32581 + Interview Questions in Mathematics Page 34 InterviewSolution

1651.	The nth term of a sequence is (2n -1). Find the 15th term of the sequence.
Answer» The nth term of a sequence is (2n -1). Find the 15th term of the sequence.

Discussion

1652.	The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another A.P. whose first term is -30 and common difference is 8 . Find n.
Answer» The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another A.P. whose first term is $- 30$ and common difference is 8 . Find n.

Discussion

1653.	If ax3 + bx3 +x - 6 has (x+2) as factor and leaves remainder 4 when divided by x-2. Find a and b.
Answer» If ax³ + bx³ +x - 6 has (x+2) as factor and leaves remainder 4 when divided by x-2. Find a and b.

Discussion

1654.

The following distribution shows the daily pocket allowance given to the children of a multistorey building. The average pocket allowance is Rs 18.00. Find the missing frequency. Class: 11−13 13−15 15−17 17−19 19−21 21−23 23−25 Frequency: 7 6 9 13 − 5 4

Answer» The following distribution shows the daily pocket allowance given to the children of a multistorey building. The average pocket allowance is Rs 18.00. Find the missing frequency.

Class:	11−13	13−15	15−17	17−19	19−21	21−23	23−25
Frequency:	7	6	9	13	−	5	4

Discussion

1655.	Division of rational number by a rational number.
Answer» Division of rational number by a rational number.

Discussion

1656.	Find the number of zeros at the end of 1098765432*1
Answer» Find the number of zeros at the end of 1098765432*1

Discussion

1657.	If x1,x2,x3 are the positive real roots of x3−9x2+kx−k=0, then the value of sin−1(1x1+1x2)+cos−1(1x2+1x3)−tan−1(1x1+1x2+1x3) is
Answer» If $x_{1}, x_{2}, x_{3}$ are the positive real roots of $x^{3} - 9 x^{2} + k x - k = 0,$ then the value of ${sin}^{- 1} (\frac{1}{x_{1}} + \frac{1}{x_{2}}) + {cos}^{- 1} (\frac{1}{x_{2}} + \frac{1}{x_{3}}) - {tan}^{- 1} (\frac{1}{x_{1}} + \frac{1}{x_{2}} + \frac{1}{x_{3}})$ is

Discussion

1658.	Yashraj invested Rs 7000 in 8% shares of Rs 50 at Rs 70. After 1 year he sold them at Rs 65 each and invested proceeds (including dividend) in Rs 100 shares at Rs 92 paying 12% calculate change in his income.
Answer» Yashraj invested Rs 7000 in 8% shares of Rs 50 at Rs 70. After 1 year he sold them at Rs 65 each and invested proceeds (including dividend) in Rs 100 shares at Rs 92 paying 12% calculate change in his income.

Discussion

1659.	Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 34 time the corresponding sides of ΔABC.
Answer» Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are $\frac{3}{4}$ time the corresponding sides of ΔABC.

Discussion

1660.	Find the H.C.F. of x3−3x2+x−3 and x4+6x2+5.
Answer» Find the H.C.F. of $x^{3} - 3 x^{2} + x - 3$ and $x^{4} + 6 x^{2} + 5.$

Discussion

1661.	Points X, Y, Z are collinear such that d(X,Y) = 17, d(Y,Z) = 8, find d(X,Z) .
Answer» Points X, Y, Z are collinear such that d(X,Y) = 17, d(Y,Z) = 8, find d(X,Z) .

Discussion

1662.	In an AP, the pth term is q, and the qth term is p. What is the nth term? ( Given p≠q)
Answer» In an AP, the $p^{t h}$ term is q, and the $q^{t h}$ term is p. What is the $n^{t h}$ term? ( Given $p \neq q$ )

Discussion

1663.	An aeroplane leaves the airport and flies due north at a speed of 600 km/hr. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 800 km/hr. What will be the distance between two planes after 1 hour?
Answer» An aeroplane leaves the airport and flies due north at a speed of 600 km/hr. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 800 km/hr. What will be the distance between two planes after 1 hour?

Discussion

1664.	The condition for the equation x2+bx+c=0 to have real roots is ____.
Answer» The condition for the equation $x^{2} + b x + c = 0$ to have real roots is ____.

Discussion

1665.	If a coin is tossed four times, what is the probability of getting exactly two heads and two tails?
Answer» If a coin is tossed four times, what is the probability of getting exactly two heads and two tails?

Discussion

1666.	Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
Answer» Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

1666.

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

Answer»

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

Discussion

1667.	A vertical tower stands on a horizontal plane and is surmounted by a flag-staff of height 6 m. From a point on the plane, the angle of elevation of the bottom of the flag-staff is 30∘ and that of the top of the flag-staff is 60∘. Find the height of the tower.
Answer» A vertical tower stands on a horizontal plane and is surmounted by a flag-staff of height 6 m. From a point on the plane, the angle of elevation of the bottom of the flag-staff is 30∘ and that of the top of the flag-staff is 60∘. Find the height of the tower.

Discussion

1668.	Question 4 (i)Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.(i) 2, 4, 8, 16, …
Answer» Question 4 (i) Which of the following are APs? If they form an A.P. find the common difference d and write three more terms. (i) 2, 4, 8, 16, …

Discussion

1669.	A vertical stick 10cm long casts a shadow 8cm long. At the same time, a tower casts a shadow 30m long. Determine the height of the tower?
Answer» A vertical stick 10cm long casts a shadow 8cm long. At the same time, a tower casts a shadow 30m long. Determine the height of the tower?

Discussion

1670.	See the given figure and write the following:(i) The point identified by the coordinates (2, - 4).(ii) The coordinates of the point L.(iii) The coordinates of the point E.
Answer» See the given figure and write the following: (i) The point identified by the coordinates (2, - 4). (ii) The coordinates of the point L. (iii) The coordinates of the point E.

Discussion

1671.	A hemispherical bowl of internal radius 9cm is full with liquid . This liquid is to be filled into cylindrical shaped bottles of diameter 3cm and height 4cm . How many bottles are necessary to empty the bowl??
Answer» A hemispherical bowl of internal radius 9cm is full with liquid . This liquid is to be filled into cylindrical shaped bottles of diameter 3cm and height 4cm . How many bottles are necessary to empty the bowl??

Discussion

1672.	A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water overflows?
Answer» A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water overflows?

Discussion

1673.	If zeros of the polynomial f(x) = x3 − 3px2 + qx − r are in A.P., then(a) 2p3 = pq − r(b) 2p3 = pq + r(c) p3 = pq − r(d) None of these
Answer» If zeros of the polynomial f(x) = x³ − 3px² + qx − r are in A.P., then (a) 2p³ = pq − r (b) 2p³ = pq + r (c) p³ = pq − r (d) None of these

Discussion

1674.	Solve \|x^2-x\|>=\|2x-1\|
Answer» Solve \|x^2-x\|>=\|2x-1\|

Discussion

1675.	6+27-3+3+1+23 when simplified is(a) positive and irrational(b) positive and rational(c) negative and irrational(d) negative and rational
Answer» $(6 + \sqrt{27}) - (3 + \sqrt{3}) + (1 + 2 \sqrt{3})$ when simplified is (a) positive and irrational (b) positive and rational (c) negative and irrational (d) negative and rational

Discussion

1676.	In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, ∠B = 90∘ and DS = 5 cm then find the radius of the circle. [CBSE 2008, 13]
Answer» In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, ∠B = 90^∘ and DS = 5 cm then find the radius of the circle. [CBSE 2008, 13]

Discussion

1677.	Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.
Answer» Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.

Discussion

1678.	Draw the graph of y=2x2+7x−9.
Answer» Draw the graph of $y = 2 x^{2} + 7 x - 9$ .

Discussion

1679.	If the equation 9x2 + 6kx + 4 = 0 has equal roots, then the roots are both equal to(a) ±23(b) ±32(c) 0(d) ±3
Answer» If the equation 9x² + 6kx + 4 = 0 has equal roots, then the roots are both equal to (a) $\pm \frac{2}{3}$ (b) $\pm \frac{3}{2}$ (c) 0 (d) ±3

Discussion

1680.	Find the smallest square number that is divisible by each of the numbers 4,9 and 25
Answer» Find the smallest square number that is divisible by each of the numbers 4,9 and 25

Discussion

1681.	Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon and measure its circumradius.
Answer» Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon and measure its circumradius.

Discussion

1682.	y=(2x^3+3)(2x^-3+1) and dy/dx at x=1 is_?
Answer» y=(2x^3+3)(2x^-3+1) and dy/dx at x=1 is_?

Discussion

1683.	In the given figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.
Answer» In the given figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

1683.

In the given figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

Answer»

In the given figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

Discussion

1684.	Fill in the blanks.(i) cot 34° – tan 56° = .......(ii) cosec 31° – sec 59° = .......(iii) cos267° + cos223° = ..........(iv) cosec254° – tan236° = .........(v) sec240° – cot250° = ..........
Answer» Fill in the blanks. (i) cot 34° – tan 56° = ....... (ii) cosec 31° – sec 59° = ....... (iii) cos²67° + cos²23° = .......... (iv) cosec²54° – tan²36° = ......... (v) sec²40° – cot²50° = ..........

Discussion

1685.	Find the probability of girls playing softball in the total class. [2 MARKS]
Answer» Find the probability of girls playing softball in the total class. [2 MARKS]

1685.

Find the probability of girls playing softball in the total class. [2 MARKS]

Answer»

Find the probability of girls playing softball in the total class. [2 MARKS]

Discussion

1686.	59. A Binary solid A+B- has rock salt structure if the edge length is 400pm and the radius of cation A +is 75pm, radius of anion B- is? A. 100pm B. 125 pm C. 250pm D. 325pm
Answer» 59. A Binary solid A+B- has rock salt structure if the edge length is 400pm and the radius of cation A +is 75pm, radius of anion B- is? A. 100pm B. 125 pm C. 250pm D. 325pm

Discussion

1687.	The table shows the Distribution of the Scores obtained by 155 shooters in a shooting competition. ScoresNo. of shooters0−101010−201220−301530−40840−502050−602460−70770−801180−903090−10018 Use a graph sheet to draw an ogive for the distribution. Using the graph estimate the Median.
Answer» The table shows the Distribution of the Scores obtained by 155 shooters in a shooting competition. $\begin{matrix} Scores & No. of shooters 0 - 10 & 10 10 - 20 & 12 20 - 30 & 15 30 - 40 & 8 40 - 50 & 20 50 - 60 & 24 60 - 70 & 7 70 - 80 & 11 80 - 90 & 30 90 - 100 & 18 \end{matrix}$ Use a graph sheet to draw an ogive for the distribution. Using the graph estimate the Median.

1687.

The table shows the Distribution of the Scores obtained by 155 shooters in a shooting competition. ScoresNo. of shooters0−101010−201220−301530−40840−502050−602460−70770−801180−903090−10018 Use a graph sheet to draw an ogive for the distribution. Using the graph estimate the Median.

Answer»

The table shows the Distribution of the Scores obtained by 155 shooters in a shooting competition.

$\begin{matrix} Scores & No. of shooters 0 - 10 & 10 10 - 20 & 12 20 - 30 & 15 30 - 40 & 8 40 - 50 & 20 50 - 60 & 24 60 - 70 & 7 70 - 80 & 11 80 - 90 & 30 90 - 100 & 18 \end{matrix}$

Use a graph sheet to draw an ogive for the distribution.

Using the graph estimate the Median.

Discussion

1688.	If 10 pens are drawn successively with replacement from a box of pens in which 10% pens are defective. The probability that at least 1 pen is defective is
Answer» If $10$ pens are drawn successively with replacement from a box of pens in which $10 %$ pens are defective. The probability that at least $1$ pen is defective is

Discussion

1689.	Prove "If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio".
Answer» Prove "If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio".

Discussion

1690.	Water drops fall at regular intervals from a roof . At an ins†an t when a drop is about to leave the roof, the separations between 3 successive drops below the roof are in the ratio (a)1:2:3 (b) 1:4:9 (c) 1:3:5 (d) 1:5:13
Answer» Water drops fall at regular intervals from a roof . At an ins†an t when a drop is about to leave the roof, the separations between 3 successive drops below the roof are in the ratio (a)1:2:3 (b) 1:4:9 (c) 1:3:5 (d) 1:5:13

Discussion

1691.	Working together,Prathap and Do keep can paint their apartment in 2 days.Working alone,Prathap can complete the job in 3days less than Do keep.How long would it take each person working alone?
Answer» Working together,Prathap and Do keep can paint their apartment in 2 days.Working alone,Prathap can complete the job in 3days less than Do keep.How long would it take each person working alone?

Discussion

1692.	In a triangle OAC, if B is the mid-point of side AC and OA→=a→, OB→=b→, then what is OC→? [CBSE 2015]
Answer» In a triangle OAC, if B is the mid-point of side AC and $\vec{OA} = \vec{a}, \vec{OB} = \vec{b}$ , then what is $\vec{OC}$ ? [CBSE 2015]

Discussion

1693.	1. Why linear equations in two variable have infinite solutions
Answer» 1. Why linear equations in two variable have infinite solutions

Discussion

1694.	Between two numbers, whose sum is157, an even number of A.Ms are inserted. The sum of these means exceeds their number by unity. Find the number of means. __
Answer» Between two numbers, whose sum is $\frac{15}{7}$ , an even number of A.Ms are inserted. The sum of these means exceeds their number by unity. Find the number of means. __

Discussion

1695.	The lines x−1a=y−23=z−34; x−23=y−34=z−45 are coplanar then, the value of a is:
Answer» The lines $\frac{x - 1}{a} = \frac{y - 2}{3} = \frac{z - 3}{4}$ ; $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}$ are coplanar then, the value of $a$ is:

Discussion

1696.

Following is the distribution of I.Q. of 100 students. Find the median I.Q. I.Q.: 55−64 65−74 75−84 85−94 95−104 105−114 115−124 125−134 135-144 No. of students: 1 2 9 22 33 22 8 2 1

Answer» Following is the distribution of I.Q. of 100 students. Find the median I.Q.

I.Q.:	55−64	65−74	75−84	85−94	95−104	105−114	115−124	125−134	135-144
No. of students:	1	2	9	22	33	22	8	2	1

Discussion

1697.	Prove that : sin A−2sin3A2cos3A−cos A=tan A.
Answer» Prove that : $\frac{s i n A - 2 s i n^{3} A}{2 c o s^{3} A - c o s A} = t a n A$ .

Discussion

1698.	Four times a certain two digit number is seven times the number obtained on interchanging its digits. If the difference between the digits is 4; find the number.
Answer» Four times a certain two digit number is seven times the number obtained on interchanging its digits. If the difference between the digits is 4; find the number.

Discussion

1699.	A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?
Answer» A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?

Discussion

1700.	Graph of the quadratic equation ax2 + bx + c = 0 , with a < 0 , D = 0 is
Answer» Graph of the quadratic equation ax² + bx + c = 0 , with a < 0 , D = 0 is

Discussion

Explore topic-wise InterviewSolutions in .

The nth term of a sequence is (2n -1). Find the 15th term of the sequence.

The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another A.P. whose first term is -30 and common difference is 8 . Find n.

If ax3 + bx3 +x - 6 has (x+2) as factor and leaves remainder 4 when divided by x-2. Find a and b.

The following distribution shows the daily pocket allowance given to the children of a multistorey building. The average pocket allowance is Rs 18.00. Find the missing frequency. Class: 11−13 13−15 15−17 17−19 19−21 21−23 23−25 Frequency: 7 6 9 13 − 5 4

Division of rational number by a rational number.

Find the number of zeros at the end of 10*9*8*7*6*5*4*3*2*1

If x1,x2,x3 are the positive real roots of x3−9x2+kx−k=0, then the value of sin−1(1x1+1x2)+cos−1(1x2+1x3)−tan−1(1x1+1x2+1x3) is

Yashraj invested Rs 7000 in 8% shares of Rs 50 at Rs 70. After 1 year he sold them at Rs 65 each and invested proceeds (including dividend) in Rs 100 shares at Rs 92 paying 12% calculate change in his income.

Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 34 time the corresponding sides of ΔABC.

Find the H.C.F. of x3−3x2+x−3 and x4+6x2+5.

Points X, Y, Z are collinear such that d(X,Y) = 17, d(Y,Z) = 8, find d(X,Z) .

In an AP, the pth term is q, and the qth term is p. What is the nth term? ( Given p≠q)

An aeroplane leaves the airport and flies due north at a speed of 600 km/hr. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 800 km/hr. What will be the distance between two planes after 1 hour?

The condition for the equation x2+bx+c=0 to have real roots is ____.

If a coin is tossed four times, what is the probability of getting exactly two heads and two tails?

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

A vertical tower stands on a horizontal plane and is surmounted by a flag-staff of height 6 m. From a point on the plane, the angle of elevation of the bottom of the flag-staff is 30∘ and that of the top of the flag-staff is 60∘. Find the height of the tower.

Question 4 (i)Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.(i) 2, 4, 8, 16, …

A vertical stick 10cm long casts a shadow 8cm long. At the same time, a tower casts a shadow 30m long. Determine the height of the tower?

See the given figure and write the following:(i) The point identified by the coordinates (2, - 4).(ii) The coordinates of the point L.(iii) The coordinates of the point E.

A hemispherical bowl of internal radius 9cm is full with liquid . This liquid is to be filled into cylindrical shaped bottles of diameter 3cm and height 4cm . How many bottles are necessary to empty the bowl??

A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water overflows?

If zeros of the polynomial f(x) = x3 − 3px2 + qx − r are in A.P., then(a) 2p3 = pq − r(b) 2p3 = pq + r(c) p3 = pq − r(d) None of these

Solve |x^2-x|>=|2x-1|

6+27-3+3+1+23 when simplified is(a) positive and irrational(b) positive and rational(c) negative and irrational(d) negative and rational

In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, ∠B = 90∘ and DS = 5 cm then find the radius of the circle. [CBSE 2008, 13]

Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.

Draw the graph of y=2x2+7x−9.

If the equation 9x2 + 6kx + 4 = 0 has equal roots, then the roots are both equal to(a) ±23(b) ±32(c) 0(d) ±3

Find the smallest square number that is divisible by each of the numbers 4,9 and 25

Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon and measure its circumradius.

y=(2x^3+3)(2x^-3+1) and dy/dx at x=1 is_?

In the given figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

Fill in the blanks.(i) cot 34° – tan 56° = .......(ii) cosec 31° – sec 59° = .......(iii) cos267° + cos223° = ..........(iv) cosec254° – tan236° = .........(v) sec240° – cot250° = ..........

Find the probability of girls playing softball in the total class. [2 MARKS]

59. A Binary solid A+B- has rock salt structure if the edge length is 400pm and the radius of cation A +is 75pm, radius of anion B- is? A. 100pm B. 125 pm C. 250pm D. 325pm

If 10 pens are drawn successively with replacement from a box of pens in which 10% pens are defective. The probability that at least 1 pen is defective is

Prove "If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio".

Water drops fall at regular intervals from a roof . At an ins†an t when a drop is about to leave the roof, the separations between 3 successive drops below the roof are in the ratio (a)1:2:3 (b) 1:4:9 (c) 1:3:5 (d) 1:5:13

Working together,Prathap and Do keep can paint their apartment in 2 days.Working alone,Prathap can complete the job in 3days less than Do keep.How long would it take each person working alone?

In a triangle OAC, if B is the mid-point of side AC and OA→=a→, OB→=b→, then what is OC→? [CBSE 2015]

1. Why linear equations in two variable have infinite solutions

Between two numbers, whose sum is157, an even number of A.Ms are inserted. The sum of these means exceeds their number by unity. Find the number of means. __

The lines x−1a=y−23=z−34; x−23=y−34=z−45 are coplanar then, the value of a is:

Following is the distribution of I.Q. of 100 students. Find the median I.Q. I.Q.: 55−64 65−74 75−84 85−94 95−104 105−114 115−124 125−134 135-144 No. of students: 1 2 9 22 33 22 8 2 1

Prove that : sin A−2sin3A2cos3A−cos A=tan A.

Four times a certain two digit number is seven times the number obtained on interchanging its digits. If the difference between the digits is 4; find the number.

A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?

Graph of the quadratic equation ax2 + bx + c = 0 , with a &lt; 0 , D = 0 is

Find the number of zeros at the end of 1098765432*1

Graph of the quadratic equation ax2 + bx + c = 0 , with a < 0 , D = 0 is