## Class/Course - Engineering Entrance

### Subject - Mathematics

#### Total Number of Question/s - 3005

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• 1. Sets, Relations and Functions - Quiz

1. Let f(x) = (x + 1)2 - 1, x ≥ 1
Statement I The set {x : f(x) = f-1(x)} = {0,-1}
Statement II f is a bijection
a) Statement I is false, Statement II is true
b) Statement I is true, Statement II is true;
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. Let f : N → Y be a function defined as f(x) = 4x + 3 where
Y = { y ∈ N : y = 4x + 3 for some x ∈ N}.
Show that f is invertible and its inverse is
a) g(y) = $\frac{y-3}{4}$
b) g(y) = $\frac{3y+4}{3}$
c) g(y) = $4 + \frac{y+3}{4}$
d) g(y) = $\frac{y+3}{4}$

• 2. Complex Numbers and Quadratic Equations - Quiz

1. If the roots of the quadratic equation x2 + px + q = 0 are tan 300 and tan 15 respectively, then the value of 2 + q - p is
a) 3
b) 0
c) 1
d) 2

2. If the cube roots of unity are 1, ω , ω2, then the roots of the equation (x-1)3 + 8 = 0, are
a) -1, 1+2ω, 1 + 2ω2
b) -1, 1 - 2ω, 1 - 2ω2
c) -1, -1, -1
d) -1, -1 + 2ω , -1 - 2ω2

• 3. Matrices and Determinants - Quiz

1. The system od equations
αx + y + z = α - 1
x + αy + z α - 1
x + y + αz = α - 1
has no solution, if α is
a) 1
b) not -2
c) either -2 or 1
d) -2

2. If (ω ≠ 1) is a cube root of unity, then $\begin{vmatrix} 1 &1 + i + \omega^{2} & 1\\ 1-i& -1 & \omega^{2}-1\\ -i & -1+\omega - i &-1 \end{vmatrix}$ equals
a) 0
b) 1
c) i
d) ω

• 4. Permutations and Combinations - Quiz

1. From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then , the middle of such arrangements is
a) at least 500 but less than 750
b) at least 750 but less than 1000
c) at least 1000
d) less than 500

2. Statement I The number of ways distributing 10 identical balls in 4 distinct boxes such that no box is empty is 9C3.
statement II The number of ways of choosing any 3 places from 9 different places is 9C3.
a) Statem,ent I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
b) Statement I is true,Statement I is false.
c) Statement I is false,Statement I is true.
d) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.

• 5. Mathematical Induction - Quiz

1. Statement I For every natural number n ≥ 2
$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + .......... + \frac{1}{\sqrt{n}}$ > $\sqrt{n}$
statement II For every natural number $n \ge 2. \sqrt{n\left (n + 1 \right )} < + 1$
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.

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. If Sn = $\sum_{r = 0}^{n}\frac{1}{^{n}C_{r}} \ and \ t_{n}= \sum_{r=0}^{n}\frac{r}{^{n}C_{r}} \ then \ \frac{t_{n}}{S_{n}}$ is equal to
a) $\frac{n}{2}$
b) $\frac{n}{2}$ - 1
c) n-1
d) $\frac{2n-1}{2}$

2. The coefficient of x5 in (1 + 2x + 3x2 + ......)-3/2 is
a) 21
b) 25
c) 26
d) None of these

• 7. Sequences and Series - Quiz

1. The sum of the series $\frac{1}{1.2} - \frac{1}{2.3} + \frac{1}{3.4} - .......$ upto ∞ is equal to
a) 2loge2
b) loge 2-1
c) log2e
d) log2$\left (\frac{4}{e} \right )$

2. The value of 21/4. 41/8, 81/16 .... is
a) 1
b) 2
c) $\frac{3}{2}$
d) 4

• 8. Limits, Continuity and Differentiabilty - Quiz

1. The values of p and q for which the function
f(x) = $\left\{\begin{matrix} \frac{sin\left (p + 1 \right )x + sinx}{x}, &x < 0 \\ q, &x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3/2}}, x > 0& \end{matrix}\right.$
is continuous for all x in R, are
a) p = $\frac{5}{2}$, q = $\frac{1}{2}$
b) p = $-\frac{3}{2}$, q = $\frac{1}{2}$
c) p = $\frac{1}{2}$, q = $\frac{3}{2}$
d) p = $\frac{1}{2}$, q = $frac{3}{2}$

2. If xmyn = (x + y)m+n, then $\frac{dy}{dx}$ is
a) $\frac{x + y}{xy}$
b) xy
c) $\frac{x}{y}$
d) $\frac{y}{x}$

• 9. Integral Calculas - Quiz

1. $lim _{n \rightarrow \infty} \sum_{r = 1}^{n}\frac{1}{n}e^{r/n}$ is
a) e
b) e - 1
c) 1 - e
d) e + 1

2. If $\int_{0}^{\pi}x f\left (sin x \right )dx$ = $A\int_{0}^{\pi/2}$ , then A is equal to
a) 0
b) π
c) $\frac{\pi}{4}$
d) 2π

• 10. Differential Equations - Quiz

1. The solution of the differential equation
$\left (1 + y^{2} \right ) + \left (x - e^{tan^{-1}y} \right )\frac{dy}{dx}$ = 0 is
a) $\left (x - 2 \right )$= $ce^{-2tan^{-1}y}$
b) $2xe^{tan^{-1}y}$ = $2^{tan^{-1}y} + c$
c) $xe^{tan^{-1}y}$ = $tan^{-1}y + c$
d) $xe^{2tan^{-1}y}$ = $e^{tan^{-1}y} + c$

2. The differential equation for the family of curves x2 + y2 - 2ay = 0, where a is an arbitrary constant, is
a) 2(x2 - y2)y' = xy
b) 2(x2 + y2)y' = xy
c) (x2 - y2)y' = 2xy
d) (x2 + y2)y' = 2xy

• 11. Coordinate Geometry - Quiz

1. A parabola has the origin as its focus and the line x = 2 as the directrix. Then , the vertex of the parabola is at
a) (2,0)
b) (0,2)
c) (1,0)
d) (0,1)

2. A triangle with vertices (4,0), (-1,-1), (3,5) is
a) isosceles and right angled
b) isosceles but not right angled
c) right angled but not isosceles
d) neither right angled nor isosceles

• 12. Three Dimensional Geometry - Quiz

1. 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

2. If a line makes an angle of $\frac{\pi}{4}$ with the positive fdirections of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is
a) $\frac{\pi}{6}$
b) $\frac{\pi}{3}$
c) $\frac{\pi}{4}$
d) $\frac{\pi}{2}$

• 13. Vector Algebra - Quiz

1. If C is the mid point of AB and P is any point outside AB, then
a) $\vec{PA} + \vec{PB} + \vec{PC}$ = $\vec{0}$
b) $\vec{PA} + \vec{PB} + 2\vec{PC}$ = $\vec{0}$
c) $\vec{PA} + \vec{PB}$ = $\vec{PC}$
d) $\vec{PA} + \vec{PB}$ = $2\overrightarrow{PC}$

2. If the vectors $\vec{a}, \vec{b}$, and $\vec{c}$ from the sides BC, CA and AB respectively of a triangle ABC, then
a) $\vec{a}.\vec{b}$ = $\vec{b}.\vec{c}$ = $\vec{c}, \vec{b}$ = 0
b) $\vec{a}\times \vec{b}$ = $\vec{b}\times \vec{c}$ = $\vec{c} \times \vec{a}$ = 0
c) $\vec{a}.\vec{b}$ = $\vec{b}.\vec{c}$ = $\vec{c}, \vec{a}$ = 0
d) $\vec{a}\times \vec{a}$ = $\vec{a}\times \vec{c}$ = $\vec{c} \times \vec{a}$ = 0

• 14. Statistics and Probabilty - Quiz

1. Consider he following statements
(1) Mode can be compound from histogram
(2) Median is not independent of change of scale
(3) Varience is independent of change of origin and scale
Which of these is/are correct?
a) Only (1)
b) Only (2)
c) Only (1) and (2)
d) (1), (2) and (3)

2. The mean and the variance of a bionomial distribution are 4 and 2 respectively. Then the probability of 2 sucesses is
a) $\frac{37}{256}$
b) $\frac{219}{256}$
c) $\frac{128}{256}$
d) $\frac{28}{256}$

• 15. Trigonometry - Quiz

1. The equation a sinx + b cos x = c where |c| > $\sqrt{a^{2} + b^{2}}$ has
a) a unique solution
b) infinite number of solutions
c) no solution
d) None of the above

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 only statement among the following that is a tautology is
a) B → ∧ ( A → B)]
b) A ∧ (A ∨ B)
c) A ∨ (A ∧ B)
d) [A ∧ (A → B)] → B

2. The statement p → (q → p) is equivalent to
a) p → (p ↔ q)
b) p → (p → q)
c) p → (p ∨ q)
d) p → (p ∧ q)