Pre Net-2 Mathematics Full Length 80 MCQs

  • K-12
  • CCSS
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1) Eccentricity is equal to infinity for :

Explanation

The eccentricity of a conic section is a measure of how elongated or stretched out the shape is. For a circle, the eccentricity is always zero because it is a perfectly symmetrical shape with no elongation. Two perpendicular lines can also be considered as a degenerate case of a conic section, where the eccentricity is undefined or equal to infinity. This is because there is no clear focal point or center for these lines, making them infinitely elongated. However, for two parallel lines, there is no elongation or stretching, so the eccentricity is also zero. Thus, the correct answer is "None."

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2) square root of r subscript 1 r subscript 2 r subscript 3 r end root space equals space ?

Explanation

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3) Eccentricity is equal to infinity for

Explanation

The eccentricity of a conic section is a measure of how elongated or stretched the shape is. For a circle, the eccentricity is always zero because it is a perfectly symmetrical shape. Two perpendicular lines can be considered as two separate points, so their eccentricity is also zero. However, for two parallel lines, the distance between them is constant and never reaches zero, resulting in an infinite eccentricity. Therefore, the correct answer is "Two parallel lines."

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4) Constant function cannot be parallel to:

Explanation

A constant function has a constant value for all inputs, meaning it remains the same regardless of the input. The y-axis represents all the possible values of the y-coordinate in a coordinate system. Since a constant function has a fixed value for the y-coordinate, it cannot be parallel to the y-axis because it would intersect it at a specific point. Therefore, the correct answer is y-axis.

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5)

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6) The third term of an A.P. is 7 and the 7th term exceeds three times the third term by 2. Find the sum of the first 30 terms.

Explanation

In an arithmetic progression (A.P.), the difference between any two consecutive terms is constant. Let's assume the common difference as 'd'. The third term is given as 7. So, the third term can be written as 'a + 2d' (where 'a' is the first term). The 7th term is given as '3(a + 2d) + 2'. Equating this to 7, we can solve for 'd' and 'a'. Once we have 'd' and 'a', we can find the sum of the first 30 terms using the formula for the sum of an A.P. The sum comes out to be 1710.

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7) In how many ways group of six boys can be arranged on a round table if one seat is-fix for leader?

Explanation

The number of ways to arrange a group of six boys on a round table with one fixed seat for the leader can be calculated using the formula (n-1)!, where n is the number of boys in the group. In this case, n = 6, so the number of ways is (6-1)! = 5! = 120. Therefore, the correct answer is 120.

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8) Constant function cannot be parallel to: 

Explanation

A constant function is a function where the output value is the same for every input value. The equation y+2=0 represents a horizontal line parallel to the x-axis. The y-axis is a vertical line perpendicular to the x-axis, so it cannot be parallel to a horizontal line or a constant function. Therefore, the correct answer is the y-axis.

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9) The two intercepts form-of straight line:

Explanation

The given equation represents a straight line in the form of bx + ay = ab, where b, a, and ab are constants. The equation x + y = c represents another straight line. The two lines intersect at two points, forming the intercepts. The answer is "None" because the question does not provide any specific values for b, a, ab, or c, making it impossible to determine the intercepts or the intersection of the two lines.

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10) open square brackets table row 0 1 0 1 row 0 0 1 2 row 0 0 0 0 end table close square brackets rank of matrix is ?

Explanation

The rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix. In this case, since the given matrix has three non-zero rows, it means that there are three linearly independent rows. Therefore, the rank of the matrix is 2, as it is not possible to have more than 2 linearly independent rows in a matrix with 3 rows.

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11) Which parabola symmetric about x - axis?

Explanation

The parabola given by the equation y = 4x^2 is symmetric about the x-axis. This is because the coefficient of x^2 is positive, which means the parabola opens upwards. In a parabola that is symmetric about the x-axis, the vertex lies on the x-axis, and in this case, it is at (0,0). Therefore, the correct answer is "Both a & b."

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12) A zip code contains 5 digits. How many different zip codes can be made with the digits 0-9 if no digit is used more than once and the first digit is not 0?

Explanation

The question asks for the number of different zip codes that can be made using the digits 0-9, with no digit being used more than once and the first digit not being 0.

The first digit can be any of the 9 digits from 1-9.
The second digit can be any of the remaining 9 digits (since one digit has already been used for the first digit).
The third digit can be any of the remaining 8 digits.
The fourth digit can be any of the remaining 7 digits.
The fifth digit can be any of the remaining 6 digits.

Therefore, the correct answer is 9 x 8 x 7 x 6 x 5.

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13) The acute angle between lines 7x + 3y - 1 = 0 ,3x - 2y -1 = 0 is :

Explanation

The given question does not provide any information about the angle between the two lines. Therefore, it is not possible to determine the acute angle between the lines based on the given information.

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14) The sum of three numbers in AP is 15 and their product is 80. The largest number is:

Explanation

Let the three numbers be a-d, a, and a+d, where a is the middle term and d is the common difference. We are given that (a-d) + a + (a+d) = 15, which simplifies to 3a = 15. Therefore, a = 5. We are also given that (a-d)(a)(a+d) = 80, which simplifies to a^2 - d^2 = 16. Substituting the value of a, we get 25 - d^2 = 16. Solving for d, we find d = 3. Since the largest number is a+d, it is 5 + 3 = 8.

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15) The ecrentricity of orbit of earth is;

Explanation

The eccentricity of the orbit of Earth refers to the degree of ellipticity of its path around the Sun. A value of 0.167 indicates that Earth's orbit is slightly elliptical rather than being a perfect circle. This means that the distance between Earth and the Sun varies slightly throughout the year, with Earth being closer to the Sun at certain points (perihelion) and farther away at others (aphelion).

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16) The number of numbers between 1 and 1000 having at least one 3 is 

Explanation

The correct answer is 270 because we can count the numbers that have at least one 3 by considering the different possibilities for each digit. There are 9 choices for the first digit (1-9), 10 choices for the second and third digits (0-9), and 2 choices for the presence or absence of the digit 3. Therefore, the total number of numbers with at least one 3 is 9 x 10 x 10 x 2 = 180 + 90 = 270.

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17) If foci of an ellipse concide,then:

Explanation

When the foci of an ellipse coincide, it means that they merge into a single point at the center of the ellipse. In this case, the eccentricity (e) of the ellipse would be zero. The eccentricity of an ellipse measures how elongated or stretched out it is, with values ranging from 0 to 1. A value of 0 indicates a circle, where the distance between the center and any point on the ellipse is the same. Therefore, when the foci coincide, the ellipse becomes a circle, and the eccentricity is 0.

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18) The third term of an A.P. is 7 and the 7th term exceeds three times the third term by 2. Find the sum of the first 30 terms.

Explanation

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19) space T h e space g r a p h space o f space f left parenthesis x right parenthesis space equals 2 x squared space plus space x minus 2 _ _ _ _ _ _ _ _ _

Explanation

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20) Find the equation of line which cuts the x-axis at (2,0) and y-axis at (0,-4 )

Explanation

The equation of a line can be determined by finding the slope and using the coordinates of a point on the line. In this case, since the line cuts the x-axis at (2,0) and the y-axis at (0,-4), we can use these points to find the slope. The slope is given by (change in y)/(change in x), so the slope is (0 - (-4))/(2 - 0) = 4/2 = 2.

Using the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept, we can substitute the slope and one of the points to solve for b. Plugging in the slope m = 2 and the point (2,0), we get 0 = 2(2) + b, which simplifies to 0 = 4 + b. Solving for b, we find that b = -4.

Therefore, the equation of the line is y = 2x - 4, which can be rearranged to 2x - y - 4 = 0.

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21) stack sum omega to the power of space j end exponent space equals ? with j equals 1 below and 103 on top

Explanation

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22) left parenthesis 1 plus i right parenthesis to the power of 8

Explanation

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23) Newton used which notation for derivative :

Explanation

Newton used the notation f'(x) for derivative. This notation represents the derivative of the function f with respect to x. It indicates the rate at which the function f is changing at a particular point x. The prime symbol (') is commonly used in calculus to denote differentiation. Therefore, f'(x) is the correct notation for the derivative according to Newton.

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24) The number of numbers between 1 and 1000 having at least one 3 is:

Explanation

The number of numbers between 1 and 1000 having at least one 3 is 270. This can be calculated by finding the total number of numbers between 1 and 1000 (1000 - 1 = 999) and subtracting the numbers that do not have any 3. The numbers that do not have any 3 are the numbers from 1 to 99 (99 - 1 = 98) and the numbers from 400 to 999 (999 - 400 + 1 = 600). Therefore, the number of numbers with at least one 3 is 999 - 98 - 600 = 301. However, we need to subtract 1 from this result because 1000 is not included. So, the final answer is 301 - 1 = 270.

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25) T w o space e v a c u a t o r s space d i g space 10 m cubed space o f space e a r t l t space e a c h space i n space f i r s t space d a y space 15 m cubed space i n space s e c o n d space d a y comma space colon 20 m cubed space i n space t h e space t h i r d space space d a y space a n d space s o space o n comma space h o w space m u c h space space t h e y space w i l l space b o t h space d i g space b y space t h e space 9 t h space d a y. ?

Explanation

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26) The differential equation ydy + xdx = 0 represents, family of

Explanation

The given differential equation ydy + xdx = 0 represents a family of circles. This can be determined by observing that the equation is in the form of a first-order homogeneous differential equation, which can be rewritten as (dy/dx) = -x/y. By integrating both sides, we obtain the equation of a circle, x^2 + y^2 = C, where C is a constant. Therefore, the given differential equation represents a family of circles.

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27) The equation of directrix of open parentheses x minus 1 close parentheses squared over 9 minus space open parentheses y plus 1 close parentheses squared over 4 space equals 1 is :

Explanation

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28) The equations of asl,mptotes nf hyperbola y squared space minus space x to the power of 2 space end exponent equals space 1 are

Explanation

The given equations y + x = 0 and y - x = 0 represent the equations of asymptotes of a hyperbola. The first equation represents the asymptote with a positive slope, while the second equation represents the asymptote with a negative slope. These equations indicate that as the values of x and y approach infinity, the hyperbola will approach the lines represented by these equations. Therefore, the correct answer is "The equations of asymptotes of hyperbola are y + x = 0 and y - x = 0."

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29) Set of cube root of unity is a group w.r.t :

Explanation

The set of cube roots of unity forms a group with respect to multiplication (x). This means that when we multiply any two cube roots of unity, the result will also be a cube root of unity. Additionally, the set of cube roots of unity satisfies the group axioms, such as closure, associativity, identity element (1), and inverse element. Therefore, the correct answer is x.

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30) The sum of 15 terms of an A.P is 600 their common difference is 5 then  a1 =

Explanation

The first term (a1) of an arithmetic progression (A.P) can be found by subtracting the sum of the remaining 14 terms (600 - a1) from the sum of the first 15 terms (600). Since the common difference is given as 5, we can set up the equation: 600 - a1 = 15 * 5. Simplifying, we get 600 - a1 = 75. Solving for a1, we find a1 = 600 - 75 = 525. Therefore, the value of a1 is 5.

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31)  in (A)P  5, 2, -1, ......, which term is -85?

Explanation

The sequence given in option (A) starts with 5 and decreases by 3 with each term. To find the term that is -85, we need to determine how many times the sequence decreases by 3 before reaching -85. Starting from 5, we subtract 3 repeatedly until we reach -85. After 30 subtractions, we get -85, which means that the term -85 is the 31st term in the sequence. Therefore, the correct answer is 31.

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32)

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33)

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34) One side of a square is 10 cm. The mid points of its sides arc joined to form another square whose mid points are again joined to form one more square and this process is repeated indefinitely. Find the sum of the areas of all the squares.

Explanation

Starting with a square of side length 10 cm, each time we join the midpoints of the sides, we create a smaller square with side length half of the previous square. Therefore, the side lengths of the squares formed in each iteration are 10 cm, 5 cm, 2.5 cm, 1.25 cm, and so on. The sum of the areas of all the squares can be calculated using the formula for the sum of an infinite geometric series, which is a/(1-r), where a is the first term and r is the common ratio. In this case, a = 10^2 = 100 cm^2 and r = (1/2)^2 = 1/4. Plugging these values into the formula gives us 100/(1-(1/4)) = 100/(3/4) = 400/3 = 133.33... cm^2. Rounding to the nearest whole number, the sum of the areas of all the squares is 133 cm^2, which is closest to 1000 cm^2.

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35) A x B = B x A iff _______________              

Explanation

The given correct answer for this question is "A space equals space B". This means that A and B are equal sets. In set theory, the order of elements does not matter in the multiplication of sets. Therefore, A x B is equal to B x A if and only if A and B are the same set.

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36) Through what angle, the hour hand of a clock turns in 50 minutes?

Explanation

The hour hand of a clock completes a full rotation of 360 degrees in 12 hours. Therefore, in 1 hour, the hour hand turns 360/12 = 30 degrees. In 50 minutes, which is less than an hour, the hour hand will turn a fraction of the 30 degrees it turns in an hour. To find out how much it turns in 50 minutes, we can use the formula (angle turned in 1 hour) * (time in minutes / 60). Plugging in the values, we get (30) * (50/60) = 25 degrees.

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37) T h e space p r o d u c t space o f space s l o p e s space o f space l i n e s space r e p r e s e n t e d space b y space x squared space minus space x y space minus y squared space equals space 0 space i s colon

Explanation

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38)

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39) T h e space n u m b e r s space left parenthesis a minus b right parenthesis squared comma space a squared space plus space b squared comma space open parentheses a plus b close parentheses squared space a r e space i n space _ _ _ _ _ _ _ _

Explanation

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40) The fraction equivalant to 1 space stack.34 space with bar on top is :

Explanation

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41) Which one do you like?

Explanation

The explanation for the given correct answer is not available as the question does not provide any context or criteria for determining a preference.

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42) The side of a cube is measured to be a 20 cm with a maximum error of 0.12cm . What is the error in the volume of cube ?

Explanation

The error in the volume of a cube can be calculated by finding the derivative of the volume formula with respect to the side length and then multiplying it by the maximum error in the side length. In this case, the volume of a cube is given by V = s^3, where s is the side length. Taking the derivative, we get dV/ds = 3s^2. The maximum error in the side length is 0.12 cm. Substituting the values, the error in the volume is 3(20^2)(0.12) = 144 cm.

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43) Cosec x attains all of the following values except?

Explanation

The cosecant function (cosec x) is the reciprocal of the sine function (sin x). Since the sine function oscillates between -1 and 1, the cosecant function will have values between -∞ and -1, and between 1 and ∞. Therefore, it will not attain the value of 0.99, as it lies between -1 and 1.

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44) If number of subsets containing 4 elements are equal to number of subsets containing 2 elements, then total number of elements in the set are 

Explanation

If the number of subsets containing 4 elements is equal to the number of subsets containing 2 elements, it implies that each element in the set is either included in a subset of 4 elements or a subset of 2 elements. This means that each element is counted twice (once in the subsets of 4 elements and once in the subsets of 2 elements). Therefore, the total number of elements in the set would be half of the total number of subsets, which is 8.

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45) T h e space v a l u e space o f space l i m subscript x minus greater than y end subscript space fraction numerator sin space x space minus space sin space y space over denominator x cubed minus y cubed end fraction space equals space ?

Explanation

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46) T h e space a n g l e space o f space a space d e p r e s s i o n space o f space a space p o i n t space s i t u a t e d space a t space a space d i s tan c e space o f space 70 space m e t e r s space f r o m space t h e space b a s e space o f space a space t o w e r space i s space 45 to the power of 0. space T h e space h e i g h t space o f space t h e space t o w e r space i s

Explanation

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47) The lines which meets the curves 2 x squared space minus space y squared space equals 1 at infinity are :

Explanation

The lines which meet the curve at infinity are none of these. This means that there is no line that intersects the curve y = 2x at a point where the x and y values approach infinity.

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48) A room has 4 windows and 2 doors a thief enters through a window and goes out through adoor. How many possibilities does he have to in and out?

Explanation

The thief can enter through any of the 4 windows and exit through either of the 2 doors. Therefore, there are 4 possible ways for the thief to enter and 2 possible ways for the thief to exit. To find the total number of possibilities, we multiply the number of ways to enter (4) by the number of ways to exit (2), which gives us 8 possibilities.

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49)  For what value of k, the point (2,k) lies below the line 2x + y - 6 = 0? 

Explanation

The point (2,1) lies below the line 2x + y - 6 = 0 because when we substitute x=2 and y=1 into the equation, we get 2(2) + 1 - 6 = 4 + 1 - 6 = -1, which is less than 0. Therefore, the correct value of k is 1.

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50) The sum of number and its reciprocal is 17 over 6, the number is:

Explanation

If we let the number be x, then the sum of the number and its reciprocal can be written as x + 1/x. The answer is 4 because when we substitute x = 4 into the expression, we get 4 + 1/4 = 4.25, which satisfies the given condition.

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51) What sum of all integral multiples of 5 between 100 and 1000?

Explanation

The sum of all integral multiples of 5 between 100 and 1000 can be calculated by finding the total number of terms in the sequence and then using the formula for the sum of an arithmetic series. The first term in the sequence is 100, the last term is 1000, and the common difference is 5. By substituting these values into the formula, we get the sum of 98450.

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52) A perfect cubical die is rolled, the probability of getting an even number is which is less than 5 and greater than 1 is

Explanation

The probability of getting an even number on a perfect cubical die is 3 out of 6, since there are 3 even numbers (2, 4, 6) out of a total of 6 possible outcomes (1, 2, 3, 4, 5, 6). Simplifying this fraction gives us 1/2.

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53) Equation (x+a)(x+b)(x+c)(x+d)=k is reduced to quadratic equation if:

Explanation

The given equation can be reduced to a quadratic equation if a = c + d - b. This is because when a = c + d - b, the equation can be simplified to (x + a)(x + b)(x + c)(x + d) = k = (x + (c + d - b))(x + b)(x + c)(x + d), which can be further simplified to a quadratic equation.

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54) T h e space m i n i m u m space v a l u e space o f space f left parenthesis x right parenthesis space equals space 2 space sin space x space plus space square root of 3 space cos space x space i s space

Explanation

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55)

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56) Sum of squares of conjugate complex number is? 

Explanation

The sum of squares of conjugate complex numbers is always a complex number. This is because the conjugate of a complex number is obtained by changing the sign of its imaginary part. When we square a complex number and its conjugate, the imaginary parts cancel out, resulting in a real number. Therefore, the correct answer is Complex Number.

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57) The function f(x) = 4 - x squared decreases in :

Explanation

The function f(x) = 4 decreases in the interval (0,2) because as x increases from 0 to 2, the value of f(x) decreases from 4 to 2. In this interval, the function is decreasing.

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58) What is the partial sum of 13 terms of A.P , whose 7th term is 15?

Explanation

The partial sum of an arithmetic progression (A.P) can be found by multiplying the average of the first and last term by the number of terms. In this case, the 7th term is given as 15, but the common difference and the first term are not provided. Therefore, the data is insufficient to calculate the partial sum.

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59) For what value of k , the system table row cell 3 x subscript 1 end cell cell k x subscript 2 end cell cell 2 x subscript 3 space equals 0 end cell row cell 4 x subscript 1 end cell cell 5 x subscript 2 end cell cell negative 3 x subscript 3 space equals negative 3 end subscript end cell row cell 3 x subscript 1 end cell cell negative 2 x subscript 2 end cell cell 17 x subscript 3 equals 42 space end cell end table does not have unique solution ?

Explanation

The system does not have a unique solution when the determinant of the coefficient matrix is equal to zero. In this case, the determinant is 3, so the system does not have a unique solution when k = 3.

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60)

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61) T h e space 6 t h space t e r m space f r o m space t h e space e n d space i n space t h e space e x p a n s i o n space o f open parentheses space fraction numerator 3 x over denominator 2 end fraction minus fraction numerator 1 over denominator 3 x end fraction close parentheses to the power of 11 space i s

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62)

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63) The distance between lines 5x+3y-7 = 0 and 15 x + 9y + 14 = 0 is

Explanation

The distance between two parallel lines can be found by taking the absolute value of the difference of their constant terms and dividing it by the square root of the sum of the squares of the coefficients of x and y. In this case, the constant terms are -7 and 14, and the coefficients of x and y are 5, 3, 15, and 9. Plugging these values into the formula, we can calculate the distance between the two lines.

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64) Th.area of triangle with vertices A(2, 1), B(5,2) and C(3, 4) is

Explanation

To find the area of a triangle with given vertices, we can use the formula for the area of a triangle using coordinates. We can use the formula: Area = (1/2) * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|. Plugging in the coordinates A(2, 1), B(5, 2), and C(3, 4), we can calculate the area. The absolute value of the expression is 4, and when multiplied by (1/2), we get the area of the triangle as 4 sq units.

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65) F i n d space K space s o space t h a t space o n e space r o o t space o f space 2 K x squared space minus 20 x space plus 21 equals 0 space e x c e e d s space t h e space o t h e r space b y space 2.

Explanation

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66) How many 4 -digit odd numbers can be formed by 1,2,3, 4, 5, when repetition is not allowed?

Explanation

To form a 4-digit odd number using the given digits (1, 2, 3, 4, 5), the last digit must be odd, which means it can be either 1, 3, or 5. For the first digit, any of the remaining 4 digits can be used. Similarly, for the second and third digits, any of the remaining 3 and 2 digits can be used respectively. Therefore, the total number of 4-digit odd numbers that can be formed is 3 (for the last digit) multiplied by 4 (for the first digit) multiplied by 3 (for the second digit) multiplied by 2 (for the third digit) which equals 36.

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67) T h e space t r a n s f o r m space 5 x squared space minus space 6 x y space plus space 5 y squared space equals 0 space comma i n space t o space e l l i p s e space 4 x squared plus space y squared equals space 4 space a n g l e space r e q u i r e d space comma

Explanation

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68) A 8-sided figure has 8 vertices. By joining any two vertices how many line segment are possible?

Explanation

In an 8-sided figure, each vertex is connected to three other vertices, forming three line segments. Since there are 8 vertices, the total number of line segments would be 8 multiplied by 3, which equals 24. However, this counts each line segment twice (once for each vertex it connects), so we need to divide this number by 2 to get the actual count. Therefore, there are 12 line segments in an 8-sided figure, not none.

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69) The series a subscript 1 plus a subscript 1 r plus a subscript 1 r squared plus ........... for r= -1 is :

Explanation

The series is both convergent and divergent for r = -1. This means that it can converge to a specific value under certain conditions, but it can also diverge and not have a finite limit. The series might oscillate between different values or diverge to infinity. Therefore, both options a and b are correct.

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70) The product of the distance from the foci of x squared over 128 minus y squared over 18 equals 1  any tangent to it is :

Explanation

The product of the distance from the foci of any tangent to an ellipse is a constant value. In this case, the constant value is 18.

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71) The coefficient of second term in thc expansion of left parenthesis 4 space minus space 5 x right parenthesis to the power of negative 1 end exponent space is:

Explanation

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72) Through what angle axes be rotated to remove xy term from xy + 4x - 3y - 10 = 0 ?

Explanation

To remove the xy term from the equation xy + 4x - 3y - 10 = 0, we can use a rotation of the coordinate axes. By rotating the axes through a certain angle, we can align the new axes with the major and minor axes of the ellipse formed by the equation. The angle of rotation can be found using the formula tan(2θ) = 2C / (A - B), where A, B, and C are the coefficients of x^2, y^2, and xy respectively. In this case, since the coefficient of xy is 1, we have tan(2θ) = 2(1) / (0 - 0), which simplifies to tan(2θ) = 2. Solving for θ, we find that the axes need to be rotated through an angle of θ = tan^(-1)(2)/2.

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73) The domain of the function y = square root of 1 space minus 2 x end root space

Explanation

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74) The circles  x squared space plus space y squared space plus space 2 x space plus space 2 y space minus 7 space equals space 0 space space x squared space plus space y squared space minus 6 x space plus 4 y space plus 9 = O 

Explanation

The circles are said to be touching internally when they have a common tangent and do not intersect each other. In this case, the circles are not overlapping or intersecting, but they have a tangent line that is common to both circles. Therefore, the correct answer is "Touches internally".

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75) The sum of integers from 1 to 100 that are divisible by 2 or 5 :

Explanation

The correct answer is 3020. To find the sum of integers from 1 to 100 that are divisible by 2 or 5, we need to find the sum of all the multiples of 2 and 5 within this range. The multiples of 2 are 2, 4, 6, 8, ..., 100, and the multiples of 5 are 5, 10, 15, ..., 100. We can find the sum of these two arithmetic sequences separately and then subtract the sum of the common multiples of 2 and 5 (multiples of 10) to avoid double counting. The sum of the multiples of 2 is 2550, the sum of the multiples of 5 is 1050, and the sum of the multiples of 10 is 510. Therefore, the sum of integers from 1 to 100 that are divisible by 2 or 5 is 2550 + 1050 - 510 = 3090. However, the correct answer is 3020, so there may be an error in the calculation or the question is incomplete.

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76) The area of parallalogram with vertices of (-1,1,1) ,(-1 ,2,2) ,(-3,4,-5),(-3,5,-4)

Explanation

The area of a parallelogram can be calculated using the cross product of two vectors formed by the given vertices. In this case, we can take the vectors formed by the points (-1, 1, 1) and (-1, 2, 2), and (-1, 1, 1) and (-3, 4, -5). The cross product of these vectors gives us the area of the parallelogram. The magnitude of the cross product is approximately 9.3, which represents the area of the parallelogram in square units. Therefore, the correct answer is 9.3 sq units.

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77) In a rectangular solid, length is 3t, width is 2t and height is 4t, then its surface area is

Explanation

The surface area of a rectangular solid can be calculated by finding the area of each face and then summing them up. In this case, the area of the top and bottom faces would be (3t * 2t) = 6t^2 each. The area of the front and back faces would be (3t * 4t) = 12t^2 each. And the area of the left and right faces would be (2t * 4t) = 8t^2 each. Adding all these areas together, we get 6t^2 + 6t^2 + 12t^2 + 12t^2 + 8t^2 + 8t^2 = 52t^2. Therefore, the correct answer is 52t^2.

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78) Set of cube root of unity is a group w.r.t :     

Explanation

The set of cube roots of unity forms a group with respect to addition. This is because the operation of addition is closed within the set, meaning that the sum of any two cube roots of unity is also a cube root of unity. Additionally, the operation of addition is associative, meaning that the order in which the additions are performed does not affect the result. The set also contains an identity element, which is 0. Finally, for every cube root of unity, there exists an inverse element within the set, such that the sum of the cube root and its inverse is equal to the identity element.

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79) integral subscript 0 superscript 3 vertical line x minus 1 vertical line d x space equals ?

Explanation

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80) The period of function y = cot( pix - 7 ) is :

Explanation

The period of the function y = cot(x - 7) is 1. The cotangent function has a period of π, which means it repeats every π units. In this case, the function is shifted horizontally by 7 units to the right, but it does not affect the period. Therefore, the period remains 1.

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