Delhi Public School, Jammu Question Bank ( )

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1 Class : XI Delhi Public School, Jammu Question Bank (07 8) Subject : Math s Q. For all sets A and B, (A B) (A B) A. LHS (A B) (A B) [(A B) A] [(A B) B] A A B A RHS Hence, given statement is true. Q. For all sets A and B, (A B) B A B. LHS (A B) B (A B) B (A B ) (B B ) (A B ) φ A B A B RHS Q3. In a class of 60 students, 5 students play cricket and 0 students play tennis and 0 students play both the games. Find the number of students who play neither. Let C be the set of students who play cricket and T be the set of students who play tennis. Then, n(u) 60, n(c) 5, n(t) 0, and n(c T) 0 n(c T) n C + n T n(c T) Number of students who play neither n(u) n(c T)

2 Q4. f(x) cos x We have, f(x) cosx - cos x - cos x 0 cos x So, f(x) is defined, if cos x 0 cosx x nπ n Z Domain of f R {nπ n z} Q5. f(x) x+ x we have, f(x) x+ x x + x x x 0, x < 0 x + x x, x 0 Hence, f(x) is defined, if x > 0 Domain of f R + Q6. If [x] 5[x] + 6 0, where [. ] denote the greatest integer function, find x We have, [x] 5[x] ([x] 3) ([x] ) 0 [x], 3 x [, 3]

3 Q7. f(x) 3 x we have, f(x) 3 x Then, y x x 3 y 3 x 3 y x y 3 y x assums real values, if y 3 0 and y > 0 y 3 Range of f 3, Q8. Find the coefficient of x in the expansion of ( 3x + 7x ) ( x) 6. Given, expansion ( 3x + 7x ) ( x) 6. ( 3x + 7x ) ( 6 C C 5 x + 6 C 4 x C 6 x 6 ) ( 3x + 7x ) ( 6x + 0x +.) Coefficient of x Q9. If p is a real number and the middle term in the expansion of find the value of p. p + 8 is 0, then Given expansion is p + 8 Here, n 8 Since, this Binomial expansion has only one middle term i.e., 8 + th 5th term T 5 T 4+ 8 C 4 p 0 8 C 4 p

4 p p4 4 p 4 p ± Q0. Find the coefficient of x 4 in the expansion of ( + x + x + x 3 ). Given expansion ( + x + x + x 3 ) [( + x) + x ( + x)] [( + x) ( + x )] ( + x).( + x ) Now, above expansion becomes ( C 0 + C x + C x + C 3 x 3 + C 4 x 4 +.) ( C 0 + C x + C x 4 +.) ( + x + 55x + 65x x 4 +.) ( + x + 55x 4 + ) Coefficient of x Q. A carpenter was hired to build 9 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job? Here, a 5 and d Let he finished the job in n days, S n 9 9 n [ 5 + n ] 9 n [8 + n] 9 4n + n n + 4n 9 0 (n ) (n + 6) 0 n, - 6

5 n Q. If t n denotes the nth term of the series find t 50 is Let S n be sum of the series t 50. S n t 50. and S n t 49 + t 50 On subtracting Eq. (ii) from Eq. (i), we get t upto 49 terms t 50 + [ upto 49 terms] + 49 [ + 48 ] + 49 [ 96] + [ x 49] + 49 x 49 + (49) Q3. If a, b and c are in GP, then the value of a b is equal to. b c Given that a, b and c are in GP. Then, b a c b r b ar c br a b a ar a( r) a( r) b c ar br r(a b) r(a ar ) a b b c r a b or b c Q4. Straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.

6 Since, the intercept form of a line is x a + y b Given that, a + b constant a + b k k a + k b So, (k,k) lies on x a + y b Hence, the passes through the fixed point Q5. P and P are points on either of the two lines y - 3 x at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from P, P on the bisector of the angle between the given lines. Given equation of lines are y - 3x and y + 3x y 3x + and y - 3x + Y 3x + 3x + D 3x 0 x 0 On putting x 0 in Eq, (i) we get So, the point of intersection of line (i) and (ii) is (0, ) P 5 (0, ) C 5 P x Here, OC 30 o 30 o 0 o In DEC, CD CE cos30o CD 5 cos 30 o X A 60 o O 60 o B X Y

7 5. 3 OD OC + CD So, the coordinates of the foot of perpendiculars are 0, Q6. The value of the λ, if the lines ( + 3y + 4) + λ 6x y + 0 are Column I Column II (i) Parallel to Y axis is (a) λ 3 4 (ii) Perpendicular to 7x + y 4 0 is (b) λ 3 (iii) Passes through (, ) is (c) λ 7 (iv) Parallel to X-axis is (d) λ 3 4 (i) Given equation of the line is (x + 3y + 4) + λ 6x y + 0 If line is parallel to Y axis i.e., it is perpendicular to X-axis Slope m tan90 o From line (i), x( + 6λ) + y 3 λ λ 0 and slope 6λ 3 λ (+6λ) 3 λ 6λ 3 λ 0 λ a (iii) If the line (i) passes through the point (, )

8 Then, ( ) + λ λ 0 λ 3 4 (iv) If the line is parallel to X axis the slope 0. Then, (+6λ) 3 λ 0 -( + 6λ) 0 λ 3 So, then correct matches are (i) d, ii c, iii a, iv (b) Q7. If one end of a diameter of the circle x + y 4x 6y + 0 is (3, 4) then find the coordinates of the other end of the diameter. Given equation of the circle is x + y 4x - 6y + 0 g - 4 and f - 6 So, the centre of the circle is (-g, -f)i.e., (, 3) Since, the mid-point of AB is (, 3) A (3, 4) C (, 3) B (x,y) Then, 3 + x x x and 3 4+ y y y So, the coordinates of other end of the diameter will be (, ) Q8. Find the coordinates of a point on the parabola y 8x, whose focal distance is 4.

9 Given parabola is y 8x 8x 4ax a Focal distance x + a 4 x + 4 x + ± 4 x, - 6 For x y 8 x y 6 y ± 4 So, the points are (, 4) and (, -4). Q9. Find the equation of the hyperbola with eccentricity 3 and foci at (±, 0). Given that eccentricity i.e., 3/ and (±, 0) b a (e ) b 6 9 b So, the equation of hyperbola is x 4 y Q0. Find the value of n, if x x n n x 80, n N. Given, x n n x 80 x n() n- 80

10 n x n- 5 x () 5- n 5 Q. Evaluate sin x sin 4x Given sin x sin 4x sin x sin x cosx sin x 4sin xcos x 4cos x 4 sin x sin (x) Q. Evaluate x+ /3 /3 x Given, x+ /3 /3 x x + /3 /3 x /3 3 /3 Q3. A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that (i) (ii) (iii) All the three balls are white. All the three balls are red One ball is red and two balls are white 5 C3 3 C 3

11 Q4. If a card is drawn from a deck of 5 cards, then find the probability of getting a king or a heart or a red card. Number of possible event 5 and favourable events 4 king + 3 heart + 6 red 3 8 Required probability Q5. If the letters of the word ASSASSINATION are arranged at random. Find the probability that (i) Four S s come consecutively in the word. Total number of letters in the word ASSASSINATION is 3 Out of which 3A s, 4S s, I s, N s, T s and 0. Number of words when all S s are together 0! 3!!! Total number of word using letter of the word ASSASSINATION 3! 3!4!!! Required probability 0! 3!!! 3! 3!4!!! 0! 4! 3! 4!

CLASS XI Maths : Sample Paper-1

CLASS XI Maths : Sample Paper-1 Allotted Time : 3 Hrs Max. Marks : 100 Instructions 1. This questionnaire consists of 30 questions divided in four sections. Please verify before attempt. 2. Section I consists