## 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) = (x-1)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 - ω2)7 equals
a) 128ω
b) -128ω
c) 128ω2
d) -128ω2

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 non-zero 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 H70 is equal to
a) H
b) 0
c) -H
d) H2

• 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 + k2. 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 - n7 - 1 is divisible by 7.
Statement II For each natural number n, n7 - 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 S1 = $\sum_{j=1}^{10}j\left (j-1 \right )^{10}C_{j}, S_{2}$ = $\sum_{j=1}^{10} j^{10}C_{j} \ and \ S_{3}$ = $\sum_{j=1}^{10}$ j2 10Cj
Statement I S3 = 55 × 29.
Statement II S1 = 90 x 28 and S2 = 10 x 28.
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 x3 and higher powers of x may be neglected, then $\frac{\left (1+3 \right )^{3/2} - \left (1 + \frac{1}{2}x \right )^{3}}{\left (1-x \right )^{1/2}}$ 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 ex
b) x
c) logxe
d) None of these

2. In p and q are positive real numbers such that p2 + q2 =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 = x2 - 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 I1 = $\int_{0}^{1}2^{x^{2}}dx$ , I2 = $\int_{0}^{1}2^{x^{3}}dx$ , I3 = $\int_{1}^{2}2^{x^{2}}dx$ and I4 = $\int_{1}^{2}2^{x^{3}}dx$ , then
a) I3 > I4
b) I3 = I4
c) I1 > I2
d) I2 > I1

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)2y'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 x-axis. 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 x-axis 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 x-axis. 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 yz-plane 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) 300
d) 900

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 non-empty 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