UPKAR PRAKASHAN, AGRA–2

ByDr. N.K. Singh

(H.O.D. of Maths)

© Publishers

Publishers

UPKAR PRAKASHAN(An ISO 9001 : 2000 Company)

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ISBN : 978-93-5013-261-6

Price : 425/-(Rs. Four Hundred Twenty Five Only)

Code No. 312

Printed at : UPKAR PRAKASHAN (Printing Unit) Bye-pass, AGRA

PREFACE

In this book short-cuts have been given to solve objective type sums in minimumpossible time. In order to solve objective type sums one should have conceptual clarityand should be able to apply the concepts properly. Almost all books containingobjective type sums do not contain subject matter. In fact they provide a collection ofproblems. That is why students required other books to study subject matter and to buildup concepts before they start solving objective type sums. Keeping this requirement ofstudents in mind, I have discussed the subject matter in detail with suitable illustrationsso that a reader of this book may not feel the need of consulting any other book.

In each chapter, subject matter has been discussed in a systematic and lucid mannerwith illustrations. Illustrative objective type examples have been given to enable thestudents to apply the concepts studied, in subject matter, in solving objective type sums.Normally, while solving objective type sums, students do not develop confidencebecause of non-availability of proper solutions from which they can verify their attempt.Keeping this in mind, solutions of objective type exercises have been given in the samesequence.

I wish to make a special mention of my son Kashish, daughter Manjory and thesupport of my wife for their involvement during the preparation of this edition.

I am confident that the readers will find the present form of the book, best suited totheir need. As there is always scope for improvement, valuable suggestions fromworthy teachers and dear students will be gratefully received.

—Dr. N. K. Singh

CONTENTS

Previous Year’s Solved Paper1. Sets, Relations and Functions ………………………………………… 3–172. Complex Numbers ………………………………………….………….. 18–353. Quadratic Equations……………………………………………………. 36–534. Sequences and Series…………………………………………………… 54–665. Logarithm, Exponential and Logarithmic Series……………………….. 67–846. Binomial Theorem……………………………………………………… 85–1037. Permutations and Combinations ……………………………………….. 104–1108. Matrices and Determinants …………………………………………….. 111–1379. Probability……………………………………………………………… 138–161

10. Inequalities and Mathematical Induction ……………………………… 162–17611. Trigonometric Ratios and Equations …………………………………... 177–20212. Properties of Triangle ………………………………………………….. 203–22713. Heights and Distances………………………………………………….. 228–24514. Inverse Trigonometric Functions ……………………………………… 246–26715. Rectangular Cartesian Coordinates……………..……………………… 268–28416. Straight Line and Pair of Straight Lines ………..……………………… 285–30717. Circles and Family of Circles ……………..…………………………… 308–33618. Conic Section …………………………….……………………………. 337–35919. Functions ……………………………………………………………… 360–37120. Limits, Continuity and Differentiability ……………………………… 372–40121. Differentiation ………………………………………………………... 402–41422. Applications of Derivatives ……………....……………………………. 415–42823. Indefinite Integration ……………...…………………………………… 429–44024. Definite Integrals ……………………...……………………………….. 441–44525. Area of Bounded Regions …………...………………………………… 446–44826. Differential Equations ………………...……………………………….. 449–46127. Vector Algebra …………………...……………………………………. 462–48628. Statistics …………………………...…………………………………… 487–49829. Statics ………………………………..………………………………… 499–51430. Dynamics ……………………………………………………………… 515–529

Level of Difficulty–1 ……………………...…………………………… 530–608Level of Difficulty–2 ………………………...………………………… 609–696

SYLLABUS

ALGEBRA : Sets relations and functions,DeMorgan’s Law, Mapping Inverse relations.Equivalence relations, Peano’s axioms, Definitionof rationals and integers through equivalencerelation, Indices and surds, Solutions ofsimultaneous and quadratic equations, A.P., G.P.and H.P., Special sums i.e., Σn2 and Σn3 (nΣN),Partial fraction, Binomial theorem for any index,exponential series, Logarithm and Logarithmicseries. Determinants and their use in solvingsimultaneous linear equations, Matrices, Algebraof matrices, Inverse of a matrix, Use of matrix forsolving equations.

PROBABILITY : Definition, dependent andindependent events, Numerical problem onaddition and multiplication theorem of proba-bility.

TRIGONOMETRY : Identities, Trigono-metric equations, Properties of triangles, solutionof triangles, heights and distances, Inversefunction, Complex numbers and their properties,Cube roots of unity, DeMoivre's theorem.

CO-ORDINATE GEOMETRY : Pair ofstraight lines, Circles, General equation of seconddegree, parabola, ellipse and hyperbola, tracing ofconics.

CALCULUS : Limits and continuity offunctions, Differentiation of function of function,tangents and normal, Simple examples of Maximaand Minima, Indeterminate forms, Integration offunction by parts, by substitution and by partialfraction, definite integral, application to volumesand surfaces of frustums of sphere, cone andcylinder. Differential equations of first order andof first degree.

VECTORS : Algebra of vectors, scalar andvector products of two and three vectors and theirapplications.

DYNAMICS : Velocity, Composition ofvelocity, relative velocity, acceleration, composi-tion of accelerations, Motion under gravity,Projectiles, Laws of motion, Principles of conser-vation of momentum and energy, direct impact ofsmooth bodies.

STATICS : Composition of coplanar,concurrent and parallel forces, moments andcouples, resultant of a set of coplanar forces andconditions of equilibrium, determination ofcentroid in simple cases, problems involvingfriction.

MathematicsState Entrance Exam.

Solved Paper(Based on Memory)

Mathematics1. A set contains n elements. The power set

contains—(A) n elements (B) 2n elements(C) n2 elements (D) None of these

2. The relation R defined on the set N of naturalnumber by—

xRy ⇔ 2x2 – 3xy + y2 = 0, ∀x, y ∈ N

then—

(A) Symmetric but not reflexive

(B) Only symmetric

(C) Not symmetric but reflexive

(D) None of the above

3. A survey shows that 64% of Americans likecheese whereas 76% like apples. If x% of theamerican like both cheese and apples, then—

(A) x = 39 (B) x = 63

(C) 39 ≤ x ≤ 63 (D) None of these

4. The value of sin ⎣⎢⎢⎡

⎦⎥⎥⎤π

2 – sin– 1 ⎝⎜

⎛ ⎠⎟⎞

– √⎯ 32

is—

(A)12

(B) – 12

(C) 1 (D) – 1

5. If a ≤ sin– 1 x + cos– 1 x + tan– 1 x ≤ b, then—

(A) a = 0, b = π

(B) a = 0, b = π2

(C) a = π2, b = π

(D) None of these

6. If p is the perpendicular length from origin to

the line xa +

yb = 1, then—

(A)1p2 =

1a2 +

1b2 (B)

1p2 =

1a2 –

1b2

(C)1p2 = –

1a2 –

1b2 (D)

1p2 = –

1a2 +

1b2

7. The condition that the straight line joining theorigin to the points of intersection of the line4x + 3y = 24 with the circle (x – 3)2 + (y – 4)2

= 25—

(A) Are coincident

(B) Are perpendicular

(C) Make equal angle with x-axis

(D) None of the above

8. The equation of circle passing throughthe origin and point of intersection of circlex2 + y2 – 2x + 4y – 20 = 0 and line x + y – 1= 0 is—

(A) x2 + y2 + 22x – 16y = 0

(B) x2 + y2 + 22x + 16y = 0

(C) x2 + y2 – 22x – 16y = 0

(D) None of the above

9. The angle between the tangents drawn fromorigin to the circle (x – 7)2 + (y + 1)2 = 25is—

(A)π3

(B)π6

(C)π2

(D)π8

10. If x – 1 = 0 is the directrix of parabola y2 – kx+ 8 = 0, then k is equal to—

(A) 1/8 (B) 8

(C) 4 (D) 1/4

11. The locus of point of intersection of tangentto an ellipse at two points, sum of whoseeccentric angle is constant, is—

(A) Parabola (B) Circle

(C) Ellipse (D) Straight line

12. The length of transverse axis of the rectangu-lar hyperbola xy = 18 is—

(A) 6 (B) 12

(C) 18 (D) 3

4M | State Engg. Ent.

13. Find the least value of n for which ( )1 + i1 – i

n

= 1—(A) 4 (B) 3(C) – 4 (D) 1

14. sin π14

sin 3π14

sin 5π14

sin 7π14

is equal to—

(A) 1 (B)14

(C)18

(D)√⎯ 27

15. If the roots of the equation ax2 + bx + c = 0are real and distinct then—

(A) Both roots are greater than – b2a

(B) Both roots are less than – b2a

(C) One of the roots exceeds – b2a

(D) None of the above

16. The number of positive integer satisfying theinequality n + 1Cn – 2 – n + 1Cn – 1 ≤ 100 is—

(A) 9 (B) 8

(C) 5 (D) None of these

17. If H is harmonic mean between P and Q.

Then the value of HP

+ HQ

is—

(A) 2 (B)PQ

P + Q

(C)P + QPQ

(D) None of these

18. Larger of 9950 + 10050 and 10150 is—

(A) 10150

(B) 9950 + 10050

(C) Both are equal

(D) None of these

19. If A is invertible matrix and B is any matrix,then—

(A) Rank (AB) = Rank (A)

(B) Rank (AB) = Rank (B)

(C) Rank (AB) > Rank (A)

(D) Rank (AB) > Rank (B)

20. Rank of the matrix A =

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤1 2 2

2 1 2

2 2 1

is—

(A) 0 (B) 1(C) 2 (D) 3

21. If y = | cos x | + | sin x |, then dydx

at x = 2π3

is—

(A)1 – √⎯ 3

2(B) 0

(C)12 ( )√⎯ 3 – 1 (D) None of these

22. The graph of the function y = f (x) issymmetrical about the line x = 2. Then,(A) f(x + 2) = f(x – 2) (B) f(2 + x) = f(2 – x)(C) f(x) = f(– x) (D) f(x) = – f(– x)

23. limx → 0

(cosec x)1/log x is equal to—

(A) 0 (B) 1

(C)1e

(D) None of these

24. ∫

xe – 1 + ex – 1

xe + ex dx is equal to—

(A)1e log (xe – ex) + c

(B)1e log (xe + ex) + c

(C)1e

log (ex – xe) + c

(D) None of the above

25. ∫0

1

dx

(√⎯⎯⎯⎯1 + x + √⎯ x) is equal to—

(A)43 ( )√⎯ 2 – 1 (B)

34 ( )√⎯ 2 – 1

(C)43 ( )1 – √⎯ 2 (D)

34 ( )1 – √⎯ 2

26. ∫0

1

|5x – 3| dx is equal to—

(A)1013

(B)3110

(C)1310

(D) None of these

State Engg. Ent. | 5M

27. For 0 ≤ x ≤ π, the area between the curvey = sin x and x-axis is—(A) 1 sq unit (B) 0 sq unit(C) 2 sq unit (D) – 1 sq unit

28. The order and power of differential equationd2ydx2 + 7

dydx

+ ∫y dx = sin x

is—(A) 1, 3 (B) 3, 1(C) 1, 2 (D) 2, 1

29. The solution of differential equation x cos2 ydx = y cos2 x dy is—(A) x tan x – y tan y – log (sec x / sec y) = c(B) y tan y – x tan x – log (sec x . sec y) = c(C) x tan x – y tan y + log (sec x . sec y) = c(D) None of the above

30. The equation of tangent of the curve y =be– x/a at the point, where the curve meet y-axis is—(A) bx + ay – ab = 0(B) ax + by – ab = 0(C) bx – ay – ab = 0(D) ax + by – ab = 0

31. The intersection angle of the curve xy = a2

and x2 – y2 = a2 is—

(A)π3

(B)π6

(C)π2

(D)5π6

32. The point of intersection of line x – 6– 1

= y + 1

0 =

z + 34

and plane x + y – z = 3 is—

(A) (2, 1, 0) (B) (7, – 1, – 7)

(C) (1, 2, – 6) (D) (5, – 1, 1)

33. At a point the addition of two active force is18 N. If the magnitude of resultant is 12 Nand meet at right angle. Then, magnitude offorces are—

(A) 5N, 13N (B) 6N, 12N

(C) 8N, 10N (D) None of these

34. The angle between two active forces P + Qand P – Q is 2α. If their resultant make angleθ with bisector of angle. Then—

(A) P cos θ = Q cos α(B) P tan θ = Q tan α(C) Q cos θ = P cos α(D) Q tan θ = P tan α

35. If the distance travel by a uniformlyaccelerated particle in pth, qth and rth secondare a, b and c respectively. Then—(A) (q – r)a + (r – p)b + (p – q)c = 1(B) (q – r)a + (r – p)b + (p – q)c = – 1(C) (q – r)a + (r – p)b + (p – q)c = 0(D) (q + r)a + (r + p)b + (p + q)c = 0

36. If the maximum height and horizontal rangeof a projectile are same. Then, projectionangle is—(A) 45° (B) 30°(C) tan– 1 3 (D) tan– 1 4

37. A particle moves by constant accelerationfrom initial position. If the distance travel by

particle in last second is 716

of total distance

travel by particle. Then, time of motion is—(A) 4s (B) 4/7 s(C) 7s (D) None of these

38. Inverse of function f(x) = 10x – 10– x

10x + 10– x is—

(A) log10 (2 – x)

(B)12 log10 ( )1 + x

1 – x

(C)12 log10 (2x – 1)

(D)14

log10 ( )2x2 – x

39. ( )1 + cos π8 ( )1 + cos

3π8 ( )1 + cos

5π8

( )1 + cos 7π8

is equal to—

(A)12

(B)18

(C) cos π8

(D)1 + √⎯ 2

2√⎯ 2

40. The solution of the equation sec θ – cosec θ =43 is—

(A)12 [nπ + (– 1)n sin– 1 3/4]

State Entrance ExaminationMathematics B.Tech/B.Pharm/B.Arch

Publisher : Upkar Prakashan ISBN : 9789350132616 Author : Dr. N.K. Singh

Type the URL : http://www.kopykitab.com/product/5090

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