOUR MARK QUESTIONS

Evaluate.
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate as a limit of sums.
Evaluate
Evaluate
Evaluate
Find the area bounded by the curve and the line x= 4y-2.
Evaluate
Evaluate
Evaluate

SIX MARK QUESTIONS

Find the area of the circle 4 which is interior to the parabola.
Find the area bounded by the curves and.
Using integration find the area of the region bounded by the triangle whose vertices are (-1, 0), (1, 3) and (3, 2).
Find the area of the smaller region bounded by the ellipse and the line.
Find the area of the region bounded by the two parabolas and.
Find the area of the region.
Using integration, find the area of the region given by.
Evaluate as limit of sums.
Using method of integration find the area of the region bounded by the lines , and.
Find the area of the region.
Evaluate
Evaluate
Find the area of the region in the first quadrant enclosed by x-axis, the line and the circle

DIFFERENTIAL EQUATIONS

ONE MARK QUESTIONS

1. / Write the degree of the differential Equation .
2. / Find the integrating factor of the differential equation.
3. / Find the order of the differential equation
4. / Find the order and degree of the differential equation
5. / Find the general solution of the differential equation
6. / What is the order of the differential equation of all circles of given radius a.
7. / Write the integrating factor of the differential equation (x(log x) +y = 2 log x.
8. / Find the general solution of the differential equation =
9. / What is the order the differential equation representing the family of parabolas =4ax.
10. / Write the general solution of differential equation +=0.
FOUR MARK QUESTIONS
1. / Solve: x(1+)dx – y(1+)dy =0.
2. / Solve:=1+x++x when y=0 and x=0.
3. / Solve:x tan dx +y tan dx=0.
4. / Solve: (3xy +) dx + ( +xy) dy =0.
5. / Solve:
6. / Solve: x= y-x tan(y/x)
7. / Solve: +(sec x)y = tan x
8. / Solve:( x + )= y.
9. / Solve: sec x( ) =y + sin x
10. / Solve: (x+y+1) =1

SIX MARK QUESTIONS

1. / Solve: (y –x) dy = (1+ )dx.
2. / Solve: x tan x +y = tan x.
3. / Solve: y dx + (x-) dy =0.
4. / Solve: (x + ) dx + (y + ) dy = 0.
5. / Find the particular solution of the differential equation - 3y cot x = sin 2x, given y=2 when x=.
6. / Find the particular solution of the differential equation log () = 3x + 4y given that y=0 when x=0.

VECTOR ALGEBRA AND 3-DIMENSIONAL GEOMETRY

ONE MARK QUESTIONS

If , and , find the angle between .
If and , find a unit vector in the direction .
If and find the direction ratios of .
What is the angle between with the vectors of magnitude and respectively and .
Write the position vector of the point dividing the line segment joining the point A and B with position vectors externally in the ratio 1:4, where and
Cartesian equations of the line AB are: . Write the Dr’s of the line parallel to AB.
If and , find the unit vector in the direction of
Let.
Write the value of .
For two non-zero vectors write when holds.
Find , if for a unit vector , .
A and B are two points with p.v’s and respectively. Write the p.v of a point P which divides the line segment AB internally in the ratio 1:2
If and are two vectors of magnitude 3 and2/3 respectively such that x is a Unitvector, write the angle between and .
Ifand , find the projection of on.
Write the distance between the parallel planes 2x+y+3z=4 and 2x+y+3z=18.
Write the value of ×+.
Write the projection of the vector on the vector
Write the vector equation of a line given by .
Find the value of such that the line is perpendicular to the plane 3x-y-2z=7.
Find the value of are perpendicular to each other
If find the projection of on .
Find the value of such that the line is perpendicular to the plane 3x-y-2z=7.
Write the distance of the following plane from the origin: .
Write a vector of magnitude 9 units in the direction of vector
Find x.

FOUR MARK QUESTIONS

If vectors are such that and , and , find the angle between .

Find the length of the perpendicular from the point (2, -1, 5) to the line .

Find the co-ordinates of the point where the line meets the plane

x+y+4z=6.

Find the vector equation of the line parallel to the line and passing through

(3 , 0 , -4). Also find the distance between these two lines.

If are the PVs of the vertices of A,B and C of a ΔABC respectively . Find an expression for the area of the ΔABC and hence deduce the condition that for the point A,B and C to be collinear.

Show that the points A,B and C with PVs and respectively , are the vertices of a right triangle. Also find the remaining angles of the triangle.

If are two unit vectors and is the angle between them, then prove that .

Find the shortest distance between the lines, whose equations are .

If are the diagonals of the parallelogram with sides , then find the area of the parallelogram in terms of and hence find the area of the parallelogram with

If are the diagonals of the parallelogram with sides , then find the area of

the parallelogram in terms of and hence find the area of the parallelogram with .

Show that the lines and

are intersecting. Hence find their point of intersection.

Find the vector equation of the plane passing through the point (2 , 1 , -1)and (-1 , 3 , 4) and

perpendicular to the plane x-2y+4z=10.

If , then find the value ofλ , so that and are perpendicular vectors.

Find the equation of the plane passing through the line of intersection of the planes

whose perpendicular distance from origin is

unity.

Let , and. Find a vector which is

perpendicular to both and and satisfying.

Find the distance between the point P(6, 5, 9) and the plane determined by the

points A(3, -1, 2), B(5, 2, 4), and C(-1, -1, 6).

Find the equation of the perpendicular drawn from the point P(2, 4, -1) to the line

. Also, write down the coordinates of the foot of the perpendicular