DCE (Delhi College of Engineering) Entrance Examination Sample Paper - Maths
PAPER : DCE (Delhi College of Engineering) Entrance Examination Sample Paper (Maths)
The question paper contains 180 questions. Four choices are given for a question out of which one choice may be correct. Each question carries 4 marks. The total marks of the Entrance Test are 720 (240 for each subject, i.e., Physics, Chemistry, Maths). You will get 4 marks for each correct response. For each incorrect response, one mark will be deducted from the total score. As such for each incorrect response, you will lose 5 marks (4 for wrong response and one mark as penalty).
121. Which of the following is not true in linear programming problem? | ||
A. A column in the simplex table that contains all of the variables in the solution is called pivot or key column. | ||
B. A basic solution which is also in the feasible region is called a basic feasible solution. | ||
C. A surplus variable is a variable subtracted from the left hand side of a greater than or equal to constraint to convert it into an equality. | ||
D. A slack variable is a variable added to the left hand side of a less than or equal to constraint to convert it into an equality. | ||
122. The equation of the circle whose diameter lies on 2x + 3y = 3 and 16x - y = 4 and which passes through (4, 6) is | |
A. x2 + y2 = 40 | B. 5(x2 + y2) - 4x - 8y = 200 |
C. x2 + y2 - 4x - 8y = 200 | D. 5(x2 + y2) - 3x - 8y = 200 |
123. Let n(A) = 4 and n(B) = 5. The number of all possible injections from A to B is | |||
A. 120 | B. 9 | C. 24 | D. none |
124. If aN = {ax : x Î N} and bN Ç cN = dN, where b, c Î N are relatively prime, then | |||
A. c = bd | B. b = cd | C. d = bc | D. none of the above |
125. A square root of 3 + 4i is | |||
A. Ö3 + i | B. 2 - i | C. 2 + i | D. none of the above |
126. Which of the following is not applicable for a complex number? | |||||
A. Inequality | B. Division | C. Subtraction | D. Addition | ||
127. | maximum amp (z) - minimum amp (z) | is equal to | |
A. sin -1 (3/5) - cos -1 (3/5) | B. p/2 + cos -1 (3/5) |
C. p - 2 cos -1 (3/5) | D. cos -1 (3/5) |
128. If e, e' be the eccentricities of two conics S and S' and if e2 + e'2 = 3, then both S and S' can be | |||
A. hyperbolas | B. ellipses | C. parabolas | D. none of the above |
129. A stick of length 'l' rests against the floor and a wall of a room. If the stick begins to slide on the floor, then the locus of its middle point is | |||
A. an ellipse | B. a parabola | C. a circle | D. a straight line |
130. The eccentricity of the ellipse which meets the straight line x/y + y/2 = 1 on the axis of x and the straight line x/3 - y/5 =1 on the axis of y and whose axes lie along the axes of co-ordinates is | |||
A. 2Ö6/7 | B. 3Ö2/7 | C. Ö6/7 | D. none of the above |
131. A and B are positive acute angles satisfying the equations 3 cos2 A + 2 cos2 B = 4 and 3 sin A/sin B = 2 cos B/cos A, then A + 2B is equal to | |||
A. p/3 | B. p/2 | C. p/6 | D. p/4 |
132. At a point 15 metres away from the base of a 15 metres high house, the angle of elevation of the top is | |||||
A. 90o | B. 60o | C. 30o | D. 45o | ||
133. If tan(p cos q) = cot(p sin q), 0 < q < 3p/4, then sin(q + p/4) equals | |||
A. 1/Ö2 | B. 1/2 | C. 1/(2Ö2) | D. Ö2 |
134. In a triangle ABC, Ð B = p/3, Ð B = p/4, and D divides BC internally in the ratio1 : 3. Then (sin Ð BAD)/(sin Ð CAD) equals | |||
A. Ö2/3 | B. 1/Ö3 | C. 1/Ö6 | D. 1/3 |
135. The straight line 5x + 4y = 0 passes through the point of intersection of the lines | |
A. x + y - 2 = 0, 3x + 4y - 7 = 0 | B. x - y = 0, x + y = 0 |
C. x + 2y - 10 = 0, 2x + y + 5 = 0 | D. none of the above |
136. The number of common tangents of the circles x2 + y2 - 2x - 1 = 0 and x2 + y2 - 2y - 7 = 0 is | |||
A. 4 | B. 1 | C. 3 | D. 2 |
137. If the product of the roots of the equation ax2 + 6x + a2 + 1 = 0 is -2, then a equals | |||
A. -2 | B. -1 | C. 2 | D. 1 |
138. If the roots of a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 are same, then | |
A. a1/a2 = b1/b2 = c1/c2 | B. a1 = b1= c1, a2 = b2 = c2 |
C. a1 = a2, b1 = b2, c1 = c2 | D. c1 = c2 |
139. The roots of the equation (3 - x)4 + (2 - x)4 = (5 - 2x)4 are | |
A. two real and two imaginary | B. all imaginary |
C. all real | D. none of the above |
141. If the 10th term of a G.P. is 9 and 4th term is 4, then its 7 th term is | |||
A. 9/4 | B. 4/9 | C. 6 | D. 36 |
142. 1 - 1/2 + 1/3 - 1/4 + ....... to ¥ equals | |||
A. log 2 | B. e | C. e -1 | D. none of the above |
143. 9/1! + 19/2! + 35/3! + 57/4! + 85/5! + ....... = | |||
A. 16e -5 | B. 7e - 3 | C. 12e - 5 | D. none of the above |
144. How many different arrangements can be made out of the letters in the expansion A2B3C4, when written in full? | |||
A. 9!/(2! + 3! + 4!) | B. 9!/(2! 3! 4!) | C. 2! + 3! + 4! (2! 3! 4!) | D. 2! 3! - 4! |
145. The numbner of straight lines that can be drawn out of 10 points of which 7 are collinear is | |||
A. 23 | B. 21 | C. 25 | D. 24 |
146. 1/n! + 1/[2! (n - 2)!] + 1/[4! (n - 4)!] + ..... is | |||
A. (2n - 1)/n! | B. 2n/[(n + 1)!] | C. 2n/n! | D. 2n - 2/[(n - 1)!] |
147. The term independent of x in (x2 - 1/x)9 is | |||
A. 1 | B. 49 | C. -1 | D. none of the above |
148. The 9th term of an A.P. is 499 and 499th term is 9. The term which is equal to zero is | |||
A. 501th | B. 502th | C. 500th | D. none of the above |
150. If AB = A and BA = B, then B2 is equal to | |||||
A. B | B. A | C. 1 | D. 0 | ||
152. The value of K so that (x - 1)/-3 = (y - 2)/2K = (z - 3)/2 and (x - 1)/3K = (y - 1)/1 = (z - 6)/-5 may be perpendicular is given by | |||
A. -7/10 | B. -10/7 | C. -10 | D. 10/7 |
156. If a, b, c, d are constants such that a and c are both negative and r is the correlation coefficient between x and y, then the correlation coefficient between (ax + b) and (cy + d) is equal to | |||
A. (a/c)r | B. c/a | C. - r | D. r |
157. A person draws a card from a pack of 52 playing cards, replaces it and shuffles the pack. He continues doing this until he draws a spade, the chance that he will fail in the first two draws is | |||
A. 1/16 | B. 9/16 | C. 9/64 | D. 1/64 |
158. In tossing 10 coins, the probability of getting exactly 5 heads is | |||
A. 193/256 | B. 9/128 | C. 1/2 | D. 63/256 |
159. Four tickets marked 00, 01, 10, 11 respectively are placed in a bag. A ticket is drawn at random five times, being replaced each time, the probability that the sum of the numbers on tickets thus drawn is 23, is | |||
A. 100/256 | B. 231/256 | C. 25/256 | D. none of the above |
161. Let f[x + (1/x)] = [x2 + (1/x2)](x ¹ 0), then f(x) is equal to | |||
A. x2 - 1 | B. x2 - 2 | C. x2 | D. none of the above |
162. Let f(x) = [tan(p/4 - x)]/cot2x, x ¹ p/4. The value which should be assigned to f at x = p/4, so that it is continous everywhere is | |||
A. 1 | B. 1/2 | C. 2 | D. none of the above |
163. If f1(x) and f2(x) are defined on domains D1 and D2 respectively, then domain of f1(x) + f2(x) is | |||
A. D1 Ç D2 | B. D1 È D2 | C. D1 - D2 | D. D2 - D1 |
164. The derivative of sin x3 with respect to cos x3 is equal to | |||
A. - tan x3 | B. - cot x3 | C. cot x3 | D. tan x3 |
165. If y = f(x) is an odd differentiable function defined on (¥, ¥) such that f'(3) = -2, then f'(-3) equals | |||
A. 4 | B. 2 | C. -2 | D. 0 |
166. The line (x/a) + (y/b) = 1 touches the curve y = be-x/a at the point | |||
A. (a, ba) | B. (a, a/b) | C. (a, b/a) | D. none of the above |
167. The least value of 'a' for which the equation (4/sin x) + [1/(1 - sin x)] = a has atleast one solution on the interval (0, p/2) is | |||
A. 4 | B. 1 | C. 9 | D. 8 |
168. The area bounded by the curve y2 = 8x and x2 = 8y is | |||
A. 32/7 | B. 24/5 | C. 72/3 | D. 64/3 |
169. The integrating factor of the differential equation [(dy/dx)(x log x)] + y = 2 log x is given by | |||
A. log (log x) | B. ex | C. log x | D. x |
170. If y = tan -1[(sin x + cos x)/(cos x - sin x)], then dy/dx is equal to | |||
A. 1/2 | B. 0 | C. 1 | D. none of the above |
171. The length of tangent from (5, 1) to the circle x2 + y2 + 6x - 4y - 3 = 0 is | |||
A. 81 | B. 29 | C. 7 | D. 21 |
172. The equation of the straight line which is perpendicular to y = x and passes through (3, 2) will be given by | |||
A. x - y = 5 | B. x + y = 5 | C. x + y = 1 | D. x - y = 1 |
173. If the imaginary part of (2z + 1)/(iz + 1) is - 2, then the locus of the point representing z in the complex plane is | |||
A. a circle | B. a straight line | C. a parabola | D. none of the above |
174. The sum of 40 terms of an A.P. whose first term is 2 and common difference 4, will be | |||
A. 3200 | B. 1600 | C. 200 | D. 2800 |
175. If a, b, c are in A.P., then a/bc, 1/c, 2/b are in | |||
A. A.P. | B. G.P. | C. H.P. | D. none of the above |
176. The term independent of x in [x2 + (1/x2)] is | |||
A. 1 | B. -1 | C. 48 | D. none of the above |
177. The equation of a line through (2, -3) parallel to y-axis is | |||
A. y = -3 | B. y = 2 | C. x = 2 | D. x = -3 |
179. The range of the function f(x) = (1 + x2)/x2 is equal to | |||
A. [0, 1] | B. [1, 0] | C. (1, ¥) | D. [2, ¥] |
180. Two vectors are said to be equal if | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
A. their magnitudes are same | B. direction is same | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
C. they meet at the same point D. they have magnitude and same sense of direction
MATHS
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