(Paper) 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).
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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. x^{2} + y^{2} = 40 
B. 5(x^{2} + y^{2})  4x  8y = 200 
C. x^{2} + y^{2}  4x  8y = 200 
D. 5(x^{2} + y^{2})  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 e^{2} + 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 coordinates 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 cos^{2} A + 2 cos^{2} 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. 90^{o} 
B. 60^{o} 
C. 30^{o} 
D. 45^{o} 
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 x^{2} + y^{2}  2x  1 = 0 and x^{2} + y^{2}  2y  7 = 0 is 

A. 4 
B. 1 
C. 3 
D. 2 
137. If the product of the roots of the equation ax^{2} + 6x + a^{2} + 1 = 0 is 2, then a equals 

A. 2 
B. 1 
C. 2 
D. 1 
138. If the roots of a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0 are same, then 

A. a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2} 
B. a_{1} = b_{1}= c_{1}, a_{2} = b_{2} = c_{2} 
C. a_{1} = a_{2}, b_{1} = b_{2}, c_{1} = c_{2} 
D. c_{1} = c_{2} 
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 A^{2}B^{3}C^{4}, 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. (2^{n  1)}/n! 
B. 2^{n}/[(n + 1)!] 
C. 2^{n}/n! 
D. 2^{n  2}/[(n  1)!] 
147. The term independent of x in (x^{2}  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 B^{2} 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)] = [x^{2} + (1/x^{2})](x ¹ 0), then f(x) is equal to 

A. x^{2}  1 
B. x^{2}  2 
C. x^{2} 
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 f_{1}(x) and f_{2}(x) are defined on domains D_{1} and D_{2} respectively, then domain of f_{1}(x) + f_{2}(x) is 

A. D_{1} Ç D_{2} 
B. D_{1} È D_{2} 
C. D_{1}  D_{2} 
D. D_{2}  D_{1} 
164. The derivative of sin x^{3} with respect to cos x^{3} is equal to 

A.  tan x^{3} 
B.  cot x^{3} 
C. cot x^{3} 
D. tan x^{3} 
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 y^{2} = 8x and x^{2} = 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. e^{x} 
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 x^{2} + y^{2} + 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 [x^{2} + (1/x^{2})] is 

A. 1 
B. 1 
C. 48 
D. none of the above 
177. The equation of a line through (2, 3) parallel to yaxis is 

A. y = 3 
B. y = 2 
C. x = 2 
D. x = 3 
179. The range of the function f(x) = (1 + x^{2})/x^{2} 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 