BIT Entrance Mathematics Questions with Answers

In the followings, the statement is

Answer: Rara lake is in Pokhara

If $S={x: x\in\mathbb{R} \text{ and } x^{2}+4=0}$ then the set $S$ is

Answer: $\emptyset$

The set is a collection of

Answer: longest rivers of Nepal

$S={x: x\ge -2, x\in\mathbb{R}}$ is represented by

Answer: $[-2,\infty)$

$|x-5|\le 3$ has values of $x=$

Answer: $[2,8]$

The ordered pair $(a,b)$ for $(2a-1,-3)=(1,b+3)$ is

Answer: $(1,-6)$

If a function $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=2x+5$ then the pre-image of $7$ in $\mathbb{R}$ is

Answer: $1$

The domain of the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\sqrt{1-x^{2}}$ is

Answer: $[-1,1]$

The base of an exponential function $y=a^{x}$ passing through $(2,4)$ is

Answer: $2$

$3^{2\log_3 x}=$

Answer: $x^{2}$

The function $f(x)=x^{2}-2$ is

Answer: an even function

A number which can be expressed in the form $p/q$, $q\ne 0$; $p,q$ are integers, is called

Answer: a rational number

The sum of an infinite geometric sequence $a, ak, ak^{2},\dots$ exists if

Answer: $-1<k<1$

If $2k+1$, $3k+2$ and $5k+1$ are in A.P., then $k=$

Answer: $2$

The matrix $\begin{pmatrix}0&1\-1&0\end{pmatrix}$ is

Answer: skew-symmetric

If $A=\begin{pmatrix}-5&3\2&-1\end{pmatrix}$ then $A^{-1}=$

Answer: $\begin{pmatrix}1&3\-2&-5\end{pmatrix}$

If $A=\begin{vmatrix}2a+1&1\3a&2\end{vmatrix}=0$, then $a=$

Answer: $-2$

A square matrix $A$ is said to be singular if

Answer: $|A|=0$

If the roots of the equation $px^{2}+qx+r=0$ are equal in magnitude but opposite in sign, then

Answer: $q=0$

If the discriminant of the equation $x^{2}+kx+4=0$ is zero then $k=$

Answer: $\pm 4$

If $i$ is the imaginary unit, then $\dfrac{1-i}{1+i}=$

Answer: $-i$

The square roots of a complex number $5+12i$ is

Answer: $\pm(3+2i)$

Which is not a rational number?

Answer: $e$

If $A$ and $B$ are non-empty sets and $A\subset B\subset U$, then $A\cap B=$

Answer: $A$

If $|x-1|\le 5$ then $x=$

Answer: $[-4,6]$

If $\log_a x=y$ then $x=$

Answer: $a^{y}$

If $f:\mathbb{R}\to\mathbb{R}$ is defined by $f(x)=3x-2$, for $x\in\mathbb{R}$, then the pre-image of $10$ in $\mathbb{R}$ is

Answer: $4$

The domain of the function $f(x)=\sqrt{16-x^2}$ is

Answer: $[-4,4]$

A sentence $x+5=4$ is

Answer: not a statement

If $p$, $q$ and $r$ are in G.P., then

Answer: $q^2=pr$

If $k+3$, $4k+7$ and $3k-1$ are in AP then the sequence is

Answer: $7,9,11$

The solution set of $3x+5>2x+1$ is

Answer: $x>-4$

If A and B are non-intersecting and disjoint sets then $n(A\cup B)$ is equal to

Answer: $n(A)+n(B)$

If $2\le y\le 3$, then

Answer: $(y-2)(y-3)\le 0$

The domain of the function $f(x)=\log(1-x)+\sqrt{x^2-1}$ is

Answer: $(-\infty,-1)$

If $f(x)=x^3+\tan x$, then $f(x)$ is

Answer: Odd function

$i+i^2+i^3+i^4=$

Answer: 0

The quadratic equation whose roots are $\frac12$ and $-\frac12$ is

Answer: $4x^2-1=0$

The sum of the series $5+\frac52+\frac54+\cdots$ to infinity is

Answer: 10

The angles of a triangle are in A.P. and the least angle is $30^\circ$. The greatest angle in radian is

Answer: $\frac{\pi}{2}$

If A is a square matrix, then $(A+A^{T})$ is a

Answer: Symmetric matrix

If the ratio of cost price and selling price of an article is 2:3 then

Answer: $\text{Profit},% = 50%$

If $a+b+c=0$ in quadratic equation $ax^2+bx+c=0$ then roots of quadratic equation is

Answer: $1$ and $\frac{c}{a}$

If $(m+n)^{th}$ term of a G.P is $p$ and $(m-n)^{th}$ term is $q$ then $m^{th}$ term is

Answer: $\sqrt{pq}$

$(a+b-c)x^2+(c+a-b)x+(b+c-a)=0$ and $a+b+c=0$ then roots of quadratic equation, if a, b, and c are rational

Answer: rational

The quadratic equation whose roots are $\frac{3}{5}$ times the roots of quadratic equation $x^2-8x+7=0$

Answer: $25x^2-120x+63=0$

$S_n=\frac{a(1-r^n)}{1-r}$, sum of geometric progression is true if

Answer: $|r|<1$

Square roots of $(3+4i)$ are

Answer: $\pm(2+i)$

$|x-a|=$

Answer: $(a-x)$ if $x<a$

Which of the following statements is are true

Answer: if $a_{ij}=-a_{ji}$ $\forall$ $i$ and $j$ and $a_{ij}=0$ for $i=j$ then matrix is called skew-symmetric matrix

Find the inverse of the function $f(x)=3x^4+2$

Answer: $\sqrt[4]{\frac{x-2}{3}}$

Domain of the function $f(x)=\frac{x^2-4}{x-2}$ is

Answer: $R-{2}$

$\log_2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}}$

Answer: $\frac{63}{64}$

$\begin{vmatrix}1&2&3\4&5&6\7&8&9\end{vmatrix}=$

Answer: $0$

If $A={x:x\in R,\sqrt{2x^2-1}=-1}$ then A is

Answer: null set

Non-intersecting lines are

Answer: Inconsistent and independent

If the price of petrol in Nepal is increased by 25%. How much percent must a Saral Shikshya student reduce his consumption of petrol so as not to increase his expenditure on petrol?

Answer: 20%

If A earns $33\frac13%$ more than B, how much percent does B earn less than A?

Answer: 25%

If $a^2+b^2+c^2-ab-bc-ca=0$ then $a:b:c$ is

Answer: 1:1:1

The speed of a boat when traveling downstream is 32km/hr, whereas when travelling upstream it is 28 km/hr, what is the speed of a boat in still water?

Answer: 30km/hr

The surface area of a cube is $150\text{cm}^2$. Its volume is

Answer: $125\text{cm}^3$

If the supply function $P_s=160+2Q^2$ and the demand function is $P_d=240-3Q^2$. The market equilibrium price is

Answer: 192

If $n(\cup)=200, n(A)=80, n(B)=60$, what is the maximum value of $n(A\cup B)$?

Answer: $140$

$-1\le x\le 11$ is equivalent to

Answer: $|x-5|\le 6$

Range of $y=x^2-4x+5$ is

Answer: $[1,\infty)$

If $\log_9 81=x$, then $x$ is

Answer: $2$

If $x=\log_a bc,\ y=\log_b ca,\ z=\log_c ab$, then $\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=$

Answer: $1$

$1.5+0.015+0.00015+\dots=$

Answer: $\frac{50}{33}$

Value of $2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot 16^{1/32}\cdots\infty$ is

Answer: $2$

The system $kx-2y=0,\ x+3y=0$ has a unique solution if

Answer: $k\ne -\frac23$

If one root of $ax^2+bx+c=0$ is the square of the other, then

Answer: $b^3+a^2c+ac^2=3abc$

The amplitude of $1-i$ is

Answer: $315^\circ$

The mean proportional between $4$ and $16$ is

Answer: $8$

A is 25% more efficient than B in working. If A takes 15 days less than B to complete a work, in how many days will they finish it working together?

Answer: $25\frac57$ days

A plane left 30 minutes later than scheduled and, to reach its destination 1500 km away in time, it had to increase its speed by 250 km/hr. Its usual speed is

Answer: $750$ km/hr

The present worth of Rs 2040 due 4 years hence at 5% p.a. is

Answer: Rs. $1700$

The interest rate at which a sum doubles in 10 years is

Answer: $10\%$

Profits in the partnership of Bikram, Bishal and Binita are shared in ratio $1:2:3$. If Bikram's share is Rs. 300, what is Binita's share?

Answer: Rs. 900

HCF of $n!,\ (n+1)!$ and $(n+2)!$ is

Answer: $n!$

Every rational number is also

Answer: a real number

If $a-ib=\frac{3-2i}{3+2i}$ then $a^{2}+b^{2}=$

Answer: $1$

Given $U={1,2,3,\ldots,15}$, $A={x: x\ge 8}$, $B={x: x\le 4}$ then $\overline{A\cap B}=$

Answer: $U$

Which are the equal ordered pairs?

Answer: (4,2),(4,2)

$S_{\infty}=\frac{a}{1-r}$, where $a, r$ and $S_{\infty}$ denote first term, common ratio and sum to infinity. If

Answer: $|r|<1$

If ${}^{21}C_{x}={}^{21}C_{y}$ then which is true?

Answer: either (b) or (c)

Which is not correct?

Answer: $(AB)^{T}=A^{T}B^{T}$

Which is not correct? (a) $\begin{vmatrix}a_1&a_2&a_3\b_1&b_2&b_3\c_1&c_2&c_3\end{vmatrix}=\begin{vmatrix}a_1&b_1&c_1\a_2&b_2&c_2\a_3&b_3&c_3\end{vmatrix}$; (b) $\begin{vmatrix}0&a_2&a_3\b_1&0&b_3\c_1&c_2&0\end{vmatrix}=0$; (c) a determinant with a row of zeros is $0$; (d) adding $k$ times one column to another leaves the determinant unchanged.

Answer: (b)

Which is correct?

Answer: all of them

The domain of $y=f(x)=\frac{1}{x}$ is

Answer: $R-{0}$

Let $f:R\to R$ be defined by $f(x)=10x-7$ then $f^{-1}(x)=$

Answer: $\frac{x+7}{10}$

If $t$ be a positive real number then the least value of $t+\frac{1}{t}$ is:

Answer: $2$

The system of linear equations $2x+3y=5$ and $4x+6y=9$ are

Answer: Inconsistent & dependent

The degree of polynomial equation $ax^{2}+bx+c=0$ is

Answer: 2

If one root of a quadratic equation in the real coefficient is $\sqrt{2}$ then equation is

Answer: $x^{2}-2=0$

If four quantities $a,b,c$ and $d$ are such that $a:b=3:4$, $b:c=6:7$ and $c:d=8:9$, then the ratio between $a$ and $d$ is

Answer: $4:7$

By selling 175 pens, a stationary store gains the selling price of 50 pens. What is his gain percent?

Answer: $40\%$

A, B, C and D enter into a partnership. A, B, C subscribe $\frac{1}{3},\frac{1}{4},\frac{1}{5}$ of the capital respectively and D, the rest. What is the share of profit of D if the total profit is Rs. 24600?

Answer: Rs. 5330

If USD 1 = Rs. 78.15, how many US dollars are equivalent to Rs. 10,000?

Answer: USD 127.96

The effective rate corresponding to 10% p.a. convertible quarterly is

Answer: 10.4%

The present value of an immediate annuity is; where symbols have their usual meaning

Answer: $\frac{p}{i}\left{1-\frac{1}{(1+i)^{n}}\right}$

If $\lim_{x\to a}\frac{x^{4}-a^{4}}{x-a}=256$, then $a=$

Answer: $4$

$\lim_{x\to 0}\frac{\tan 3x}{x}=$

Answer: $3$

A function $f(x)=2x+1$ is continuous at $x=1$, if

Answer: $\lim_{x\to0}f(x)=f(1)$

If $y=e^{\sin x^{2}}$, then $\frac{dy}{dx}=$

Answer: $2xy\cos x^{2}$

If $y=x^{3}-x^{2}$, then for $\frac{dy}{dx}=0$, $x=$

Answer: $0,2/3$

The function $f(x)=2x+5$ on $R$ is

Answer: increasing

$\int\frac{dx}{ax+b}=$

Answer: $\frac1a\log|ax+b|+c$

$\int\frac{dx}{2\cos^{2}x\sin^{2}x}=$

Answer: $-\cot2x+c$

$\int_1^3\frac{\sin(\ln t)}{t}dt=$

Answer: $1-\cos(\ln3)$

The area under the curve $y=x^{2}$ bounded by the x axis and the ordinates $x=1$ and $x=4$ is

Answer: $21$

If $\lim_{x\to3}\frac{x^n-3^n}{x-3}=27$, then $n=$

Answer: $3$

$\lim_{x\to0}\frac{\sin3x}{x}=$

Answer: $3$

The function $f(x)=\begin{cases}\frac{x^2-x}{x-1}&x\ne1\k&x=1\end{cases}$ is continuous at $x=1$ for $k=$

Answer: $1$

The differential coefficient of $\sin\theta$ with respect to $\cos\theta$ is

Answer: $-\cot\theta$

$\lim_{n\to\infty}\frac{\sum n^2}{n^3} =$

Answer: $\frac{1}{3}$

$\lim_{x\to 1}(x-1)\tan\frac{\pi x}{2} =$

Answer: $-\frac{2}{\pi}$

If $y=\sqrt{\frac{1+\cos 2x}{1-\cos 2x}}$ then $\frac{dy}{dx}$ at $x=\frac{\pi}{2}$ is

Answer: $-1$

Derivative of $e^{\sqrt{x^2+a^2}}$ w.r.t $\sqrt{x^2+a^2}$

Answer: $e^{\sqrt{x^2+a^2}}$

The point of inflection of quadratic function is

Answer: does not exist

$\int_{-1}^{1}\sin^7x,dx=$

Answer: $0$

$\int\frac{x}{x+1}dx=$

Answer: $x-\log(1+x)+c$

$\lim_{x\to64}\frac{\sqrt[6]{x}-2}{\sqrt[3]{x}-4}$

Answer: $\frac14$

$\lim_{x\to0}\frac{e^{\sin x}-1}{x}$

Answer: $1$

$\frac{d}{dx}\cos^{-1}(\sin x)$

Answer: $-1$

If $x^py^q=(x+y)^{p+q}$ then $\frac{dy}{dx}=$

Answer: $\frac yx$

Derivative of $\sin^{-1}x$ is

Answer: $\frac1{\sqrt{1-x^2}}$

At a critical point, $f(x)$ is

Answer: stationary

The value of $\lim_{x\to\infty}\frac{\sqrt{x+1}}{\sqrt{x}+1}$ is

Answer: $1$

The point of discontinuity of the function $f(x)=\frac{x^{3}-27}{x-3}$ is

Answer: $3$

If $x+y=xy$ then $\frac{dx}{dy}=$

Answer: $\frac{1-x}{y-1}$

If $x=2at$, $y=at^{2}$ then $\frac{dy}{dx}=$

Answer: $t$

If the total cost function $C=50x+5000$. The marginal cost function MC is

Answer: $50$

The function $f(x)=2-3x$ is

Answer: Decreasing

$\int \log x,dx=$

Answer: $x\log x-x+c$

$\int_{-2}^{2}|x|,dx$ is equal to

Answer: $4$

If the marginal cost of a product is given by $MC=8+6x-3x^{2}$ and the fixed cost is Rs. 150. The cost function is

Answer: $8x+3x^{2}-x^{3}+150$

The slope of the line $px+qy+c=0$ is

Answer: $-p/q$

If the length of the perpendicular from the point $(k,2)$ to $3x-4y+6=0$ is $2$, then $k=$

Answer: $4$

The homogeneous equation is

Answer: $ax^{2}+xy+by^{2}=0$

The two lines represented by $kx^{2}+xy-y^{2}=0$ are perpendicular to each other if $k=$

Answer: $1$

The distance of the point $(1,2,3)$ from origin is

Answer: $\sqrt{14}$

The direction ratios of the line $AB$ joining the points $A(4,5,0)$ and $B(2,3,4)$ are

Answer: $2,2,-4$

Equation of a line perpendicular to the line $x-3y+7=0$ is

Answer: $3x+y+10=0$

Equation of tangent at $(0,1)$ to the circle $x^2+y^2=1$ is

Answer: $y=1$

In a manufacturing company the production cost of $50$ items is Rs $5000$ in one day and Rs $12000$ to produce $150$ items in one day. If $x$ represents the number of items and $y$ represents the cost, the model equation is

Answer: $y=70x+1500$

A company finds the variable cost of Rs $.5$ per unit output and fixed cost of Rs $25$, where $Q$ is the number of outputs, then the total cost function, $T$ is

Answer: $T=5Q+25$

The circumcenter of an acute angled triangle lies

Answer: inside the triangle

The area of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,4)$ is

Answer: 0

The equation of the line making an angle of $135^\circ$ with the positive x-axis and cutting an intercept 3 from the negative y-axis is

Answer: $x+y+3=0$

The area of triangle with vertices $(1,2),(2,3)$ and $(3,4)$ is

Answer: 0 sq. unit

The polar coordinate of point $(-1,-\sqrt{3})$ is

Answer: $(2,\frac{4\pi}{3})$

The distance between parallel lines $3x+2y=6$ and $3x+2y=12$ is

Answer: $\frac{6}{\sqrt{13}}$

Angle between the lines represented by $2x^2-7xy+6y^2=0$ is

Answer: $\tan^{-1}(\pm\frac18)$

Length of the y-intercept made by the circle $x^2+y^2-8x+y-20=0$ is

Answer: $9$

Radius of the circle $x^2+y^2+2x+3y+k=0$ is $\frac12$, then $k=$

Answer: $3$

The equation $x=0$ represents

Answer: y-axis

The angle between the lines $xy=0$ is

Answer: $90^{0}$

The area of the circle $x^{2}+y^{2}=16$ is

Answer: $16\pi$

The equation $x^{2}+y^{2}=0$ represent

Answer: degenerate circle

$\sin(\cos^{-1}x)=$

Answer: $\sqrt{1-x^{2}}$

The general solution of $4\sin^{2}x=1$, for $n\in Z$ is

Answer: $x=n\pi\pm\frac{\pi}{6}$

The function $f(x)=\cos x$ is

Answer: an even function

In a triangle $ABC$, if $a=2\sqrt3$, $b=2$, $c=4$ then $A=$

Answer: $60^\circ$

The perimeter $2s$ of the triangle $ABC$ is

Answer: $a+b+c$

Area of the triangle $ABC$ is

Answer: $abc/4R$

$\sin(270^\circ-\theta)=$

Answer: $-\cos\theta$

$\tan 15^\circ=$

Answer: $2-\sqrt3$

$\sin^{-1}\frac{1-x^2}{1+x^2}+\cos^{-1}\frac{2x}{1+x^2}$

Answer: none of these

$\cos x=$

Answer: All of these

Period of the function $\cos 2x$ is

Answer: $\pi$

If $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\frac{\pi}{2}$ then $xy+yz+zx=$

Answer: $1$

Solution for $\sin\theta+\cos\theta=\sqrt2$ is

Answer: $2n\pi+\frac\pi4$

In any $\triangle ABC$, $2\cos A\sin C=\sin B$, then the triangle is

Answer: Isosceles

The maximum value of $\sin\theta\cos\theta$ is

Answer: $\frac{1}{2}$

The equation $\sin x+\cos x=2$ has

Answer: no solution

$\sin^{-1}x+\cos^{-1}x=$

Answer: $\frac{\pi}{2}$

If $y=e^{ax^2+b}$ then $\frac{dy}{dx}=$

Answer: $2axy$

If $x^2+y^2=25$ then $\frac{dy}{dx}$ at $(0,5)$ is

Answer: $0$

The stationary point of the function $f(x)=x^2-6x+3$ is $x=$

Answer: $3$

$\int e^{\log_e x},dx=$

Answer: $\frac{x^2}{2}+c$

$\int_0^1\frac{2x}{x^2+2},dx=$

Answer: $\log_e\frac32$

$\int\frac{1}{\cos^2x},dx=$

Answer: $\tan x+c$

Area bounded by the x-axis and the curve $y=x+2$ and the lines $x=0$ and $x=2$ is

Answer: $6$ sq. units

If $f(x)=2x+1$ then the approximate value of $f'(2)$ with $h=0.1$ is

Answer: $2.0$

$\int\cot x,dx=$

Answer: $\ln(\sin x)+c$

$\int\frac{e^x-1}{e^x+1}dx=$

Answer: $2\log\left(e^{x/2}+e^{-x/2}\right)$

$\int\log x,dx=$

Answer: $x\log x-x$

Area of the region bounded by the curves $y^2=4ax$ and $x^2=4by$ is

Answer: $\frac{16ab}{3}$

The standard deviation of $3,6,9,12,15$ is

Answer: $3\sqrt{2}$

The probability of getting an ace of club or a king of spade from a pack of 52 cards is

Answer: $1/26$

The probability of selecting a vowel from the English alphabet is

Answer: $5/26$

The standard deviation of the data: $5, 10, 15, 20$ and $25$ is

Answer: $5\sqrt2$

Probability of an event E(x) is

Answer: $0\le p(\varepsilon)\le1$

If A and B are two independent events with $p(A)=\frac23$ and $p(B)=\frac35$ then $p(A\cup B)=$

Answer: $\frac{13}{15}$

For negative skewness

Answer: Mean<median<Mode

Which is the measure of dispersion?

Answer: range

Two brothers Mr. X and Mr. Y appear in an interview for a scholarship. The probability of Mr. X getting it is $\frac{1}{7}$ and of Mr. Y is $\frac{1}{5}$. What is the probability that both of them will get scholarship?

Answer: $\frac{1}{35}$

The direction of the vector $a\mathbf{i}+b\mathbf{j}$ is

Answer: $\tan^{-1}\left(\frac{b}{a}\right)$

The unit vector along $x\mathbf{i}-y\mathbf{j}$ is

Answer: $\frac{x\mathbf{i}-y\mathbf{j}}{\sqrt{x^{2}+y^{2}}}$

The unit vector of $i+j$ is

Answer: $\frac{i}{\sqrt2}+\frac{i}{\sqrt2}$

The direction of vector $v=ai+bj$ is

Answer: $\tan^{-1}\frac ba$

$\vec a\times\vec b=|\vec a||\vec b|\sin\theta,\hat\eta$, for any non-zero vectors $\vec a$ and $\vec b$

Answer: $\hat\eta$ is the unit vector along $\vec a\times\vec b$

If three equal forces $P$, $Q$ and $R$ are acting at a point of rigid body are in equilibrium, then the angle between any two forces is

Answer: $120^\circ$

A force equal to $4N$ acting vertically upwards is resolved into forces, one being horizontal and equal to $3N$. The magnitude of the other remaining force is

Answer: $5N$

If the acceleration of a particle, starting from rest is $5,m/sec^{2}$, then the velocity at the end of 3 seconds is

Answer: $15,m/sec^{2}$

A stone is thrown vertically upwards with a velocity of $30$ m/sec from the top of a tower and hits the ground after $10$ secs, then the height of the tower is

Answer: $200,m$

If $\begin{pmatrix}0&6\5+k&0\end{pmatrix}$ is a skew symmetric matrix then $k=$

Answer: $-11$

The value of $x$ for the determinant $\begin{vmatrix}x+3&0\1&1\end{vmatrix}=0$ is

Answer: $-3$

The value of $\sqrt{-8}\sqrt{-2}$ is

Answer: $-4$

The modulus of $\frac{1+i}{1-i}$ is

Answer: $1$

The general formula for Newton-Raphson method is

Answer: $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

The approximate value of $\int_0^2x^{2}dx$ using midpoint rule with one interval is

Answer: $2$

The following table gives two kinds of food and two kinds of vitamin: V1 needs 4 units of F1 and 8 of F2 with daily requirement 80; V2 needs 6 of F1 and 4 of F2 with requirement 100; cost per unit is Rs. 3 for F1 and Rs. 5 for F2. If qty of F1 $=x$, qty of F2 $=y$ then formulate the following problem to minimizing the cost.

Answer: min(z) = 3x+5y subject to 4x+8y >= 80, 6x+4y >= 100

The two lines represented by $ax^2+xy+by^2=0$ are perpendicular to each other if

Answer: $a+b=0$

The focus of the parabola $y^2=4x$ is

Answer: $(1,0)$

The pole of the line $y=mx+c$ with respect to the parabola $y^2=4ax$ is

Answer: $\left(\frac{c}{m},\frac{2a}{m}\right)$

If a ball is projected vertically upwards with a velocity of $24.9$ m/s, then the maximum height described by the ball is

Answer: $44.1$ m