The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Concept: Elementary Operation (Transformation) of a Matrix

The equation of tangent to the curve y=`y=x^2+4x+1` at

(-1,-2) is...............

(a) 2x -y = 0 (b) 2x+y-5 = 0

(c) 2x-y-1=0 (d) x+y-1=0

Concept: Conics > Tangents and normals - equations of tangent and normal at a point

Given that X ~ B(n= 10, p). If E(X) = 8 then the value of

p is ...........

(a) 0.6

(b) 0.7

(c) 0.8

(d) 0.4

Concept: Bernoulli Trials and Binomial Distribution

Find the area bounded by the curve y^{2} = 4ax, x-axis and the lines x = 0 and x = a.

Concept: Area of the Region Bounded by a Curve and a Line

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`

Concept: Methods of Integration - Integration by Substitution

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

Concept: Methods of Integration - Integration by Substitution

The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.

Concept: Elementary Operation (Transformation) of a Matrix

A fair coin is tossed five times. Find the probability that it shows exactly three times head.

Concept: Conditional Probability

Find the inverse of the matrix, `A=[[1,3,3],[1,4,3],[1,3,4]]`by using column transformations.

Concept: Elementary Operation (Transformation) of a Matrix

Find the area of the region bounded by the parabola y^{2} = 4ax and its latus rectum.

Concept: Area of the Region Bounded by a Curve and a Line

Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

Concept: Methods of Integration - Integration by Substitution

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

Concept: Pair of Straight Lines > Pair of Lines Passing Through Origin - Combined Equation

Solve the following equations by the method of reduction :

2x-y + z=1, x + 2y +3z = 8, 3x + y-4z=1.

Concept: Elementary Operation (Transformation) of a Matrix

Evaluate :`intxlogxdx`

Concept: Methods of Integration - Integration by Substitution

The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive.

Concept: Conditional Probability

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

Concept: Area of the Region Bounded by a Curve and a Line

Show that the function defined by f(x) =|cosx| is continuous function.

Concept: Introduction of Continuity

A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?

Concept: Statistics > Bivariate Frequency Distribution

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

Concept: Mean Value Theorem

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`

Concept: Methods of Integration - Integration by Substitution