1
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1 The curve $$y(x)=a x^{3}+b x^{2}+c x+5$$ touches the $$x$$-axis at the point $$\mathrm{P}(-2,0)$$ and cuts the $$y$$-axis at the point $$Q$$, where $$y^{\prime}$$ is equal to 3 . Then the local maximum value of $$y(x)$$ is:

A
$$\frac{27}{4}$$
B
$$\frac{29}{4}$$
C
$$\frac{37}{4}$$
D
$$\frac{9}{2}$$
2
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + $$\alpha$$, 0) and (0, 50 + $$\alpha$$), $$\alpha$$ > 0, then (x, y) also lies on the line :

A
y = 4x
B
x = 4y
C
y = 4x + $$\alpha$$
D
x = 4y $$-$$ $$\alpha$$
3
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

Let $$f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R$$. Then f :

A
has a local minina at $$x = {1 \over 2}$$
B
has a local minima at $$x = {3 \over 4}$$
C
is increasing in $$\left( {{1 \over 2},{3 \over 4}} \right)$$
D
is decreasing in $$\left( {{1 \over 2},{4 \over 3}} \right)$$
4
JEE Main 2022 (Online) 29th June Morning Shift
+4
-1 A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

A
$${{22} \over {9 + 4\sqrt 3 }}$$
B
$${{66} \over {9 + 4\sqrt 3 }}$$
C
$${{22} \over {4 + 9\sqrt 3 }}$$
D
$${{66} \over {4 + 9\sqrt 3 }}$$
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