Class/Course  Engineering Entrance
Subject  Mathematics
Total Number of Question/s  3005
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1. Sets, Relations and Functions  Quiz
1. Let R be the set of real numbers.
Statement I A = {(x,y) ε R x R : y  x is an integer} is an equivalent relation on R.
Statement II B = {(x,y) ε R x R : x = αy for some rational number α } is an equivalence relation on R.
a) Statement I is true, Statement II is true ; Statement II is not a correct explantion for Statement I.
b) Statement I is true, Statement II is false
c) Statement I is false, Statement II is true.
d) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
2. Let f be a function defined by f(x) = (x1)^{2} + 1, (x ≥ 1)
Statement I
The set { x : f(s) = f^{1}(x)} = {1,2}
Statement II
f = f is bijection and f^{1}(s) = 1 + $\sqrt{x  1}$, x ≥ 1
a) Statement I is false, Statement II is true.
b) Statement I is true , Statement II is true; Statement II is a correct explanation for Statement I.
c) Statement I is true, Statement II is true, Statement II is not a correct explanation for explanation I.
d) Statement I is true, Statement I is true, Statement II is false.

2. Complex Numbers and Quadratic Equations  Quiz
1. If ω is an imaginary cube root of unity, then (1 + ω  ω^{2})^{7} equals
a) 128ω
b) 128ω
c) 128ω^{2}
d) 128ω^{2}
2. If z+4 ≤ 3, then maximum value of z+1 is
a) 4
b) 10
c) 6
d) 4

3. Matrices and Determinants  Quiz
1. If the system of linear equations
x + 2ay + az = 0
x + 3by + bz = 0
and x + 4cy + cz = 0
has a nonzero solution, then a,b,c
a) are in AP
b) are in GP
c) are in HP
d) satisfy a + 2b + 3c = 0
2. If ω ≠ 1 is the complex cube root of unity and matrix H  $\begin{bmatrix} \omega & 0\\ 0& \omega \end{bmatrix}$ , then H^{70} is equal to
a) H
b) 0
c) H
d) H^{2}

4. Permutations and Combinations  Quiz
1. The number of ways in which 6 men and 15 women can dine at a round table, if no women are to sit together, is given by
a) 6! X 5!
b) 30
c) 5! X 4!
d) 7! X 5!
2. How many ways are there to arrange the letter in the word GARDEN with the vowels in apphabetical order.
a) 120
b) 240
c) 360
d) 480

5. Mathematical Induction  Quiz
1. Let s(k) = 1 + 3 + 5 + ...... + (2k  1) = 3 + k^{2}. Then which of the following is true?
a) S(1) is correct
b) s(k) ⇒ S(k+1)
c) S(k) ⇒ S(k+1)
d) Principle of mathematical induction can be used to prove the formula.
2. Statement I For each natural number n,(n+1)^{7}  n^{7}  1 is divisible by 7.
Statement II For each natural number n, n^{7}  n is divisible by 7.
a) Statement I is false , Statement II is true.
b) Statement I is true, Statement II is true; Statement II is correct explanation for Statement I.
c) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
d) Statement I is true, Statement II is false.

6. Binomial Theorem  Quiz
1. Let S_{1} = $\sum_{j=1}^{10}j\left (j1 \right )^{10}C_{j}, S_{2}$ = $\sum_{j=1}^{10} j^{10}C_{j} \ and \ S_{3}$ = $\sum_{j=1}^{10}$ j^{2} ^{10}C_{j}
Statement I S_{3} = 55 × 2^{9}.
Statement II S_{1} = 90 x 2^{8} and S_{2} = 10 x 2^{8}.
a) Statement I is false, Statement II is true.
b) Statement I is true, Statement II is true; Statement II is a correct explanations of Statement I.
c) Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I.
d) Statement I is true, Statement II is false.
2. If x is so small that x^{3} and higher powers of x may be neglected, then $\frac{\left (1+3 \right )^{3/2}  \left (1 + \frac{1}{2}x \right )^{3}}{\left (1x \right )^{1/2}}$ may be approximated as
a) $\frac{x}{2}  \frac{3}{8}x^{2}$
b) $\frac{3}{8}x^{2}$
c) $3x + \frac{3}{8}x^{2}$
d) $1  \frac{3}{8}x^{2}$

7. Sequences and Series  Quiz
1. $\sum_{n=0}^{\infty}\frac{\left (log_{e} x \right )^{n}}{n!}$ is equal to
a) log _{e}x
b) x
c) log_{x}e
d) None of these
2. In p and q are positive real numbers such that p^{2} + q^{2} =1, then the maximum value of (p + q) is
a) 2
b) $\frac{1}{2}$
c) $\frac{1}{\sqrt{2}}$
d) $\sqrt{2}$

8. Limits, Continuity and Differentiabilty  Quiz
1. The normal to the curve x = a(1+ cosθ ), y = a sinθ at θ always passes through the fixed point
a) (a,0)
b) (0,a)
c) (0,0)
d) (a.a)
2. Angle between the tangents to the curve y = x^{2}  5x + 6 at the points (2,0) and (3,0) is
a) $\frac{\pi}{2}$
b) $\frac{\pi}{6}$
c) $\frac{\pi}{4}$
d) $\frac{\pi}{3}$

9. Integral Calculas  Quiz
1. If I_{1} = $\int_{0}^{1}2^{x^{2}}dx$ , I_{2} = $\int_{0}^{1}2^{x^{3}}dx$ , I_{3} = $\int_{1}^{2}2^{x^{2}}dx$ and I_{4} = $\int_{1}^{2}2^{x^{3}}dx$ , then
a) I_{3} > I_{4}
b) I_{3} = I_{4}
c) I_{1} > I_{2}
d) I_{2} > I_{1}
2. $\lim_{n \rightarrow \infty} \left [\frac{1}{n^{2}}sec^{2}\frac{1}{n^{2}} sec^{2}\frac{4}{n^{2}} + ..... + \frac{n}{n^{2}} sec^{2}1 \right ]$ equals
a) $\frac{1}{2}tan 1$
b) tan 1
c) $\frac{1}{2}cosec 1$
d) $\frac{1}{2}sec 1$

10. Differential Equations  Quiz
1. The differential equation of the family of circles with fixed radius 5 unit and centre on the line y = 2 is
a) (x  2)^{2y'2 = 25  (y  2)2}
b) (x  2)y'^{2} = 25  (y  2)^{2}
c) (y  2)y'^{2} = 25  (y  2)^{2}
d) (y  2)^{2}y'^{2} = 25  (y  2)^{2}
2. If $\frac{dy}{dx}$ = y + 3 > 0 and y(0) = 2, then y (log 2) is equal to
a) 5
b) 13
c) 2
d) 7

11. Coordinate Geometry  Quiz
1. Consider a family of circles which are passing through the point (1,1) and are tangent to through the point (1,1) and are tangent to xaxis. If (h,k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval
a) $0 < k < \frac{1}{2}$
b) $k \ge \frac{1}{2}$
c) $\frac{1}{2} \le k \le \frac{1}{2}$
d) $k \le \frac{1}{2}$
2. A square of sides a lies above the xaxis and has one vertex at the origin. The side passing through at the origin. The side passing through the origin makes an angle $\alpha\left (0 < \alpha < \frac{\pi}{4} \right )$ with the positive direction of xaxis. The equation of its diagonal not passing through the origin is
a) y(cosα  sinα)  x(sinα  cosα) = a
b) y(cosα + sinα) + x(sinα  cosα) = a
c) y(cosα + sinα) + x(sinα + cosα) = a
d) y(cosα + sinα) + x(cosα + sinα) = a

12. Three Dimensional Geometry  Quiz
1. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
a) $\frac{3}{2}$
b) $\frac{5}{2}$
c) $\frac{7}{2}$
d) $\frac{9}{2}$
2. The line passing through the points (5,1,a) and (3,b,1) crosses the yzplane at the point $\left (0, \frac{17}{2}, \frac{13}{2} \right )$. Then,
a) a = 8, b = 2
b) a = 2, b = 8
c) a = 4, b = 6
d) a = 6, b = 4

13. Vector Algebra  Quiz
1. A tetrahedron has vertices at (0,0,0), A(1,2,1) , B(2,1,3) and C (1,1,2). Then , the angle between the faces OAB and ABC will be
a) $cos^{1}\left (\frac{19}{35} \right )$
b) $cos^{1}\left (\frac{17}{31} \right )$
c) 30^{0}
d) 90^{0}
2. The vector $\hat{i} + x\hat{j} + 3\hat{k}$ is rotated through an angle θ and doubled in magnitude, then it becomes $4\hat{i} + \left (4x  2 \right )\hat{j} + 2\hat{k}$ . The value of x are
a) $\left \{ \frac{2}{3},2 \right \}$
b) $\left (\frac{1}{3}, 2 \right )$
c) $\left \{ \frac{2}{3},0 \right \}$
d) {2,7}

14. Statistics and Probabilty  Quiz
1. A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not a prize. The probability that they will not win a prize in a single trial, is
a) $\frac{1}{25}$
b) $\frac{24}{25}$
c) $\frac{2}{25}$
d) None of these
2. Events A, B, C are mutually exclusive events such that P(A) = $\frac{3x + 1}{3}$ , P(B) = $\frac{1  x}{4}$ and P(C) = $\frac{1  2x}{2}$ . The set of possible values of x are in the interval.
a) $\left [\frac{1}{3}, \frac{1}{2} \right ]$
b) $\left [\frac{1}{3}, \frac{2}{3} \right ]$
c) $\left [\frac{1}{3}, \frac{1}{13} \right ]$
d) [0,1]

15. Trigonometry  Quiz
1. The value of cot $\left (cosec^{1}\frac{5}{3} + tan^{1}\frac{2}{3} \right )$ is
a) $\frac{5}{17}$
b) $\frac{6}{17}$
c) $\frac{3}{17}$
d) $\frac{4}{17}$
2. In a triangle ABC, medians AD and BE are drawn, IF AD = 4, ∠DAB = $\frac{\pi}{6}$ and ∠ABE = $\frac{\pi}{3}$>, then the area of the δABC is
a) $\frac{8}{3}$ sq unit
b) $\frac{16}{3}$ sq unit
c) $\frac{32}{\sqrt{3}}$ sq unit
d) $\frac{64}{3}$ sq unit

16. Mathematical Reasoning  Quiz
1. The statement p → (q → p) is equivalent to
a) p → (p ↔ q)
b) p → (p → q)
c) p → (p ∨ q)
d) p → (p ∧ q)
2. Let S be a nonempty subset of R. Consider the following statement.
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P ?
a) There is a rational number x ∈ S such that x ≤ 0
b) There is no rational number x ∈ S such that x ≤ 0
c) Every rational number x ≤ S satisfies x ≤ 0
d) x ∈ S and x ≤ 0 ⇒ x is not rational