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? |
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A. A column in the simplex table that contains all of the variables in the solution is called pivot or key column. |
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B. A basic solution which is also in the feasible region is called a basic feasible solution. |
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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. |
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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. |
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122. The equation of the circle whose diameter lies on 2x + 3y = 3 and 16x - y = 4 and which passes through (4, 6) is |
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A. x2 + y2 = 40 |
B. 5(x2 + y2) - 4x - 8y = 200 |
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C. x2 + y2 - 4x - 8y = 200 |
D. 5(x2 + y2) - 3x - 8y = 200 |
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123. Let n(A) = 4 and n(B) = 5. The number of all possible injections from A to B is |
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A. 120 |
B. 9 |
C. 24 |
D. none |
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124. If aN = {ax : x Î N} and bN Ç cN = dN, where b, c Î N are relatively prime, then |
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A. c = bd |
B. b = cd |
C. d = bc |
D. none of the above |
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125. A square root of 3 + 4i is |
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A. Ö3 + i |
B. 2 - i |
C. 2 + i |
D. none of the above |
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126. Which of the following is not applicable for a complex number? |
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A. Inequality |
B. Division |
C. Subtraction |
D. Addition |
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127. | maximum amp (z) - minimum amp (z) | is equal to |
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A. sin -1 (3/5) - cos -1 (3/5) |
B. p/2 + cos -1 (3/5) |
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C. p - 2 cos -1 (3/5) |
D. cos -1 (3/5) |
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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 |
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A. hyperbolas |
B. ellipses |
C. parabolas |
D. none of the above |
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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 |
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A. an ellipse |
B. a parabola |
C. a circle |
D. a straight line |
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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 |
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A. 2Ö6/7 |
B. 3Ö2/7 |
C. Ö6/7 |
D. none of the above |
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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 |
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A. p/3 |
B. p/2 |
C. p/6 |
D. p/4 |
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132. At a point 15 metres away from the base of a 15 metres high house, the angle of elevation of the top is |
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A. 90o |
B. 60o |
C. 30o |
D. 45o |
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133. If tan(p cos q) = cot(p sin q), 0 < q < 3p/4, then sin(q + p/4) equals |
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A. 1/Ö2 |
B. 1/2 |
C. 1/(2Ö2) |
D. Ö2 |
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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 |
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A. Ö2/3 |
B. 1/Ö3 |
C. 1/Ö6 |
D. 1/3 |
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135. The straight line 5x + 4y = 0 passes through the point of intersection of the lines |
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A. x + y - 2 = 0, 3x + 4y - 7 = 0 |
B. x - y = 0, x + y = 0 |
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C. x + 2y - 10 = 0, 2x + y + 5 = 0 |
D. none of the above |
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136. The number of common tangents of the circles x2 + y2 - 2x - 1 = 0 and x2 + y2 - 2y - 7 = 0 is |
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A. 4 |
B. 1 |
C. 3 |
D. 2 |
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137. If the product of the roots of the equation ax2 + 6x + a2 + 1 = 0 is -2, then a equals |
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A. -2 |
B. -1 |
C. 2 |
D. 1 |
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138. If the roots of a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 are same, then |
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A. a1/a2 = b1/b2 = c1/c2 |
B. a1 = b1= c1, a2 = b2 = c2 |
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C. a1 = a2, b1 = b2, c1 = c2 |
D. c1 = c2 |
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139. The roots of the equation (3 - x)4 + (2 - x)4 = (5 - 2x)4 are |
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A. two real and two imaginary |
B. all imaginary |
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C. all real |
D. none of the above |
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141. If the 10th term of a G.P. is 9 and 4th term is 4, then its 7 th term is |
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A. 9/4 |
B. 4/9 |
C. 6 |
D. 36 |
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142. 1 - 1/2 + 1/3 - 1/4 + ....... to ¥ equals |
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A. log 2 |
B. e |
C. e -1 |
D. none of the above |
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143. 9/1! + 19/2! + 35/3! + 57/4! + 85/5! + ....... = |
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A. 16e -5 |
B. 7e - 3 |
C. 12e - 5 |
D. none of the above |
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144. How many different arrangements can be made out of the letters in the expansion A2B3C4, when written in full? |
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A. 9!/(2! + 3! + 4!) |
B. 9!/(2! 3! 4!) |
C. 2! + 3! + 4! (2! 3! 4!) |
D. 2! 3! - 4! |
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145. The numbner of straight lines that can be drawn out of 10 points of which 7 are collinear is |
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A. 23 |
B. 21 |
C. 25 |
D. 24 |
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146. 1/n! + 1/[2! (n - 2)!] + 1/[4! (n - 4)!] + ..... is |
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A. (2n - 1)/n! |
B. 2n/[(n + 1)!] |
C. 2n/n! |
D. 2n - 2/[(n - 1)!] |
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147. The term independent of x in (x2 - 1/x)9 is |
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A. 1 |
B. 49 |
C. -1 |
D. none of the above |
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148. The 9th term of an A.P. is 499 and 499th term is 9. The term which is equal to zero is |
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A. 501th |
B. 502th |
C. 500th |
D. none of the above |
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150. If AB = A and BA = B, then B2 is equal to |
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A. B |
B. A |
C. 1 |
D. 0 |
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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 |
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A. -7/10 |
B. -10/7 |
C. -10 |
D. 10/7 |
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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 |
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A. (a/c)r |
B. c/a |
C. - r |
D. r |
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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 |
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A. 1/16 |
B. 9/16 |
C. 9/64 |
D. 1/64 |
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158. In tossing 10 coins, the probability of getting exactly 5 heads is |
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A. 193/256 |
B. 9/128 |
C. 1/2 |
D. 63/256 |
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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 |
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A. 100/256 |
B. 231/256 |
C. 25/256 |
D. none of the above |
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161. Let f[x + (1/x)] = [x2 + (1/x2)](x ¹ 0), then f(x) is equal to |
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A. x2 - 1 |
B. x2 - 2 |
C. x2 |
D. none of the above |
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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 |
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A. 1 |
B. 1/2 |
C. 2 |
D. none of the above |
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163. If f1(x) and f2(x) are defined on domains D1 and D2 respectively, then domain of f1(x) + f2(x) is |
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A. D1 Ç D2 |
B. D1 È D2 |
C. D1 - D2 |
D. D2 - D1 |
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164. The derivative of sin x3 with respect to cos x3 is equal to |
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A. - tan x3 |
B. - cot x3 |
C. cot x3 |
D. tan x3 |
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165. If y = f(x) is an odd differentiable function defined on (¥, ¥) such that f'(3) = -2, then f'(-3) equals |
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A. 4 |
B. 2 |
C. -2 |
D. 0 |
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166. The line (x/a) + (y/b) = 1 touches the curve y = be-x/a at the point |
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A. (a, ba) |
B. (a, a/b) |
C. (a, b/a) |
D. none of the above |
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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 |
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A. 4 |
B. 1 |
C. 9 |
D. 8 |
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168. The area bounded by the curve y2 = 8x and x2 = 8y is |
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A. 32/7 |
B. 24/5 |
C. 72/3 |
D. 64/3 |
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169. The integrating factor of the differential equation [(dy/dx)(x log x)] + y = 2 log x is given by |
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A. log (log x) |
B. ex |
C. log x |
D. x |
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170. If y = tan -1[(sin x + cos x)/(cos x - sin x)], then dy/dx is equal to |
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A. 1/2 |
B. 0 |
C. 1 |
D. none of the above |
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171. The length of tangent from (5, 1) to the circle x2 + y2 + 6x - 4y - 3 = 0 is |
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A. 81 |
B. 29 |
C. 7 |
D. 21 |
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172. The equation of the straight line which is perpendicular to y = x and passes through (3, 2) will be given by |
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A. x - y = 5 |
B. x + y = 5 |
C. x + y = 1 |
D. x - y = 1 |
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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 |
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A. a circle |
B. a straight line |
C. a parabola |
D. none of the above |
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174. The sum of 40 terms of an A.P. whose first term is 2 and common difference 4, will be |
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A. 3200 |
B. 1600 |
C. 200 |
D. 2800 |
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175. If a, b, c are in A.P., then a/bc, 1/c, 2/b are in |
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A. A.P. |
B. G.P. |
C. H.P. |
D. none of the above |
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176. The term independent of x in [x2 + (1/x2)] is |
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A. 1 |
B. -1 |
C. 48 |
D. none of the above |
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177. The equation of a line through (2, -3) parallel to y-axis is |
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A. y = -3 |
B. y = 2 |
C. x = 2 |
D. x = -3 |
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179. The range of the function f(x) = (1 + x2)/x2 is equal to |
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A. [0, 1] |
B. [1, 0] |
C. (1, ¥) |
D. [2, ¥] |
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180. Two vectors are said to be equal if |
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A. their magnitudes are same |
B. direction is same |
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C. they meet at the same point D. they have magnitude and same sense of direction
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