QUESTION BANK. Class : 10+1 & (Mathematics)

Transcription

1 QUESTION BANK Class : + & + (Mathematics) Question Bank for + and + students for the subject of Mathematics is hereby given for the practice. While preparing the questionnaire, emphasis is given on the concepts, students, from the eamination point of view. We hope that you might appreciate this question bank. We welcome suggestions to improve the question bank. Rumkeet Kaur Subject Epert Maths SCERT, Punjab (M) : Jaspreet Kaur Lect. Maths GSSS Lohgarh (Mohali) (M) : Jagjit Singh GPS Mauli Baidwan (Mohali) (M) :

2 CLASS + MATHEMATICS 9

3 CLASS + CONTENTS S.No. Chapter. Sets. Relations & Functions. Trigonometric Functions. Principle of Mathematical Induction. Comple numbers and Quadratic Equations 6. Linear Inequalities 7. Permutations & Combinations 8. Binomial Theorem 9. Sequence & Series. Straight lines. Conic Sections. Introduction to Three-dimensional Geometry. Limits & Derivatives. Mathematical Reasoning. Statistics 6. Probability

4 Sets (Marks - ) Q.. ( A B) C is equation to (a) (c) C C C A B (b) A C C A B (d) None of these B C Q.. Q.. The set of girls in a boy's school is (a) a null set (b) singleton set (c) a finite set (d) not a well defined collection The set of principals in a school is (a) a null set (b) a singleton set (c) an infinite (d) None of these Q.. Solution set of equation + 6 in roster form is (a) {-, -} (b) {, } (c) {-, } (d) {-, } Q.. Set of even prime numbers is (a) a Null set (b) a Singleton set (c) a finite set (d) an infinite set Q.6. The se { u: u R, u 9, u } A is (a) φ (b) {7} (c) {-7} (d) {-7,7} Q.7. Q.8. In a college of students, every student reads newspapers and every newspaper is read by 6 students. The number of newspapers is (a) At least (b) at most (c) eactly (d) None of the above If S {,,,,7} then the total number of subsets of S is (a) 6 (b) (c) (d)

5 Q.9. if A < B then B A equals (a) A (b) B A (c) B (d) None of these Q.. Set of odd natural numbers divisible by is (a) null set (b) a singleton set (c) a finite set (d) an infinite set Q.. The set of { : R, 6 and 6} A is equals (a) φ (b) {,,} (c) {} (d) {} Q.. If A & B are any two sets then A ( A B) equals (a) A (b) B (c) A C (d) B C

6 CHAPTER SETS ( Marks Questions). Let A {,,},{,},{ 6,7,8 } Determine which of the following is true or false : (a) A (b) {,, } A (c) {6, 7, 8} A (d) {{, }} A (e) φ A (f) φ A (g) {6, 7, 8} A (h) A. If A {, }, B { : is a root of }, then find (i) (ii) A B A B (iii) Are they equal sets? (iv) Are they equivalent sets?. Let U {,,,,, 6, 7, 8,9}, A {,,, }, B {,, 6, 8}, Find : (a) C A (b) C B C (c) ( ) C A (d) ( A B) C. If A {,,}, B {,, 6} and C {7, 8, 9}, verify that : A (B C) (A B) (A C). Out of members in a family, like to take tea and like coffee. Assume that each one likes at least one of the two drinks. How many like : (a) both tea and coffee (b) only tea and not coffee (d) only coffee and not tea 6.(i) Write the following sets in set builder form : A {,,, 7, 9}, E {,,, } (ii) Write the following sets in Roster Form : A { : is an integer and < <7}, B { : is a natural number less than 6}

7 7. Let A{,}, B {,,,}, C{,6} and D{,6,7,8}. Verify that (a) A ( B C) ( A B) ( A C) (b) (A C) (B D) 8. Out of members in a family, like to take tea and like coffee. Assume that each one likes at least one of the two drinks. How many like (i) (ii) Both tea and coffee. Only tea and not coffee. 9. Prove that (i) A B B C A C (ii) B A A BA. Prove that (i) (ii) A C -B C B-A B-A B A C

8 CHAPTER-RELATIONS AND FUNCTIONS ( mark question) Q. Let R {(,),(,),(,),(,),(,) } be a relation on the set A {,,,}. The relation R is (A) a function (B) transitive (C) not symmetric (D) refleive. Q. Let R {(,),( 6,6), ( 9,9),(, ), ( 6, ), (,9), (, ), (,6) } be a relation on the set A {,6,9, }. The relation is (A) refleive only (B) refleive and transitive only (C) refleive and symmetric only (D) an equivalence relation Q. Let R be the real number. Consider the following subsets of RR S {(, y) : y + and< < } T {(, y ): y is an integer } which one of the following is true? (A) T is an equivalence relation on R but S is not. (B) Neither S nor T is an equivalence relation on R. (C) (D) Both S and T are equivalence relations on R S is an equivalence relation on R but T is not. Q. Let f() [] then f is equal to : (a) (b) (c). (d) None of these

9 Q. Range of f() +, where is a real number, is : (a) [, ) (b) (, ] (c) (, ) (d) [, ] Q.6 The domain of f ( ) + log e( ) is : (a) < (b) (c) < (d) e e e Q.7 For real, let f ( ) + + then : (a) f is onto R but not one-one (c) f is neither one-one nor onto R (b) f is one-one and onto R (d) f is one-one but not onto R Q.8, then find the value of f ( ) + f If f ( ) (a) (b) (c) (d) Q.9 Find the domain of the function f ( ) 7 + a) (,) b) [, ) c) (, ] [, ) d) (,) [, ) Q.. Let (a) (b) (c) (d) f : R R be defined as f ( ). Then f is one-one onto, f is many one-onto f is one one but not onto f is neither one-one nor onto 6

10 ( marks question) Q. If G { 7,8} and {,,} Q. If {,} P form the set P P P. Q. Let A {,,,} and B {,7,9} (i) Determine : A Band represent it graphically. H find G H and H G. (ii) B A and represent it graphically. (iii) Is n ( A B) n( B A ) Q. Let A {,,, } B {,,} andc {,} verify that ( B C ) ( A B) ( A C ) A Q. If A {,9,6, }, B {,,,} and R is the relation "is square of" from A to B. Write down the set corresponding to R. Also find the domain and range of R. Q.6 If R is a relation "is divisor of" from the set A {,,} to B {,,} set of ordered pairs corresponding to R. Q.7 Let A {,} and B {,} Q.8 Let N, write down the. Find the number of relations from A to B. N be defines by f ( ). Show that f is not an onto function. + Q.9 If 'f' is a real function defined by f ( ) then prove that f ( ) f f ( ) + ( ) + Q. If f ( ), +, then show that f ( f ( ) ) +, + 7

11 Q. The function 't' which maps temperature in Celsius into temperature in Fahrenheit is 9 c t + find (i) t() (ii) t(8) (iii) t(-) (iv) the value of c when defined by ( c) t(c) Q. Find the domain of the function f ( ) Q. The function f is defined by: f ( ),, +, < > Draw the graph of f() Q. Let A{9,,,,} and f : A N be defined by f () The highest prime factor of n. Find the range of f.. Radian measure of ' is : CHAPTER TRIGONOMETRIC FUNCTIONS (a) radians (b) radians (c) π radians. Radian measure of is : (a) π 6 (b) 9. Value of sin 76 is : (c) ( Mark Questions) (d) None of these 6 (d) None of these 9π (a) (b) (c) (d) 76. The principal solution of tan is : (a) π (b) π (c) π (d) π 8

12 . The most general solution of tan θ, cos θ is : (a) θ π + 7π n (b) θ nπ + ( ) 7π (c) θ n π + (d) None of these n 7π 6. The value of cos + cos 68 + cos7 is : (a) (b) (c) (d) 7. The equation sin + cos has : (a) only one solution (b) two solutions (c) infinite many solutions (d) no solution 8. The value of cos cos7 sin7 is : (a) (b) 8 (c) (d) 6 9. Sin 7 + sin 6 sin sin (a) sin 7 (b) cos 7 (c) sin 6 (d) cos 6. The period of the function sin is (a) π (b) π (c) π (d) None of these 9 ( Marks Questions). If in two circles, arcs of the same length subted angles of 6 and 7 at the centre, find the ratio of their radii.. Find the angle between the minute hand and the hour hand of a clock when the time is :.. Prove that : (a) sec A sec A tan A + tan A (b) tan θ sin θ tan θ sin θ. cosec θ cosec θ Prove that : + sec θ cosec θ cosec θ+

13 . If cot functions. 6. Prove that : 7. Prove that : θ and θ lies in the second quadrant, find the values of other five sin cos π + cos 6 π tan π π π + sec + tan π 8. Prove that : tan 7 tan + tan 9 9. Find the principal solutions of the following : (a) tan (b) sec. Prove that the equation cosθ + is impossible if be real. o o o o. Prove that sin cos + cos sin 66. Simplify the following sin9 ( + θ) tan( 7 + θ) cot9 ( + θ) cosec ( 7 + θ). If +φ prove that ( tanθ )( + tan ) θ + φ. Prove that sin 7 cos cos 7 sin. Prove that tan 7 tan + tan 6. Prove that 7. Prove that cos sin π + π sec + tan π π + cos tan 6 π π 9 8. sin A cos B + cos A sin B + cos A cos B + sin A sin B + sinθ 9. secθ+ tanθ sinθ. cosθ sinθ sinθ + cosθ. Prove that sin 6 sin sin sin. Write down the values of cos 68 cos 8 + sin 68 sin 8 6

14 ( Marks Question). In any triangle ABC, prove that : sin B C b c cos a A. In any triangle ABC, prove that : cos A + a cos B b + cos C C a + b + c abc. Prove that : tan 9 tan 7 tan 6 + tan 8. Show that cos 8θ cosθ. Prove that ( cosα + cosβ) + ( sinα + sinβ) α cos β 6. Prove that sin sin sin6 sin7 6 tanθ+ tanθ 7. Prove that cosθcosθ tanθ tanθ 8. Prove that ( tanθ) tanθ tanθ 6tanθ+ tanθ 9. If A + B+ C π prove that sin A + sin B + sin C sin Asin Bsin C. Prove that π π π π sin sin sin sin 6. Find the values of other trigonometric function (i) (ii) cot, lies in third quadrant tan, lies in second quadrant 6

15 CHAPTER PRINCIPLE OF MATHEMATICAL INDUCTION ( Marks Questions). By using the Principle of mathematical induction n - is divisible by 8 for all n N. By Principle of Mathematical Induction, prove that : n n n 6 ( n + )( + ). Prove that n- + is divisible by for all n N n n. For every positive integer n, prove that 7 is divisible by.. By principle of mathematical Induction, prove that : n( n + ) n n + 6. By Principle of Mathematical induction, Prove that +... n ( n + ) n + (n 6 7. By principle of Mathematical induction, prove that n ) n ( n + ) n + for all n > for all n. 8. By Principle of Mathematical induction. Prove that Prove the rule of eponents : (ab) n a n b n by using principle of mathematical Induction for every natural number. n n +... for all n N. i is : CHAPTER COMPLEX NUMBERS AND QUADRATIC EQUATIONS (a) i (b) (c) (d) i. Solution of + is : (a) (b) (c) ± (d) ± i ( mark Question) 6

16 . Comple conjugate of i is : (a) i + (b) i (c) i + (d) None of these. Additive inverse of comple number 7i is: (a) + 7i (b) + 7i (c) 7i (d) None of these. The imaginary part of (a) zero (b) i + is : 6 6. The value of i + i + i + i is : (c) (a) i (b) i (c) zero (d) (d) None of these 7 7. i + equals : i (a) (b) i (c) i (d) z + i 8. The comple number z + iy, which satisfies the equation, lies on : z i (a) The line y (c) the ais (b) a circle through the origin (d) None of these 9. The modulus of i i + + i is : (a) units (b) units (c) units (d) units. The conjugate of a comple number is (a) i (b) i. Then that comple number is : i (c) i + (d) i + ( Mark Questions). Solve the equation + +. Solve the equation 7i 6

17 . Find the conjugate of ( i )( + i ) ( + i)( i). Prove that + l + l + i is real number. Epress the comple no. 9 9 i + i in the form of a+ ib 6. Find the multiplicative inverse of -i 7. Epress (+7i) in the form a+ib. 8. Evaluate i 8 + i 9. Find the modulus of + i i i i + i. If + i i m then fine the least +ve integral value of m. (6 Mark Questions) i + i. Show that a real value of will satisfy the equation a ib where a, b are real. a + ib c + id. If + iy, show that + y a + b c + d. If ( + iy) u + iv, then show that : u v + ( y ) y if a +b,. Find the modulus and the argument of the comple number z + i. If z, z are comple numbers, such that z z + z z + 7i 6. Convert into polar form : ( ) i 7. Solve : ( + ) + 6 i i z z is purely imaginary number, find 8. z prove that ( z ) If, z z + is purely imaginary number. What will you 6

18 conclude if z. 9. Convert into polar form : z i π π cos + i sin a + ib c + id a ib c id. If ( + iy ), then prove : ( iy ), and + y a c + b + d 6

19 CHAPTER LINEAR IN EQUATIONS (6 Marks Questions). Solve the following inequations and show the graph on number line : (a) 6 < (b) + 9 (c) 7 + > (d). Solve the following inequations and show that graph on number line : > (a) (b) 7 <. Solve the following system of inequations : + > and + <. Solve the following system of inequations : ( + ) < 6( ) and. Solve graphically : (i) < (ii) y Find the region enclosed by the following inequations + y, + y,, y 7. Find the region for following inequation : + y, + y, and y 8. Solve the following system of inequalities graphically: + y 6, y,,, y 66

20 CHAPTER PERMUTATION AND COMBINATION Multiple Choice Questions :. 7!! is :. (a) 7! (b)! (c) (d) The value of is! :!! (a) (b) 66 (c) 76 (d). The value of C C is : ( Mark Questions) (a) (b) (c) (d). If P n P, then n is : (a) 8 (b) 6 (c) 7 (d). If n 7 and r, then the value of n C r is : (a) (b) (c) (d) 7 6. If n 8 and r then the value of n P r is : (a) (b) 6 (c) (d) 8 7. Evaluate : C + C + C +... C (a) (b) (c) (d) 8. The number of ways in which 6 men and women can sit at a round table if no two women are to sit together is given by : (a) (b)!! (c) 7!! (d) 6!! n+ n 9. If P : P :, then the value of n equal : n n (a) (b) (c) (d) (e). If n n then the value of n equal : c c (a) - (b) (c) 7 (d) -7 67

21 n P. Find n such that, n >. n P ( Marks Questions). In how many ways can 9 eamination papers be arranged so that the best and the worst papers never come together?. The letters of the word RANDOM are written in all possible orders and these words are written out as in dictionary. Find the rank of the word RANDOM.. How many natural numbers less than can be formed with the digits,,, and if (a) no digit is repeated (b) repetition of digits is allowed.. Find out how many arrangements can be made with the letters of the word MATHEMATICS. In how many ways can consonants occur together? 6. In how many ways can persons A, B, C, D and E sit around a circular table if : (a) B and D sit net to each other. (b) A and D do not sit net to each other. 7. How many triangles can be obtained by joining points, five of which are collinear? 8. If m parallel lines in a plane are intersected by a family of n parallel lines, find to number of parallelograms formed. 9. What is the number of ways of choosing, cards from a pack of playing cards? In how many of these : (a) four cards are of the same suits (b) are face cards. Prove that n (... ( ) ) n! n n C n 68

22 CHAPTER BIONOMIAL THEOREM ( Marks Questions) y + z. Find the number of terms in the epansion of ( ) n. Epand +. Determine the two middle terms in the epansion of ( ). Find the term containing, if any, in 8 + a. Find the term, which is independent of in the epansion of For what value of m, the coefficients of (m+) th and (m+) th terms in the epansion of ( + ) are equal. 7. Which term is independent of in the epansion of Evaluate n r n cr r 9. What is the fourth term in the epansion of? 6 7. Find the middle term in the epansion of 9. Find the middle term in the epansion of + 9 y +. Find the positive value of m for which the coefficient of in the epansion of ( ) m is 6.. Find the r th term in the epansion of r +. 69

23 . If p is a real number and if the middle term in the epansion of find the values of p. P + 8 is, then. If n is even and the middle term in the epansion of value of n. n + is 9 6, then find the 6. Find the co-efficient of in the epansion of. 7. Find the term independent of in the epansion of 8. Find the epansion of + 9. Find the term independent of in the epansion of 6. Find the coefficient of in the epansion of ( + ) 6. CHAPTER SEQUENCE AND SERIES. th term of a G.P. is, then the product of first 9 terms is : (a) 6 (b) 8 (c) (d) None of these. If a, b, c are in A.P., then : (a + b c) (b + c a) (c + a b) equals : (a) abc (b) abc (c) abc (d) abc ( Mark Questions). Sum of the series is n : n (a) ( n ) n (b) ( n + )( n + ). The sum of the first n odd numbers is (a) n (b) n (c) n( n ) (d) n (c) ( n + )( n ) n( n +) n (d) ( n + ) 7

24 . If the third term of a G.P. is, then the product of its first terms is : (a) (b) 8 (c) (d) Cannot be determined. 6. th term of a G.P. is, then the product of its 9 terms is : (a) 6 (b) (c) (d) None of these 7. If the p th, q th and r th terms of G.P. are a, b and c respectively. Then a q r b r p c p q is equal to (a) (b) (c) (d) - 8. Find the number of terms between and which are divisible by 7. (a) 8 (b) (c) 9 (d) 7 9. Which term in the A.P.,,-,... is -? (a) (b) (c) (d) 9 ( Marks Questions). Determine nd term and r th term of an A.P. whose 6 th term is and 8 th term is.. Sum of the first p, q and r terms of an A.P. are a, b and c respectively. Prove that q p b q c r ( q r ) + ( r p ) + ( p q ). If the th term of an A.P. is and the sum of the first four terms is, what is the sum of the first terms?. Insert A.M s between and 9.. The sum of three numbers in A.P. is and their product is 8. Find the numbers. 6. The digits of a positive integer having three digits are in A.P. and their sum is. The number obtained by reversing the digits is 9 less than the original number. Find the number. 7

25 7. If a, b, c are in A.P., prove that :,, b + c c + a a + b are also in A.P. 8. Find a G.P. for which sum of the first two terms is and fifth term is times 9. the third term. The value of n so that a a n + n + n + b + b n may be the geometric mean between a and b.. Determine the number n in a geometric progression { a n}, if a, an 96and s n 89.. Sum to n terms : Find the sum of terms of a sequence : 7, 7.7, 7.77, 7.777,. The arithmetic mean between two numbers is and their geometric mean is 8. Find the numbers.. The first term of a G.P. is and the sum to infinity is 6. Find the common ratio.. Evaluate :. 6. Find the sum of n terms of the series : Sum to n terms the series : CHAPTER-STRAIGHT LINES ( mark question) Q. Find the distance of the point (,) from line -y-9 (A) (B) (C) Q. The equation of straight line passing through the point (,) and perpendicular to the line y is (A) + y (B) + y (C) + y 8 (D) + y (D) 7

26 Q.. Q.. Q.. Q.6. Find the value of for which the points (,-) (,) and (,) are collinear. (A) (B) - (C) (D) Find the distance between the parallel lines -y+7 and -y+ (A) (B) (C) (D) Find the angle between the lines +y+7 and -y+ (A) (B) (C) 9 (D) 7 Find the values of k for the line ( ) + ( k ) y + k 7k + 6 k which is parallel to the -ais (A) + (B) (C) - (D) Q.7. The lines a n+ b y + c and a n+ b y + c are perpendicular to each other if (A) a b ab (B) a a b b (C) a a + bb (D) a b + a b Q.8. Find the equation of a line passing through the point (,) and parallel to y + (A) + y + 6 (B) y + (C) + y (D) y + 9 Q.9. Find the slope and y-intercept of st. line + 6 y 7 (A) (C) 6, 7, (B) 6, (D) 6 7, Q.. Find the equation of the line perpendicular to the line y + 7 and having - intercept is. (A) + y (C) (B) y y + 7 (D) y 7

27 Q.. A line has slope m and y intercept, the distance between the origin and the line is equal to (A) (E) m m m (B) m (C) Q.. Find the distance between st. line + y (A) 6 (B) 9 (C) (D) m + m + m and the point (, ) (D). Find a point on ais, which is equidistant from (7, 6) and (, ). ( Marks Questions). Show that the points (, ), (, ) and (, ) are the vertices of a right-triangle.. Find the coordinates of the points, which divide internally and eternally the line joining (, ) and (, 9) in the ratio :.. Find the centroid by the triangle with vertices at (, ), (, ) and (8, ).. Find the coordinates of incentre of the triangle whose vertices are ( 6, 7); (, 7) and (, 8). 6. A point moves so that the sum of its distances from the points (ae, o) and ( ae, o) is a. Prove that its locus is : y + where ( b a e ) b a 7. State whether the two lines are parallel, perpendicular or neither parallel nor perpendicular: (a) Through (, 6) and (, ); through (9, ) and (6, ). (b) Through (, ) and (, ); through (6, ) and (, ). 8. Find equation of the line bisecting the segment joining the points (, ), (, ) and making an angle with the -ais. 7

28 9. The perpendicular from the origin to a line meets it at the point (, 9), find the equation of the line.. Write the equation of the line for which the line and (i) y-intercept is tan θ. (ii) intercept is., where θis the inclination of. Find the perpendicular form of the equation of the lines from the given values of p and α : (i) p and α, (ii) p, α. Find the slope and y intercept of the straight line + 6y 7.. Two lines passing through the point (, ) make an angle of. If the slope of one of the lines is, find the slope of other.. Determine the angle B of the triangle with vertices A(, ), B(, ) and C(, ).. Find the equation of the straight line through the origin making angle of 6 with the straight line + y Find the equation of a line passing through the point (, ) and parallel to y If by + and 9 + y + a represent the same straight line, find the values of a and b. 8. Find the co-ordinates of the orthocentre of the triangle whose angular points are (,) (, ) and (, ). 9. Prove that these lines : 7 7, y and 8 y meet in a point.. Find the equation of the line passing through the point of intersection of + y and y 7, (b) (, ) and passing through the point : (a) (, ); 7

29 CHAPTER - CONIC SECTION ( mark question) Q. If the eq. of the circle is + y + 8 y + 8 then its centre is (A) (8,-), (B) (-8,), (C) (-,) (D) (,-) Q.. Find the equation of the circle whose centre is (, ) and radius. (A) + y + 6 y + (B) + y + 6 y (C) + y 6 + y + (D) + y 6+ y Q. The directri of the Parabola y a is (A) a (B) a (C) (D) None of the these Q. The foci of the ellipse 9 + y 6 are (A) (, ) (B) (, ± ) (C) ( ±, ) (D) (, ) Q. The eccentricity of the parabola y 8 is (A) - (B) (C) - (D) Q.6 The eccentricity of the ellipse + y + 8y + are (A) (B) (C) Q.7 If in a Hyper-bola, the distance between the foci is and the transverse ais has length 8, than the length of its latus rectum is (A) 9 (B) 9 (C) (D) (D) 6 76

30 Q.8. The focus of the parabola y 6 is (A) (6,) (B) (,6) (C) (-6,) (D) (,) Q.9. The eccentricity of circle is (A) e< (B) e > (C) e (D) e Q.. The eccentricity of Hyperbola is (A) e < (B) e > (C) e (D) e ( Marks Questions). Find the equation of the circle whose radius is and which touches the circle y y eternally at the point (, ).. Find the parametric representation of the circle : + y + y.. Show that the point : rt r( t ) values of t such that < t <., y (r constant) lies on a circle for all + t + t. Find the equation of the circle, the co-ordinates of the end-points of whose diameter are (, ) and (, ).. For the parabola y, find the verte, the ais and the focus. 6. Show that the equation y 8y + 9 represents a parabola. Find its verte, focus and directri. 7. Find the lengths of the major and minor aes, co-ordinates of the foci, vertices, the eccentricity and equations of the directrices for the ellipse 9 + 6y. 8. Find the equation of the ellipse with e, foci on y-ais, centre at the origin, and passing through the point (6, ). 77

31 9. Find the lengths of the transverse and conjugate aes, co-ordinates of the foci, vertices and eccentricity for the hyperbola 9 6y.. Find the equation of the parabola satisfying the following conditions : Vertices at, ±, foci at (, ± ). CHAPTER INTRODUCTION TO -D GEOMETRY ( Marks Questions). Show that the triangle with vertices (6,, ) (,, ) and (6,, ) is a right angled triangle.. Using section formula, prove that (, 6, ) (,, 6) and (,, ) are collinear.. Show that the points A (,, ), B(,, ) and C(,, ) are vertices of right angled isosceles triangle.. Show that the points (,, ), (,, ), (,, ) and (, 6, ) are vertices of parallelogram.. Find the third verte of triangle whose centroid is (7,, ) and whose other vertices are (, 6, ) and (,, ). 6. Find the point in XY-plane which is equidistant from three points A(,,), B(,,) and C(,,) through A. 7. Find lengths of the medians of the triangle with vertices A(,,6), B(,,) and C(6,,) 78

32 8. Find the ratio in which the line joining the points (,,) and (-,,-) is divided by the XY-plane. Also, find the co-ordinates of the point of division. 9. Find the ratio in which the plane +y z divides the line joining the points (-,, -6) and (, -, 8).. Using section formula, show that the points A(, -, ), B(-,, ) and C(, /, ) are collinear. CHAPTER LIMIT AND DERIVATIVES ( Mark Questions). lim is (A) (B) (C) does not eist (D) none of these. The value of lim sin b sin a is equal to (A) (B) (C) b/a (D) a/b. sin θ lim θ θ is (A) (B) / (C) (D) none of these. lim is (A) (B) - (C) (D) Does not eist. The value of the derivatives of ( ) h at / and are (A) Different (B) Same (C) Negative (D) Positive 79

33 6. lim a sin b is a (A) b/a (B) a/b (C) b b (D) a 7. The derivative of sin cos w.r.t is (A) sin (B) cos (C) sin (D) cos 8. The derivative of tan π is equation to (A) π sec (B) - cos ec (C) cosec (D) None of these 9. lim sin sin is : (A) (B) (C) (D). lim π (A) sin π is (B) π (C) (D) None of these. The value of lim sin is (A) zero (B) (C) (D) Does not eist ( Mark Questions). Evaluate (a) cos lim (b) a lim b 8

34 . Evaluate using factor method : (a) lim (b) lim /. Find the derivative of the function : f() +- at -. Also prove that f ()+ f (-). For each of the following functions, evaluate the derivative at the indicated value (s) : (a) s.9 t ; t, t (b) s 8 ;. Evaluate Lt f ( ) where f ( ),, 6. Find dy, when y + + d + 7. Find d 8. Find d 9. Find d dy, when ( ) ( ) y + dy, when y ( ) + + dy, when y sin.cos( ),. Evaluate : (6 Mark Questions) (a) lim (b) lim h + h h. Evaluate :. lim h h Find f ( ) + h lim, where f ( ),, > 8

35 . Evaluate : (a) lim. Evaluate : 6. Evaluate : 7. Prove : cos cos cos cos lim sin Lt 8. Evaluate : a + cos b sin log lim sin ( + ) (b) cos lim sin lim e cos 9. Given f ( ), >, find f ' () by delta method.. Given f( ) sin, find f ' ( ). bydeltamethod CHAPTER MATHEMATICAL REASONING ( Marks Questions). Write the negation of the following statements: a) Both the diagonals of the rectangle have same length. b) 7 is rational. Identify the quantifies in the following statement and write the negation of the statements. i) There eists a number, which is equal to its square. ii) For every real number, is less than +.. Write the converse of the following statements: i) If a number is odd, then is odd. 8

36 ii) If two integers a and b are such that a > b, then (a-b) is always a +ve integer.. Let p : He is rich and q: He is happy be the given statements, write each of the following statements in the symbolic form, using p and q. i) If he is rich, then he is unhappy. ii) It is necessary to be poor is order to be happy.. Determine the truth value of the following : i) +9 iff 8-6 ii) Apple is a fruit iff Delhi is in Japan. 6. Show that the following statement is true by the method of contrapositive p : if is an integer and is even, then is also even. 7. Verify by the method of contraction : p : 7 is irrational 8. Given below the two statements : p : is a multiple of q : is a multiple of 8. Connecting, these two statements with 'And' and 'Or'. In both cases check the validity of the compound statement. 9. Which of the following are statements and which are not? Give reasons for your answers. (i) The number 6 has three prime factors (ii) Rajendra Prasad was the first President of India.. Write the negation of the following statements. (i) The number is greater than 7. (ii) All triangles are not equilateral triangles.. Write the negation of following statement. (i) Australia is a continent. (ii) Every natural number is greater than.. Find the component statements of the following compound statements. (i) is a multiple of and 8. 8

37 (ii) The sun shines or it rains.. Find the component statements of the following and check whether they are true or not. (i) All prime numbers are either even or odd. (ii) India is a democracy and a monarchy.. Write each of the statements in the form 'if P, then q' (i) P : It is necessary to have a password in log on to the server. (ii) q : There is traffic jam whenever it rains. Write the contra positive of the following statements (i) If a number is divisible by 9, then it is divisible by. (ii) If you are born in India, then you are a citizen of India. 6. By giving a counter eample, show that the following statements are not true. (i) If n is an odd integer, then n is prime. (ii) The equation does not have a root lying between and. 7. Show by the method of contradiction P : is irrational. 8. Show that the statement "Given a positive number, there eists a rational number r such that < r < is true 9. Determine the truth value of each of the following statements. (i) +6 iff + (ii) + 7 iff +6. Given below are pairs of statements combine them using 'if and only if' (i) P : If two lines are parallel, then their slopes are equal q : if the slopes of two lines are equal, then they are parallel 8

38 CHAPTER - STATISTICS (6 Marks questions) Q. If is the mean and Mean Deviation from mean is MD( ), then find the number of observations lying between -MD( ) and +MD( ) from the following data :,,, 7, 9,,, 8,,. Q. Calculate the mean deviation about median for the following data. Class Frequency Q. Calculate the mean, variance and standard deviation for the following distribution : Class Frequency 7 8 Q. The mean and variance of 8 observations are 9 and 9. respectively. If si observations are 6,7,,,,, find the remaining two observations. Q. Calculate the mean and variance for the following data : Income (in Rs.) No. of families Q.6 Find the mean and variance for the data. i y i 7 8 8

39 CHAPTER PROBABILITY (One mark questions). In a single through of two dice, the probability of getting a total other than 9 or is : (a) 6 (b) 9 (c) 8 (d) 8. Two numbers are chosen from {,,,,, 6} one after another without replacement. Find the probability that one of the smaller value of two is less than : (a) (b) (c) (d). Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting the other. The probability that all apply for the same house is : (a) 9 (b) 9 (c) 9 7 (d) 9 8. If distinct numbers are chosen randomly from the first natural numbers then the probability that all of them are divisible by and is : (a) (b) (c) (d). What is the chance that a leap year, selected at random, will contain Sundays? (a) (b) (c) (d) Find the probability that in a random arrangement at the word 'Society' all the three vowels come together. (a) (b) (c) (d)

40 7. One card is drawn from a well shuffled deck of cards. If each outcome is equally likely, calculate the probability that the card will be a diamond : (a) (b) (c) (d) 6 8. In a single throw of three dice, find the probability of getting a total of atmost. 7 (a) 8 (b) 8 (c) 8 (d) 6 9. From an urn containing white and 6 green balls, a ball is drawn at random. The probability of not a green ball is : (a) (b) (c) (d) ( marks questions). A letter is chosen at random from the ward 'ASSASSINATION'. Find the probability that letter is (i) a vowel (ii) a consonant.. A coin is tossed three times. Consider the following events : A : No head appears. B : Eactly one head appears. C : At least two heads appear. Do they form a set of mutually eclusive and ehaustive events?. Two dice are thrown and the sums of numbers which come up on the dice are noted. Consider the following events : A : the sum is even. B : the sum is a multiple of. C : the sum is less than. D : the sum is greater than.. A die is thrown, find the probability of the following events : (a) A prime number will appear (b) A number less than 6 will appear 87

41 . One card is drawn from a well shuffled deck of cards. If each outcome is equally likely, calculate the probability that the card will be : (a) a diamond (b) not an ace (c) a black card (d) not a black card 6. In a throw of coins, find the probability of getting both heads or both tails. 7. A bag contains 8 red, white and 9 blue balls. Three balls are drawn at random from the bag. Determine the probability that none of the balls is white. 8. Find the probability of turning for at least once in two tosses of a fair die. 9. A and B are two mutually eclusive events, for which P(A)., P(B) p and P(AUB).. Find p.. In a class of students with roll numbers to, a student is picked up at random to answer a question. Find the probability that the roll number of the selected student is either a multiple of or 7. 88

42 SAMPLE PAPER I CLASS XI MATHEMATICS Time : hrs. Theory : 9 marks CCE : marks Total : marks. All questions are compulsory.. Q.. will consist of parts and each part will carry one [] marks.. Q. to Q. 9 each will be of marks.. Q. to Q. 9 each will be of marks.. Q. to Q. each will be of 6 marks. 6. There will be no overall choice. There will be an internal choice in any questions of marks each and all questions of 6 marks [Total of 7 internal choices] 7. Use of calculator is not allowed. Q..(i) If S {,,,,7} then the total number of subsets of S is equal to : () (a) 6 (b) (c) (d) (ii) Let f ( ) [ ] then f is equal to () (a) - (b) - (c) -. (d) none of these (iii) Value of sin 76 is () (a) (b) (c) (iv) ( ) (d) 76 i in the form of a + ib can be written as () (v) (a) If 98 i (b) i (c) ( ) i (d) i + then the value of will be () 8! 9!! (a) (b) (c) 8 (d) 9 89

43 (vi) The th term of the sequence defined by () ( n )( n)( n) a n + is equal to (a) 6768 (b) (c) 6678 (d) (vii) The slope of the line passing through the points (,-) and (,) is () (a) (b) not defined (c) (d) (viii) The equation of the circle with centre (-,) and radius is () (a) ( + ) + ( y ) 6 (b) ( ) + ( y + 7) (d) + y y (c) ( ) + ( y + ) 6 + (i) The value of lim is equal to () + () (a) (b) (c) (d) Two coins (a one rupee coin and a two rupee coin) are tossed once the sample space will be () (a) {HH,HT,TH,TT} (b) {HH, TT} (c) {HT, TH} (d) {HH, HT, TH} Q.. Prove that sec A sec A tan A + tan A () Q.. Write down the values of cos 68 cos8 + sin68. sin8 () Q.. If a + ib + iy prove that + y () a ib Q.. Epand + () 7 Q.6. Find the middle term in the epansion of? 6 () Q.7. Show that the points P(-,, ) Q (,,) and R (7,,-) are collinear. () Q.8. Write the negation of the following statement (a) Both the diagonals of the rectangle have same length. () 9

44 (b) 7 is rational Q.9. Find the component statements of the following compound statements. () (i) is a multiple of and 8. (ii) The sun Shines or it rains Q.. If {,}, B { : is a root of } (i) (ii) (iii) (iv) A then find () A B A B Are they equal sets? Are they equivalent sets? Q.. If G { 7,8 } and H {,,} find G H and H G () Q.. Prove that π π π π sin sin sin sin Q.. By principle of mathematical Induction, prove that () n n n 6 n P Q.. Find n such that, n > n P 6 ( n + ) ( + ) Q.. Determine nd term and r th term of an A.P. whose 6 th term is and 8 th. OR () () () term is How many triangles can be obtained by joining Points, of which are collinear? Q.6. The perpendicular from the origin to a line meets it at the point (-, 9), find the equation of the line. () Q.7. Find the equation of the ellipse whose vertices are (+,) and foci are (+,) OR () Find the equation of the circle whose radius is and which touches the circle y y eternally at the point (,) 9

45 Q.8. Evaluate using factor method () (a) lim (b) lim Q.9. A coin is tossed three times. Consider the following events () A : B : C : No head appears Eactly one head appears At least two heads appear Do they form a set of mutually eclusive and ehaustive events? OR A and B are two mutally eclusive events, for which P(A)., P(B)P and P(A B). find 'P'. + 7i i Q.. Convert into polar form ( ) (6) OR Solve : ( + i) + 6 i Q.. Solve the following in equations and show the graph on number line. (6) (a) 6 < (b) + 9 (c) 7 + > (d) OR Solve the following system of in equations. + > and + < Q.. Evaluate lim (6) h h + h OR lim, where f ( ) Find f ( ),, > 9

46 Q.. The mean and variance of 8 observations are 9 and 9. respectively. If si observations are 6, 7,,,, find the remaining two observations. (6) OR Calculate the mean deviation about median for the following data. Class Frequency SAMPLE PAPER II CLASS XI MATHEMATICS Time : hrs. Theory : 9 marks CCE : marks Total : marks. All questions are compulsory.. Q.. will consist of parts and each part will carry one [] marks.. Q. to Q. 9 each will be of marks.. Q. to Q. 9 each will be of marks.. Q. to Q. each will be of 6 marks. 6. There will be no overall choice. There will be an internal choice in any questions of marks each and all questions of 6 marks [Total of 7 internal choices] 7. Use of calculator is not allowed. Q..(i) ( A B) C is equal to () (a) (c) C C C A B (b) A C B C C A B (d) None of these (ii) Let f ( ) [ ], then f is equal to () (a) -, (b) -, (c) -. (d) None of these 9

47 (iii) Value of sin 8 is () (a) (b) (c) (d) (iv) Comple conjugate of i- is () (a) i+, (b) -i- (c) -i+, (d) None of these (v) The value of C C is () (a) (b) (c) (d) (vi) Which term in the A.P.,, -... is -? () (a) (b) (c) (d) 9 (vii) Find the distance, of the point (,) from line -y-9 () (a) (b) (c) (viii) The eccentricity of circle is () (i) () (d) (a) e< (b) e> (c) e (d) e/ lim is (a) (b) - (c) (d) Does not eist. In a single through of two dice, the probability of getting a total sum is (a) 6 (b) (c) 8 (d) 9 + sinθ Q.. Prove that secθ+ tanθ sinθ () Q.. Prove that sin7 cos cos7 sin () Q.. Solve the equation. () 7i 9

48 Q.. Write the th term in the epansion of, > 6 Q.6. Find the coefficient of in the epansion of (+) 6. () Q.7. Show that the points A (,,), B (,-,) and C (,-,) are vertices of right angles isosceles triangle. () Q.8. Write the negative of following statements. () (i) (ii) Australia is a continents. Every natural number is greater than zero. Q.9. Determine the truth value of each of the following statements. () (i) +6 off + (ii) +7 off +6 Q.. Let U{,,,,,6,7,8,9}, A{,,,}, B{,,6,8}. Find () (a) A C (b) B C (c) (A C ) C (d) (A B) C Q.. Let A {,,}, B{,,}, C{,} verify that () ( A C) A (B C) (A B) Q.. Show that () cosθ cosθ Q.. By Principle of Mathematical Induction, prove that 7 () n n 7 is divisible by, for all n N () Q.. In how many ways can persons- A, B, C, D and E sit around a circular () table if (a) B and D sit net to each other. (b) A and D do not sit net to each other. Q.. The sum of three numbers in A.P. is - and their product is 8. Find the numbers. Or Prove that tan9 - tan7 - tan6 o +tan8 o Q.6. Find the equation of a line passing through the point (,) and parallel to - y+ () 9

49 Q.7. For the parabola y, Find the verte, the ais and the focus. () OR Find the centroid by the triangle with vertices at (-,), (,-) and (8,) Q.8. Find d dy, when ( ) ( ) y + () Q.9. A die is thrown, find the probability of the following events. () (a) (b) A prime number will appear. A number less than 6 will appear. OR Evaluate Lt f ( ) where f ( ),, Q.. If ( iy) u + iv, + then show that : (6) u + v y ( ) y OR Convert into polar form : i z π π cos + i sin. Find the region enclosed by the following in equations (6). + y, + y,, y OR Solve the following system of inequations : ( + ) < 6 ( ) and log Lt sin ( + ) OR Given f ( ) sin, Find f '( ) by delta method. (6) 96

50 . Calculate the mean, variance and standard deviation for the following distribution: (6) Class Frequency 7 8 OR Find the mean and variance for the data : i y i

51 CLASS - + MATHEMATICS 98

52 CONTENTS S.No. Chapter. Relations and Functions. Inverse Trigonometric Functions. Matrices. Determinants. Continuity and Differentiation 6. Applications of Derivatives 7. Integrals 8. Applications of Integrals 9. Differential Equation. Vectors. Three-Dimensional Geometry. Linear Programming. Probability 99

53 CHAPTER RELATIONS AND FUNCTIONS ( Mark Questions) ) If number of elements in set A and B are m and n respectively, then the number of relations from A to B is (a) m+n (b) mn (c) m+n (d) mn ) Let A {,,,} and Let R{(,), (,), (,), (,)} be relation in A, then R is (a) Refleive (b) Symmetric (c) Transitive (d) None of these. ) Let A{a,b,c} and B{,}. Consider a relation R defined from Set A to set B. Then, R is equal to subset of (a) A (b) B (c) A X B (d) B X A ) Let A {,,}. The total number of distinct relations that can be defined over A is (a) 9 (b) 6 (c) 8 (d) None of these ) R is a relation on N given by N {(,y): +y}. Which of the following belongs to R? (a) (-, ) (b) (, ) (c) (,) (d) (,) 6) Let X be a family of sets and R be a relation in X, defined by 'A is disjoint from B'. Then, R is (a) Refleive (b) Symmetric (c) Anti-Symmetric (d) Transitive. 7) For an onto function f: {,,} {,,} is always (a) into (b) one-one (c) not one-one (d) Many one 8) Function f: R R defined by f() is (a) one-one (b) onto (c) one-one onto (d) Neither one-one nor onto.

54 9) The function f: R R defined by f() cos is (a) into (b) onto (c) one-one (d) many-one onto ) If f: R R is defined by f() ( ) /, then fof () is (a) (b) / (c) (d) 9 ) Number of all one-one functions from Set A{,,} to itself is (a) (b) 6, (c) 8 (d) 9 ) If f: A B and g: B C are onto then gof : A C is (a) onto (b) one-one (c) not onto (d) one one but not onto ) Function f: y is invertible it (a) f is one one (b) f is onto (c) f is one-one onto (d) f is one-one but not onto. ) Let a * b a+b, '*' be a binary opration, then * equals (a) 7 (b) 9 (c) (d) None of these ) If, f(), and g(), then gof () is (a) - (b) (c) (d) ( Mark Questions) ) Check the following functions for one-one and onto. (a) f :R R, f ( ) 7 (c) f :R R,f ( ) + ) Prove that the Greatest integer function,given by f(x)[ ],is neither one-one nor onto, where[ ] denotes the Greatest integer less than or equal to

55 + ) Consider y : R [, ) given by y +. Show that f is both one-one and onto, where R + is the set of all non-negative real numbers. Epress in terms of y. ) Check the function for one-one and onto f: R R, f()9 ) Consider f : R R given by f() +. Show that f is invertible. Find inverse of f. 6) Let AR-{}and BR-{ {},consider the function f : A B defined by f() show that f is one-one and onto and hence find f - 7) Show that the modulus function 8) Check the function f : R R 9) Let f : X Y and g: Y Z be two invertible function, then show that (gof) - f - og -. ) If f : X Y andg : Y Z are onto functions, then show that g of ) If L is the set of all lines in the plane and R is the relation in L defined by R {(l, l ): l is parallel to l }. Show that the relation R is equivalence relation. ) Show that the relation R, defined in a set A of all triangles as {(T, T ): T is similar triangle to T }, is equivalence relation. ) Show that the relation Q in symmetric. ) Show that the relation R in the set A {a, b, c} given by R {(b, c), (c, b)} is symmetric but neither refleive nor transitive. ) State the reason for the relation R in the set {,, } given by R {(, ) ), (, )} not to be transitive 6) Let * be a binary operation on Q defined by Show that is commutative as well as associative. Also find its identity element, if it eists. 7) Consider the binary operation * on N defined by a * b LCM (a, b). for all a, b belongs to N. Write the multiplication table for binary operation *. Also find * 7. 8) Consider the binary operation * on the set {,,,, } defined by a * b min (a, b). Write the multiplication table for binary operation *. Also find ( * ) * ( * ). 9). If A N N and binary operation * is defined on A as (a, b) * (c, d) (ac, bd). (i) (ii) Check * for commutativity and associativity. Find the identity element for * in A (If eists). f : R R defined by f() is neither one-one nor onto. given by ( ) f is one-one or not. R defined as Q {(a, b) : b ab a * b - X Z is also onto. a}, is refleivee and transitive but not

56 CHAPTER- INVERSE TRIGONOMETRIC FUNCTIONS ( Mark Questions) ) The principal value of sin is (A) (B) π π (C) π (D) π + ) If sin cos sin C, then C is (A) 6 (B) 66 (C) 6 6 (D) ) If A tan, then the value of sin A is (A) (B) (C) + (D) None of these sin + + sin ) The value of cot is sin + sin (A) π (B) π (C) (D) π + a a + b + b ) If sin sin tan, then equals (A) a b a+ ab (B) b + ab (C) b ab (D) a + b ab 6) The value of (A) tan cos 6 (B) 7 + tan is 6 (C) 7 7 (D) None of these 6

57 π 7) sin sin (A) (B) 8) tan ( ) cot ( ) (A) π (B) 9) tan sec ( ) (A) π (B) is equal to (C) is equal to is equal to (D) π (C) (D) π (C) π (D) π. The value of tan cos sin (A) π (B) π is (C) π (D) π. The number of real solution of tan ( + ) π + sin + + is (A) zero (B) one (C) both (D) infinite. The value of which satisfies the equation tan sin is (A) (B) - (C) (D). ( ) sin ( ) π sin then is equal to (A), (B), (C) (D). The value of tan sin is (A) 7 (B) 7 (C) 7 (D) 7

58 ( Mark Questions) Question ). Find the Principal values of following inverse trigonometric functions : (i) tan ( ) (ii) cos (iii) sin (iv) cot Question. Prove the following g : Question Prove the following : Question Write the principal value of cos sin Question Prove that : Question 6 Prove that Question 7 Solve for Question 8 Prove that Question 9 Prove that : Question Prove that

59 Question Prove that : if <, y > and y > Question Prove that : Question Prove that : Question Prove that Question Prove that tan + cos + + cos cos π + cos, where, < < π Question Prove that : sin [cot {cos (tan )}] 6

60 CHAPTER & MATRICES AND DETERMINANTS ( Mark Questions). Let A be a square matri of order than KA is equal to : (A) k A (B) K A (C) k A (D) k A. If a, b, c are in A.P. then determinant a b is c (A) (B) (C) (D) Tp Tq Tr. T p, Tq, Tr are the p th, q th and r th terms of an A.P. then p q r equals (A) (B) - (C) (D) p+q+r ω ω. The value of ω ω, ω being a cube root of unity is ω ω (A) (B) (C) ω (D) ω. If a+b+c, one root of a c b c b a b a c (A) (B) (C) a + b + c (D) 6. The roots of the equation a b c b are b c (A) a and b (B) b and c (C) a and c (D) a,b and c 7

61 a 7. Value of b c a b c is (A) (a-b) (b-c) (c-a) (B) (a -b ) (b -c ) (c -a ) (C) (a-b+c) (b-c+a) (c-a+b) (D) None of these 8. If A and B are any matrices, then det (A+B) implies (A) det A + det B (B) det A or det B (C) det A and det B (D) None of these 9. If A and B are matrices then AB implies (A) A and B (B) A and B (C) Either A or B (D) A or B. The value of λ for which the system of equations :- + y + 6, + y + z, + y + λ z has a unique solution is (A) λ 7 (B) λ 7 (C) λ 7 (D) λ 7. If the system of the equation : ky z, k y z, + y z has a non-zero solution, then the possible values of k are : (A) -, (B), (C), (D) -,. If A is a non singular matri than det [adj. (A)] is equal to (A) ( det A ) (B) ( det A ) (C) det A (D) ( ) deta. If A is an invertible matri of order n, then the determinant of Adj. A (A) (B) n+ n A A (C) A n (D) n+ A 8

62 9. The value of b a c a c b c b a is (A) a+b+c (B) (C) (D) abc. If A -A+I then the inverse of A is (A) A (B) A+I (C) I-A (D) A-I ( Mark Questions). Find a matri X such that where + X B A, B A. Find and y, if y. Solve [ ]. Show that A+A T is symmetric Matri 8 7 A. Where A T is the transpose of A.. If α α α α sin cos cos sin A, then prove that A'AI

63 6. Show that the matri A is skew symmetric, where c b c a b a A 7. Find the value of k if area of the triangle is square units and vertices are (-, ) (,), (, k) 8. If A, then show that A 7 A 9. Without actual epansion, Prove that the determinant A Vanish. Where b a c a c b c b a A If A Find (adj. A). Find the inverse of matri. If A, show that I A A ( Mark Questions). Construct a matri A [a ij ] whose elements are given by a ij [ ] < + j i if j i if j i j i a ij < + j i if j i j i if j i a ij

64 . If A and B [- - -], very that (AB)' B' A'. Epress the matri P + Q matri. where P is a symmetric and Q is a skew-symmetric cosθ sinθ cosnθ sin nθ. If A verify prove that A n sinθ cosθ sin nθ cosnθ where n is a natural number.. Let A B C find a matri D such that CD-ABO Find the value of such that cos θ cosθsinθ cos φ cosφ sinφ 7 Prove that the product of the matrices and cosθ sinθ sin θ cosφ sinφ sin φ the null matri, when and differ by an odd multiple of. 8. If A show that A A I. Hence find A If A, find and y such that A A + yi.. If A and B then show that (AB) B A.. Test the consistency of the following system of equations by matri method : y ; 6 y

65 6. Using elementary row transformations, find the inverse of the matri A, if possible.. By using elementary column transformation, find the inverse of A.. If cos sin A and A + A I, then find the general value of. sin cos Using properties of determinants, prove the following : Q. to Q

66 . a b c if a, b, c are in A.P. 6. sin cos sin sin cos sin sin cos sin 7. a ab b + bc c ac +c. a ab b ac a b c. + aab b b c bc c. a a + ab bc b c a + b c a + b + c + d a b c a a b +b + c c + a + b +c a b c + ac ac a b c 8. ( ) 9. Show that : y z y z z y yz z y. yz z y. (i) If the points (a, b) (a, b ) and (a a, b b ) are collinear. Show that ab a b. (ii) If A and B verify that AB A B

67 Solve the following equation for. a + a a a a a a a a a. a + a a a a a a a + LONG ANSWER TYPE QUESTIONS (6 MARKS). Obtain the inverse of the following matri using elementary row operations A.. Using matri method, solve the following system of linear equations : y + z, y z, y + z.. Solve the following system of equations by matri method, where, y, z,,, y z y z y z. where, y, z. Find A, where A, hence solve the system of linear equations : + y z + y + z y z. The sum of three numbers is. If we subtract the second number from twice the first number, we get. By adding double the second number and the third number we get. Represent it algebraically and find the numbers using matri method. 6. Compute the inverse of the matri. A 6 and verify that A I. A 6 and verify that A A I

68 7. ( AB If the matric A and ) B, then compute 7. If the matri A and B -, then compute (AB) Using matri method, solve the following system of linear equations : y, y + z, z Find the inverse of the matri A by using elementary column transformations.. Let A and f() + 7. Show that f (A). Use this result to find A. If A cos sin sin cos, verify that A. (adj A) (adj A). A A I.. For the matri A, verify that A 6A + 9A I, hence find A.. By using properties of determinants prove the following : a b ab b ab a b a b a a a b b. y z y z. y z yz z yz y yz y z.

69 CHAPTER- CONTINUITY AND DIFFERENTIATION ( Mark Questions). If f ( ) then ( ) f is (A) Continuous for all (B) Continuous only at certain points (C) Differentiable at all points (D) None of these. Find k, f () k, if is continuous at, if > (A) / (B) / (C) 6 (D). Lt h (A) (C) ( + h) cos cos is equal to h cos (B) sin sin cos (D) sin. If f ( ) sin π a π is continuous at π if if π then, 'a' equals (A) (B) (C) (D). The derivative of log ( e ) is (A) e (B) e (C) (D) None of these 6