CHAPTER NO # 02 THE STRAIGHT LINE
SHORT QUESTION
(i) The line joining the points A \((2, -3)\) and B \((-4, 5)\) is trisected. Find the coordinate of the points of trisection.
(ii) The line through \((2, 5)\) and \((-3, -2)\) is perpendicular to the line through \((4, -1)\) and \((x, 3)\). Find x. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(iii) Find the ratio in which the point P \((0,-1)\) divides the join of Q\((1,2)\) and R\((2,5)\).
(iv) Determine the equation of the line which passes through the point \((-2,-4)\) and has sum of intercepts equal to 3.
(v) The centroid of a triangle, whose two vertices are \((2,4)\) and \((3,-4)\) is found to be \((3,1)\) Find its third vertex.
(vi) Find the ratio in which the y-axis divides the join of \((-5 , 3)\) and \((8 , 6)\). Also, find the coordinate of the point of division. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(vii) Prove that the points, whose coordinates are \((5 , 1)\), \((1 , -1)\) and \((11, 4)\) lies on a straight line. Find the intercepts made by this line on axes.
(viii) Find the measure of the angle from a line with slope -2/3 to: (a) y-axis (b) x-axis.
(ix) Using Slope, Prove that \((12 , 8)\), \((-2, 6)\) and \((6, 0)\) are the vertices of right traingle.
(x) Determine the equation of the line which passes through the point \((-3,-4)\) and has sum of intercepts equal to 1.(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(xi) Determine the equation of the line which passes through the point \((-1,2)\) and has sum of intercepts equal to 2.
(xii) A straight line passes through the points A\((-12, -13)\), B\((-2, -5)\). Find the point on line whose ordinate is -1. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
DETAILED QUESTION
(i) A is two-third the way from \((1, 10)\) and \((-8, 4)\) and is the mid point of \((0, -7)\) and \((6, -11)\). find the distance \(|\overline{AB}|\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(ii) Find the equation of the straight line which passes through the point \((-3, 2)\) and is such that the portion of it between the axis is divided by the point in the ratio 1:2.
(iii) The vertices A, B, and C of a triangle are \((2, 1)\), \((5, 2)\), and \((3, 4)\) respectively. Find the coordinates of the circumcenter and radius of the circumcircle of the triangle ABC.
CHAPTER NO # 03 THE GENERAL EQUATION OF STRAIGHT LINE
SHORT QUESTION
(i) Find the equation of the line through the intersection of the lines \(2x-2y+4=0, 3x+3y-5=0\) and parallel to the y-axis. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(ii) The point \((2, -5)\) is a vertex of a square, one of which sides lies on the line \(x - 2y -7=0\). Calculate the area of the square.
(iii) In what ratio is the line segment joining \((1, 3)\) and \((2, 7)\) divides by \(3x + y =9\)?
(iv) Find the equation of the line which is perpendicular to the line \(2x + 3y +4=0\) and passing through the point \((2, -1)\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(v) A line whose y-intercept is 1 less than its x-intercept, form with the coordinates axis a triangle of area 6 square units. What is its equation?
(vi) Find the coordinate of the foot perpendicular from \((-2, 5)\) to \(3x + y +11=0\).
(vii) Find the value of k when the vertices of the triangle are the points \((2, 9)\), \((-2, 1)\), and \((k, 3)\) and its area are 28 square units.
(viii) The gradient of one of the lines \(ax^{2}+ 2hxy+ by^{2}=0\) is five times that of the other. Show that \(5h^{2}=9ab\).(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(ix) The gradient of one of the lines \(ax^{2}+ 2hxy+ by^{2}=0 \) is two times that of the other. Show that \(8h^{2}=9ab\).
(x) Find the value of k for which two lines \((k-1)x + ky -50 =0\) , \(kx + (2k-1)y+7=0\) intersect at a point lying on the axis of x.(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(xi) The area of a triangle is 8 square units. Two of its vertices are the points A \((1,-2)\) and
B \((2,3)\). and the third vertex C lies on the line \(2x + y -2 =0\) Find the co-ordinate of vertex of C.
(xii) Find the combined equation of the pair of lines through the origin which is perpendicular to the lines represented by \(6x^{2}-13xy+ 16y^{2}=0\).
(xiii) Find the distance between two parallel lines \(5x-12y+10=0\) and \(5x-12y-16=0\).
(xiv) Find the equation of the locus of the points which are equidistance from the point \((0, 3)\) and the line \(y+3=0\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(xv) Find the equation of the line passing through the intersecting of the line \(2x+3y+1=0\), \(3x-4y-5=0\) and passing through the point \((2,1)\).
(xvi) Find the value of k when the vertices of the triangle are the points \((2,6)\), \((6,3)\) & \((4,k)\) and its area are 15 sq. units.
(xvii) Find the equation of a line through the intersection of the lines \(7x-13y+46=0\), \(19x+ 11y-41=0\) and passing through the point \((3,1)\) by using K-Method.
(xviii) The point \((-2,1)\) is a vertex of a rectangle whose two sides lie on the lines \(3x-2y-5=0, 2x + 3y +7=0\). Find the area of the rectangle.(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
DETAILED QUESTION
(i) The coordinates of two points A and B are \((3,4)\) and \((5,-2)\)respectively. Find the coordinates of any point P if \(|\overline{PA}| = |\overline{PB}|\) and the area of triangle PAB is 10 square units.
(ii) Find the measures of the angles of the triangle, the equations of whose sides are \(x+y-5=0, x-y +1=0\) and \(y=1\).
(iii) Find the centroid of the triangle the equations of whose sides are \(12x^{2}-20xy+ 7y^{2}=0\) and \(2x-3y+4=0\).(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(iv) Find the equation of the locus of moving points such that the point to A \((1,3)\) is three times the slope of the line joining the point to B\((3,1)\).
(v) The point \((2, -5)\) is a vertex of a square, one of whose sides lies on the line \(x -2y -7 =0. \) Calculate the area of the square.
(vi) Find the equation of the two straight lines passing through \((3, -2)\) and inclined at \(60\degree\) to the line \(\sqrt{3}x+y =1\).
CHAPTER NO # 04 DIFFERENTIATION OR DIFFERENTIABILITY
SHORT QUESTION
(i) Find the differentiation by the first principle method at any point x in the domain D(f) of the following function
(a) \(f(x) = 3x^{3}-x \) (b) \(f(x)=tanx\) (c) \(f(x)=sinx^{2}\)
(d) \(f(x)=x^{\frac{2}{3}}\) (e)\(f(x)= cos^{2}x\) (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) (f) \(f(x)= sin2x\) (g) \(f(x)= cosecx\) (h) \(f(x)= cotx\)
(i) \(f(x) = 3x^{2}+ x \) (j) \(f(x) = 3x^{2}-x \) (k) \(f(x)= cosx^{2}\)
(ii) Find \(\frac{\text{d}y}{\text{d}x}\) of the following. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(a) \(x^{3}+y^{3}+ 3xy=0\) (b) \(sin(x+y) =ln(x-y)\) (c) \(x=lnt+sint , y =e^{t} +cost\) OR \(x=lnt+cost , y =e^{t} +sint\) (d) \(y= \frac{1}{2} tan^{2}x+ln cosx\) (e) \(y = (lnx)^{sinx}\) (f) \( 2x^{2}-3xy+y^{2}=5\) (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(g) \(x^{y}\cdot y^{x}=5\) (h) \(y=ln\left(\frac{e^{x}}{1+e^{x}}\right)\) (i) \(x=acos^{2}3\theta , y=asin^{2}3\theta\) (j) \(2x^{3}-2xy+y^{3}=5\) OR \(2x^{2}-3xy+y^{2}=5\) (k) \(e^{x}lny=sin^{-1}y\) (l) \(\sqrt{x^{2}+y^{2}}=ln(x^{2}-y^{2})\) (m) \(e^{sinx+cosx}\) (n) \(y=(sin^{-1}x)^{3}\) (o) \(\sqrt[5]{x^{2}+2x+3}\) (p) \((lnx)^{tan^{-1}x}\) (q) \(x=sint^{3}+cost^{3} , y =sint+2cos^{-1}t\) (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
DETAILED QUESTION
(i) Find \(\frac{\text{d}y}{\text{d}x}\) of the following.
(a) \(y=ln\left(\frac{1-x^{2}}{1+x^{2}}\right)\) (b) \(y=\sqrt{a^{2}-x^{2}}+ln\sqrt{1+x^{2}}\) (c) \(x=lnt + sin^{-1}t , y =e^{t}+cost\) (d) \(y= cos^{-1}\left(\frac{2x}{1-x^{2}}\right)\) (e) \(x=a(\theta-sin\theta) , y=a(1-cos\theta)\) (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
(ii)
CHAPTER NO # 05 APPLICATION OF DIFFERENTIAL CALCULUS
SHORT QUESTION
(i) Using differentials, calculate the approximate value of \(cos 46^{0}) or \(sin 46^{0}\).
(ii) Find the approximate value of \( sin 46^{0}) using differentials.
SHORT QUESTION
Q: Evaluate the Following:
(i) \(\int\frac{secxtanx}{a+bsecx}dx\)
(ii) \(\int sin^{2}\theta cos^{2}\theta d\theta\)
(iii) \(\int3x\sqrt{1-2x^{2}}dx\)
(iv) Solve the differential equation dy/dx=cosec 2y cos y
(v) Find the area above the x-axis, under the curve \(y=x-5x^{-2}\) , between the coordinates \(y = 2\) and \(x=4\).
(vi) Solve the differential equation: \(2+2y\frac{dy}{dx} =1+3x^{2} \)
CHAPTER NO # 07 : CIRCLE
SHORT QUESTION
- Find the equation of the circle having \((-5,6)\) and \((3,-4)\) are the endpoints of a diameter.
- Prove that the curves \(3x^{2}-y^{2} = 12\) and \(x^{2}+3 y^{2} -24=0\) intersect at right angle at the point \((\sqrt{6} , \sqrt{6})\)
- Find the equation of the circle passing through the points \((0, 3)\), \(( 2, 1)\), and \(( 1, 0 )\).
- Find the equation of the circle with radius \(\sqrt{a^{2}+b^{2}}\) which passes through the two points \((a,0)\) and \((-a, 0)\).
- find the equation of the circle of radius a which passes through the two points on the x-axis which are at a distance b from the origin.
- Find the equation of the circle concentric with the circle \(x^{2}+y^{2}+6x-10y+33=0\) and touching line \(y = 0\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
- Prove that the product of abscissas of the point where the straight line \(y = mx\) meets the circle \(x^{2}+y^{2}+2gx+2fy+c=0\) is equal to \(\frac{c}{1+ m^{2}}\)
- Show that the lines \(x =5\) and \y = 7\) both touches the circle \(x^{2}+y^{2}-4x-8y+11=0\).
- Prove that the point \((5, -7.5)\) lies outside the circle whose equation is \(x^{2}+y^{2}-4x+2y=44\).
- Find the equation of a circle with center at the point \((1, -1)\) and touching the straight line \(5x + 12y = 7\).
- Prove that the two circle \(x^{2}+y^{2}+2gx+c=0\) and \(x^{2}+y^{2}+2fy+c=0\) touch each other if \(\frac{1}{f^{2}}+\frac{1}{g^{2}}=\frac{1}{c}\)
- Find the equation of the circle which is concentric with the circle \(x^{2}+y^{2}-8x+12y -12=0\) and passing through the point \((5,4)\).
- Find the equation of the circle touching each axis in the 4th quadrant at a distance of \(5\) units from the origin.
- Find the equation of circle concentric with the circle \(x^{2}+y^{2}+6x-10y+33=0\) and touching the y-axis.
- Prove that the straight line \(y=x+c\sqrt{2}\) touches the circle \(x^{2}+y^{2}=c^{2}\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED)
DETAILED QUESTION
- Find the condition that conic \(ax^{2}+by^{2}=1\) and \(\acute{a}x^{2}+\acute{b}y^{2}=1\) cut each other orthogonally.
- Find the equation of the circle containing the points \((-1, -1)\) and \((3,1)\) and with the center on the line \(y - x + 10 = 0\).
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